From bf70ecccaf7d64e2a97436a09293b8bafc9f91d4 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 28 Jun 2026 11:15:32 +0200 Subject: [PATCH 01/34] Initial draft --- _posts/2026-06-28-todo.md | 122 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 122 insertions(+) create mode 100644 _posts/2026-06-28-todo.md diff --git a/_posts/2026-06-28-todo.md b/_posts/2026-06-28-todo.md new file mode 100644 index 0000000..3a31b03 --- /dev/null +++ b/_posts/2026-06-28-todo.md @@ -0,0 +1,122 @@ +--- +title: "TODO" +date: 2026-06-28 +description: "TODO" +tags: ["theoretical mathematics", "first principles"] +--- + +/-- If $x$ and $y$ are numbers, then either $x \leq y$ or $y \leq x$. -/ + +```lean +Statement le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by + sorry +``` + + +## INTRO + + + assumptiosn: reader is familiar with discrete mathematic[118;1:3us + +### MOTIVATION + I don't like assumptions + I don't like bullshit + How do we know if a given stastement is real? + Why just "hope it works" + First principles + Start with Axioms and build our way up + + Some definitions first: + Axiom := most fundamental assumptions + Proof := + Theorem := + + How do I verify that a given proof is correct? + How do I verify that I didn't bullshit my way to QED. + + + My main struggle with discrete mathematics is the lack of feedback to my work. Analogy: checking if code compiles and runs correclty when only on paper is difficult. same with proofs. one misstep. GOAL: reduce human error + +### 99 VARIATIONS OF A PROOF + + 99 variations of a proofs proposes the idea that proofs are but mere logical arguments. + + we start with theorem: + (x : ℝ) - x^3-6x^2+11x-6=2x-2 => x=1∨x=4 + + and proove it in 99 ways + (find 99 logical reasonings) + my favourite is : [TODO] + + // Most logical arguments are difficult to systematically veirfy. + // I love machines - they do exactly what they have been told to do. - not always what i want them to but at least they are deterministic. WHY can't we use them for mathematics? WE CAN + + // statement -> irrefutable proof + + +## LEAN + + // LEAN -> proof asistant + // > quote LEAN + // TOPIC: today we will prove that ℕ are totally orderred + // [image of a number line] + + // assumptions: ℕ has been defined as either 0 or succ + // a <= b has been defined as ∃ c : b = a + c (numberline example svg) + + + // Or (∨) + // a/b | False True | + -------+--------------+ + // False | False True | + // True | True True | + + + // our assumption: (x y : ℕ) : x <= y ∨ y <= x + // reads as: given two natural numbers, one of them is greater or equal than the other. Intuitively makes sense. example: 6,7. so here (6≤7) ∨ (7≤6) -> True ∨ False -> True. + Or given same numbers: (1≤1) ∨ (1≤1) -> True ∨ True -> True + // It is easy to show that the statement holds for two _specific_ numbers but we need to prove that it holds for _all_ numbers. We focus on ℕ for simplicity. + + + my version of the x <= y ∨ y <= x proof has inspired by Kevin Buzzard. + https://profiles.imperial.ac.uk/k.buzzard/about + + x <= y ∨ y <= x + + // here we can apply induction on y for example. + // we can split this into two cases. + // ind y with d hd + + hd: (x <= y ∨ y <= x) + + induction y = 0 + x <= 0 ∨ 0 <= x // right one has been proven + 0 <= x // exactly zero_le + induction y = succ(d) + x <= succ(d) ∨ succ(d) <= x + + // let's focus on our hypothesis from earlier (hd) + // let's break it up into parts + + (x <= y) ∨ (y <= x) + | | + | right: hr := (y <= x) + left: hl := (x <= y) + + + // here we can apply cases as per definition + + // > If h : P ∨ Q is a hypothesis, then cases h with hp hq will turn one goal into two goals, one with a hypothesis hp : P and the other with a hypothesis hq : Q. + +.. + + + + + + // Additional stuff + + - https://adam.math.hhu.de/#/ + - + - https://github.com/julian/lean.nvim + - https://leanprover-community.github.io/1000.html From c179f4bb744ba8b18795048972b3994b51cc7834 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 28 Jun 2026 12:03:41 +0200 Subject: [PATCH 02/34] wip --- _posts/2026-06-28-todo.md | 84 ++++++++++++++++++++++++++++++++++++--- 1 file changed, 78 insertions(+), 6 deletions(-) diff --git a/_posts/2026-06-28-todo.md b/_posts/2026-06-28-todo.md index 3a31b03..d3cac93 100644 --- a/_posts/2026-06-28-todo.md +++ b/_posts/2026-06-28-todo.md @@ -3,6 +3,7 @@ title: "TODO" date: 2026-06-28 description: "TODO" tags: ["theoretical mathematics", "first principles"] +math: true --- /-- If $x$ and $y$ are numbers, then either $x \leq y$ or $y \leq x$. -/ @@ -42,9 +43,18 @@ Statement le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by 99 variations of a proofs proposes the idea that proofs are but mere logical arguments. we start with theorem: - (x : ℝ) - x^3-6x^2+11x-6=2x-2 => x=1∨x=4 - - and proove it in 99 ways +$$ + (x : \mathbb{N})) - x^3-6x^2+11x-6=2x-2 \implies x=1 \lor x=4 +$$ + +Twoliner (Proof by factorization (Proof 1 - Oneline)): +$$ +x^3-6x^2+9x-4=0 \\ +(x-1)^2(x-4)=0 +\Box +$$ + + and proove it in 98 other ways (find 99 logical reasonings) my favourite is : [TODO] @@ -71,15 +81,31 @@ Statement le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by // False | False True | // True | True True | +| | $a = False$ | $b = True$ | +| --- | --- | --- | +| $b = False$ | $False \lor False \implies False$ | $a \lor b \implies True$ | +| $b = True$ | $a \lor b \implies True$ | $a \lor b \implies True$ | + + +// our assumption: + +$$ +\fbox{ + $(x, y \in \mathbb{N}) : x <= y ∨ y <= x$ +} + +\Box +$$ + + - // our assumption: (x y : ℕ) : x <= y ∨ y <= x // reads as: given two natural numbers, one of them is greater or equal than the other. Intuitively makes sense. example: 6,7. so here (6≤7) ∨ (7≤6) -> True ∨ False -> True. Or given same numbers: (1≤1) ∨ (1≤1) -> True ∨ True -> True // It is easy to show that the statement holds for two _specific_ numbers but we need to prove that it holds for _all_ numbers. We focus on ℕ for simplicity. my version of the x <= y ∨ y <= x proof has inspired by Kevin Buzzard. - https://profiles.imperial.ac.uk/k.buzzard/about + x <= y ∨ y <= x @@ -112,9 +138,55 @@ Statement le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by +## Final Proof + +Prereqs / Assumptions + +```lean +Statement succ_eq_add_one n : succ n = n + 1 := by + rw [one_eq_succ_zero] + rw [add_succ] + rw [add_zero] + rfl + +Statement zero_le (x : ℕ) : 0 ≤ x := by + use x + rw [zero_add] + rfl +``` + +Now our proof: + +```lean +Statement le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by + induction y with d hd + right + exact zero_le x + + cases hd with h1 h2 + left + cases h1 with e h1 + rw [h1] + use e + 1 + + + rw [succ_eq_add_one, add_assoc] + rfl + cases h2 with e he + cases e with a + rw [he] + left + rw [add_zero] + use 1 + exact succ_eq_add_one d + right + use a + rw [add_succ] at he + rw [succ_add] + exact he +``` - // Additional stuff - https://adam.math.hhu.de/#/ - From eb8c9948cc53da0b73216bfd40aaf06b89b75501 Mon Sep 17 00:00:00 2001 From: cb341 Date: Mon, 29 Jun 2026 17:23:41 +0200 Subject: [PATCH 03/34] wip --- _posts/2026-06-28-todo.md | 133 ++++++++++++++++++++++++++++++-------- 1 file changed, 106 insertions(+), 27 deletions(-) diff --git a/_posts/2026-06-28-todo.md b/_posts/2026-06-28-todo.md index d3cac93..f87f35a 100644 --- a/_posts/2026-06-28-todo.md +++ b/_posts/2026-06-28-todo.md @@ -14,6 +14,12 @@ Statement le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by ``` +## scraps + +Underwood Dudley +"In mathematics problems can be solved, using reason, and the solutions can be checked and shown to be correct." + + ## INTRO @@ -54,9 +60,34 @@ x^3-6x^2+9x-4=0 \\ \Box $$ +// thanks Josua for gifting me the book + and proove it in 98 other ways (find 99 logical reasonings) - my favourite is : [TODO] + my favourite is : Proof #6 - axiomatic -> first principles. first we define notation, then definitions and then solve find proor by means of applying axioms + + +// 20 - defintiional is quite similar to 6 +// - no hiddewn assumptions, everything clearly defined + +// some other proofs that spark joy to me are: +// 9 - monosyllabic - sma words +// 10 - wordless by showing pricutres of cube +// 27 - algorithmic + + +// fun ones: +// 11 - exam +// 15 - matrix +// 25 - open collaborative +// 28 - flowchart +// 29 - model + +// 26 - auditory + +// hate: +// 19 - jargon (reminds of LinkedIn) +/ [insert tier list] // Most logical arguments are difficult to systematically veirfy. // I love machines - they do exactly what they have been told to do. - not always what i want them to but at least they are deterministic. WHY can't we use them for mathematics? WE CAN @@ -64,6 +95,9 @@ $$ // statement -> irrefutable proof +made me realize LEAN exists: +- https://arxiv.org/abs/2605.22763 + ## LEAN // LEAN -> proof asistant @@ -140,6 +174,46 @@ $$ ## Final Proof +Axioms +```lean + +/- BEGIN PROVE: x ≤ y ∨ y ≤ x -/ +induction y with +| zero => + right /- 0 ≤ x -/ + exact zero_le x +| succ d hd => + /- ind y=0 -/ + /- BEGIN PROVE: x ≤ 0 ∨ 0 ≤ x -/ + + /- END PROVE: x ≤ 0 ∨ 0 ≤ x -/ + + /- ind y=d -/ + cases hd with hl hr /- hd = hl ∨ hr + | hr := d ≤ x + hl := x ≤ d + -/ + cases hl with c hc /- hc := d = x + c -/ + left /- x ≤ succ d -/ + rw[hc] + use c + 1 + rw[succ_eq_add_one] + rw[← add_assoc] + rfl + + /- trial and error -/ + cases hr with e he /- + hr : d ≤ x + he : x = d + e + -/ + cases e with a /- e -> {0, succ(a)} -/ + /- e=0 ; he : x = d + 0 -/ + rw[add_zero] at he /- he : x = d -/ + left + rw[he] + exact le_succ_self d /- le_succ_self x - x ≤ succ x-/ + /- he : x = d + succ a -/ +``` Prereqs / Assumptions @@ -156,35 +230,40 @@ Statement zero_le (x : ℕ) : 0 ≤ x := by rfl ``` -Now our proof: +Now MY proof: ```lean Statement le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by - induction y with d hd - right - exact zero_le x - - cases hd with h1 h2 - left - cases h1 with e h1 - rw [h1] - use e + 1 - - - rw [succ_eq_add_one, add_assoc] - rfl - cases h2 with e he - cases e with a - rw [he] - left - rw [add_zero] - use 1 - exact succ_eq_add_one d - right - use a - rw [add_succ] at he - rw [succ_add] - exact he + induction y with + | zero => + right + exact zero_le x + | succ d hd => + cases hd with + | inl hl => + cases hl with c hc + rw[hc] + left + rw[succ_eq_add_one] + use c + 1 + rw[← add_assoc] + rfl + | inr hr => + cases hr with c hc + cases c with + | zero => + rw[zero_eq_0] at hc + rw[add_zero d] at hc + left + rw[hc] + exact le_succ_self d + | succ => + rw[add_succ] at hc + right + rw[hc] + use a + rw[succ_add] + rfl ``` From 37c22f0b545b4da656e2d988dab17a7d99ff2bc1 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 5 Jul 2026 22:16:42 +0200 Subject: [PATCH 04/34] wip --- _posts/2026-06-28-todo.md | 32 ++++++++++++++++++++++++++++++-- 1 file changed, 30 insertions(+), 2 deletions(-) diff --git a/_posts/2026-06-28-todo.md b/_posts/2026-06-28-todo.md index f87f35a..b1b3f8c 100644 --- a/_posts/2026-06-28-todo.md +++ b/_posts/2026-06-28-todo.md @@ -1,11 +1,19 @@ --- -title: "TODO" +title: "Doing Mathematics the HARD way" date: 2026-06-28 description: "TODO" tags: ["theoretical mathematics", "first principles"] math: true --- +alternate titles: + +Functional Programming Meets Rigoporous Mathematics +Mathematics but cut the bullshit +Mathematics but not handwavey +Mathematics but done right +Dipping my toes into rigorous mathematics + /-- If $x$ and $y$ are numbers, then either $x \leq y$ or $y \leq x$. -/ ```lean @@ -19,6 +27,11 @@ Statement le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by Underwood Dudley "In mathematics problems can be solved, using reason, and the solutions can be checked and shown to be correct." ++ Recently completed the natrual numbers game as an intro to LEAN and i think i really like it +https://adam.math.hhu.de/#/g/leanprover-community/nng4 + + +First principles (like [my 8bit CPU](projects/8bit-cpu/)) ## INTRO @@ -33,17 +46,29 @@ Underwood Dudley First principles Start with Axioms and build our way up - Some definitions first: + **Some definitions first:** Axiom := most fundamental assumptions Proof := Theorem := + Statement := + Proposition := + How do I verify that a given proof is correct? How do I verify that I didn't bullshit my way to QED. + + Some semesters back I have completed the discrete mathematics (https://eventoweb.zhaw.ch/Evt_Pages/Brn_ModulDetailAZ.aspx?node=2901247e-aa27-4f84-a5d6-d6b33b234dbd&IDAnlass=1456265&IdLanguage=133&clearcache=true) course at ZHAW. Now i am here to re-visit it with a different lense upon learning about proof asistants. + My main struggle with discrete mathematics is the lack of feedback to my work. Analogy: checking if code compiles and runs correclty when only on paper is difficult. same with proofs. one misstep. GOAL: reduce human error + I feel like my understanding of mathematics is too fuzzy. I want to be less hand wavey and want to see my gaps explicitly. + + explicitly out of scope: (type system of Lean, how lean works, RCOQ, full 99 variations of a proof review) + explicitly mention: NO AI has been used to aid with LEAN + NO AI has been used in this article except for language / phrasing + ### 99 VARIATIONS OF A PROOF 99 variations of a proofs proposes the idea that proofs are but mere logical arguments. @@ -89,6 +114,9 @@ $$ // 19 - jargon (reminds of LinkedIn) / [insert tier list] +full book review and +tier list comes later.. + // Most logical arguments are difficult to systematically veirfy. // I love machines - they do exactly what they have been told to do. - not always what i want them to but at least they are deterministic. WHY can't we use them for mathematics? WE CAN From 70147e199e95561923993a927d7a8761d3178d3a Mon Sep 17 00:00:00 2001 From: cb341 Date: Tue, 7 Jul 2026 13:57:06 +0200 Subject: [PATCH 05/34] wip --- _posts/2026-06-28-todo.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/_posts/2026-06-28-todo.md b/_posts/2026-06-28-todo.md index b1b3f8c..e9d492a 100644 --- a/_posts/2026-06-28-todo.md +++ b/_posts/2026-06-28-todo.md @@ -33,6 +33,10 @@ https://adam.math.hhu.de/#/g/leanprover-community/nng4 First principles (like [my 8bit CPU](projects/8bit-cpu/)) +ETH Discrete Mathematik Chapter 6 + +> Definition 6.1. A proof system is a quadruple Π = (S,P,τ,φ), as above. + ## INTRO From 5c44b46f6f762d26e66d793e58f378d32906dea4 Mon Sep 17 00:00:00 2001 From: cb341 Date: Thu, 9 Jul 2026 20:39:00 +0200 Subject: [PATCH 06/34] progress --- _posts/2026-06-28-lean.md | 205 +++++++++++++++++++++++ _posts/2026-06-28-todo.md | 50 +++++- assets/blog/dependency_graph.svg | 2 + assets/blog/lean_graph.png | Bin 0 -> 99704 bytes assets/blog/lean_graph.svg | 272 +++++++++++++++++++++++++++++++ assets/blog/lean_thing.png | Bin 0 -> 152532 bytes assets/blog/numberline.png | Bin 0 -> 51213 bytes proof.dot | 109 +++++++++++++ 8 files changed, 632 insertions(+), 6 deletions(-) create mode 100644 _posts/2026-06-28-lean.md create mode 100644 assets/blog/dependency_graph.svg create mode 100644 assets/blog/lean_graph.png create mode 100644 assets/blog/lean_graph.svg create mode 100644 assets/blog/lean_thing.png create mode 100644 assets/blog/numberline.png create mode 100644 proof.dot diff --git a/_posts/2026-06-28-lean.md b/_posts/2026-06-28-lean.md new file mode 100644 index 0000000..75c94c7 --- /dev/null +++ b/_posts/2026-06-28-lean.md @@ -0,0 +1,205 @@ +--- +title: "Mathematics but not handwavey (real)" +date: 2026-06-28 +description: "First steps in formal mathematics with Lean" +tags: ["theoretical mathematics", "first principles"] +math: true +--- + +## Intro + +### Discrete Mathematics +### ZHAW Background -> different approach to mathematics +### Vocabulary + + + +∃: there exists +≤: less than or equal to + +Today we will be proving the total order of natural numbers. + +$$ +\Huge \forall x,y \in \mathbb{N} : x \le y \lor y \le x +$$ + + +$$ +\boxed{\displaystyle \quad \forall x,y \in \mathbb{N} : x \le y \lor y \le x \quad} +$$ + +$$ +\boxed{\forall x,y \in \mathbb{N} : x \le y \lor y \le x} +$$ + +$$ +\fbox{\(\displaystyle \quad \forall x,y \in \mathbb{N} : x \le y \lor y \le x \quad\)} +$$ + +$$ +\colorbox{#fff3cd}{$\displaystyle \forall x,y \in \mathbb{N} : x \le y \lor y \le x$} +$$ + + +$$ +\fbox{ + $\forall (x,y : ℕ) : x ≤ y ∨ y ≤ x$ +} +$$ + + +![graph](/assets/blog/dependency_graph.svg) + + + +Before we do that we need to get some definitions out of the way. + +## Definitions + +### Natural Number? + +First: + +What even is a Natural number? +Natural numbers are defined to be all "positive" "whole" numbers. + +$$ +\mathbb{N} = \{0,1,2,3,4,5, \ldots \} +$$ + +But this isn't yet a concrete definiton. +Natural numbers can be seen as a sequeence. The sequence starts at 0. +Each number is either 0 or a successor. + +$$ +\mathbb{N} + \stackrel{\mathrm{def}}{=} \begin{cases} + 0 &\text{base} \\ + \mathrm{succ} &\text{else} \\ +\end{cases} +$$ + +### Addition? + +Now we can define addition. (Using axioms) + +It isn't enough to define addition for speicfic numbers, like let's say `6 + 1 = 7`. +We need to define addition for ALL numbers. We do this using induction. + +$$ +\forall a,b \in \mathbb{N} : a + b = ? +$$ + +Here we fix the variable $a$ and check the our definition of natrual numbers from before. +B is either zero, or a successor of a natural number. + +We branch: + +**Zero:** + +$$ +\forall a \in \mathbb{N} : a + 0 \stackrel{\mathrm{def}}{=} a +$$ + +**Succ:** + +$$ +(a, d : \mathbb{N}) : a + succ(d) \stackrel{\mathrm{def}}{=} succ (a + d) +$$ + +Example: $\fbox{$1 + 2 = 3$}$ + +Axioms: + +$$ +\begin{array}{rl} +\mathrm{i.} & 1 \stackrel{\mathrm{def}}{=} succ(0) \\ +\mathrm{ii.} & 2 \stackrel{\mathrm{def}}{=} succ(1) \\ +\mathrm{iii.} & 3 \stackrel{\mathrm{def}}{=} succ(2) \\ +\end{array} +$$ + +$$ + \fbox{$1 + 2 = 3$} \\ + + \begin{array}{l} + a. 1 \stackrel{\mathrm{def}}{=} succ(0) \\ + b. 2 \stackrel{\mathrm{def}}{=} succ(1) \\ + c. 3 \stackrel{\mathrm{def}}{=} succ(2) \\ + \end{array} \\ + + \begin{array}{l|l} + 1 + 2 & \text{Given} \\ + 1 + \operatorname{succ}(1) & \mathrm{i.} \\ + \operatorname{succ}(1 + 1) & \mathrm{i.} \\ + \operatorname{succ}(1 + \operatorname{succ}(0)) & \mathrm{i.} \\ + \operatorname{succ}(\operatorname{succ}(1 + 0)) & \mathrm{i.} \\ + \operatorname{succ}(\operatorname{succ}(1)) & \mathrm{i.} \\ + \operatorname{succ}(2) & \mathrm{i.} \\ + 3 & QED. \Box \\ + \end{array} +$$ + +$$ +\begin{array}{l|l} +\textbf{Statements} & \textbf{Reasons} \\ +\hline +1. \ 1 + 2 = 3 & 1. \ \text{Given} \\ +2. \ 1 + succ(1) = succ(2) & 2. \ \text{Definition of congruent angles} \\ +3. \ succ(1 + 1) = succ(2) & 3. \ \text{Given} \\ +4. \ & 4. \ \text{Transitive Property of Equality} \\ +\end{array} +$$ + +We don't proove that $a + 0 = a$, we define it. + +### Inequalities? + +What we can do next is define inequalities. + +$$ +(a,b : \mathbb{N} ) a ≤ b \stackrel{\mathrm{def}}{\implies} ∃ (c : ℕ), b = a + c +$$ + +![graph](/assets/blog/numberline.png) + +## Prooving + +```lean +theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by + induction y with + | zero => + right + exact zero_le x + | succ d hd => + cases hd with + | inl hl => + cases' hl with c hc + rewrite[hc] + left + rewrite[succ_eq_add_one] + use c + 1 + rewrite[← add_assoc] + rfl + | inr hr => + cases' hr with c hc + cases c with + | zero => + rewrite[zero_eq_0] at hc + rewrite[add_zero d] at hc + left + rewrite[hc] + exact le_succ_self d + | succ a => + rewrite[add_succ] at hc + right + rewrite[hc] + use a + rewrite[succ_add] + rfl +``` + +![graph](/assets/blog/lean_thing.png) + + + diff --git a/_posts/2026-06-28-todo.md b/_posts/2026-06-28-todo.md index e9d492a..8f632d9 100644 --- a/_posts/2026-06-28-todo.md +++ b/_posts/2026-06-28-todo.md @@ -1,5 +1,5 @@ --- -title: "Doing Mathematics the HARD way" +title: "Mathematics but not handwavey" date: 2026-06-28 description: "TODO" tags: ["theoretical mathematics", "first principles"] @@ -8,11 +8,12 @@ math: true alternate titles: -Functional Programming Meets Rigoporous Mathematics -Mathematics but cut the bullshit -Mathematics but not handwavey -Mathematics but done right -Dipping my toes into rigorous mathematics +* Functional Programming Meets Rigoporous Mathematics +* Mathematics but cut the bullshit +* Mathematics but done right +* Dipping my toes into rigorous mathematics +* Mathematics but without gaps +* Let's PROVE IT! /-- If $x$ and $y$ are numbers, then either $x \leq y$ or $y \leq x$. -/ @@ -22,6 +23,9 @@ Statement le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by ``` +Explicitly use ≤ instead of <= +Use Latex where possible + ## scraps Underwood Dudley @@ -37,8 +41,28 @@ ETH Discrete Mathematik Chapter 6 > Definition 6.1. A proof system is a quadruple Π = (S,P,τ,φ), as above. + +Excerpt + +from ETH Zürich +Departement Informatik +Diskrete +Mathematik +Ueli Maurer +Herbstsemester 2024 + +> **6.2.4 Proof Systems in Theoretical Computer Science**\* +> +> An important extension of the concept of proof systems are so-called interactive proofs. $^{16}$ In such a system, the proof is not a bit-string, but it consists of an interaction (a protocol) between the prover and the verifier, where one tolerates an immensely small (e.g. exponentially small) probability that a verifier accepts a “proof” for a false state- ment. The reason for considering such interactive proofs are: +> +> * Such interactive proofs can exist for statements for which a classical (non-interactive) proof does not exist. For example, there exists an interactive proof system for the non-Hamiltonicity of graphs. +> * Such interactive proofs can have a special property, called *zero-knowledge*, which means that the verifier learns absolutely nothing (in a well-defined sense) during the protocol, except that the statement is true. In particular, the verifier cannot prove the statement to somebody else. +> * Zero-knowledge proofs (especially non-interactive versions, so-called NIZK’s) are of crucial importance in a large number of applications, for example in sophisticated block-chain systems. + ## INTRO +![Path from Axioms to QED](/assets/blog/lean_graph.svg) + assumptiosn: reader is familiar with discrete mathematic[118;1:3us @@ -132,6 +156,17 @@ made me realize LEAN exists: ## LEAN +``` +ℕ +├── add_zero ───────┬── zero_add ──────┬── zero_le ───────┐ +│ │ ├── add_comm │ +│ │ └── add_assoc ─────┤ +├── add_succ ───────┼── succ_add ──────┬── add_comm ├── le_total +│ │ └── add_assoc ─────┤ +└── one_eq_succ_zero┴── succ_eq_add_one┬── le_succ_self ──┘ + └──────────────────┘ +``` + // LEAN -> proof asistant // > quote LEAN // TOPIC: today we will prove that ℕ are totally orderred @@ -141,12 +176,15 @@ made me realize LEAN exists: // a <= b has been defined as ∃ c : b = a + c (numberline example svg) + + | | $a = False$ | $b = True$ | | --- | --- | --- | | $b = False$ | $False \lor False \implies False$ | $a \lor b \implies True$ | diff --git a/assets/blog/dependency_graph.svg b/assets/blog/dependency_graph.svg new file mode 100644 index 0000000..9efe3e4 --- /dev/null +++ b/assets/blog/dependency_graph.svg @@ -0,0 +1,2 @@ + +a + succ(d) = succ(a + d)a + 0 = a1 = succ(0)succ(n) = n + 10 + n = nsucc(a) + b = succ(a + b)a + b = b + aa + b + c = a + (b + c)0 ≤ xx ≤ succ(x)x ≤ y ∨ y ≤ x \ No newline at end of file diff --git a/assets/blog/lean_graph.png b/assets/blog/lean_graph.png new file mode 100644 index 0000000000000000000000000000000000000000..966ee83832cf119a665e2a36223db35d777eb048 GIT binary patch literal 99704 zcmeFZbyQXD7c~k9N{ApSh=8bcNlG6?8l*v5q@{#I_W=Q=r8%@TNSCyVN_TSzm2RY) z`y5{de&f62{(r~#{!n>&&)NIgPpmc9oOA8BicckRvB|MfP*8BCrJg9Gpj{c2O*X|9E2jOxi?V9)%J7jERC8V2*;0{0R6%4*r0zr=p@< z0so>Rf0v5(@BdygOTF^%XLM8K2ZM=J%27}tDAG?vRh>~+CUCap9Zt`Fhn@BLk?Tq= 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ℕ) : succ n = n + 1 + + + +add_zero->succ_eq_add_one + + + + + +zero_add + +zero_add +(n : ℕ) : 0 + n = n + + + +add_zero->zero_add + + + + + +succ_add + +succ_add +(a b : ℕ) : succ a + b = succ (a + b) + + + +add_zero->succ_add + + + + + +add_comm + +add_comm +(a b : ℕ) : a + b = b + a + + + +add_zero->add_comm + + + + + +add_assoc + +add_assoc +(a b c : ℕ) : a + b + c = a + (b + c) + + + +add_zero->add_assoc + + + + + +le_total + +le_total +(x y : ℕ) : x ≤ y ∨ y ≤ x + + + +add_zero->le_total + + + + + +add_succ->succ_eq_add_one + + + + + +add_succ->zero_add + + + + + +add_succ->succ_add + + + + + +add_succ->add_comm + + + + + +add_succ->add_assoc + + + + + +add_succ->le_total + + + + + +one_eq_succ_zero->succ_eq_add_one + + + + + +le_succ_self + +le_succ_self +(x : ℕ) : x ≤ succ x + + + +succ_eq_add_one->le_succ_self + + + + + +succ_eq_add_one->le_total + + + + + +zero_add->add_comm + + + + + +zero_add->add_assoc + + + + + +zero_le + +zero_le +(x : ℕ) : 0 ≤ x + + + +zero_add->zero_le 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z+jwlT>6KeCk5k%lMVTzq-m;lT{dj6h7u1&Bd-pW>)aP5rla^fX5B@y=iZtId?eAjq z?4l)k_8ob5c!Pa>`PcK?5B-yfwypoM-twX`4lgTwkY$SB^w;P?zxNjBe;@y*>CCHM z{SvtResg_M%KdqJ G0; + G0 -> Z0 [label="induction y (0)"]; + G0 -> start_succ [label="induction y (succ d, hd)", lhead=cluster_succ]; + Z0 -> Z1 [label="right"]; + Z1 -> end_top [label="exact zero_le x"]; + + // ── succ branch ── + subgraph cluster_succ { + label="succ d (hd : x≤d ∨ d≤x)"; + style=rounded; fontname="Helvetica"; + start_succ [shape=circle, width=.2, style=filled, fillcolor=black, label=""]; + end_succ [shape=doublecircle, width=.15, style=filled, fillcolor=black, label=""]; + + S0 [label="x ≤ succ d ∨ succ d ≤ x"]; + + start_succ -> S0; + S0 -> start_inl [label="cases hd (inl hl)", lhead=cluster_inl]; + S0 -> start_inr [label="cases hd (inr hr)", lhead=cluster_inr]; + + // inl sub-scope + subgraph cluster_inl { + label="inl"; + style=rounded; fontname="Helvetica"; + start_inl [shape=circle, width=.2, style=filled, fillcolor=black, label=""]; + end_inl [shape=doublecircle, width=.15, style=filled, fillcolor=black, label=""]; + + IL0 [label="…∨…\n(hl : x≤d)"]; + IL1 [label="…∨…\n(hc : d = x+c)"]; + IL2 [label="x ≤ succ(x+c) ∨ succ(x+c) ≤ x"]; + IL3 [label="x ≤ succ(x+c)"]; + IL4 [label="x ≤ (x+c)+1"]; + IL5 [label="(x+c)+1 = x+(c+1)"]; + IL6 [label="(x+c)+1 = (x+c)+1"]; + + start_inl -> IL0; + IL0 -> IL1 [label="cases' hl with c hc"]; + IL1 -> IL2 [label="rewrite[hc]"]; + IL2 -> IL3 [label="left"]; + IL3 -> IL4 [label="rewrite[succ_eq_add_one]"]; + IL4 -> IL5 [label="use c+1"]; + IL5 -> IL6 [label="rewrite[← add_assoc]"]; + IL6 -> end_inl [label="rfl"]; + } + + end_inl -> end_succ [ltail=cluster_inl]; + + // inr sub-scope + subgraph cluster_inr { + label="inr"; + style=rounded; fontname="Helvetica"; + start_inr [shape=circle, width=.2, style=filled, fillcolor=black, label=""]; + end_inr [shape=doublecircle, width=.15, style=filled, fillcolor=black, label=""]; + + IR0 [label="…∨…\n(hr : d≤x)"]; + IR1 [label="…∨…\n(hc : x = d+c)"]; + + // c = 0 + RZ0 [label="…∨…\n(hc : x = d+0)"]; + RZ2 [label="…∨…\n(hc : x = d)"]; + RZ3 [label="x ≤ succ d"]; + RZ4 [label="d ≤ succ d"]; + + // c = succ a + RS0 [label="…∨…\n(hc : x = d + succ a)"]; + RS1 [label="…∨…\n(hc : x = succ(d+a))"]; + RS2 [label="succ d ≤ x"]; + RS3 [label="succ d ≤ succ(d+a)"]; + RS4 [label="succ(d+a) = succ d + a"]; + RS5 [label="succ(d+a) = succ(d+a)"]; + + start_inr -> IR0; + IR0 -> IR1 [label="cases' hr with c hc"]; + IR1 -> RZ0 [label="cases c (0)"]; + IR1 -> RS0 [label="cases c (succ a)"]; + + RZ0 -> RZ2 [label="rewrite[add_zero d] at hc"]; + RZ2 -> RZ3 [label="left"]; + RZ3 -> RZ4 [label="rewrite[hc]"]; + RZ4 -> end_inr [label="exact le_succ_self d"]; + + RS0 -> RS1 [label="rewrite[add_succ] at hc"]; + RS1 -> RS2 [label="right"]; + RS2 -> RS3 [label="rewrite[hc]"]; + RS3 -> RS4 [label="use a"]; + RS4 -> RS5 [label="rewrite[succ_add]"]; + RS5 -> end_inr [label="rfl"]; + } + + end_inr -> end_succ [ltail=cluster_inr]; + } + + end_succ -> end_top [ltail=cluster_succ]; +} From e3bf3120ea1b3843722778bde6c6beef03a03daf Mon Sep 17 00:00:00 2001 From: cb341 Date: Thu, 9 Jul 2026 20:54:30 +0200 Subject: [PATCH 07/34] wip --- _posts/.2026-06-28-lean.md.swp | Bin 0 -> 12288 bytes _posts/2026-06-28-lean.md | 53 ++++++++++++++------------------- 2 files changed, 23 insertions(+), 30 deletions(-) create mode 100644 _posts/.2026-06-28-lean.md.swp diff --git a/_posts/.2026-06-28-lean.md.swp b/_posts/.2026-06-28-lean.md.swp new file mode 100644 index 0000000000000000000000000000000000000000..23eba83d4e5878a2c67d5a27ad3f0861f84a09fc GIT binary patch literal 12288 zcmeI2&x;&I6vt~l`70`V7GGu;vx_~wvzr)`(HOH?C4uZJ=9k71Q{7!NQ`w%bv8#G# zCd(!ee_%koc@Y#85B@j`3W10L5pSLZK~da;F(>^4MEzFvOx#Tr$#J0;KHF0@Rd3$= zzV`}-si=1B-~#OqXBqzPVk~;>$y2YN`g80;KE~o``;Hkq|3828Jd?V*z~eckma0$X zDoj<()9ZGfaR#TmxjH3cqo>_4Y)R8DqEM>o7rIJgdbT#Zvs&9(o!wPU1<%4xa-AhV z;T7--+>io=X;yb_VcTbCYIfgvWh>ot_t6_N;TL-ayaHYUuYgy;E8rFI3U~#)0-IX_ 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zjG)tD1#zu2SqZ4pwAn!TGK*ywAyuAdiNJ@*uI<-4Rb0AIdZMe*$Ao6@Y_tLmO`8tk metd-`+=x2;b8Vg(X;REo*7-{j(v~ShkS^)*WihBYzyAh*Lt-KT literal 0 HcmV?d00001 diff --git a/_posts/2026-06-28-lean.md b/_posts/2026-06-28-lean.md index 75c94c7..a0b7499 100644 --- a/_posts/2026-06-28-lean.md +++ b/_posts/2026-06-28-lean.md @@ -107,50 +107,43 @@ $$ (a, d : \mathbb{N}) : a + succ(d) \stackrel{\mathrm{def}}{=} succ (a + d) $$ -Example: $\fbox{$1 + 2 = 3$}$ +$$ +\forall a,b \in \mathbb{N} : a + b \stackrel{\mathrm{def}}{=} +\begin{cases} +0 & \text{0} \\ +\mathrm{succ} & \operatorname{succ}(d) \\ +\end{cases} +$$ + +$$ +\Large{\text{Example: } \underline{1 + 2 = 3}} +$$ + -Axioms: $$ \begin{array}{rl} + & \textbf{Axioms} \\ \mathrm{i.} & 1 \stackrel{\mathrm{def}}{=} succ(0) \\ \mathrm{ii.} & 2 \stackrel{\mathrm{def}}{=} succ(1) \\ \mathrm{iii.} & 3 \stackrel{\mathrm{def}}{=} succ(2) \\ +\mathrm{iv.} & a + 0 \stackrel{\mathrm{def}}{=} a \\ +\mathrm{v.} & a + \operatorname{succ}(b) \stackrel{\mathrm{def}}{=} \operatorname{succ}(a+b) \\ \end{array} -$$ - -$$ - \fbox{$1 + 2 = 3$} \\ - - \begin{array}{l} - a. 1 \stackrel{\mathrm{def}}{=} succ(0) \\ - b. 2 \stackrel{\mathrm{def}}{=} succ(1) \\ - c. 3 \stackrel{\mathrm{def}}{=} succ(2) \\ - \end{array} \\ - \begin{array}{l|l} +\textbf{Statements} & \textbf{Reasons} \\ +\hline 1 + 2 & \text{Given} \\ - 1 + \operatorname{succ}(1) & \mathrm{i.} \\ - \operatorname{succ}(1 + 1) & \mathrm{i.} \\ + 1 + \operatorname{succ}(1) & \mathrm{ii.} \\ + \operatorname{succ}(1 + 1) & \mathrm{ii.} \\ \operatorname{succ}(1 + \operatorname{succ}(0)) & \mathrm{i.} \\ - \operatorname{succ}(\operatorname{succ}(1 + 0)) & \mathrm{i.} \\ - \operatorname{succ}(\operatorname{succ}(1)) & \mathrm{i.} \\ - \operatorname{succ}(2) & \mathrm{i.} \\ - 3 & QED. \Box \\ + \operatorname{succ}(\operatorname{succ}(1 + 0)) & \mathrm{v.} \\ + \operatorname{succ}(\operatorname{succ}(1)) & \mathrm{iv.} \\ + \operatorname{succ}(2) & \mathrm{iii.'} \\ + 3 & \Box \\ \end{array} $$ -$$ -\begin{array}{l|l} -\textbf{Statements} & \textbf{Reasons} \\ -\hline -1. \ 1 + 2 = 3 & 1. \ \text{Given} \\ -2. \ 1 + succ(1) = succ(2) & 2. \ \text{Definition of congruent angles} \\ -3. \ succ(1 + 1) = succ(2) & 3. \ \text{Given} \\ -4. \ & 4. \ \text{Transitive Property of Equality} \\ -\end{array} -$$ - We don't proove that $a + 0 = a$, we define it. ### Inequalities? From 743531ebb0c51cef3168a3cd6956c60f9aa9805a Mon Sep 17 00:00:00 2001 From: cb341 Date: Thu, 9 Jul 2026 21:21:56 +0200 Subject: [PATCH 08/34] lean --- _posts/2026-06-28-lean.md | 43 ++++++++++++--------------------------- 1 file changed, 13 insertions(+), 30 deletions(-) diff --git a/_posts/2026-06-28-lean.md b/_posts/2026-06-28-lean.md index a0b7499..08f5ad3 100644 --- a/_posts/2026-06-28-lean.md +++ b/_posts/2026-06-28-lean.md @@ -74,8 +74,8 @@ Each number is either 0 or a successor. $$ \mathbb{N} \stackrel{\mathrm{def}}{=} \begin{cases} - 0 &\text{base} \\ - \mathrm{succ} &\text{else} \\ + 0 &\text{Base case} \\ + \mathrm{succ} &\text{Else} \\ \end{cases} $$ @@ -91,27 +91,14 @@ $$ $$ Here we fix the variable $a$ and check the our definition of natrual numbers from before. -B is either zero, or a successor of a natural number. - -We branch: - -**Zero:** - -$$ -\forall a \in \mathbb{N} : a + 0 \stackrel{\mathrm{def}}{=} a -$$ -**Succ:** - -$$ -(a, d : \mathbb{N}) : a + succ(d) \stackrel{\mathrm{def}}{=} succ (a + d) -$$ +B is either zero, or a successor of a natural number. $$ \forall a,b \in \mathbb{N} : a + b \stackrel{\mathrm{def}}{=} \begin{cases} -0 & \text{0} \\ -\mathrm{succ} & \operatorname{succ}(d) \\ +a & b = \text{0} \\ +succ(a + d) & b = \operatorname{succ}(d) \\ \end{cases} $$ @@ -133,23 +120,19 @@ $$ \begin{array}{l|l} \textbf{Statements} & \textbf{Reasons} \\ \hline - 1 + 2 & \text{Given} \\ - 1 + \operatorname{succ}(1) & \mathrm{ii.} \\ - \operatorname{succ}(1 + 1) & \mathrm{ii.} \\ - \operatorname{succ}(1 + \operatorname{succ}(0)) & \mathrm{i.} \\ - \operatorname{succ}(\operatorname{succ}(1 + 0)) & \mathrm{v.} \\ - \operatorname{succ}(\operatorname{succ}(1)) & \mathrm{iv.} \\ - \operatorname{succ}(2) & \mathrm{iii.'} \\ - 3 & \Box \\ + 1 + 2 & \text{Given} \\ + 1 + \operatorname{succ}(1) & \mathrm{ii.} \\ + \operatorname{succ}(1 + 1) & \mathrm{v.} \\ + \operatorname{succ}(1 + \operatorname{succ}(0)) & \mathrm{i.} \\ + \operatorname{succ}(\operatorname{succ}(1 + 0)) & \mathrm{v.} \\ + \operatorname{succ}(\operatorname{succ}(1)) & \mathrm{iv.} \\ + \operatorname{succ}(2) & \mathrm{ii.'} \\ + 3 & \mathrm{iii.'} \\ \end{array} $$ -We don't proove that $a + 0 = a$, we define it. - ### Inequalities? -What we can do next is define inequalities. - $$ (a,b : \mathbb{N} ) a ≤ b \stackrel{\mathrm{def}}{\implies} ∃ (c : ℕ), b = a + c $$ From 44f94309231ce862f82cb6607aac823dde753eb8 Mon Sep 17 00:00:00 2001 From: cb341 Date: Thu, 9 Jul 2026 21:29:07 +0200 Subject: [PATCH 09/34] do stuff --- _posts/2026-06-28-lean.md | 17 ++++++++++------- 1 file changed, 10 insertions(+), 7 deletions(-) diff --git a/_posts/2026-06-28-lean.md b/_posts/2026-06-28-lean.md index 08f5ad3..0a10cd1 100644 --- a/_posts/2026-06-28-lean.md +++ b/_posts/2026-06-28-lean.md @@ -86,14 +86,14 @@ Now we can define addition. (Using axioms) It isn't enough to define addition for speicfic numbers, like let's say `6 + 1 = 7`. We need to define addition for ALL numbers. We do this using induction. -$$ -\forall a,b \in \mathbb{N} : a + b = ? -$$ - Here we fix the variable $a$ and check the our definition of natrual numbers from before. B is either zero, or a successor of a natural number. +$$ +\underline{\Large{\forall a,b \in \mathbb{N} : a + b = \text{?}}} +$$ + $$ \forall a,b \in \mathbb{N} : a + b \stackrel{\mathrm{def}}{=} \begin{cases} @@ -103,7 +103,7 @@ succ(a + d) & b = \operatorname{succ}(d) \\ $$ $$ -\Large{\text{Example: } \underline{1 + 2 = 3}} +\text{Example: } \underline{1 + 2 = 3} $$ @@ -131,10 +131,13 @@ $$ \end{array} $$ -### Inequalities? $$ -(a,b : \mathbb{N} ) a ≤ b \stackrel{\mathrm{def}}{\implies} ∃ (c : ℕ), b = a + c +\underline{\Large{\forall a,b \in \mathbb{N} : a \le b \text{ ?}}} +$$ + +$$ +\forall a, b \in \mathbb{N} : a \le b \stackrel{\mathrm{def}}{\iff} ∃ (c : ℕ), b = a + c $$ ![graph](/assets/blog/numberline.png) From 61d97d6de28bc7f04a638f9c25e318b24d57f224 Mon Sep 17 00:00:00 2001 From: cb341 Date: Thu, 9 Jul 2026 21:48:47 +0200 Subject: [PATCH 10/34] wip --- _posts/2026-06-28-lean.md | 28 ++++++++++++++++++++++++++++ 1 file changed, 28 insertions(+) diff --git a/_posts/2026-06-28-lean.md b/_posts/2026-06-28-lean.md index 0a10cd1..4f125f4 100644 --- a/_posts/2026-06-28-lean.md +++ b/_posts/2026-06-28-lean.md @@ -144,6 +144,34 @@ $$ ## Prooving +In the following proof we use the following tactics: + +rewrite := X = Y +induction := break statement down into base case, successor + +CASES + +$$ + +h : \text{left} \lor \text{right} + +\xrightarrow{\mathrm{cases}} + +\begin{cases} + \text{inl} : \text{left} \\ + \text{inr} : \text{right} +\end{cases} + + + \\ + +$$ + +$\text{left} \lor \text{right}$ into $\text{left}$ and $\text{right}$ + +$\forall a, b \in \mathbb{N} : a \le b \stackrel{\mathrm{def}}{\iff} ∃ (c : ℕ), b = a + c$ + + ```lean theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by induction y with From 515166e566a75134e1c694231c500bfafa3bc1ad Mon Sep 17 00:00:00 2001 From: cb341 Date: Thu, 9 Jul 2026 22:45:30 +0200 Subject: [PATCH 11/34] refactor --- _posts/.2026-06-28-lean.md.swp | Bin 12288 -> 0 bytes _posts/2026-06-28-lean.md | 89 ++++++++++++++++++++++++--------- 2 files changed, 65 insertions(+), 24 deletions(-) delete mode 100644 _posts/.2026-06-28-lean.md.swp diff --git a/_posts/.2026-06-28-lean.md.swp b/_posts/.2026-06-28-lean.md.swp deleted file mode 100644 index 23eba83d4e5878a2c67d5a27ad3f0861f84a09fc..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 12288 zcmeI2&x;&I6vt~l`70`V7GGu;vx_~wvzr)`(HOH?C4uZJ=9k71Q{7!NQ`w%bv8#G# zCd(!ee_%koc@Y#85B@j`3W10L5pSLZK~da;F(>^4MEzFvOx#Tr$#J0;KHF0@Rd3$= zzV`}-si=1B-~#OqXBqzPVk~;>$y2YN`g80;KE~o``;Hkq|3828Jd?V*z~eckma0$X zDoj<()9ZGfaR#TmxjH3cqo>_4Y)R8DqEM>o7rIJgdbT#Zvs&9(o!wPU1<%4xa-AhV z;T7--+>io=X;yb_VcTbCYIfgvWh>ot_t6_N;TL-ayaHYUuYgy;E8rFI3U~#)0-IX_ z(;Z`H5yDNwFrFSh$F6w}hyLLe@CtYZyaHYUuYgy;E8rFI3U~#)0$u^Hz$R3HFI@k>eT)u;`@4#2!WAG995TsxUw80Z# zKbQjy{I!L#-@$L-S8y484PFJWfODV$o(0cjFs#x8*`z(w#Lco)0@`k)7vKpV8c zBj921%WaIk1KtL2fmLuC48VQh%B}DLtbw1wkKhvc99#sSflt9F;2by$&VWU*0JZ`G z_ky3G;}76_;56NYHQ`IMw%zj}z`+s%4qA7{lBo4eG&V>N4ZBCwW5TMagpE>g0= z$0>~L9$m!oWHN=F_7zRqxn#pssdJt1<1rE}J6fxD(6r_-1UNagqx zYguTG*Q$wO#!Zp)l(M1|i9BQnL@un)P1#2?Vk|MUggzTWrnazvHRMBexmp$DTGH%LvWHg{uE(uD7*``VJ0iE256wJWA zhfX4bMb*1n8D%+)ybEdasPlpMrRwM@m#K!xl9iZip;sAm8NI(@P);i%)nb(z4PrGK zPDNN&PO){hyN4Q)Xvu73#n$KJJZ=yDU6;)cv>SI?0kEks6K+e$zsV-Smw zjv&)+h!JVSbAttjY9X5|Oet|G=Rg?jj8zur!VD{08mqWX-$u_@YQan}8*C4D1P=z( zND~FCja8TDYn|h$y0T4N?%^j-Q>x&8G?Kq8DpcvURf=a>r)WLbreYbvApe{;8r+NT zUozJQZGZz}-sZ{`kw#m%a&&E}5Lwylqnc|WJgquc14AuoQjh?}ZM&hz2g4rj2ZQ1H zfk)pRFaw+`9}xcIu<5%fkEjGSgl_6dM1!DkO4yUi*)qzYRaBm!Kg-@(*NFCPVK7uj zEoyE3M1$qz&K?vq!wFWmRv??`OI#vHoa1y1x6q>^<@vx3PaN2HoaT7E)XG(X)~!BD ziEK8pDMJ>!aK@GrTFoAo0{>)O+L@QS%{EeGm|UEs?{p9d*DJ|2xfRo*Xg+m zjG)tD1#zu2SqZ4pwAn!TGK*ywAyuAdiNJ@*uI<-4Rb0AIdZMe*$Ao6@Y_tLmO`8tk metd-`+=x2;b8Vg(X;REo*7-{j(v~ShkS^)*WihBYzyAh*Lt-KT diff --git a/_posts/2026-06-28-lean.md b/_posts/2026-06-28-lean.md index 4f125f4..b4d1f2b 100644 --- a/_posts/2026-06-28-lean.md +++ b/_posts/2026-06-28-lean.md @@ -107,31 +107,40 @@ $$ $$ - $$ -\begin{array}{rl} - & \textbf{Axioms} \\ -\mathrm{i.} & 1 \stackrel{\mathrm{def}}{=} succ(0) \\ -\mathrm{ii.} & 2 \stackrel{\mathrm{def}}{=} succ(1) \\ -\mathrm{iii.} & 3 \stackrel{\mathrm{def}}{=} succ(2) \\ -\mathrm{iv.} & a + 0 \stackrel{\mathrm{def}}{=} a \\ -\mathrm{v.} & a + \operatorname{succ}(b) \stackrel{\mathrm{def}}{=} \operatorname{succ}(a+b) \\ +\newcommand{\mathcolorbox}[2]{\colorbox{#1}{$\displaystyle #2$}} + +\begin{array}{ll} +& \textbf{Axioms} \\ +\colorbox{#fff3cd}{i.} & 1 \stackrel{\mathrm{def}}{=} \operatorname{succ}(0) \\ +\colorbox{#cfe2ff}{ii.} & 2 \stackrel{\mathrm{def}}{=} \operatorname{succ}(1) \\ +\colorbox{#e2d9f3}{iii.} & 3 \stackrel{\mathrm{def}}{=} \operatorname{succ}(2) \\ +\colorbox{#f8d7da}{iv.} & a+0 \stackrel{\mathrm{def}}{=} a \\ +\colorbox{#d1e7dd}{v.} & a+\operatorname{succ}(b) \stackrel{\mathrm{def}}{=} \operatorname{succ}(a+b) \end{array} - \begin{array}{l|l} +\qquad +\begin{array}{l|l} \textbf{Statements} & \textbf{Reasons} \\ \hline - 1 + 2 & \text{Given} \\ - 1 + \operatorname{succ}(1) & \mathrm{ii.} \\ - \operatorname{succ}(1 + 1) & \mathrm{v.} \\ - \operatorname{succ}(1 + \operatorname{succ}(0)) & \mathrm{i.} \\ - \operatorname{succ}(\operatorname{succ}(1 + 0)) & \mathrm{v.} \\ - \operatorname{succ}(\operatorname{succ}(1)) & \mathrm{iv.} \\ - \operatorname{succ}(2) & \mathrm{ii.'} \\ - 3 & \mathrm{iii.'} \\ - \end{array} + 1 + 2 + & \text{Given} \\ + 1 + \mathcolorbox{#cfe2ff}{\operatorname{succ}(1)} + & \mathcolorbox{#cfe2ff}{\mathrm{\rightarrow ii.}} \\ + \mathcolorbox{#d1e7dd}{\operatorname{succ}(1 + 1))} + & \mathcolorbox{#d1e7dd}{\mathrm{\rightarrow v.}} \\ + \operatorname{succ}(1 + \mathcolorbox{#fff3cd}{\operatorname{succ}(0)}) + & \mathcolorbox{#fff3cd}{\mathrm{\rightarrow i.}} \\ + \operatorname{succ}(\operatorname{succ}(\mathcolorbox{#d1e7dd}{1 + 0})) + & \mathcolorbox{#d1e7dd}{\mathrm{\rightarrow v.}} \\ + \operatorname{succ}(\operatorname{succ}(\mathcolorbox{#f8d7da}{1})) + & \mathcolorbox{#f8d7da}{\mathrm{\rightarrow iv.}} \\ + \operatorname{succ}(\mathcolorbox{#cfe2ff}{2}) + & \mathcolorbox{#cfe2ff}{\mathrm{\leftarrow ii.}} \\ + \mathcolorbox{#e2d9f3}{3} + & \mathcolorbox{#e2d9f3}{\mathrm{\leftarrow iii.}} +\end{array} $$ - $$ \underline{\Large{\forall a,b \in \mathbb{N} : a \le b \text{ ?}}} $$ @@ -144,22 +153,48 @@ $$ ## Prooving +### Or + +| | $a = false$ | $b = true$ | +| --- | --- | --- | +| $b = false$ | false | true | +| $b = true$ | true | true | + + +### Exists +### Induction +### Rewriting + In the following proof we use the following tactics: rewrite := X = Y induction := break statement down into base case, successor -CASES +$$ +h: a = b, g: a \xrightarrow{\mathrm{rewrite[h]}} g: b +$$ $$ +h : y -h : \text{left} \lor \text{right} +\xrightarrow{\mathrm{induction(d,hd)}} + +\begin{cases} + \text{hd} : y = 0 \\ + \text{hd} : y = succ(d), d \in \mathbb{N} +\end{cases} + +$$ + +$$ + +h : \text{a} \lor \text{b} \xrightarrow{\mathrm{cases}} \begin{cases} - \text{inl} : \text{left} \\ - \text{inr} : \text{right} + \text{inl} : \text{a} \\ + \text{inr} : \text{b} \end{cases} @@ -167,7 +202,13 @@ h : \text{left} \lor \text{right} $$ -$\text{left} \lor \text{right}$ into $\text{left}$ and $\text{right}$ +$$ +h: \text{a} \lor \text{b} \xrightarrow{\mathrm{left}} h: a +$$ + +$$ +h: \text{a} \lor \text{b} \xrightarrow{\mathrm{right}} h: b +$$ $\forall a, b \in \mathbb{N} : a \le b \stackrel{\mathrm{def}}{\iff} ∃ (c : ℕ), b = a + c$ From fa0f94ad3f4cc0dcf2384b0206813f0f8bef21db Mon Sep 17 00:00:00 2001 From: cb341 Date: Thu, 9 Jul 2026 22:47:02 +0200 Subject: [PATCH 12/34] update --- _posts/2026-06-28-lean.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2026-06-28-lean.md b/_posts/2026-06-28-lean.md index b4d1f2b..bd7d2b5 100644 --- a/_posts/2026-06-28-lean.md +++ b/_posts/2026-06-28-lean.md @@ -126,7 +126,7 @@ $$ & \text{Given} \\ 1 + \mathcolorbox{#cfe2ff}{\operatorname{succ}(1)} & \mathcolorbox{#cfe2ff}{\mathrm{\rightarrow ii.}} \\ - \mathcolorbox{#d1e7dd}{\operatorname{succ}(1 + 1))} + \mathcolorbox{#d1e7dd}{\operatorname{succ}(1 + 1)} & \mathcolorbox{#d1e7dd}{\mathrm{\rightarrow v.}} \\ \operatorname{succ}(1 + \mathcolorbox{#fff3cd}{\operatorname{succ}(0)}) & \mathcolorbox{#fff3cd}{\mathrm{\rightarrow i.}} \\ From f10937a4c2d7ccb161c2c44e37fcb4f27b7a8b04 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 11:35:34 +0200 Subject: [PATCH 13/34] wip --- _posts/2026-06-28-lean-todos.md | 75 +++++++++++++++++++++++++++++++++ _posts/2026-06-28-lean.md | 40 ++++++++++++++++-- 2 files changed, 112 insertions(+), 3 deletions(-) create mode 100644 _posts/2026-06-28-lean-todos.md diff --git a/_posts/2026-06-28-lean-todos.md b/_posts/2026-06-28-lean-todos.md new file mode 100644 index 0000000..fd4ca96 --- /dev/null +++ b/_posts/2026-06-28-lean-todos.md @@ -0,0 +1,75 @@ +--- +title: "TITLE" +date: 2026-09-05 +description: "First steps in formal mathematics with Lean" +tags: ["theoretical mathematics", "first principles"] +math: true +--- + +Context: I am studying computer science at ZHAW. +I have noticed that I enjoy mathematics a lot but that I am missing foundations to do understand mathematics at a deeper level. + +I'll be studying at mathematics in hagen in parallel to my part time zhaw curiculum instead of working as SWE. + +I have started learning lean in June 2026. +I have never had the time to do a writeup. + +My first steps in lean were some months ago but I still wanted to share my first steps. + +At zhaw i have really enjoyed linear algebra and analysis and would like to push it further by starting at the very very basics. + +I am folloiwng "How to prove it", "Elyssa intro to metaphysciss', "Axler linear algebra", "ETH discrete maths script", NNG4, set theory game, lin alg game.. + + +Seam a bit futther along now buyt that shouldn't be of much relevance. + +Around end of june 2026 i have completed natural numbers game +Wanted to share my findings, deriving a property of natural numbers form first princieples in Lean. + +What i want to show is a short introduction to Lean form the perspective of computers cience , how even elementary concepts can be approached rigorously + + +THE PROOF + +- We need to go over syntacrc things first, define notation + +Logical connectives +- \or , ∧ , → + +Quantors +- ∀ : allquantor +- ∃ : existence quantor + +Then we probably need to go over the basics of proofs no? + +Like in HTPi + + +Proofs of form + +``` +Let x be arbitrary . + Suppose P (x) is true. + [Proof of Q(x) goes here.] + Thus, if P (x) then Q(x) . +Thus, for all x, if P (x) then Q(x) . +``` + + + +MY SINGLE FILE LEAN SOLUTION +INCLUDING ALL THOREMS DERIVED FROM FIRST PRINCIPLES +- https://tinyurl.com/4tr5uc7c + + +FURTHER READING +- Von neuman universe +- Metaphysics introduction +- https://leanprover-community.github.io/learn.html +- How to Prove it supplementary lean course in additoon to paper: https://djvelleman.github.io/HTPIwL/ +- Peano ETH: https://people.math.ethz.ch/~halorenz/4students/LogikGT/Ch08.pdf +- Peano https://www.maths.tcd.ie/~odunlain/u11602/online_notes/pdf_peano.pdf +- Original Peano arithmetic paper (1889), Latin - i don't actually know latin but if I could I'd read this: https://archive.org/details/arithmeticespri00peangoog/page/n10/mode/2up +- 1-1 translation of the 1889 paper to english: https://www.scribd.com/document/678192145/Peano +- Ueli Maurer ETH Diskrete Mathematik Script: https://crypto.ethz.ch/teaching/DM23/ln/DM23_LNss-tablet.pdf + diff --git a/_posts/2026-06-28-lean.md b/_posts/2026-06-28-lean.md index bd7d2b5..aed0c1b 100644 --- a/_posts/2026-06-28-lean.md +++ b/_posts/2026-06-28-lean.md @@ -6,14 +6,28 @@ tags: ["theoretical mathematics", "first principles"] math: true --- + +# PROPER OUTLINE + +INTRO +- ASSUMPTIONS + + +--- + + + + ## Intro + + ### Discrete Mathematics ### ZHAW Background -> different approach to mathematics ### Vocabulary - +$\forall$: for all ∃: there exists ≤: less than or equal to @@ -40,7 +54,6 @@ $$ \colorbox{#fff3cd}{$\displaystyle \forall x,y \in \mathbb{N} : x \le y \lor y \le x$} $$ - $$ \fbox{ $\forall (x,y : ℕ) : x ≤ y ∨ y ≤ x$ @@ -49,7 +62,7 @@ $$ ![graph](/assets/blog/dependency_graph.svg) - +*A transient dependency graph showing Before we do that we need to get some definitions out of the way. @@ -102,6 +115,27 @@ succ(a + d) & b = \operatorname{succ}(d) \\ \end{cases} $$ +
+ + There are many alternate ways to represnet natural numbers. Most notably the Zermelo / von Neuman ordinals [^1]: + + + + Zermelo: + $0=\emptyset,\quad 1=\lbrace\emptyset\rbrace,\quad 2=\lbrace\lbrace\emptyset\rbrace\rbrace,\quad\ldots$ + + von Neumann: + $0=\emptyset,\quad 1=\lbrace\emptyset\rbrace,\quad 2=\lbrace\emptyset,\lbrace\emptyset\rbrace\rbrace,\quad\ldots$ + + + +
+ + +[^1]: https://www.researchgate.net/publication/228574851_von_Neumann_universe_A_perspective + + + $$ \text{Example: } \underline{1 + 2 = 3} $$ From 37c64a14f060ad8b2a4559653d803f02434406fd Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 11:42:43 +0200 Subject: [PATCH 14/34] wip --- _posts/2026-06-28-lean-todos.md | 24 +++++++++++++++++++----- 1 file changed, 19 insertions(+), 5 deletions(-) diff --git a/_posts/2026-06-28-lean-todos.md b/_posts/2026-06-28-lean-todos.md index fd4ca96..8af6d7c 100644 --- a/_posts/2026-06-28-lean-todos.md +++ b/_posts/2026-06-28-lean-todos.md @@ -1,5 +1,5 @@ --- -title: "TITLE" +title: "So You Think You Know ≤ ?" date: 2026-09-05 description: "First steps in formal mathematics with Lean" tags: ["theoretical mathematics", "first principles"] @@ -29,13 +29,16 @@ Wanted to share my findings, deriving a property of natural numbers form first p What i want to show is a short introduction to Lean form the perspective of computers cience , how even elementary concepts can be approached rigorously -THE PROOF +THE PROOF - PREFACE - We need to go over syntacrc things first, define notation Logical connectives - \or , ∧ , → +- Deifnition of <= +- Defintion of N, defintion of include in N + Quantors - ∀ : allquantor - ∃ : existence quantor @@ -47,6 +50,8 @@ Like in HTPi Proofs of form +_Proof structure has been introduced in the preface of how to prove it (footnote insert) : + ``` Let x be arbitrary . Suppose P (x) is true. @@ -56,10 +61,19 @@ Thus, for all x, if P (x) then Q(x) . ``` +THE THEOREM + + +$$ +\forall x,y \in \mathbb{N} : x \le y \lor y \le x +$$ + + + + + -MY SINGLE FILE LEAN SOLUTION -INCLUDING ALL THOREMS DERIVED FROM FIRST PRINCIPLES -- https://tinyurl.com/4tr5uc7c +MY SINGLE FILE LEAN SOLUTION INCLUDING ALL THOREMS DERIVED FROM FIRST PRINCIPLES: FURTHER READING From 7c5701eae90580203e9c1e3b09984da12be6c4e0 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 13:19:15 +0200 Subject: [PATCH 15/34] w --- _posts/2026-06-28-lean-todos.md | 139 +++++++++++++++- assets/blog/lean_dependency_graph.svg | 2 + assets/blog/lean_state_overview.png | Bin 0 -> 152532 bytes single_file.lean | 230 ++++++++++++++++++++++++++ 4 files changed, 370 insertions(+), 1 deletion(-) create mode 100644 assets/blog/lean_dependency_graph.svg create mode 100644 assets/blog/lean_state_overview.png create mode 100644 single_file.lean diff --git a/_posts/2026-06-28-lean-todos.md b/_posts/2026-06-28-lean-todos.md index 8af6d7c..4108cf9 100644 --- a/_posts/2026-06-28-lean-todos.md +++ b/_posts/2026-06-28-lean-todos.md @@ -6,6 +6,8 @@ tags: ["theoretical mathematics", "first principles"] math: true --- +STYLE: Stick to writing style of all other blogs, threads. concise, lowercase. Make sure to use footnotes, cite sources diligently. Precision, less fluff. Find inspiration in ~dani/.codex/skills/ + Context: I am studying computer science at ZHAW. I have noticed that I enjoy mathematics a lot but that I am missing foundations to do understand mathematics at a deeper level. @@ -26,7 +28,13 @@ Seam a bit futther along now buyt that shouldn't be of much relevance. Around end of june 2026 i have completed natural numbers game Wanted to share my findings, deriving a property of natural numbers form first princieples in Lean. -What i want to show is a short introduction to Lean form the perspective of computers cience , how even elementary concepts can be approached rigorously +What i want to show is a short introduction to Lean form the perspective of computers cience , how even elementary concepts can be approached rigorously. + + +The blog article should show a breif introduction to prepositional logic, lean and proof strucutre. + +there are other drafts in this repo (lean/todo..) +they include colorful latex constructions, look into them first. THE PROOF - PREFACE @@ -36,6 +44,54 @@ THE PROOF - PREFACE Logical connectives - \or , ∧ , → +We introuce tautology: T, contradiction: (flipped) T + +→ ocnnectieve is not explained well purely by truth table. there are many philosophical questions one may ask aboiut the truth or falsenss of \r . + +ETH discrete math i don't like. +Many profos are just "compare truth tables and see that expressions are the same" + +De Morgan’s laws +¬ +(P ∧ Q) is equivalent to ¬ P ∨ ¬Q . +¬ +(P ∨ Q) is equivalent to ¬ P ∧ ¬Q . +Commutative laws +P ∧ Q is equivalent to Q ∧ P. +P ∨ Q is equivalent to Q ∨ P. +Associative laws +P ∧(Q ∧ R) is equivalent to(P ∧ Q) ∧ R. +P ∨(Q ∨ R) is equivalent to(P ∨ Q) ∨ R. +Idempotent laws +P ∧ P is equivalent to P. +P ∨ P is equivalent to P. +Distributive laws +Absorption laws +P ∧(Q ∨ R) is equivalent to(P ∧ Q) ∨(P ∧ R) . +P ∨(Q ∧ R) is equivalent to(P ∨ Q) ∧(P ∨ R) . +P ∨(P ∧ Q) is equivalent to P. +P ∧(P ∨ Q) is equivalent to P. +Double Negation law +¬¬ P is equivalent to P. + +DEFINE: converse, contrapositive,statement,theorem,empty set, set, element, bound variable, substitution +PHILLOSPHY: number,greater,value + + +How to prove it - P52 (Sentential Logic) + +Statements that mean P → Q come up very often in mathematics, but +sometimes they are not written in the form “If P then Q . +” Here are a few other +ways of expressing the idea P → Q that are used often in mathematics: +P implies Q . +Q , if P. +P only if Q . +P is a sufficient condition for Q . +Q is a necessary condition for P. + + + - Deifnition of <= - Defintion of N, defintion of include in N @@ -60,6 +116,35 @@ Let x be arbitrary . Thus, for all x, if P (x) then Q(x) . ``` +What is a tacitc? We probably want lexygraphic defintiosn here. + +https://lean-lang.org/theorem_proving_in_lean4/Tactics/ +> A proof term is a representation of a mathematical proof; tactics are commands, or instructions, that describe how to build such a proof. + + +We need to define how proofs work based on logical connectives explained before. + +We solve proofs with Givens, GOal table. + +In case of Goal: P->Q, we can extract P as givens, Q as goal. + +P->Q iff ¬P∨Q iff Q∨¬P iff ¬Q→¬P + +So we can proove contrapositive +Goal:P->Q ; Givens: \¬Q, Goal: \notP + + +If given goal: ∀ x - Px, we can apply universal instantiation, Givens: x0 Goal: Px +If given givens ∃ x - Px we can givens: x, Px +If goal ∃x we can `use` to pass arg +If givens is \All we can use choose any x statisfying condiiton. + + + +The prof should traverse the dependnecy graph one step at a time. questioning what is a number, what makes a number natural. what ways are there to express / define natural numbers? zf/van neuman. we do zero/succ. + +Explain tactics (rw as substitute, use as existential instantiation) + THE THEOREM @@ -70,6 +155,58 @@ $$ +MY PROOF + +```lean +theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by + induction y with + | zero => + right + exact zero_le x + | succ d hd => + cases hd with + | inl hl => + cases' hl with c hc + rewrite[hc] + left + rewrite[succ_eq_add_one] + use c + 1 + rewrite[← add_assoc] + rfl + | inr hr => + cases' hr with c hc + cases c with + | zero => + rewrite[zero_eq_0] at hc + rewrite[add_zero d] at hc + left + rewrite[hc] + exact le_succ_self d + | succ a => + rewrite[add_succ] at hc + right + rewrite[hc] + use a + rewrite[succ_add] + rfl +``` + + + +_ASSETS_ + +lean_dependency_graph shows the all the theorems that are required to prove the theorem. +Blue are the defintions +Green are the arithmetic +Orange are what follows + +we start at N, we end at theorem. + + +lean_state_overview is how the lean state evolves with each tactic applied. + + + diff --git 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theorems along the +-- way and ending at QED. +-- +-- I use the natural number definitions from the natural numbers +-- Lean game as a basis +-- +-- https://adam.math.hhu.de/#/g/leanprover-community/nng4 +-- +-- And proove theorems on top of that step by step. +import Mathlib.Tactic.Have +import Mathlib.Tactic.Contrapose +import Mathlib.Tactic.ApplyAt +import Mathlib.Tactic.Cases +import Mathlib.Tactic.NthRewrite +import Mathlib.Tactic.Tauto +import Lean.Elab.Tactic.Basic +import Lean.Elab.Tactic.Induction +import Batteries.Tactic.OpenPrivate +import Batteries.Data.List.Basic +import Mathlib.Lean.Expr.Basic +import Mathlib.Tactic.Cases +import Lean.Meta.Tactic.Refl +import Lean.Elab.Tactic.Basic +import Mathlib.Lean.Expr.Basic +import Lean.Elab.Tactic.Basic +import Lean.Elab.Tactic.Rewrite +import Mathlib.Tactic.Use +import Mathlib.Lean.Meta.Simp + +-- The following definitions and axioms were taken over / modified from +-- the Apache Licensed NNG4 environment I am familiar with. +-- I do not take ownership over the folllowing definitions +-- but I do acknowledge modifications. +-- +-- https://github.com/leanprover-community/NNG4 +-- +-- Disclaimer: This is not the entirity of the NNG4 env. +-- DX features such as pretty printing, +-- formatting a[118;1:3und more intuitive tactics were not taken over. + +-- BEGIN NNG4 -- +inductive MyNat where +| zero : MyNat +| succ : MyNat → MyNat +attribute [pp_nodot] MyNat.succ +notation (name := MyNatNotation) (priority := 1000000) "ℕ" => MyNat + +namespace MyNat + +instance : Inhabited MyNat where + default := MyNat.zero + +def ofNat (x : Nat) : MyNat := + match x with + | Nat.zero => MyNat.zero + | Nat.succ b => MyNat.succ (ofNat b) + +def toNat (x : MyNat) : Nat := + match x with + | MyNat.zero => Nat.zero + | MyNat.succ b => Nat.succ (toNat b) + +instance instofNat {n : Nat} : OfNat MyNat n where + ofNat := ofNat n + +instance : ToString MyNat where + toString p := toString (toNat p) + +theorem zero_eq_0 : MyNat.zero = 0 := rfl + +def one : MyNat := MyNat.succ 0 + +def pred : ℕ → ℕ +| 0 => 37 -- random (garbage) value according to NNG4 +| succ n => n + +lemma pred_succ (n : ℕ) : pred (succ n) = n := rfl + +def is_zero : ℕ → Prop +| 0 => True +| succ _ => False + +lemma is_zero_zero : is_zero 0 = True := rfl +lemma is_zero_succ (n : ℕ) : is_zero (succ n) = False := rfl + +theorem zero_ne_succ (a : ℕ) : 0 ≠ succ a := by + intro h + rewrite[← is_zero_succ a] + rewrite[← h] + rewrite[is_zero_zero] + trivial + +theorem one_eq_succ_zero : 1 = succ 0 := by rfl +theorem two_eq_succ_one : 2 = succ 1 := by rfl +theorem three_eq_succ_two : 3 = succ 2 := by rfl +theorem four_eq_succ_three : 4 = succ 3 := by rfl + +-- addition +opaque add : MyNat → MyNat → MyNat + +instance instAdd : Add MyNat where + add := MyNat.add + +axiom add_zero (a : MyNat) : a + 0 = a +axiom add_succ (a d : MyNat) : a + (succ d) = succ (a + d) + +-- inequality +def le (a b : ℕ) := ∃ (c : ℕ), b = a + c +instance : LE MyNat := ⟨MyNat.le⟩ +-- END NNG4 -- + +-- What follow are all the theorems required to proove the totality of ℕ. +-- The Lean4Game environment provided me with placeholders: +-- +-- theorem succ_eq_add_one n : succ n = n + 1 := by +-- sorry +-- +-- The theorems were proven by me as an excercise. + +-- BEGIN DANI -- + +theorem succ_eq_add_one n : succ n = n + 1 := by + rewrite[one_eq_succ_zero] + rewrite[add_succ] + rewrite[add_zero] + rfl + +theorem zero_add (n : ℕ) : 0 + n = n := by + induction n with + | zero => + rewrite[zero_eq_0] + rewrite[add_zero] + rfl + | succ d hd => + rewrite[add_succ] + rewrite[hd] + rfl + +theorem succ_add (a b : ℕ) : succ a + b = succ (a + b) := by + induction b with + | zero => + rewrite[zero_eq_0] + rewrite[add_zero] + rewrite[add_zero] + rfl + | succ d hb => + rewrite[add_succ] + rewrite[hb] + rewrite[add_succ] + rfl + +theorem add_comm (a b : ℕ) : a + b = b + a := by + induction b with + | zero => + rewrite[zero_eq_0] + rewrite[add_zero] + rewrite[zero_add] + rfl + | succ n hn => + rewrite[add_succ] + rewrite[succ_add] + rewrite[hn] + rfl + +theorem add_assoc (a b c : ℕ) : a + b + c = a + (b + c) := by + induction b with + | zero => + rewrite[zero_eq_0] + rewrite[add_zero] + rewrite[zero_add] + rfl + | succ n hn => + rewrite[add_succ] + rewrite[succ_add] + rewrite[succ_add] + rewrite[add_succ] + rewrite[hn] + rfl + +theorem zero_le (x : ℕ) : 0 ≤ x := by + use x + rewrite[zero_add] + rfl + +theorem le_succ_self (x : ℕ) : x ≤ succ x := by + use 1 + exact succ_eq_add_one x + +-- This is the climax - the proof we have been approaching thus far +theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by + induction y with + | zero => + right + exact zero_le x + | succ d hd => + cases hd with + | inl hl => + cases' hl with c hc + rewrite[hc] + left + rewrite[succ_eq_add_one] + use c + 1 + rewrite[← add_assoc] + rfl + | inr hr => + cases' hr with c hc + cases c with + | zero => + rewrite[zero_eq_0] at hc + rewrite[add_zero d] at hc + left + rewrite[hc] + exact le_succ_self d + | succ a => + rewrite[add_succ] at hc + right + rewrite[hc] + use a + rewrite[succ_add] + rfl + +end MyNat From d2b2af8bc51400c3674da655f50afdbcd22fb2c8 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 13:51:41 +0200 Subject: [PATCH 16/34] wip --- _posts/2026-06-28-lean_real.md | 418 ++++++++++++++++++++++ assets/blog/lean_numberline_two_cases.svg | 73 ++++ 2 files changed, 491 insertions(+) create mode 100644 _posts/2026-06-28-lean_real.md create mode 100644 assets/blog/lean_numberline_two_cases.svg diff --git a/_posts/2026-06-28-lean_real.md b/_posts/2026-06-28-lean_real.md new file mode 100644 index 0000000..447e678 --- /dev/null +++ b/_posts/2026-06-28-lean_real.md @@ -0,0 +1,418 @@ +--- +title: "Mathematics but not handwavey ?" +date: 2026-06-28 +description: "Proving that any two natural numbers compare, from the definition of a natural number upwards, in Lean." +tags: ["theoretical mathematics", "first principles"] +math: true +--- + +paper proofs have no compiler. i went looking for one and found Lean. + +no AI was used for the Lean here, and none for the maths either: the definitions, the axioms i picked, the proof strategy and the dependency graph are mine. AI was used for phrasing. + +## The theorem + +$$ +\forall x,y \in \mathbb{N} : x \le y \lor y \le x +$$ + +any two natural numbers compare. one of them is at most the other. + +take 6 and 7. then $(6 \le 7) \lor (7 \le 6)$ is $T \lor F$, so $T$. take 1 twice. then $(1 \le 1) \lor (1 \le 1)$ is $T \lor T$, so $T$. + +you already believe this. the article is about what believing it costs. + +those two examples cover two pairs, and $\mathbb{N}^2$ is infinite. the $\forall$ is the whole difficulty. + +right now $\le$, $+$ and $\mathbb{N}$ are all undefined, so that line is notation. the rest of the article pays the debt in order. + +## What the proof rests on + +![Dependency graph for le_total](/assets/blog/lean_dependency_graph.svg) + +lavender nodes are definitions: $\mathbb{N}$ itself, the two clauses of addition, and $1 = \operatorname{succ}(0)$. mint nodes are the arithmetic that follows, including commutativity and associativity. peach nodes are the order results, ending in the theorem. + +the graph is the table of contents. we start at $\mathbb{N}$ at the top and walk down to `le_total`, and every node below is a section or a paragraph. nothing gets used before its node has been visited, so you can check the article against the picture as you go. + +## Notation + +three symbols appear in the proof. + +$\lor$ is disjunction. $P \lor Q$ holds when at least one side holds. + +| $P$ | $Q$ | $P \lor Q$ | +| --- | --- | --- | +| false | false | false | +| false | true | true | +| true | false | true | +| true | true | true | + +$\exists$ is existence. $\exists x, P(x)$ is discharged by producing one witness and showing $P$ holds of it. + +$\forall$ is universal quantification. $\forall x, P(x)$ needs a proof that works for an arbitrary $x$, and the proof may not look at which value it got. + +the rest of propositional logic is assumed. [^velleman] a refresher, if you want one: + +
+Prerequisites + +$P \land Q$ holds when both sides hold. + +$\lnot P$ holds when $P$ does not. + +$P \to Q$ holds unless $P$ holds and $Q$ fails. the two rows where $P$ is false both come out true, which is worth its own article. + +$P \leftrightarrow Q$ is $P \to Q$ together with $Q \to P$. it is the $:\Leftrightarrow$ used to define $\le$ below. + +$\top$ is the statement that always holds, $\bot$ the statement that never does. + +the converse of $P \to Q$ is $Q \to P$, a different statement. the contrapositive is $\lnot Q \to \lnot P$, the same statement. + +the standard equivalences are the De Morgan, commutative, associative, idempotent, distributive and absorption laws, plus double negation. the two De Morgan laws: + +$$ +\lnot(P \land Q) \Leftrightarrow \lnot P \lor \lnot Q +\qquad +\lnot(P \lor Q) \Leftrightarrow \lnot P \land \lnot Q +$$ + +
+ +[^velleman]: Daniel J. Velleman, *How To Prove It: A Structured Approach*, 3rd edition, Cambridge University Press, 2019, sections 1.1 and 1.2. Maurer's ETH script covers the same ground from the truth-table side in chapter 2: + +## Definitions + +### Natural numbers + +$\mathbb{N} = \lbrace 0,1,2,3,\ldots \rbrace$ is a listing. the ellipsis carries the definition, which means there is no definition yet. + +a natural number is either zero or the successor of a natural number. [^peano] + +$$ +n \in \mathbb{N} \stackrel{\mathrm{def}}{=} +\begin{cases} +0 & \text{base case} \\ +\operatorname{succ}(d) & d \in \mathbb{N} +\end{cases} +$$ + +[^peano]: Giuseppe Peano, *Arithmetices principia, nova methodo exposita*, 1889. The Latin original is on archive.org: + +there is more than one way to build the naturals out of sets. [^ordinals] + +
+Other encodings of the same numbers + +Zermelo: + +$$ +0=\emptyset,\quad 1=\lbrace\emptyset\rbrace,\quad 2=\lbrace\lbrace\emptyset\rbrace\rbrace,\quad\ldots +$$ + +von Neumann: + +$$ +0=\emptyset,\quad 1=\lbrace\emptyset\rbrace,\quad 2=\lbrace\emptyset,\lbrace\emptyset\rbrace\rbrace,\quad\ldots +$$ + +the von Neumann version has the property that each number is the set of all smaller numbers, so $n < m$ becomes $n \in m$. order comes for free. i am using zero and succ instead, which is what the Natural Number Game uses. + +
+ +[^ordinals]: + +### Our zero and Lean's zero + +the definition above introduces a constructor, written `MyNat.zero` in Lean. the character `0` is a numeral, which is what a person types. they denote the same natural number and they are not the same term. + +this matters mechanically. `rfl` closes a goal when both sides are literally the same term, and `rewrite` matches on the shape of a term. a goal reading `MyNat.zero + x` does not match a lemma stated about `0 + x`, so the two have to be connected first. `zero_eq_0` is that connection, and it is a theorem you prove rather than something the definition gives you. + +the same split shows up between `succ n` and `n + 1`, bridged by `succ_eq_add_one`. both are nodes in the dependency graph. a definition fixes which terms exist, notation is a separate layer, and the two get connected by proof. + +the proof below carries `rewrite[zero_eq_0] at hc` for exactly this reason. + +### Addition + +$6 + 1 = 7$ defines one sum. $\mathbb{N}^2$ has infinitely many, so addition is defined by recursion on the second argument, mirroring the definition of $\mathbb{N}$. + +$$ +\forall a,b \in \mathbb{N} : a + b \stackrel{\mathrm{def}}{=} +\begin{cases} +a & b = 0 \\ +\operatorname{succ}(a + d) & b = \operatorname{succ}(d) +\end{cases} +$$ + +worked on $1 + 2 = 3$. the right column names the axiom used and which direction it was applied in. $(\rightarrow)$ unfolds, replacing a name by its definition. $(\leftarrow)$ folds, recognising a definition and naming it. + +$$ +\begin{array}{ll} +& \textbf{Axioms} \\ +\colorbox{#fff3cd}{i.} & 1 \stackrel{\mathrm{def}}{=} \operatorname{succ}(0) \\ +\colorbox{#cfe2ff}{ii.} & 2 \stackrel{\mathrm{def}}{=} \operatorname{succ}(1) \\ +\colorbox{#e2d9f3}{iii.} & 3 \stackrel{\mathrm{def}}{=} \operatorname{succ}(2) \\ +\colorbox{#f8d7da}{iv.} & a+0 \stackrel{\mathrm{def}}{=} a \\ +\colorbox{#d1e7dd}{v.} & a+\operatorname{succ}(b) \stackrel{\mathrm{def}}{=} \operatorname{succ}(a+b) +\end{array} +\qquad +\begin{array}{l|l} +\textbf{Statement} & \textbf{Reason} \\ +\hline +1 + 2 & \text{given} \\ +1 + \colorbox{#cfe2ff}{$\operatorname{succ}(1)$} & \colorbox{#cfe2ff}{$\text{ii.}$} \; (\rightarrow) \\ +\colorbox{#d1e7dd}{$\operatorname{succ}(1 + 1)$} & \colorbox{#d1e7dd}{$\text{v.}$} \; (\rightarrow) \\ +\operatorname{succ}(1 + \colorbox{#fff3cd}{$\operatorname{succ}(0)$}) & \colorbox{#fff3cd}{$\text{i.}$} \; (\rightarrow) \\ +\operatorname{succ}(\operatorname{succ}(\colorbox{#d1e7dd}{$1 + 0$})) & \colorbox{#d1e7dd}{$\text{v.}$} \; (\rightarrow) \\ +\operatorname{succ}(\operatorname{succ}(\colorbox{#f8d7da}{$1$})) & \colorbox{#f8d7da}{$\text{iv.}$} \; (\rightarrow) \\ +\operatorname{succ}(\colorbox{#cfe2ff}{$2$}) & \colorbox{#cfe2ff}{$\text{ii.}$} \; (\leftarrow) \\ +\colorbox{#e2d9f3}{$3$} & \colorbox{#e2d9f3}{$\text{iii.}$} \; (\leftarrow) +\end{array} +$$ + +the same axiom gets used in both directions. which direction you pick is a choice, and picking wrong is how a rewrite fails to terminate. + +### Less than or equal + +$$ +\forall a, b \in \mathbb{N} : a \le b \stackrel{\mathrm{def}}{\iff} \exists (c : \mathbb{N}), b = a + c +$$ + +there is a gap, and the gap is itself a natural number. the second half carries the content, since there are no negative naturals available to serve as gaps. + +![Number line showing the gap c in both cases](/assets/blog/lean_numberline_two_cases.svg) + +this is the definition the Natural Number Game uses. [^nng] mathlib defines $\le$ differently, so code pasted from here into a mathlib project will not typecheck unchanged. + +[^nng]: Natural Number Game 4, by Kevin Buzzard and Mohammad Pedramfar: + +### The tactics + +a Lean proof is written as a list of tactics. the ones in this article: + +| tactic | what it does | +| --- | --- | +| `induction` | splits a natural into the zero case and the successor case, and hands you the induction hypothesis | +| `cases` | splits a disjunction hypothesis into two branches, `inl` and `inr` | +| `cases'` | unpacks an existential hypothesis into a witness and an equation | +| `left` / `right` | picks which side of a disjunction goal to prove | +| `use` | supplies a witness for an existential goal | +| `rewrite[h]` | replaces occurrences of the left side of `h` with the right side | +| `rewrite[← h]` | the same equation applied right to left | +| `rfl` | closes a goal whose two sides are the same term | +| `exact` | closes a goal with something already proved | + +`rw` is the short form of `rewrite`. `zero_eq_0`, `succ_eq_add_one` and `cases'` are Natural Number Game spellings, so pasting this into a fresh mathlib project gets you errors on the names before anything interesting. + +## One lemma, three ways + +before the theorem, a smaller one: $0 \le x$ for every natural $x$. it sits in the dependency graph as `zero_le`, and the theorem's base case consumes it. + +unfolding the definition, $0 \le x$ means $\exists c, x = 0 + c$. take $c := x$. the goal becomes $x = 0 + x$, which is `zero_add`. + +by induction on $x$: the base case is $0 \le 0$ with $c := 0$, and the step goes from $0 \le d$ to $0 \le \operatorname{succ}(d)$. longer, and it pulls more lemmas into the graph. + +in Lean, the first route is three lines. + +```lean +theorem zero_le (x : ℕ) : 0 ≤ x := by + use x + rw [zero_add] + rfl +``` + +one theorem, several proofs, each a different path through the dependency graph. Ording's *99 Variations on a Proof* takes this to its conclusion with 99 proofs of a single cubic. [^ording] + +[^ording]: Philip Ording, *99 Variations on a Proof*, Princeton University Press, 2019. + +## Tactics as state transitions + +a Lean proof has a state: the hypotheses you have, and the goal you owe. a tactic changes that state. the proof is the sequence of changes. + +the Lean documentation puts it this way: [^tactics] + +> A proof term is a representation of a mathematical proof; tactics are commands, or instructions, that describe how to build such a proof. + +[^tactics]: + +written as two columns, with `zero_le` as the example: + +| Givens | Goal | +| --- | --- | +| `x : ℕ` | `0 ≤ x` | +{: .table-equal-2} + +after unfolding the definition of $\le$: + +| Givens | Goal | +| --- | --- | +| `x : ℕ` | `∃ c, x = 0 + c` | +{: .table-equal-2} + +after `use x`: + +| Givens | Goal | +| --- | --- | +| `x : ℕ` | `x = 0 + x` | +{: .table-equal-2} + +after `rw [zero_add]`: + +| Givens | Goal | +| --- | --- | +| `x : ℕ` | `x = x` | +{: .table-equal-2} + +after `rfl`, no goals remain. + +the left column only grows and the right column only shrinks. the proof is finished when the right column is empty, which is a condition rather than a judgement call. + +the same shape, for the tactics that split the state in two. `induction` on `y`: + +| Givens | Goal | +| --- | --- | +| `x : ℕ` | `x ≤ 0 ∨ 0 ≤ x` | +{: .table-equal-2} + +| Givens | Goal | +| --- | --- | +| `x d : ℕ`, `hd : x ≤ d ∨ d ≤ x` | `x ≤ succ d ∨ succ d ≤ x` | +{: .table-equal-2} + +`cases hd` on a disjunction hypothesis, again two states: + +| Givens | Goal | +| --- | --- | +| `hl : x ≤ d` | `x ≤ succ d ∨ succ d ≤ x` | +{: .table-equal-2} + +| Givens | Goal | +| --- | --- | +| `hr : d ≤ x` | `x ≤ succ d ∨ succ d ≤ x` | +{: .table-equal-2} + +`cases' hr with c hc` on an existential hypothesis, which names the witness: + +| Givens | Goal | +| --- | --- | +| `c : ℕ`, `hc : x = d + c` | `x ≤ succ d ∨ succ d ≤ x` | +{: .table-equal-2} + +and `right`, which picks a side of the goal and discards the other: + +| Givens | Goal | +| --- | --- | +| `c : ℕ`, `hc : x = d + c` | `succ d ≤ x` | +{: .table-equal-2} + +`left` and `right` are the only tactics here that can lose you the proof. everything else preserves provability, and those two commit to a disjunct before you have checked it holds. + +## Tests and proofs + +the obvious way for a programmer to check `le_total` is to assert it. + +``` +assert le_total(6, 7) +assert le_total(1, 1) +assert le_total(0, 255) +``` + +green suite, three pairs, out of infinitely many. property-based testing does better. QuickCheck, Hypothesis and proptest generate values of `a` and `b` and assert the property on each, which samples more widely and still samples. [^quickcheck] + +[^quickcheck]: Koen Claessen and John Hughes, *QuickCheck: A Lightweight Tool for Random Testing of Haskell Programs*, ICFP 2000: + +the parts that do correspond: + +| Testing | Lean | +| --- | --- | +| test suite | theorem statement | +| running the suite | type-checking the proof | +| green | no goals remaining | +| a failing assert | a goal that will not close | +| coverage | no counterpart | + +coverage has no counterpart because a proof holds for the whole domain or it is not a proof. + +where the analogy does hold is the feedback loop. tests are worth writing because the machine answers immediately and without sympathy, and the goal window gives that same answer after every step. on paper, "clearly" is a sentence a person can write. Lean has no such sentence. + +`sorry` completes the correspondence at the other end. it stands in for an unfinished proof, compiles with a warning, and lets you fill in a skeleton piece by piece the way a test suite grows. + +## The proof + +```lean +theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by + induction y with + | zero => + right + exact zero_le x + | succ d hd => + cases hd with + | inl hl => + cases' hl with c hc + rewrite[hc] + left + rewrite[succ_eq_add_one] + use c + 1 + rewrite[← add_assoc] + rfl + | inr hr => + cases' hr with c hc + cases c with + | zero => + rewrite[zero_eq_0] at hc + rewrite[add_zero d] at hc + left + rewrite[hc] + exact le_succ_self d + | succ a => + rewrite[add_succ] at hc + right + rewrite[hc] + use a + rewrite[succ_add] + rfl +``` + +the induction is on `y`, which gives two cases. + +in the zero case the goal is `x ≤ 0 ∨ 0 ≤ x`. the right disjunct is `zero_le`, proved above, so `right` followed by `exact zero_le x` closes it. + +in the successor case the goal is `x ≤ succ d ∨ succ d ≤ x`, with `hd : x ≤ d ∨ d ≤ x` available. splitting `hd` gives two branches. + +in the `inl` branch, `x ≤ d`, so there is a gap `c` with `d = x + c`. the same gap extended by one witnesses `x ≤ succ d`, which is `use c + 1`. + +in the `inr` branch, `d ≤ x`, so there is a gap `c` with `x = d + c`. this branch needs a second split, on `c` itself, because the two cases close differently. + +when `c` is zero, `x = d`, so `x ≤ succ d` follows from `le_succ_self`. when `c` is `succ a`, then `x = succ(d + a)` and `succ d ≤ x` holds with witness `a`. + +![Lean proof state overview](/assets/blog/lean_state_overview.png) + +every node in that diagram is one of the two-column states from earlier, and every edge is a tactic. + +## In English + +TODO + +## What it cost + +eleven supporting results and twenty-eight lines of tactics, for a statement that needs no defending to anyone who has counted to seven. + +the gaps are visible instead of assumed. that is what the exercise buys. + +the full single-file solution, with every supporting theorem derived from the definitions above, is at . + +next is linear algebra at the [FernUniversität in Hagen](https://www.fernuni-hagen.de/mi/studium/module/lin_alg.shtml), alongside part-time studies at ZHAW. first course where proofs are the work rather than a step inside it, which is why i wanted this done now. + +## Further reading + +- Natural Number Game 4: +- the set theory and linear algebra games, same site: +- Ueli Maurer, *Diskrete Mathematik*, ETH Zürich. proof systems are 6.1, interactive proofs 6.2.4: +- Velleman, *How To Prove It*, and its Lean companion: +- Alyssa Ney, *Metaphysics: An Introduction*. its treatment of first and second order predicate logic ties the notation to philosophical questions instead of deriving it from truth tables, which makes it a good counterweight to the ETH script. +- Axler, *Linear Algebra Done Right* +- Peano's original 1889 paper, in Latin: +- an English translation of the same: +- Peano axioms, ETH: +- Peano axioms, Trinity College Dublin: +- and the list of 1000 theorems: +- Kevin Buzzard, whose version of this proof mine follows: diff --git a/assets/blog/lean_numberline_two_cases.svg b/assets/blog/lean_numberline_two_cases.svg new file mode 100644 index 0000000..444f3e8 --- /dev/null +++ b/assets/blog/lean_numberline_two_cases.svg @@ -0,0 +1,73 @@ + + + + + + + + + + + (a) + + + + + + + + + + + + 0 + 1 + 2 + · · · + + + a = b + + + (b) + + + + + + + + + + + 0 + 1 + 2 + · · · + + + + + a + a+1 + a+c = b + + + + c + From 013552acc70cf97407acafd71ca4893fff7c57b4 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 14:22:44 +0200 Subject: [PATCH 17/34] Fix table, katex display --- _layouts/base.html | 30 ++++++++++++++++++++++++++++++ 1 file changed, 30 insertions(+) diff --git a/_layouts/base.html b/_layouts/base.html index c49d74c..a41a948 100644 --- a/_layouts/base.html +++ b/_layouts/base.html @@ -107,6 +107,23 @@ } blockquote { margin: 1.2em 2em; font-style: italic; } + /* Keep wide display math inside the article instead of widening the page. */ + .katex-display { + max-width: 100%; + overflow-x: auto; + overflow-y: hidden; + } + + .katex-display > .katex { + max-width: none; + } + + /* Allow long URLs and inline code to wrap before widening the viewport. */ + main a, + main code { + overflow-wrap: anywhere; + } + hr { border: 0; border-top: 1px solid #e6e6e2; margin: .6em 0; } table { @@ -114,6 +131,19 @@ border-collapse: collapse; } + table[class*="table-equal-"] { + table-layout: fixed; + } + + table.table-equal-2 th, + table.table-equal-2 td { width: 50%; } + + table.table-equal-3 th, + table.table-equal-3 td { width: 33.333%; } + + table.table-equal-4 th, + table.table-equal-4 td { width: 25%; } + th, td { padding: .1em .5em; From 4dfbadda41ee92689c9f7209ef0a950bdf84fe7c Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 14:22:52 +0200 Subject: [PATCH 18/34] Polish --- _posts/2026-06-28-lean_real.md | 191 +++++++++++++++++++++++---------- 1 file changed, 136 insertions(+), 55 deletions(-) diff --git a/_posts/2026-06-28-lean_real.md b/_posts/2026-06-28-lean_real.md index 447e678..915f7fd 100644 --- a/_posts/2026-06-28-lean_real.md +++ b/_posts/2026-06-28-lean_real.md @@ -6,23 +6,21 @@ tags: ["theoretical mathematics", "first principles"] math: true --- -paper proofs have no compiler. i went looking for one and found Lean. +paper proofs have no compiler. i went looking for one and found Lean, by way of the [Natural Number Game](https://adam.math.hhu.de/#/g/leanprover-community/nng4), which builds the naturals from nothing and makes you prove your way back out. [^nng] i finished it end of june and wanted to write up what the last level actually took. no AI was used for the Lean here, and none for the maths either: the definitions, the axioms i picked, the proof strategy and the dependency graph are mine. AI was used for phrasing. ## The theorem $$ -\forall x,y \in \mathbb{N} : x \le y \lor y \le x +\Large \forall x,y \in \mathbb{N} : x \le y \lor y \le x $$ any two natural numbers compare. one of them is at most the other. take 6 and 7. then $(6 \le 7) \lor (7 \le 6)$ is $T \lor F$, so $T$. take 1 twice. then $(1 \le 1) \lor (1 \le 1)$ is $T \lor T$, so $T$. -you already believe this. the article is about what believing it costs. - -those two examples cover two pairs, and $\mathbb{N}^2$ is infinite. the $\forall$ is the whole difficulty. +checking pairs by hand settles those two pairs. $\mathbb{N}^2$ is infinite, so no amount of checking gets through it, and the $\forall$ has to be discharged some other way. right now $\le$, $+$ and $\mathbb{N}$ are all undefined, so that line is notation. the rest of the article pays the debt in order. @@ -42,31 +40,40 @@ $\lor$ is disjunction. $P \lor Q$ holds when at least one side holds. | $P$ | $Q$ | $P \lor Q$ | | --- | --- | --- | -| false | false | false | -| false | true | true | -| true | false | true | -| true | true | true | +| $F$ | $F$ | $F$ | +| $F$ | $T$ | $T$ | +| $T$ | $F$ | $T$ | +| $T$ | $T$ | $T$ | -$\exists$ is existence. $\exists x, P(x)$ is discharged by producing one witness and showing $P$ holds of it. +$\exists$ is existential quantification. $\exists x \in S, P(x)$ holds when at least one $w \in S$ has $P(w)$. to prove it you produce such a $w$, called the witness. -$\forall$ is universal quantification. $\forall x, P(x)$ needs a proof that works for an arbitrary $x$, and the proof may not look at which value it got. +$\forall$ is universal quantification. $\forall x \in S, P(x)$ holds when every $x \in S$ has $P(x)$. to prove it you take an arbitrary $x$ and derive $P(x)$ without using anything specific about it. the rest of propositional logic is assumed. [^velleman] a refresher, if you want one:
@@ -327,7 +349,7 @@ after `use x`: | `x : ℕ` | `x = 0 + x` | {: .table-equal-2} -after `rw [zero_add]`: +after `rewrite[zero_add]`: | Givens | Goal | | --- | --- | @@ -336,7 +358,7 @@ after `rw [zero_add]`: after `rfl`, no goals remain. -the left column only grows and the right column only shrinks. the proof is finished when the right column is empty, which is a condition rather than a judgement call. +in this example the left column only grows and the right column only shrinks. that is not a law of Lean proofs in general, since `induction` and `cases` replace one goal with several and a rewrite can make a goal larger before it gets smaller. what does hold everywhere is the stopping condition: the proof is finished when no goals remain, which the machine decides rather than the author. some tactics split the state in two instead of changing it. those are the ones that make a proof branch. @@ -383,7 +405,7 @@ and `right`, which picks a side of the goal and discards the other: -`left` and `right` are the only tactics here that can lose you the proof. everything else preserves provability, and those two commit to a disjunct before you have checked it holds. +`left`, `right` and `use` commit to a choice the goal did not force. the first two pick a disjunct and discard the other, `use` picks a witness and discards every other candidate. pick wrong and the remaining goal is unprovable even though the original was fine, so you undo and try again. the rewriting tactics do not have this property: they transform a goal into an equivalent one, so a provable goal stays provable. ## Tests and proofs @@ -411,11 +433,13 @@ the parts that do correspond: coverage has no counterpart because a proof holds for the whole domain or it is not a proof. -where the analogy does hold is the feedback loop. tests are worth writing because the machine answers immediately and without sympathy, and the goal window gives that same answer after every step. on paper, "clearly" is a sentence a person can write. Lean has no such sentence. +where the analogy does hold is the feedback loop. tests are worth writing because the machine answers immediately and without sympathy, and the goal window gives that same answer after every step. on paper, "clearly" is a sentence a person can write and nothing checks it. Lean's equivalent is `sorry`, which compiles but marks the file with a warning you have to look at. -`sorry` completes the correspondence at the other end. it stands in for an unfinished proof, compiles with a warning, and lets you fill in a skeleton piece by piece the way a test suite grows. +that makes `sorry` the counterpart of a stubbed test: it lets you write the shape of a proof first and fill in the parts one at a time, with the compiler tracking what is still owed. -## The proof +## The last step + +what follows is the final theorem only, the bottom node of the dependency graph. it is thirty lines, and it is thirty lines rather than more because everything it stands on has already been proved: the definition of $\mathbb{N}$, both clauses of addition, the definition of $\le$, and the seven theorems above it, another fifty-three lines that are not in this snippet. the whole development is in the artefact. ```lean theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by @@ -451,6 +475,8 @@ theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by rfl ``` +[open the whole development in the Lean web editor](https://tinyurl.com/4tr5uc7c) to see the definitions and the seven supporting theorems this rests on, and to click through the states below yourself. + the induction is on `y`, which gives two cases. in the zero case the goal is `x ≤ 0 ∨ 0 ≤ x`. the right disjunct is `zero_le`, proved above, so `right` followed by `exact zero_le x` closes it. @@ -465,7 +491,13 @@ case (I), `c` is zero, so `x = d` and `x ≤ succ d` follows from `le_succ_self` ![Lean proof state overview](/assets/blog/lean_state_overview.png) -every node in that diagram is one of the two-column states from earlier, and every edge is a tactic. +every node in that diagram is one of the two-column states from earlier, and every edge is a tactic. three things it shows that the linear listing hides. + +the labelled frames are where the state splits. `induction y` opens the `succ d` frame, `cases hd` opens `inl` and `inr` inside it, and `cases c` splits again inside `inr`. each frame starts at its own filled dot, so the nesting on the page is the nesting of the proof. + +`left` and `right` discard a disjunct. the top right branch goes from `x ≤ 0 ∨ 0 ≤ x` to `0 ≤ x` under `right`, and the left half never appears again. the same happens inside the frames, where the goal is written `…∨…` while both halves are still live and collapses to a single inequality the moment `left` or `right` fires. + +four goals get closed, one per leaf, and this proof uses two tactics to do it. `rfl` closes the two that end in an equation whose sides are the same term, `(x+c)+1 = (x+c)+1` and `succ(d+a) = succ(d+a)`, both rewritten until the two halves are literally identical. `exact` closes the other two by naming a result proved earlier, `zero_le x` on the far right and `le_succ_self d` in the middle. every rewrite above them exists to reach one of those two endings. the remaining circles lower down are merge points where the branches rejoin. ## In English @@ -473,9 +505,11 @@ TODO ## What it cost -eleven supporting results and twenty-eight lines of tactics, for a statement that needs no defending to anyone who has counted to seven. +eight theorems and eighty-three lines of tactics, for a statement that needs no defending to anyone who has counted to seven. + +`le_total` itself is thirty of those lines. the other fifty-three are the seven theorems underneath it: `succ_eq_add_one` at four lines, `zero_add` at nine, `succ_add` at eleven, `add_comm` at eleven, `add_assoc` at thirteen, `zero_le` at three, `le_succ_self` at two. five of the seven are about addition, and none of them mention $\le$ at all. proving that two numbers compare turns out to be mostly a matter of proving that addition behaves. -this article shows three of those eleven. the other eight are commutativity, associativity, `zero_add`, `succ_add`, `add_succ`, `add_zero`, `one_eq_succ_zero` and `zero_eq_0`, and every one of them is proved in the artefact from the two clauses of addition and nothing else. no step is assumed, quoted from a library, or left to the reader. that is the part i could not have got from a paper proof: `sorry` is the only way to skip a step, and it shows up as a warning every time you compile. +what the artefact assumes is small and explicit: the inductive definition of `MyNat`, `add_zero` and `add_succ` as the defining equations of `+`, and the definition of `≤`. everything after that is derived. `sorry` is the only way to skip a step, and it shows up as a warning every time you compile. the full single-file solution is at . it opens in the Lean 4 web editor with the whole development in it, so you can click any line and watch the givens and goal in the right-hand panel, exactly the two columns from earlier. put the cursor inside the `inr` branch and you can see the case split on `c` open up. no install, and it typechecks end to end. diff --git a/_posts/2026-06-28-lean-todos.md b/_posts/2026-06-28-lean-todos.md deleted file mode 100644 index 4108cf9..0000000 --- a/_posts/2026-06-28-lean-todos.md +++ /dev/null @@ -1,226 +0,0 @@ ---- -title: "So You Think You Know ≤ ?" -date: 2026-09-05 -description: "First steps in formal mathematics with Lean" -tags: ["theoretical mathematics", "first principles"] -math: true ---- - -STYLE: Stick to writing style of all other blogs, threads. concise, lowercase. Make sure to use footnotes, cite sources diligently. Precision, less fluff. Find inspiration in ~dani/.codex/skills/ - -Context: I am studying computer science at ZHAW. -I have noticed that I enjoy mathematics a lot but that I am missing foundations to do understand mathematics at a deeper level. - -I'll be studying at mathematics in hagen in parallel to my part time zhaw curiculum instead of working as SWE. - -I have started learning lean in June 2026. -I have never had the time to do a writeup. - -My first steps in lean were some months ago but I still wanted to share my first steps. - -At zhaw i have really enjoyed linear algebra and analysis and would like to push it further by starting at the very very basics. - -I am folloiwng "How to prove it", "Elyssa intro to metaphysciss', "Axler linear algebra", "ETH discrete maths script", NNG4, set theory game, lin alg game.. - - -Seam a bit futther along now buyt that shouldn't be of much relevance. - -Around end of june 2026 i have completed natural numbers game -Wanted to share my findings, deriving a property of natural numbers form first princieples in Lean. - -What i want to show is a short introduction to Lean form the perspective of computers cience , how even elementary concepts can be approached rigorously. - - -The blog article should show a breif introduction to prepositional logic, lean and proof strucutre. - -there are other drafts in this repo (lean/todo..) -they include colorful latex constructions, look into them first. - - -THE PROOF - PREFACE - -- We need to go over syntacrc things first, define notation - -Logical connectives -- \or , ∧ , → - -We introuce tautology: T, contradiction: (flipped) T - -→ ocnnectieve is not explained well purely by truth table. there are many philosophical questions one may ask aboiut the truth or falsenss of \r . - -ETH discrete math i don't like. -Many profos are just "compare truth tables and see that expressions are the same" - -De Morgan’s laws -¬ -(P ∧ Q) is equivalent to ¬ P ∨ ¬Q . -¬ -(P ∨ Q) is equivalent to ¬ P ∧ ¬Q . -Commutative laws -P ∧ Q is equivalent to Q ∧ P. -P ∨ Q is equivalent to Q ∨ P. -Associative laws -P ∧(Q ∧ R) is equivalent to(P ∧ Q) ∧ R. -P ∨(Q ∨ R) is equivalent to(P ∨ Q) ∨ R. -Idempotent laws -P ∧ P is equivalent to P. -P ∨ P is equivalent to P. -Distributive laws -Absorption laws -P ∧(Q ∨ R) is equivalent to(P ∧ Q) ∨(P ∧ R) . -P ∨(Q ∧ R) is equivalent to(P ∨ Q) ∧(P ∨ R) . -P ∨(P ∧ Q) is equivalent to P. -P ∧(P ∨ Q) is equivalent to P. -Double Negation law -¬¬ P is equivalent to P. - -DEFINE: converse, contrapositive,statement,theorem,empty set, set, element, bound variable, substitution -PHILLOSPHY: number,greater,value - - -How to prove it - P52 (Sentential Logic) - -Statements that mean P → Q come up very often in mathematics, but -sometimes they are not written in the form “If P then Q . -” Here are a few other -ways of expressing the idea P → Q that are used often in mathematics: -P implies Q . -Q , if P. -P only if Q . -P is a sufficient condition for Q . -Q is a necessary condition for P. - - - -- Deifnition of <= -- Defintion of N, defintion of include in N - -Quantors -- ∀ : allquantor -- ∃ : existence quantor - -Then we probably need to go over the basics of proofs no? - -Like in HTPi - - -Proofs of form - -_Proof structure has been introduced in the preface of how to prove it (footnote insert) : - -``` -Let x be arbitrary . - Suppose P (x) is true. - [Proof of Q(x) goes here.] - Thus, if P (x) then Q(x) . -Thus, for all x, if P (x) then Q(x) . -``` - -What is a tacitc? We probably want lexygraphic defintiosn here. - -https://lean-lang.org/theorem_proving_in_lean4/Tactics/ -> A proof term is a representation of a mathematical proof; tactics are commands, or instructions, that describe how to build such a proof. - - -We need to define how proofs work based on logical connectives explained before. - -We solve proofs with Givens, GOal table. - -In case of Goal: P->Q, we can extract P as givens, Q as goal. - -P->Q iff ¬P∨Q iff Q∨¬P iff ¬Q→¬P - -So we can proove contrapositive -Goal:P->Q ; Givens: \¬Q, Goal: \notP - - -If given goal: ∀ x - Px, we can apply universal instantiation, Givens: x0 Goal: Px -If given givens ∃ x - Px we can givens: x, Px -If goal ∃x we can `use` to pass arg -If givens is \All we can use choose any x statisfying condiiton. - - - -The prof should traverse the dependnecy graph one step at a time. questioning what is a number, what makes a number natural. what ways are there to express / define natural numbers? zf/van neuman. we do zero/succ. - -Explain tactics (rw as substitute, use as existential instantiation) - - -THE THEOREM - - -$$ -\forall x,y \in \mathbb{N} : x \le y \lor y \le x -$$ - - - -MY PROOF - -```lean -theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by - induction y with - | zero => - right - exact zero_le x - | succ d hd => - cases hd with - | inl hl => - cases' hl with c hc - rewrite[hc] - left - rewrite[succ_eq_add_one] - use c + 1 - rewrite[← add_assoc] - rfl - | inr hr => - cases' hr with c hc - cases c with - | zero => - rewrite[zero_eq_0] at hc - rewrite[add_zero d] at hc - left - rewrite[hc] - exact le_succ_self d - | succ a => - rewrite[add_succ] at hc - right - rewrite[hc] - use a - rewrite[succ_add] - rfl -``` - - - -_ASSETS_ - -lean_dependency_graph shows the all the theorems that are required to prove the theorem. -Blue are the defintions -Green are the arithmetic -Orange are what follows - -we start at N, we end at theorem. - - -lean_state_overview is how the lean state evolves with each tactic applied. - - - - - - -MY SINGLE FILE LEAN SOLUTION INCLUDING ALL THOREMS DERIVED FROM FIRST PRINCIPLES: - - -FURTHER READING -- Von neuman universe -- Metaphysics introduction -- https://leanprover-community.github.io/learn.html -- How to Prove it supplementary lean course in additoon to paper: https://djvelleman.github.io/HTPIwL/ -- Peano ETH: https://people.math.ethz.ch/~halorenz/4students/LogikGT/Ch08.pdf -- Peano https://www.maths.tcd.ie/~odunlain/u11602/online_notes/pdf_peano.pdf -- Original Peano arithmetic paper (1889), Latin - i don't actually know latin but if I could I'd read this: https://archive.org/details/arithmeticespri00peangoog/page/n10/mode/2up -- 1-1 translation of the 1889 paper to english: https://www.scribd.com/document/678192145/Peano -- Ueli Maurer ETH Diskrete Mathematik Script: https://crypto.ethz.ch/teaching/DM23/ln/DM23_LNss-tablet.pdf - diff --git a/_posts/2026-06-28-lean.md b/_posts/2026-06-28-lean.md deleted file mode 100644 index aed0c1b..0000000 --- a/_posts/2026-06-28-lean.md +++ /dev/null @@ -1,287 +0,0 @@ ---- -title: "Mathematics but not handwavey (real)" -date: 2026-06-28 -description: "First steps in formal mathematics with Lean" -tags: ["theoretical mathematics", "first principles"] -math: true ---- - - -# PROPER OUTLINE - -INTRO -- ASSUMPTIONS - - ---- - - - - -## Intro - - - -### Discrete Mathematics -### ZHAW Background -> different approach to mathematics -### Vocabulary - - -$\forall$: for all -∃: there exists -≤: less than or equal to - -Today we will be proving the total order of natural numbers. - -$$ -\Huge \forall x,y \in \mathbb{N} : x \le y \lor y \le x -$$ - - -$$ -\boxed{\displaystyle \quad \forall x,y \in \mathbb{N} : x \le y \lor y \le x \quad} -$$ - -$$ -\boxed{\forall x,y \in \mathbb{N} : x \le y \lor y \le x} -$$ - -$$ -\fbox{\(\displaystyle \quad \forall x,y \in \mathbb{N} : x \le y \lor y \le x \quad\)} -$$ - -$$ -\colorbox{#fff3cd}{$\displaystyle \forall x,y \in \mathbb{N} : x \le y \lor y \le x$} -$$ - -$$ -\fbox{ - $\forall (x,y : ℕ) : x ≤ y ∨ y ≤ x$ -} -$$ - - -![graph](/assets/blog/dependency_graph.svg) -*A transient dependency graph showing - - -Before we do that we need to get some definitions out of the way. - -## Definitions - -### Natural Number? - -First: - -What even is a Natural number? -Natural numbers are defined to be all "positive" "whole" numbers. - -$$ -\mathbb{N} = \{0,1,2,3,4,5, \ldots \} -$$ - -But this isn't yet a concrete definiton. -Natural numbers can be seen as a sequeence. The sequence starts at 0. -Each number is either 0 or a successor. - -$$ -\mathbb{N} - \stackrel{\mathrm{def}}{=} \begin{cases} - 0 &\text{Base case} \\ - \mathrm{succ} &\text{Else} \\ -\end{cases} -$$ - -### Addition? - -Now we can define addition. (Using axioms) - -It isn't enough to define addition for speicfic numbers, like let's say `6 + 1 = 7`. -We need to define addition for ALL numbers. We do this using induction. - -Here we fix the variable $a$ and check the our definition of natrual numbers from before. - -B is either zero, or a successor of a natural number. - -$$ -\underline{\Large{\forall a,b \in \mathbb{N} : a + b = \text{?}}} -$$ - -$$ -\forall a,b \in \mathbb{N} : a + b \stackrel{\mathrm{def}}{=} -\begin{cases} -a & b = \text{0} \\ -succ(a + d) & b = \operatorname{succ}(d) \\ -\end{cases} -$$ - -
- - There are many alternate ways to represnet natural numbers. Most notably the Zermelo / von Neuman ordinals [^1]: - - - - Zermelo: - $0=\emptyset,\quad 1=\lbrace\emptyset\rbrace,\quad 2=\lbrace\lbrace\emptyset\rbrace\rbrace,\quad\ldots$ - - von Neumann: - $0=\emptyset,\quad 1=\lbrace\emptyset\rbrace,\quad 2=\lbrace\emptyset,\lbrace\emptyset\rbrace\rbrace,\quad\ldots$ - - - -
- - -[^1]: https://www.researchgate.net/publication/228574851_von_Neumann_universe_A_perspective - - - -$$ -\text{Example: } \underline{1 + 2 = 3} -$$ - - -$$ -\newcommand{\mathcolorbox}[2]{\colorbox{#1}{$\displaystyle #2$}} - -\begin{array}{ll} -& \textbf{Axioms} \\ -\colorbox{#fff3cd}{i.} & 1 \stackrel{\mathrm{def}}{=} \operatorname{succ}(0) \\ -\colorbox{#cfe2ff}{ii.} & 2 \stackrel{\mathrm{def}}{=} \operatorname{succ}(1) \\ -\colorbox{#e2d9f3}{iii.} & 3 \stackrel{\mathrm{def}}{=} \operatorname{succ}(2) \\ -\colorbox{#f8d7da}{iv.} & a+0 \stackrel{\mathrm{def}}{=} a \\ -\colorbox{#d1e7dd}{v.} & a+\operatorname{succ}(b) \stackrel{\mathrm{def}}{=} \operatorname{succ}(a+b) -\end{array} -\qquad -\begin{array}{l|l} -\textbf{Statements} & \textbf{Reasons} \\ -\hline - 1 + 2 - & \text{Given} \\ - 1 + \mathcolorbox{#cfe2ff}{\operatorname{succ}(1)} - & \mathcolorbox{#cfe2ff}{\mathrm{\rightarrow ii.}} \\ - \mathcolorbox{#d1e7dd}{\operatorname{succ}(1 + 1)} - & \mathcolorbox{#d1e7dd}{\mathrm{\rightarrow v.}} \\ - \operatorname{succ}(1 + \mathcolorbox{#fff3cd}{\operatorname{succ}(0)}) - & \mathcolorbox{#fff3cd}{\mathrm{\rightarrow i.}} \\ - \operatorname{succ}(\operatorname{succ}(\mathcolorbox{#d1e7dd}{1 + 0})) - & \mathcolorbox{#d1e7dd}{\mathrm{\rightarrow v.}} \\ - \operatorname{succ}(\operatorname{succ}(\mathcolorbox{#f8d7da}{1})) - & \mathcolorbox{#f8d7da}{\mathrm{\rightarrow iv.}} \\ - \operatorname{succ}(\mathcolorbox{#cfe2ff}{2}) - & \mathcolorbox{#cfe2ff}{\mathrm{\leftarrow ii.}} \\ - \mathcolorbox{#e2d9f3}{3} - & \mathcolorbox{#e2d9f3}{\mathrm{\leftarrow iii.}} -\end{array} -$$ - -$$ -\underline{\Large{\forall a,b \in \mathbb{N} : a \le b \text{ ?}}} -$$ - -$$ -\forall a, b \in \mathbb{N} : a \le b \stackrel{\mathrm{def}}{\iff} ∃ (c : ℕ), b = a + c -$$ - -![graph](/assets/blog/numberline.png) - -## Prooving - -### Or - -| | $a = false$ | $b = true$ | -| --- | --- | --- | -| $b = false$ | false | true | -| $b = true$ | true | true | - - -### Exists -### Induction -### Rewriting - -In the following proof we use the following tactics: - -rewrite := X = Y -induction := break statement down into base case, successor - -$$ -h: a = b, g: a \xrightarrow{\mathrm{rewrite[h]}} g: b -$$ - -$$ -h : y - -\xrightarrow{\mathrm{induction(d,hd)}} - -\begin{cases} - \text{hd} : y = 0 \\ - \text{hd} : y = succ(d), d \in \mathbb{N} -\end{cases} - -$$ - -$$ - -h : \text{a} \lor \text{b} - -\xrightarrow{\mathrm{cases}} - -\begin{cases} - \text{inl} : \text{a} \\ - \text{inr} : \text{b} -\end{cases} - - - \\ - -$$ - -$$ -h: \text{a} \lor \text{b} \xrightarrow{\mathrm{left}} h: a -$$ - -$$ -h: \text{a} \lor \text{b} \xrightarrow{\mathrm{right}} h: b -$$ - -$\forall a, b \in \mathbb{N} : a \le b \stackrel{\mathrm{def}}{\iff} ∃ (c : ℕ), b = a + c$ - - -```lean -theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by - induction y with - | zero => - right - exact zero_le x - | succ d hd => - cases hd with - | inl hl => - cases' hl with c hc - rewrite[hc] - left - rewrite[succ_eq_add_one] - use c + 1 - rewrite[← add_assoc] - rfl - | inr hr => - cases' hr with c hc - cases c with - | zero => - rewrite[zero_eq_0] at hc - rewrite[add_zero d] at hc - left - rewrite[hc] - exact le_succ_self d - | succ a => - rewrite[add_succ] at hc - right - rewrite[hc] - use a - rewrite[succ_add] - rfl -``` - -![graph](/assets/blog/lean_thing.png) - - - diff --git a/_posts/2026-06-28-todo.md b/_posts/2026-06-28-todo.md deleted file mode 100644 index 8f632d9..0000000 --- a/_posts/2026-06-28-todo.md +++ /dev/null @@ -1,343 +0,0 @@ ---- -title: "Mathematics but not handwavey" -date: 2026-06-28 -description: "TODO" -tags: ["theoretical mathematics", "first principles"] -math: true ---- - -alternate titles: - -* Functional Programming Meets Rigoporous Mathematics -* Mathematics but cut the bullshit -* Mathematics but done right -* Dipping my toes into rigorous mathematics -* Mathematics but without gaps -* Let's PROVE IT! - -/-- If $x$ and $y$ are numbers, then either $x \leq y$ or $y \leq x$. -/ - -```lean -Statement le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by - sorry -``` - - -Explicitly use ≤ instead of <= -Use Latex where possible - -## scraps - -Underwood Dudley -"In mathematics problems can be solved, using reason, and the solutions can be checked and shown to be correct." - -+ Recently completed the natrual numbers game as an intro to LEAN and i think i really like it -https://adam.math.hhu.de/#/g/leanprover-community/nng4 - - -First principles (like [my 8bit CPU](projects/8bit-cpu/)) - -ETH Discrete Mathematik Chapter 6 - -> Definition 6.1. A proof system is a quadruple Π = (S,P,τ,φ), as above. - - -Excerpt - -from ETH Zürich -Departement Informatik -Diskrete -Mathematik -Ueli Maurer -Herbstsemester 2024 - -> **6.2.4 Proof Systems in Theoretical Computer Science**\* -> -> An important extension of the concept of proof systems are so-called interactive proofs. $^{16}$ In such a system, the proof is not a bit-string, but it consists of an interaction (a protocol) between the prover and the verifier, where one tolerates an immensely small (e.g. exponentially small) probability that a verifier accepts a “proof” for a false state- ment. The reason for considering such interactive proofs are: -> -> * Such interactive proofs can exist for statements for which a classical (non-interactive) proof does not exist. For example, there exists an interactive proof system for the non-Hamiltonicity of graphs. -> * Such interactive proofs can have a special property, called *zero-knowledge*, which means that the verifier learns absolutely nothing (in a well-defined sense) during the protocol, except that the statement is true. In particular, the verifier cannot prove the statement to somebody else. -> * Zero-knowledge proofs (especially non-interactive versions, so-called NIZK’s) are of crucial importance in a large number of applications, for example in sophisticated block-chain systems. - -## INTRO - -![Path from Axioms to QED](/assets/blog/lean_graph.svg) - - - assumptiosn: reader is familiar with discrete mathematic[118;1:3us - -### MOTIVATION - I don't like assumptions - I don't like bullshit - How do we know if a given stastement is real? - Why just "hope it works" - First principles - Start with Axioms and build our way up - - **Some definitions first:** - Axiom := most fundamental assumptions - Proof := - Theorem := - Statement := - Proposition := - - - How do I verify that a given proof is correct? - How do I verify that I didn't bullshit my way to QED. - - - Some semesters back I have completed the discrete mathematics (https://eventoweb.zhaw.ch/Evt_Pages/Brn_ModulDetailAZ.aspx?node=2901247e-aa27-4f84-a5d6-d6b33b234dbd&IDAnlass=1456265&IdLanguage=133&clearcache=true) course at ZHAW. Now i am here to re-visit it with a different lense upon learning about proof asistants. - - - My main struggle with discrete mathematics is the lack of feedback to my work. Analogy: checking if code compiles and runs correclty when only on paper is difficult. same with proofs. one misstep. GOAL: reduce human error - - I feel like my understanding of mathematics is too fuzzy. I want to be less hand wavey and want to see my gaps explicitly. - - explicitly out of scope: (type system of Lean, how lean works, RCOQ, full 99 variations of a proof review) - explicitly mention: NO AI has been used to aid with LEAN - NO AI has been used in this article except for language / phrasing - -### 99 VARIATIONS OF A PROOF - - 99 variations of a proofs proposes the idea that proofs are but mere logical arguments. - - we start with theorem: -$$ - (x : \mathbb{N})) - x^3-6x^2+11x-6=2x-2 \implies x=1 \lor x=4 -$$ - -Twoliner (Proof by factorization (Proof 1 - Oneline)): -$$ -x^3-6x^2+9x-4=0 \\ -(x-1)^2(x-4)=0 -\Box -$$ - -// thanks Josua for gifting me the book - - and proove it in 98 other ways - (find 99 logical reasonings) - my favourite is : Proof #6 - axiomatic -> first principles. first we define notation, then definitions and then solve find proor by means of applying axioms - - -// 20 - defintiional is quite similar to 6 -// - no hiddewn assumptions, everything clearly defined - -// some other proofs that spark joy to me are: -// 9 - monosyllabic - sma words -// 10 - wordless by showing pricutres of cube -// 27 - algorithmic - - -// fun ones: -// 11 - exam -// 15 - matrix -// 25 - open collaborative -// 28 - flowchart -// 29 - model - -// 26 - auditory - -// hate: -// 19 - jargon (reminds of LinkedIn) -/ [insert tier list] - -full book review and -tier list comes later.. - - // Most logical arguments are difficult to systematically veirfy. - // I love machines - they do exactly what they have been told to do. - not always what i want them to but at least they are deterministic. WHY can't we use them for mathematics? WE CAN - - // statement -> irrefutable proof - - -made me realize LEAN exists: -- https://arxiv.org/abs/2605.22763 - -## LEAN - -``` -ℕ -├── add_zero ───────┬── zero_add ──────┬── zero_le ───────┐ -│ │ ├── add_comm │ -│ │ └── add_assoc ─────┤ -├── add_succ ───────┼── succ_add ──────┬── add_comm ├── le_total -│ │ └── add_assoc ─────┤ -└── one_eq_succ_zero┴── succ_eq_add_one┬── le_succ_self ──┘ - └──────────────────┘ -``` - - // LEAN -> proof asistant - // > quote LEAN - // TOPIC: today we will prove that ℕ are totally orderred - // [image of a number line] - - // assumptions: ℕ has been defined as either 0 or succ - // a <= b has been defined as ∃ c : b = a + c (numberline example svg) - - - - -| | $a = False$ | $b = True$ | -| --- | --- | --- | -| $b = False$ | $False \lor False \implies False$ | $a \lor b \implies True$ | -| $b = True$ | $a \lor b \implies True$ | $a \lor b \implies True$ | - - -// our assumption: - -$$ -\fbox{ - $(x, y \in \mathbb{N}) : x <= y ∨ y <= x$ -} - -\Box -$$ - - - - // reads as: given two natural numbers, one of them is greater or equal than the other. Intuitively makes sense. example: 6,7. so here (6≤7) ∨ (7≤6) -> True ∨ False -> True. - Or given same numbers: (1≤1) ∨ (1≤1) -> True ∨ True -> True - // It is easy to show that the statement holds for two _specific_ numbers but we need to prove that it holds for _all_ numbers. We focus on ℕ for simplicity. - - - my version of the x <= y ∨ y <= x proof has inspired by Kevin Buzzard. - - - x <= y ∨ y <= x - - // here we can apply induction on y for example. - // we can split this into two cases. - // ind y with d hd - - hd: (x <= y ∨ y <= x) - - induction y = 0 - x <= 0 ∨ 0 <= x // right one has been proven - 0 <= x // exactly zero_le - induction y = succ(d) - x <= succ(d) ∨ succ(d) <= x - - // let's focus on our hypothesis from earlier (hd) - // let's break it up into parts - - (x <= y) ∨ (y <= x) - | | - | right: hr := (y <= x) - left: hl := (x <= y) - - - // here we can apply cases as per definition - - // > If h : P ∨ Q is a hypothesis, then cases h with hp hq will turn one goal into two goals, one with a hypothesis hp : P and the other with a hypothesis hq : Q. - -.. - - - -## Final Proof - -Axioms -```lean - -/- BEGIN PROVE: x ≤ y ∨ y ≤ x -/ -induction y with -| zero => - right /- 0 ≤ x -/ - exact zero_le x -| succ d hd => - /- ind y=0 -/ - /- BEGIN PROVE: x ≤ 0 ∨ 0 ≤ x -/ - - /- END PROVE: x ≤ 0 ∨ 0 ≤ x -/ - - /- ind y=d -/ - cases hd with hl hr /- hd = hl ∨ hr - | hr := d ≤ x - hl := x ≤ d - -/ - cases hl with c hc /- hc := d = x + c -/ - left /- x ≤ succ d -/ - rw[hc] - use c + 1 - rw[succ_eq_add_one] - rw[← add_assoc] - rfl - - /- trial and error -/ - cases hr with e he /- - hr : d ≤ x - he : x = d + e - -/ - cases e with a /- e -> {0, succ(a)} -/ - /- e=0 ; he : x = d + 0 -/ - rw[add_zero] at he /- he : x = d -/ - left - rw[he] - exact le_succ_self d /- le_succ_self x - x ≤ succ x-/ - /- he : x = d + succ a -/ -``` - -Prereqs / Assumptions - -```lean -Statement succ_eq_add_one n : succ n = n + 1 := by - rw [one_eq_succ_zero] - rw [add_succ] - rw [add_zero] - rfl - -Statement zero_le (x : ℕ) : 0 ≤ x := by - use x - rw [zero_add] - rfl -``` - -Now MY proof: - -```lean -Statement le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by - induction y with - | zero => - right - exact zero_le x - | succ d hd => - cases hd with - | inl hl => - cases hl with c hc - rw[hc] - left - rw[succ_eq_add_one] - use c + 1 - rw[← add_assoc] - rfl - | inr hr => - cases hr with c hc - cases c with - | zero => - rw[zero_eq_0] at hc - rw[add_zero d] at hc - left - rw[hc] - exact le_succ_self d - | succ => - rw[add_succ] at hc - right - rw[hc] - use a - rw[succ_add] - rfl -``` - - - - https://adam.math.hhu.de/#/ - - - - https://github.com/julian/lean.nvim - - https://leanprover-community.github.io/1000.html From e53d9f704637b0c20341f07d1ef5f253a6732d85 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 17:17:04 +0200 Subject: [PATCH 22/34] Drop base table style --- _layouts/base.html | 14 ++------------ 1 file changed, 2 insertions(+), 12 deletions(-) diff --git a/_layouts/base.html b/_layouts/base.html index a41a948..fbd8346 100644 --- a/_layouts/base.html +++ b/_layouts/base.html @@ -147,13 +147,12 @@ th, td { padding: .1em .5em; - border-top: 1px solid black; border: 0; text-align: left; } - tr { - border-top: 1px solid black; + thead { + border-bottom: 1px solid black; } @media (max-width: 600px) { @@ -163,15 +162,6 @@ max-width: 100%; } - th, - td { - min-width: 9rem; - } - - th:last-child, - td:last-child { - min-width: 18rem; - } } code { From 917f4855274dc3114b46ad4ca4a58391da4b4919 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 19:54:19 +0200 Subject: [PATCH 23/34] wip --- _posts/2026-06-28-lean-intro.md | 78 ++++++++++++++++++++++++++++++++- 1 file changed, 77 insertions(+), 1 deletion(-) diff --git a/_posts/2026-06-28-lean-intro.md b/_posts/2026-06-28-lean-intro.md index 12ecc8c..704b87f 100644 --- a/_posts/2026-06-28-lean-intro.md +++ b/_posts/2026-06-28-lean-intro.md @@ -501,7 +501,83 @@ four goals get closed, one per leaf, and this proof uses two tactics to do it. ` ## In English -TODO +### Definitions. + +two rules describe $\mathbb{N}$. zero is a natural number, and the successor $\operatorname{succ}(d)$ of a natural number $d$ is a natural number. nothing else is a natural number. from this follows the induction principle: a property that holds of $0$, and holds of $\operatorname{succ}(d)$ whenever it holds of $d$, holds of every natural number. + +from here on the universe of discourse is $\mathbb{N}$. + +each numeral abbreviates iterated successors: $1 = \operatorname{succ}(0)$, $2 = \operatorname{succ}(1)$, $3 = \operatorname{succ}(2)$, through to $7 = \operatorname{succ}(6)$. + +### Axioms. + +**Peano axiomatic arithmetic.** addition satisfies two equations. + +$$ +\begin{array}{rcll} +a + 0 &=& a & \qquad (\texttt{add\_zero}) \\[6pt] +a + \operatorname{succ}(d) &=& \operatorname{succ}(a + d) & \qquad (\texttt{add\_succ}) +\end{array} +$$ + +**Inequality.** $a \le b$ when some natural number $c$ satisfies $b = a + c$, with $c$ being the gap. the definition is an equivalence, so a gap proves an inequality and an inequality yields a gap. + +### Lemmas. + +**Lemma (`succ_eq_add_one`).** $\operatorname{succ}(n) = n + 1$. + +Proof. by the numeral definitions $1 = \operatorname{succ}(0)$, so $n + 1$ is $n + \operatorname{succ}(0)$, which the successor equation rewrites as $\operatorname{succ}(n + 0)$, and the zero equation reduces $n + 0$ to $n$. $\Box$ + +**Lemma (`zero_add`).** $0 + n = n$. + +Proof. by induction on $n$. + +- *Base case:* $n = 0$. the claim is $0 + 0 = 0$, the zero equation. +- *Inductive step:* let $d$ be arbitrary and take[118;1:3u $n = \operatorname{succ}(d)$. inductive hypothesis: $0 + d = d$. the successor equation gives $0 + \operatorname{succ}(d) = \operatorname{succ}(0 + d)$, and the hypothesis rewrites the inner sum as $d$. $\Box$ + +**Lemma (`succ_add`).** $\operatorname{succ}(a) + b = \operatorname{succ}(a + b)$. + +Proof. let $a$ be arbitrary and fixed. by induction on $b$. + +- *Base case:* $b = 0$. both sides reduce to $\operatorname{succ}(a)$ by the zero equation. +- *Inductive step:* let $d$ be arbitrary and take $b = \operatorname{succ}(d)$. inductive hypothesis: $\operatorname{succ}(a) + d = \operatorname{succ}(a + d)$. the left side becomes $\operatorname{succ}(\operatorname{succ}(a) + d)$, then $\operatorname{succ}(\operatorname{succ}(a + d))$ by the hypothesis. the right side becomes the same, by the successor equation under the outer successor. $\Box$ + +**Lemma (`add_comm`).** $a + b = b + a$. + +Proof. let $a$ be arbitrary and fixed. by induction on $b$. + +- *Base case:* $b = 0$. both sides equal $a$, by the zero equation and by `zero_add`. +- *Inductive step:* let $d$ be arbitrary and take $b = \operatorname{succ}(d)$. inductive hypothesis: $a + d = d + a$. the left side is $\operatorname{succ}(a + d)$, hence $\operatorname{succ}(d + a)$ by the hypothesis. the right side is $\operatorname{succ}(d + a)$ by `succ_add`. $\Box$ + +**Lemma (`add_assoc`).** $(a + b) + c = a + (b + c)$. + +Proof. let $a$ and $c$ be arbitrary and fixed. by induction on the middle summand $b$, which occurs under a successor on both sides. + +- *Base case:* $b = 0$. both sides equal $a + c$, by the zero equation and by `zero_add`. +- *Inductive step:* let $d$ be arbitrary and take $b = \operatorname{succ}(d)$. inductive hypothesis: $(a + d) + c = a + (d + c)$. the left side becomes $\operatorname{succ}((a + d) + c)$, the right side $\operatorname{succ}(a + (d + c))$, and the hypothesis equates the inner sums. $\Box$ + +**Lemma (`zero_le`).** $0 \le x$. + +Proof. the gap is $x$, since $0 + x = x$ by `zero_add`. $\Box$ + +**Lemma (`le_succ_self`).** $x \le \operatorname{succ}(x)$. + +Proof. the gap is one, since $\operatorname{succ}(x) = x + 1$ by `succ_eq_add_one`. $\Box$ + +### The theorem. + +**Theorem (`le_total`).** for all $x$ and $y$, either $x \le y$ or $y \le x$. + +Proof. let $x$ be arbitrary and fixed. by induction on $y$. + +- *Base case:* $y = 0$. the right half holds, since $0 \le x$ by `zero_le`. +- *Inductive step:* let $d$ be arbitrary and take $y = \operatorname{succ}(d)$. inductive hypothesis: $x \le d$ or $d \le x$. goal: $x \le \operatorname{succ}(d)$ or $\operatorname{succ}(d) \le x$. the hypothesis is a disjunction, and the labels name which half is assumed. + - *Case 1 (left):* $x \le d$. the gap $c$ satisfies $d = x + c$. then $\operatorname{succ}(d) = (x + c) + 1 = x + (c + 1)$ by `succ_eq_add_one` and `add_assoc`, so $c + 1$ is a gap and $x \le \operatorname{succ}(d)$. + - *Case 2 (right):* $d \le x$. the gap $c$ satisfies $x = d + c$, and is zero or a successor. this is (I) against (II) from the number line. + - *Case 2a (gap zero):* $c = 0$. then $x = d$ by the zero equation, and $d \le \operatorname{succ}(d)$ by `le_succ_self`, so $x \le \operatorname{succ}(d)$. + - *Case 2b (gap a successor):* $c = \operatorname{succ}(a)$. then $x = \operatorname{succ}(d + a)$ by the successor equation, which is $\operatorname{succ}(d) + a$ by `succ_add`, so $a$ is a gap and $\operatorname{succ}(d) \le x$. + +each case establishes one half of the goal, so the goal holds at $\operatorname{succ}(d)$. both cases of the induction are now proved, and by the induction principle the statement holds for every $y$. $\blacksquare$ ## What it cost From f1f3effb6550cd971a9a8b5a5c3e2c69704619af Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 20:12:22 +0200 Subject: [PATCH 24/34] 67 --- ...8-lean-intro.md => 2026-06-28-six-less-than-seven.md} | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) rename _posts/{2026-06-28-lean-intro.md => 2026-06-28-six-less-than-seven.md} (98%) diff --git a/_posts/2026-06-28-lean-intro.md b/_posts/2026-06-28-six-less-than-seven.md similarity index 98% rename from _posts/2026-06-28-lean-intro.md rename to _posts/2026-06-28-six-less-than-seven.md index 704b87f..50e156e 100644 --- a/_posts/2026-06-28-lean-intro.md +++ b/_posts/2026-06-28-six-less-than-seven.md @@ -501,6 +501,11 @@ four goals get closed, one per leaf, and this proof uses two tactics to do it. ` ## In English +the same development written out as a mathematician would write it, with the tactics replaced by prose. the Lean above and the proof below are the same argument. + +
+the whole thing, in prose + ### Definitions. two rules describe $\mathbb{N}$. zero is a natural number, and the successor $\operatorname{succ}(d)$ of a natural number $d$ is a natural number. nothing else is a natural number. from this follows the induction principle: a property that holds of $0$, and holds of $\operatorname{succ}(d)$ whenever it holds of $d$, holds of every natural number. @@ -533,7 +538,7 @@ Proof. by the numeral definitions $1 = \operatorname{succ}(0)$, so $n + 1$ is $n Proof. by induction on $n$. - *Base case:* $n = 0$. the claim is $0 + 0 = 0$, the zero equation. -- *Inductive step:* let $d$ be arbitrary and take[118;1:3u $n = \operatorname{succ}(d)$. inductive hypothesis: $0 + d = d$. the successor equation gives $0 + \operatorname{succ}(d) = \operatorname{succ}(0 + d)$, and the hypothesis rewrites the inner sum as $d$. $\Box$ +- *Inductive step:* let $d$ be arbitrary and take $n = \operatorname{succ}(d)$. inductive hypothesis: $0 + d = d$. the successor equation gives $0 + \operatorname{succ}(d) = \operatorname{succ}(0 + d)$, and the hypothesis rewrites the inner sum as $d$. $\Box$ **Lemma (`succ_add`).** $\operatorname{succ}(a) + b = \operatorname{succ}(a + b)$. @@ -579,6 +584,8 @@ Proof. let $x$ be arbitrary and fixed. by induction on $y$. each case establishes one half of the goal, so the goal holds at $\operatorname{succ}(d)$. both cases of the induction are now proved, and by the induction principle the statement holds for every $y$. $\blacksquare$ +
+ ## What it cost eight theorems and eighty-three lines of tactics, for a statement that needs no defending to anyone who has counted to seven. From bfc91ca13e7ef1980a8818ff687f61e2b6b3117f Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 20:12:28 +0200 Subject: [PATCH 25/34] w --- single_file.lean | 100 +++++++++++++++++++++++++++++++++++++---------- 1 file changed, 80 insertions(+), 20 deletions(-) diff --git a/single_file.lean b/single_file.lean index 012d00f..c984e96 100644 --- a/single_file.lean +++ b/single_file.lean @@ -58,7 +58,11 @@ namespace MyNat instance : Inhabited MyNat where default := MyNat.zero -def ofNat (x : Nat) : MyNat := +-- We define ℕ by induction. +-- Base case. '0' (zero). +-- Inductive step. Successor of x ∈ ℕ +-- From now on the universe of discourse is ℕ. +def ofNat (x : Nat) : MyNat := match x with | Nat.zero => MyNat.zero | Nat.succ b => MyNat.succ (ofNat b) @@ -76,32 +80,22 @@ instance : ToString MyNat where theorem zero_eq_0 : MyNat.zero = 0 := rfl +-- we define '1' to be the successor of '0' def one : MyNat := MyNat.succ 0 -def pred : ℕ → ℕ -| 0 => 37 -- random (garbage) value according to NNG4 -| succ n => n - -lemma pred_succ (n : ℕ) : pred (succ n) = n := rfl - def is_zero : ℕ → Prop | 0 => True | succ _ => False -lemma is_zero_zero : is_zero 0 = True := rfl -lemma is_zero_succ (n : ℕ) : is_zero (succ n) = False := rfl - -theorem zero_ne_succ (a : ℕ) : 0 ≠ succ a := by - intro h - rewrite[← is_zero_succ a] - rewrite[← h] - rewrite[is_zero_zero] - trivial - -theorem one_eq_succ_zero : 1 = succ 0 := by rfl -theorem two_eq_succ_one : 2 = succ 1 := by rfl -theorem three_eq_succ_two : 3 = succ 2 := by rfl +-- Number definitions. +-- we define '7' to be the successor of '6' which is the successor of '5' which is the succ of '6' ... +theorem one_eq_succ_zero : 1 = succ 0 := by rfl +theorem two_eq_succ_one : 2 = succ 1 := by rfl +theorem three_eq_succ_two : 3 = succ 2 := by rfl theorem four_eq_succ_three : 4 = succ 3 := by rfl +theorem five_eq_succ_four : 5 = succ 4 := by rfl +theorem six_eq_succ_five : 6 = succ 5 := by rfl +theorem seven_eq_succ_six : 7 = succ 6 := by rfl -- addition opaque add : MyNat → MyNat → MyNat @@ -109,10 +103,15 @@ opaque add : MyNat → MyNat → MyNat instance instAdd : Add MyNat where add := MyNat.add +-- Inductive addition: +-- Axiom 1 (add_zero). a + 0 = a +-- Axiom 2 (add_succ). a + (succ d) = succ (a + d) axiom add_zero (a : MyNat) : a + 0 = a axiom add_succ (a d : MyNat) : a + (succ d) = succ (a + d) -- inequality +-- Inequality: +-- Axiom 3 (le). if there exists c, such that a + c = b, then a ≤ b. def le (a b : ℕ) := ∃ (c : ℕ), b = a + c instance : LE MyNat := ⟨MyNat.le⟩ -- END NNG4 -- @@ -127,12 +126,24 @@ instance : LE MyNat := ⟨MyNat.le⟩ -- BEGIN DANI -- +-- Lemma. (succ_eq_ad_one). succ n = n + 1. +-- Proof. In 'Number definitions' we defined '1' to be the successor of '0'. +-- By the axiom 2 of addition, n + succ ( 0 ) iff succ( n + 0 ). +-- By unfolding the axiom 1 of addition we get succ ( n ), showing that succ n = n + 1. QED theorem succ_eq_add_one n : succ n = n + 1 := by rewrite[one_eq_succ_zero] rewrite[add_succ] rewrite[add_zero] rfl +-- Lemma. (zero_add). 0 + n = n. +-- Proof. By mathematical induction. +-- Base case. n = zero. By the axiom 1 (add_zero) of addition, we show that 0 + n = n. +-- Induction step. Let d be arbitrary. +-- n = succ ( d ). By axiom 2 (add_succ) of addition, we show that 0 + succ (d) = succ ( 0 + d). +-- By axiom 1 (add_zero), we conclude that 0 + succ d = succ d. +-- As the induction is exhaustive, we have proven that 0 + n = n. QED +-- theorem zero_add (n : ℕ) : 0 + n = n := by induction n with | zero => @@ -144,6 +155,11 @@ theorem zero_add (n : ℕ) : 0 + n = n := by rewrite[hd] rfl +-- Lemma. (succ_add). succ a + b = succ (a + b). +-- Proof. By mathematical induction on b, fixing a. +-- Base case: b = zero. Goal: succ a + b +-- Inductive step: Let d be arbitrary. b = succ d. +-- Then theorem succ_add (a b : ℕ) : succ a + b = succ (a + b) := by induction b with | zero => @@ -194,6 +210,48 @@ theorem le_succ_self (x : ℕ) : x ≤ succ x := by use 1 exact succ_eq_add_one x +-- Theorem. (le_total) for all natural numbers x and y, either x is greater or equal to y or y is greater than or equal to x. +-- +-- +-- Definitions. +-- +-- NUMBERS +-- We define ℕ by induction. +-- Base case. '0' (zero). +-- Inductive step. Successor of x ∈ ℕ +-- From now on the universe of discourse is ℕ. +-- +-- Number definitions. +-- we define '7' to be the successor of '6' which is the successor of '5' which is the succ of '6' ... +-- Specifically number strings in the order ('0',1','2','3','4','5','6','7') +-- +-- Peano arithmetic: +-- Axiom 1 (add_zero). a + 0 = a +-- Axiom 2 (add_succ). a + (succ d) = succ (a + d) +-- +-- Inequality. +-- Axiom 3 (le). if there exists c, such that a + c = b, then a ≤ b. +-- +-- LEMMAS. +-- +-- Lemma. (succ_eq_ad_one). succ n = n + 1. +-- Proof. In 'Number definitions' we defined '1' to be the successor of '0'. +-- By the axiom 2 of addition, n + succ ( 0 ) iff succ( n + 0 ). +-- By unfolding the axiom 1 of addition we get succ ( n ), showing that succ n = n + 1. QED +-- +-- Lemma. (zero_add). 0 + n = n. +-- Proof. By mathematical induction. +-- Base case. n = zero. By the axiom 1 (add_zero) of addition, we show that 0 + n = n. +-- Induction step. Let d be arbitrary. +-- n = succ ( d ). By axiom 2 (add_succ) of addition, we show that 0 + succ (d) = succ ( 0 + d). +-- By axiom 1 (add_zero), we conclude that 0 + succ d = succ d. +-- As the induction is exhaustive, we have proven that 0 + n = n. QED +-- +-- Proof. we use mathematical induction. +-- Base case: [] +-- Induction step: [] +-- + -- This is the climax - the proof we have been approaching thus far theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by induction y with @@ -227,4 +285,6 @@ theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by rewrite[succ_add] rfl +#check le_total 6 7 + end MyNat From 1398fc06ba300ddb7b5fe0d08ea9789959808a18 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 20:49:44 +0200 Subject: [PATCH 26/34] Update title for accuracy --- _posts/2026-06-28-six-less-than-seven.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2026-06-28-six-less-than-seven.md b/_posts/2026-06-28-six-less-than-seven.md index 50e156e..dc4deef 100644 --- a/_posts/2026-06-28-six-less-than-seven.md +++ b/_posts/2026-06-28-six-less-than-seven.md @@ -1,5 +1,5 @@ --- -title: "Proving 6 ≤ 7 the hard way" +title: "Proving comparability of 6 and 7, the hard way" date: 2026-09-05 description: "Proving that any two natural numbers compare, from the definition of a natural number upwards, in Lean." tags: ["theoretical mathematics", "first principles"] From ee20e3388fa26eff97b70dbdea8e1f24d41c2177 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 21:00:55 +0200 Subject: [PATCH 27/34] Drop lean --- single_file.lean | 290 ----------------------------------------------- 1 file changed, 290 deletions(-) delete mode 100644 single_file.lean diff --git a/single_file.lean b/single_file.lean deleted file mode 100644 index c984e96..0000000 --- a/single_file.lean +++ /dev/null @@ -1,290 +0,0 @@ --- Author: cb341 (Dani) --- --- This document aims to show the steps required in prooving --- the totality property of natural numbers, namely --- --- theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x --- --- from first principles, starting at the inductive definition --- of natural numbers - proving all mediary theorems along the --- way and ending at QED. --- --- I use the natural number definitions from the natural numbers --- Lean game as a basis --- --- https://adam.math.hhu.de/#/g/leanprover-community/nng4 --- --- And proove theorems on top of that step by step. -import Mathlib.Tactic.Have -import Mathlib.Tactic.Contrapose -import Mathlib.Tactic.ApplyAt -import Mathlib.Tactic.Cases -import Mathlib.Tactic.NthRewrite -import Mathlib.Tactic.Tauto -import Lean.Elab.Tactic.Basic -import Lean.Elab.Tactic.Induction -import Batteries.Tactic.OpenPrivate -import Batteries.Data.List.Basic -import Mathlib.Lean.Expr.Basic -import Mathlib.Tactic.Cases -import Lean.Meta.Tactic.Refl -import Lean.Elab.Tactic.Basic -import Mathlib.Lean.Expr.Basic -import Lean.Elab.Tactic.Basic -import Lean.Elab.Tactic.Rewrite -import Mathlib.Tactic.Use -import Mathlib.Lean.Meta.Simp - --- The following definitions and axioms were taken over / modified from --- the Apache Licensed NNG4 environment I am familiar with. --- I do not take ownership over the folllowing definitions --- but I do acknowledge modifications. --- --- https://github.com/leanprover-community/NNG4 --- --- Disclaimer: This is not the entirity of the NNG4 env. --- DX features such as pretty printing, --- formatting a[118;1:3und more intuitive tactics were not taken over. - --- BEGIN NNG4 -- -inductive MyNat where -| zero : MyNat -| succ : MyNat → MyNat -attribute [pp_nodot] MyNat.succ -notation (name := MyNatNotation) (priority := 1000000) "ℕ" => MyNat - -namespace MyNat - -instance : Inhabited MyNat where - default := MyNat.zero - --- We define ℕ by induction. --- Base case. '0' (zero). --- Inductive step. Successor of x ∈ ℕ --- From now on the universe of discourse is ℕ. -def ofNat (x : Nat) : MyNat := - match x with - | Nat.zero => MyNat.zero - | Nat.succ b => MyNat.succ (ofNat b) - -def toNat (x : MyNat) : Nat := - match x with - | MyNat.zero => Nat.zero - | MyNat.succ b => Nat.succ (toNat b) - -instance instofNat {n : Nat} : OfNat MyNat n where - ofNat := ofNat n - -instance : ToString MyNat where - toString p := toString (toNat p) - -theorem zero_eq_0 : MyNat.zero = 0 := rfl - --- we define '1' to be the successor of '0' -def one : MyNat := MyNat.succ 0 - -def is_zero : ℕ → Prop -| 0 => True -| succ _ => False - --- Number definitions. --- we define '7' to be the successor of '6' which is the successor of '5' which is the succ of '6' ... -theorem one_eq_succ_zero : 1 = succ 0 := by rfl -theorem two_eq_succ_one : 2 = succ 1 := by rfl -theorem three_eq_succ_two : 3 = succ 2 := by rfl -theorem four_eq_succ_three : 4 = succ 3 := by rfl -theorem five_eq_succ_four : 5 = succ 4 := by rfl -theorem six_eq_succ_five : 6 = succ 5 := by rfl -theorem seven_eq_succ_six : 7 = succ 6 := by rfl - --- addition -opaque add : MyNat → MyNat → MyNat - -instance instAdd : Add MyNat where - add := MyNat.add - --- Inductive addition: --- Axiom 1 (add_zero). a + 0 = a --- Axiom 2 (add_succ). a + (succ d) = succ (a + d) -axiom add_zero (a : MyNat) : a + 0 = a -axiom add_succ (a d : MyNat) : a + (succ d) = succ (a + d) - --- inequality --- Inequality: --- Axiom 3 (le). if there exists c, such that a + c = b, then a ≤ b. -def le (a b : ℕ) := ∃ (c : ℕ), b = a + c -instance : LE MyNat := ⟨MyNat.le⟩ --- END NNG4 -- - --- What follow are all the theorems required to proove the totality of ℕ. --- The Lean4Game environment provided me with placeholders: --- --- theorem succ_eq_add_one n : succ n = n + 1 := by --- sorry --- --- The theorems were proven by me as an excercise. - --- BEGIN DANI -- - --- Lemma. (succ_eq_ad_one). succ n = n + 1. --- Proof. In 'Number definitions' we defined '1' to be the successor of '0'. --- By the axiom 2 of addition, n + succ ( 0 ) iff succ( n + 0 ). --- By unfolding the axiom 1 of addition we get succ ( n ), showing that succ n = n + 1. QED -theorem succ_eq_add_one n : succ n = n + 1 := by - rewrite[one_eq_succ_zero] - rewrite[add_succ] - rewrite[add_zero] - rfl - --- Lemma. (zero_add). 0 + n = n. --- Proof. By mathematical induction. --- Base case. n = zero. By the axiom 1 (add_zero) of addition, we show that 0 + n = n. --- Induction step. Let d be arbitrary. --- n = succ ( d ). By axiom 2 (add_succ) of addition, we show that 0 + succ (d) = succ ( 0 + d). --- By axiom 1 (add_zero), we conclude that 0 + succ d = succ d. --- As the induction is exhaustive, we have proven that 0 + n = n. QED --- -theorem zero_add (n : ℕ) : 0 + n = n := by - induction n with - | zero => - rewrite[zero_eq_0] - rewrite[add_zero] - rfl - | succ d hd => - rewrite[add_succ] - rewrite[hd] - rfl - --- Lemma. (succ_add). succ a + b = succ (a + b). --- Proof. By mathematical induction on b, fixing a. --- Base case: b = zero. Goal: succ a + b --- Inductive step: Let d be arbitrary. b = succ d. --- Then -theorem succ_add (a b : ℕ) : succ a + b = succ (a + b) := by - induction b with - | zero => - rewrite[zero_eq_0] - rewrite[add_zero] - rewrite[add_zero] - rfl - | succ d hb => - rewrite[add_succ] - rewrite[hb] - rewrite[add_succ] - rfl - -theorem add_comm (a b : ℕ) : a + b = b + a := by - induction b with - | zero => - rewrite[zero_eq_0] - rewrite[add_zero] - rewrite[zero_add] - rfl - | succ n hn => - rewrite[add_succ] - rewrite[succ_add] - rewrite[hn] - rfl - -theorem add_assoc (a b c : ℕ) : a + b + c = a + (b + c) := by - induction b with - | zero => - rewrite[zero_eq_0] - rewrite[add_zero] - rewrite[zero_add] - rfl - | succ n hn => - rewrite[add_succ] - rewrite[succ_add] - rewrite[succ_add] - rewrite[add_succ] - rewrite[hn] - rfl - -theorem zero_le (x : ℕ) : 0 ≤ x := by - use x - rewrite[zero_add] - rfl - -theorem le_succ_self (x : ℕ) : x ≤ succ x := by - use 1 - exact succ_eq_add_one x - --- Theorem. (le_total) for all natural numbers x and y, either x is greater or equal to y or y is greater than or equal to x. --- --- --- Definitions. --- --- NUMBERS --- We define ℕ by induction. --- Base case. '0' (zero). --- Inductive step. Successor of x ∈ ℕ --- From now on the universe of discourse is ℕ. --- --- Number definitions. --- we define '7' to be the successor of '6' which is the successor of '5' which is the succ of '6' ... --- Specifically number strings in the order ('0',1','2','3','4','5','6','7') --- --- Peano arithmetic: --- Axiom 1 (add_zero). a + 0 = a --- Axiom 2 (add_succ). a + (succ d) = succ (a + d) --- --- Inequality. --- Axiom 3 (le). if there exists c, such that a + c = b, then a ≤ b. --- --- LEMMAS. --- --- Lemma. (succ_eq_ad_one). succ n = n + 1. --- Proof. In 'Number definitions' we defined '1' to be the successor of '0'. --- By the axiom 2 of addition, n + succ ( 0 ) iff succ( n + 0 ). --- By unfolding the axiom 1 of addition we get succ ( n ), showing that succ n = n + 1. QED --- --- Lemma. (zero_add). 0 + n = n. --- Proof. By mathematical induction. --- Base case. n = zero. By the axiom 1 (add_zero) of addition, we show that 0 + n = n. --- Induction step. Let d be arbitrary. --- n = succ ( d ). By axiom 2 (add_succ) of addition, we show that 0 + succ (d) = succ ( 0 + d). --- By axiom 1 (add_zero), we conclude that 0 + succ d = succ d. --- As the induction is exhaustive, we have proven that 0 + n = n. QED --- --- Proof. we use mathematical induction. --- Base case: [] --- Induction step: [] --- - --- This is the climax - the proof we have been approaching thus far -theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by - induction y with - | zero => - right - exact zero_le x - | succ d hd => - cases hd with - | inl hl => - cases' hl with c hc - rewrite[hc] - left - rewrite[succ_eq_add_one] - use c + 1 - rewrite[← add_assoc] - rfl - | inr hr => - cases' hr with c hc - cases c with - | zero => - rewrite[zero_eq_0] at hc - rewrite[add_zero d] at hc - left - rewrite[hc] - exact le_succ_self d - | succ a => - rewrite[add_succ] at hc - right - rewrite[hc] - use a - rewrite[succ_add] - rfl - -#check le_total 6 7 - -end MyNat From ebfc3926906e7e7abfdcff28514be6ec30d91fee Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 21:09:57 +0200 Subject: [PATCH 28/34] Polish --- _posts/2026-06-28-six-less-than-seven.md | 132 +++++++++++++---------- 1 file changed, 74 insertions(+), 58 deletions(-) diff --git a/_posts/2026-06-28-six-less-than-seven.md b/_posts/2026-06-28-six-less-than-seven.md index dc4deef..58dee38 100644 --- a/_posts/2026-06-28-six-less-than-seven.md +++ b/_posts/2026-06-28-six-less-than-seven.md @@ -1,26 +1,27 @@ --- title: "Proving comparability of 6 and 7, the hard way" date: 2026-09-05 -description: "Proving that any two natural numbers compare, from the definition of a natural number upwards, in Lean." +description: "Proving that any two natural numbers compare, from an inductive definition of the naturals and two axioms for addition, in Lean." tags: ["theoretical mathematics", "first principles"] math: true --- paper proofs have no compiler. i went looking for one and found Lean, by way of the [Natural Number Game](https://adam.math.hhu.de/#/g/leanprover-community/nng4), which builds the naturals from nothing and makes you prove your way back out. [^nng] i finished it end of june and wanted to write up what the last level actually took. -no AI was used for the Lean here, and none for the maths either: the proofs, the proof strategy and the dependency graph are mine. AI was used for phrasing. the definitions of `MyNat`, `+` and `≤` are taken from the Natural Number Game, which is Apache licensed, and modified. [^nng4src] +no AI was used for the Lean or the maths: the tactic proofs, proof strategy and dependency graph are mine. the statements of `le_total` and the seven supporting theorems come from the Natural Number Game. so do the definitions of `MyNat`, `+` and `≤`, which are Apache licensed and were modified. [^nng4src] AI was used for phrasing, including condensing the English proof, and for review. -[^nng4src]: . the artefact carries the same attribution in its header. the seven supporting theorems and `le_total` are proved by me from those definitions. +[^nng4src]: . the artefact carries the same attribution in its header. ## The theorem $$ -\Large \forall x,y \in \mathbb{N} : x \le y \lor y \le x +\Large \forall x,y \in \mathbb{N},\; x \le y \lor y \le x $$ any two natural numbers compare. one of them is at most the other. -take 6 and 7. then $(6 \le 7) \lor (7 \le 6)$ is $T \lor F$, so $T$. take 1 twice. then $(1 \le 1) \lor (1 \le 1)$ is $T \lor T$, so $T$. +take 6 and 7. then $(6 \le 7) \lor (7 \le 6)$ is $T \lor F$, so $T$. +take 1 twice. then $(1 \le 1) \lor (1 \le 1)$ is $T \lor T$, so $T$. checking pairs by hand settles those two pairs. $\mathbb{N}^2$ is infinite, so no amount of checking gets through it, and the $\forall$ has to be discharged some other way. @@ -30,9 +31,11 @@ right now $\le$, $+$ and $\mathbb{N}$ are all undefined, so that line is notatio ![Dependency graph for le_total](/assets/blog/lean_dependency_graph.svg) -lavender nodes are definitions: $\mathbb{N}$ itself, the two clauses of addition, and $1 = \operatorname{succ}(0)$. mint nodes are the arithmetic that follows, including commutativity and associativity. peach nodes are the order results, ending in the theorem. +lavender nodes are the definitions: $\mathbb{N}$ itself, the two axioms for addition, and $1 = \operatorname{succ}(0)$. mint nodes are the arithmetic that follows, including commutativity and associativity. peach nodes are the order results, ending in the theorem. -the graph is the table of contents. we start at $\mathbb{N}$ at the top and walk down to `le_total`, and every node below is a section or a paragraph. nothing gets used before its node has been visited, so you can check the article against the picture as you go. +the graph is the table of contents and a dependency map for this development. + +_side note: `add_comm` is in the file and in the graph, but `le_total` does not depend on it. every other theorem shown has a path to `le_total`._ ## Notation @@ -67,7 +70,7 @@ each connective is fixed by its truth table. the two rows of $\to$ where $P$ is false both come out true, which is worth its own article. -$P \leftrightarrow Q$ is $(P \to Q) \land (Q \to P)$. it is the $:\Leftrightarrow$ used to define $\le$ below. +$P \leftrightarrow Q$ is $(P \to Q) \land (Q \to P)$. the `def` over the $\iff$ used below says that the equivalence defines $\le$; it is still the same logical connective. $\top$ is the statement that always holds, $\bot$ the statement that never does. @@ -103,17 +106,17 @@ $$ \frac{d \in \mathbb{N}}{\;\operatorname{succ}(d) \in \mathbb{N}\;} $$ -zero is a natural. the successor of a natural is a natural. every natural is reached by applying the second rule to the first some finite number of times, and nothing else is in $\mathbb{N}$. written as cases: +zero is a natural. the successor of a natural is a natural. every natural is reached by applying the second rule to the first some finite number of times, and nothing else is in $\mathbb{N}$. therefore every $n \in \mathbb{N}$ has one of two forms: $$ -n \in \mathbb{N} \stackrel{\mathrm{def}}{=} -\begin{cases} -0 & \text{base case} \\ -\operatorname{succ}(d) & d \in \mathbb{N} -\end{cases} +n = 0 +\qquad\text{or}\qquad +n = \operatorname{succ}(d) \text{ for some } d \in \mathbb{N}. $$ -the same shape gives us induction for free. to prove something about every natural, prove it for $0$ and prove that $d$ having it forces $\operatorname{succ}(d)$ to have it. that is the `induction` tactic later, and it is the reason this definition is worth the trouble. +Lean's inductive definition also gives us induction. to prove something about every natural, prove it for $0$ and prove that $d$ having it forces $\operatorname{succ}(d)$ to have it. that is the `induction` tactic later, and it is the reason this definition is worth the trouble. + +the Lean artefact names this inductive type `MyNat`; i write $\mathbb{N}$ in the prose. [^peano]: Giuseppe Peano, *Arithmetices principia, nova methodo exposita*, 1889. The Latin original is on archive.org: @@ -138,9 +141,9 @@ $$ simpler to write, but $n < m$ is no longer $n \in m$, so order has to be defined separately. [^zermelo] -**Church numerals.** a number is a function that applies another function that many times. $n$ is $\lambda f. \lambda x. f^n(x)$, so $3$ is $\lambda f. \lambda x. f(f(f(x)))$. addition is function composition. this is how the naturals appear in untyped lambda calculus. [^church] +**Church numerals.** a number is a function that applies another function that many times. $n$ is $\lambda f. \lambda x. f^n(x)$, so $3$ is $\lambda f. \lambda x. f(f(f(x)))$. addition composes the two iterates of $f$. this is how the naturals appear in untyped lambda calculus. [^church] -**Peano axioms as first-order theory.** rather than constructing the naturals, state the properties they must have: $0$ is not a successor, $\operatorname{succ}$ is injective, and induction holds. this leaves the objects unspecified and constrains them instead. [^pa] +**Peano axioms as first-order theory.** rather than constructing the naturals, state the properties they must have: $0$ is not a successor, $\operatorname{succ}$ is injective, and the induction schema is included. this leaves the objects unspecified and constrains them instead. [^pa] i am using zero and succ, which is what the Natural Number Game uses and what Lean's own `Nat` is. @@ -154,40 +157,41 @@ i am using zero and succ, which is what the Natural Number Game uses and what Le [^church]: Alonzo Church, *An Unsolvable Problem of Elementary Number Theory*, American Journal of Mathematics 58 (1936), 345–363. Barendregt, *The Lambda Calculus*, section 6.4 gives the arithmetic. -[^pa]: The first-order theory is usually attributed to Peano 1889 by way of Dedekind. Hájek and Pudlák, *Metamathematics of First-Order Arithmetic* (1998), chapter I, is the reference treatment. The distinction that matters here: the first-order schema quantifies over formulas, so it does not pin down $\mathbb{N}$ up to isomorphism, whereas the inductive definition above does. +[^pa]: The first-order theory is usually attributed to Peano 1889 by way of Dedekind. Hájek and Pudlák, *Metamathematics of First-Order Arithmetic* (1998), chapter I, is the reference treatment. First-order induction is an axiom schema, with one instance per formula, and the theory has nonstandard models. the inductive type here has only the constructors stated above. ### Our zero and Lean's zero -the definition above introduces a constructor, written `MyNat.zero` in Lean. the character `0` is a numeral, which is what a person types. they denote the same natural number and they are not the same term. +the definition above introduces a constructor, written `MyNat.zero` in Lean. the character `0` is a numeral, which is what a person types. they denote the same natural number. they are not syntactically the same term, but Lean reduces them to the same term, so they are definitionally equal. -this matters mechanically. `rfl` closes a goal when both sides are literally the same term, and `rewrite` matches on the shape of a term. a goal reading `MyNat.zero + x` does not match a lemma stated about `0 + x`, so the two have to be connected first. `zero_eq_0` is that connection, and it is a theorem you prove rather than something the definition gives you. +this matters mechanically. `rfl` closes a goal up to definitional equality. `zero_eq_0` makes that equality available as an equation for `rewrite`. -the same split shows up between `succ n` and `n + 1`, bridged by `succ_eq_add_one`. both are nodes in the dependency graph. a definition fixes which terms exist, notation is a separate layer, and the two get connected by proof. +`succ n = n + 1` is different: it is not definitional here. `succ_eq_add_one` proves it from the two addition axioms. the proof below carries `rewrite[zero_eq_0] at hc` for exactly this reason. ### Addition -$6 + 1 = 7$ defines one sum. $\mathbb{N}^2$ has infinitely many, so addition is defined by recursion on the second argument, mirroring the definition of $\mathbb{N}$. +$6 + 1 = 7$ gives one sum. $\mathbb{N}^2$ has infinitely many, so we need rules for all pairs. in the artefact, `add` is an opaque function and NNG4 supplies two axioms that characterise recursion on the second argument: $$ -\forall a,b \in \mathbb{N} : a + b \stackrel{\mathrm{def}}{=} -\begin{cases} -a & b = 0 \\ -\operatorname{succ}(a + d) & b = \operatorname{succ}(d) -\end{cases} +\begin{aligned} +a + 0 &= a && (\texttt{add\_zero}) \\ +a + \operatorname{succ}(d) &= \operatorname{succ}(a + d) && (\texttt{add\_succ}) +\end{aligned} $$ -worked on $1 + 2 = 3$. the reason column gives the direction, then the definition used. $(\rightarrow)$ unfolds, replacing a name by its definition. $(\leftarrow)$ folds, recognising a definition and naming it. +these are axioms in the Lean file, not definitional reductions: `add` is opaque, so it cannot be unfolded to prove them. + +worked on $1 + 2 = 3$. the reason column names the rule and how it is used. $(\rightarrow)$ means forwards, as written; $(\leftarrow)$ means reversed. $$ \begin{array}{rl} -& \textbf{Definitions} \\[2pt] +& \textbf{Rules used} \\[2pt] \colorbox{#fff3cd}{$\vphantom{Ag}\text{i.}$} & 1 \stackrel{\mathrm{def}}{=} \operatorname{succ}(0) \\[2pt] \colorbox{#cfe2ff}{$\vphantom{Ag}\text{ii.}$} & 2 \stackrel{\mathrm{def}}{=} \operatorname{succ}(1) \\[2pt] \colorbox{#e2d9f3}{$\vphantom{Ag}\text{iii.}$} & 3 \stackrel{\mathrm{def}}{=} \operatorname{succ}(2) \\[2pt] -\colorbox{#f8d7da}{$\vphantom{Ag}\text{iv.}$} & a+0 \stackrel{\mathrm{def}}{=} a \\[2pt] -\colorbox{#d1e7dd}{$\vphantom{Ag}\text{v.}$} & a+\operatorname{succ}(b) \stackrel{\mathrm{def}}{=} \operatorname{succ}(a+b) +\colorbox{#f8d7da}{$\vphantom{Ag}\text{iv.}$} & a+0 = a \\[2pt] +\colorbox{#d1e7dd}{$\vphantom{Ag}\text{v.}$} & a+\operatorname{succ}(b) = \operatorname{succ}(a+b) \end{array} \quad \begin{array}{l|l} @@ -204,15 +208,15 @@ $$ \end{array} $$ -the same definition gets used in both directions. which direction you pick is a choice, and picking wrong is how a rewrite fails to terminate. +rules ii and iii appear in both directions: first to expand the numerals, then reversed to recover them. ### Less than or equal $$ -\forall a, b \in \mathbb{N} : a \le b \stackrel{\mathrm{def}}{\iff} \exists (c : \mathbb{N}), b = a + c +\forall a, b \in \mathbb{N},\quad a \le b \stackrel{\mathrm{def}}{\iff} \exists (c : \mathbb{N}), b = a + c $$ -there is a gap, and the gap is itself a natural number. the second half carries the content, since there are no negative naturals available to serve as gaps. +there is a gap, and the gap is itself a natural number. $c$ cannot be negative because its type is $\mathbb{N}$. the gap is a natural, so it is either zero or a successor, which gives two pictures. in (I) the gap is $c = 0$ and $a = b$, the case where $\le$ holds because the two numbers are equal. in (II) the gap is nonzero and $a + c = b$ with $a$ strictly below $b$. @@ -230,7 +234,7 @@ protected inductive Nat.le (n : Nat) : Nat → Prop both say the same thing about the same numbers. the existential version hands you a gap to compute with, the inductive version hands you a chain of steps to recurse on. code pasted from here into a mathlib project will not typecheck unchanged. -[^natle]: `Init/Prelude.lean`, line 2022 in the Lean 4 source: . mathlib inherits this definition rather than replacing it. +[^natle]: Lean 4 API documentation for `Init.Prelude`, entry `Nat.le`: . mathlib inherits this definition rather than replacing it. [^nng]: Natural Number Game 4, by Kevin Buzzard and Mohammad Pedramfar: @@ -241,22 +245,22 @@ a Lean proof is written as a list of tactics. the ones in this article: | tactic | what it does | | --- | --- | | `induction` | splits a natural into the zero case and the successor case, and hands you the induction hypothesis | -| `cases` | splits a disjunction hypothesis into two branches, `inl` and `inr` | -| `cases'` | unpacks an existential hypothesis into a witness and an equation | +| `cases` | splits a value or hypothesis by its constructors; here those are zero/successor or `inl`/`inr` | +| `cases'` | Mathlib tactic used here to unpack an existential hypothesis into a witness and an equation | | `left` / `right` | picks which side of a disjunction goal to prove | | `use` | supplies a witness for an existential goal | -| `rewrite[h]` | replaces occurrences of the left side of `h` with the right side | -| `rewrite[← h]` | the same equation applied right to left | -| `rfl` | closes a goal whose two sides are the same term | +| `rewrite[h]` | uses `h` forwards, as written | +| `rewrite[← h]` | uses `h` in reverse | +| `rfl` | closes an equality when both sides are definitionally equal | | `exact` | closes a goal with something already proved | -`rw` is the usual short form of `rewrite`, though the artefact spells it out everywhere. `zero_eq_0`, `succ_eq_add_one` and `cases'` are Natural Number Game spellings, so pasting this into a fresh mathlib project gets you errors on the names before anything interesting. +`rw` is the usual short form of `rewrite`, though the artefact spells it out everywhere. `zero_eq_0` and `succ_eq_add_one` are names from this NNG-style development; `cases'` comes from Mathlib. ## The smaller lemmas before the theorem, a smaller one: $0 \le x$ for every natural $x$. it sits in the dependency graph as `zero_le`, and the theorem's base case consumes it. -unfolding the definition, $0 \le x$ means $\exists c, x = 0 + c$. take $c := x$. the goal becomes $x = 0 + x$, which is `zero_add`. in Lean that is three lines. +unfolding the definition, $0 \le x$ means $\exists c, x = 0 + c$. take $c := x$. the goal becomes $x = 0 + x$; rewriting the right side with `zero_add` leaves $x = x$. in Lean that is three lines. ```lean theorem zero_le (x : ℕ) : 0 ≤ x := by @@ -283,7 +287,7 @@ theorem zero_add (n : ℕ) : 0 + n = n := by rfl ``` -addition recurses on its second argument, so `a + 0 = a` is true by definition and `0 + n = n` is not. the two look symmetric and only one of them is free. this asymmetry is why `succ_add`, `add_comm` and `add_assoc` all need their own inductive proofs. +the two axioms describe addition through its second argument. `a + 0 = a` is available directly as `add_zero`, while `0 + n = n` has to be derived as `zero_add`. the two look symmetric and only one is assumed. this asymmetry is why `succ_add`, `add_comm` and `add_assoc` all need their own inductive proofs. @@ -291,7 +295,7 @@ one theorem often admits several proofs, each a different path through the depen [^ording]: Philip Ording, *99 Variations on a Proof*, Princeton University Press, 2019. -the main proof calls two more results by name. both are proved the same way, from the definitions above. +two small results are short enough to show in full.
succ_eq_add_one and le_succ_self @@ -316,11 +320,23 @@ theorem le_succ_self (x : ℕ) : x ≤ succ x := by
+three arithmetic theorems remain: + +$$ +\begin{aligned} +\operatorname{succ}(a) + b &= \operatorname{succ}(a+b) && (\texttt{succ\_add}) \\ +(a+b)+c &= a+(b+c) && (\texttt{add\_assoc}) \\ +a+b &= b+a && (\texttt{add\_comm}) +\end{aligned} +$$ + +`le_total` depends on `succ_add` and `add_assoc`. it does not depend on `add_comm`, though the artefact proves that too. + ## Tactics as state transitions a Lean proof has a state: the hypotheses you have, and the goal you owe. a tactic changes that state. the proof is the sequence of changes. -the tables below are static copies of something you can drive yourself. [the whole development runs in the Lean web editor](https://tinyurl.com/4tr5uc7c), and clicking a line shows its state in the panel on the right. +the tables below are static copies of something you can drive yourself. [the whole development runs in the Lean web editor][lean-live], and clicking a line shows its state in the panel on the right. the Lean documentation puts it this way: [^tactics] @@ -409,12 +425,12 @@ and `right`, which picks a side of the goal and discards the other: ## Tests and proofs -the obvious way for a programmer to check `le_total` is to assert it. +a programmer might test a boolean version on a few examples. this is pseudocode: `comparable` returns a boolean, while Lean's `le_total` produces a proof. ``` -assert le_total(6, 7) -assert le_total(1, 1) -assert le_total(0, 255) +assert comparable(6, 7) +assert comparable(1, 1) +assert comparable(0, 255) ``` green suite, three pairs, out of infinitely many. property-based testing does better. QuickCheck, Hypothesis and proptest generate values of `a` and `b` and assert the property on each, which samples more widely and still samples. [^quickcheck] @@ -439,7 +455,7 @@ that makes `sorry` the counterpart of a stubbed test: it lets you write the shap ## The last step -what follows is the final theorem only, the bottom node of the dependency graph. it is thirty lines, and it is thirty lines rather than more because everything it stands on has already been proved: the definition of $\mathbb{N}$, both clauses of addition, the definition of $\le$, and the seven theorems above it, another fifty-three lines that are not in this snippet. the whole development is in the artefact. +what follows is the final theorem only, the bottom node of the dependency graph. it uses the definition of $\mathbb{N}$, the two addition axioms, the definition of $\le$, and six earlier theorems. the artefact also proves `add_comm`, but `le_total` does not use it. ```lean theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by @@ -475,7 +491,7 @@ theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by rfl ``` -[open the whole development in the Lean web editor](https://tinyurl.com/4tr5uc7c) to see the definitions and the seven supporting theorems this rests on, and to click through the states below yourself. +[open the whole development in the Lean web editor][lean-live] to click through the proof states yourself. it contains the six supporting theorems and the separate proof of `add_comm`. the induction is on `y`, which gives two cases. @@ -516,7 +532,7 @@ each numeral abbreviates iterated successors: $1 = \operatorname{succ}(0)$, $2 = ### Axioms. -**Peano axiomatic arithmetic.** addition satisfies two equations. +**Arithmetic axioms.** the opaque addition function satisfies two equations. $$ \begin{array}{rcll} @@ -590,11 +606,11 @@ each case establishes one half of the goal, so the goal holds at $\operatorname{ eight theorems and eighty-three lines of tactics, for a statement that needs no defending to anyone who has counted to seven. -`le_total` itself is thirty of those lines. the other fifty-three are the seven theorems underneath it: `succ_eq_add_one` at four lines, `zero_add` at nine, `succ_add` at eleven, `add_comm` at eleven, `add_assoc` at thirteen, `zero_le` at three, `le_succ_self` at two. five of the seven are about addition, and none of them mention $\le$ at all. proving that two numbers compare turns out to be mostly a matter of proving that addition behaves. +`le_total` itself takes thirty lines. six theorems on its dependency path take another forty-two: `succ_eq_add_one` at four lines, `zero_add` at nine, `succ_add` at eleven, `add_assoc` at thirteen, `zero_le` at three, `le_succ_self` at two. `add_comm` takes the remaining eleven lines, though `le_total` does not use it. four of the six dependencies are about addition, and none of those four mention $\le$ at all. proving that two numbers compare turns out to be mostly a matter of proving that addition behaves. -what the artefact assumes is small and explicit: the inductive definition of `MyNat`, `add_zero` and `add_succ` as the defining equations of `+`, and the definition of `≤`. everything after that is derived. `sorry` is the only way to skip a step, and it shows up as a warning every time you compile. +what the artefact assumes is small and explicit: the inductive definition of `MyNat`, an opaque `add`, the axioms `add_zero` and `add_succ`, and the definition of `≤`. everything after that is derived. `sorry` is the explicit escape hatch for an unfinished proof, and Lean reports its presence. -the full single-file solution is at . it opens in the Lean 4 web editor with the whole development in it, so you can click any line and watch the givens and goal in the right-hand panel, exactly the two columns from earlier. put the cursor inside the `inr` branch and you can see the case split on `c` open up. no install, and it typechecks end to end. +the full single-file solution is in the [Lean web editor][lean-live]. it opens in the Lean 4 web editor with the whole development in it, so you can click any line and watch the givens and goal in the right-hand panel, exactly the two columns from earlier. put the cursor inside the `inr` branch and you can see the case split on `c` open up. no install, and it typechecks end to end. next is linear algebra at the [FernUniversität in Hagen](https://www.fernuni-hagen.de/mi/studium/module/lin_alg.shtml), alongside part-time studies at ZHAW. first course where proofs are the work rather than a step inside it, which is why i wanted this done now. @@ -607,10 +623,10 @@ next is linear algebra at the [FernUniversität in Hagen](https://www.fernuni-ha - Alyssa Ney, *Metaphysics: An Introduction*. its treatment of first and second order predicate logic ties the notation to philosophical questions instead of deriving it from truth tables, which makes it a good counterweight to the ETH script. - Axler, *Linear Algebra Done Right* - Peano's original 1889 paper, in Latin: -- an English translation of the same: +- an English translation of Peano's 1889 paper: Jean van Heijenoort (ed.), *From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931*, Harvard University Press, 1967, pp. 83–97 - Peano axioms, ETH: - Peano axioms, Trinity College Dublin: - and the list of 1000 theorems: -- Kevin Buzzard, whose version of this proof mine follows: +- Kevin Buzzard, co-author of Natural Number Game 4: -## Notes +[lean-live]: https://live.lean-lang.org/#codez=LTAEEEFcBcAsHsBOAuUBjARgZgCwEZQAKAEQEMA7ASwEoAoEesAFVkoGdQATeNSAWwCm5aKFKU%2BHaPFBsEAd1BwBM6AIAOHRAICOkSls6hK5UGsTx4AN2MBzRotjKknAYluh4AM1DlS0SIikADY%2B%2FBiubAA0PqSCQQCejPagDgJIAnygQQIA%2BlLQwUQAHqDxoKiAqITU5aAlgCZEpaCAFESNDUVJIKCe5plqAhTSpEWU8BLRbAWI0O5%2BqUbknJBo05bKLp7GlNPw5PZeMf6BIeRhEdFmVjNBIaSQw0GUpIhl2Xx8pBzBOzap9nKkZQohiEnBmIgAigBRYgAOg6YAAkqBIGxlEoDgFCic%2BOFEFwBBsqNtyBxuqM5r5DljTog2PYADL9Ew2WLKD6iUAYD7seGgWDQaAaZAAemFpE4sRh7zgMNgsEgMJcwoAxMKbMLshQLqtEMA0KM%2BJAifFheRyDYcLRxGokCIALJ%2BWAPDAwpikZaUNAwgASpFWVr4NqmoAdcGdrvd0y9AGEdtBAjbUQGg%2FbHeG3R6veA1GoEuBoMnbSG05QXRmozDox8BHTrUXQ07SxHMzCAHJwABKAjkblUheDDfTkc9EZg8H7IkZFBhkKCpDLw69ACFuWgJ6Ap%2BQZ3OFy2EQslsT1yuBa5KDXmxWAPJ9cgABTclj8AmPflUbgvZAKMPp7GgMJXNhPXXQcm03GcijMADVxAktdwrKtUVrQMi3Au0BG%2FcsRy7TwgnXcDZ3nS8R0A4C6wHOCfyZCCoNItdyMnajCPgkiYIYjcmJ3YivS7HsthfdjQJYr0AFUk0Eyi0Iw0gYQAZWtWh7BYZRPHga54Dkdx1k2YlPgWURhlGDg5FcNFSAAayEDwdVAYVQD4eBQQ2ARDDJPh7HRbN3UcDdPSEVFDFbVsAHEcFAIRrHMchBGEUAkViLpYkoB4nlATSZXsJFuB8eARAKSyPDkcgIlYNRrNcOZVOuIINK0gkdJGEl7AwGA4q4QY0HM8gNOyTgbGUBynM9PxGrYOEGE6flBTYEU1S2eUXX1PgNSZbVXD1A0jS2E0gtC3liHYNA53EVxUBYdgjA4bq8p8oRpl7Mp9nRXawoi8awGIAANLp%2BkOGsZCWWBRA4MwMOgMozGMaZzUiexVMQaVoZ%2BIF7PSeZ%2FC2ShVkURdjNMnK8osqyrFcOF7CXSFgoRVtQBe0AGGMRYPWxu14lbWY5EcLRaAAH1AAAvVxpFQVn2YLPm2CWNAalF2ZACTCEM2b8Wg3zcFrVFAABtHMcm67hoAAXUVsWYUltA12ukadiIXxBHKABeY2%2FFbXKrfIapCEhpBtod0A8AABkDoPqgAIgqEPQHtgA%2BJ2C1oW2azUd1lFluPjAmCg0GUVB91ged%2BMMVO0q5l8UnWW4ghEZBHdTmFBfMRT1g8TwxeKGoxeqEWlar%2B3aBSaU0CBkp0tgPvQD5k36%2BkFJo9juuhbHie%2FFNqXOUjmPa7N6XCC8VuMDoWgm6kVvCBKLuO%2Fb2Zq7Hgeh7S%2BbF7nqeUnX2nl6nx%2FN9XjBX5NreiGPrMfeil04FHIFneYExd6zAAN4mFQGLAAvjUK8LdZhFxMJzUyY9oE92bq3XYVoSRgIgWdeAsl4zuCLlg7mKQpAULcOaUwvt6GUKYYQQBIg1AHyUOkTIU8cg6ByP7GW3d57mEjqAER1dQCIFwo3AkHhiqiNbjIr%2B5spEKO8OwHIz9KigAVg%2BeAaheZSNfkwRAkAXwS1XjkV%2BAAxYISZ7CthpPiQkmMdhjVoLwrQmQdi5CEVvXRQsX6oAII7f%2B0jHYYDKHIvCviMiKDkPAQR2gcjBICS%2FGoAAmSR%2F8CAyNibI%2BRiTMhwC0IE9JwToApJSKgLA%2BTV55KKXE0pjg%2BFdHgAENJGSpZ5FgJUmoYVImr0aa0kpCSOl%2BK6FjKpfTzY5FUgEepoAACsTSNFhQmfEnx0yklASKL04JGxsarIAGybOlhsnZ7S0gzNRKscgxz%2BmHNWQAdiuaAS5ty8L2HFKCI8xjSC6DZJwQw595axwMbHEBxDM7KFAeAcFNRkWF27sXbBKQAW%2B1rgCxSQwRiZABSEiRhBSAqL8J3DkABqMxjtSAq0MsS8FCzt4UohbHalFK6WEH%2FpwaooyNHktAHSgVilOjGB0JAYI21D6KOyEQClP9Kid0dqAQAwEREGlqq6IP8GWivQEQjO4Ds4bkhNCmRgAL8lrtkQAl%2BT2EhK2YgtMQphQYPYAA6nnEQVUaoKCeGya4cwymaGlfoFyihpAXBJnMZwZ4mH7AqO9UAykOIUBwMFVk4VyCRR2DFLh5hrAuEMHbEepg5xZwQEEFwtJkC8jKQDRZQiSVZPgU26WJhHYmDpYUmJiROgpDYEgZ4vI02hrSvjNaJhil23ZBQcKRQs6IDQOwAQZNOgUypjTYg4BWxIg9a8d4HbemtuUe2%2F%2BXafCGr7ZyRIKQtB8VUJrAJLzFlTwNmPR9vYBCaxJVvT9D7uw%2Fr%2Fayj9X75G0CPRSgROLCDtqqDUERdKr3wP7WPRmh5GrXpHo%2FZ%2B0cx5Aafb%2BgRQj%2FaAZft%2B%2FioHOCkvgBRh98iUg2I0YYWAhgCPZKo8%2B%2F9UsGOyOA9R9j%2FHdlQYyMe4JcHlU1EQ6gf%2BPK15CvZYa%2FevtYkYYPB6a2P9cPMYFqEzjlHBPPtI%2Bk8jhGBPEZo3RkTxnf0kvA1xpj48O1cD5PqqOFnuP2dZQBrzdnNawAwLZqzvHzYicg9B0QrKlqZBFSq0AsnaWKbXnSilrSNNM2JGvXTLn8Oea4wF0zwiQsgYc0LUr1HYPgoi3hPTl6%2BRdoK0Z0Lvm%2BP%2Fas5JmrHWQOwHILVxSUWSUfGHcpn%2BOrEvcpU4a6WBreU%2FzpWgNVd7MtYe0%2FfOAeGDPNaIyB4r5nCutdo45lre2hY5ABQN%2Brq8TB9fXj16jYW0CVefV1zgL3f1vY%2B9Zvzh3ev9a85F8TMHzuKtPjJ6lIi2hqfvciVEtQv1FfO5diDfyovZDZRkgQQRvDg9VTUeormz7oZSCiZQeAx4CCKJGE9LbWVZPaB5HyjaTJ8j9MocIVlSA5nMN5dwcAUSJUQGJt4FKMf5EKODso%2BPUCE7KC0Mo0OMspEw1pkwZRct83y15ygNh%2BQWapzTgRir2jXdY3yDjO30DVg4Oxjbo9sl82MCEJ093skpDQDbgA5HyEI5bpawDXO7yzvXnsWZSNkTwBZg%2Fec1sEuntGAn8dJ%2FD6Wvbw8h%2Bo4ABMJou0ZGzwZPkyLNO%2FIHiQZbv3ee6Qj78v%2Fu%2BRB8rzb9A9uM9a%2B2xn3bVXztkaNrMQPHfM88bA6E97ogRD9%2BDxHgk0fJ%2Bx8D4XlIhvlhZFyME1EOOuCt9cxSwzMeAtPd7%2BPhvMfdf68n4P398%2BB9k9EAP2PX27%2BQeVIPAQnUV95FdiES57zFIgljkAA From 741624733a69344d82336fa8a390933065222880 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 21:49:36 +0200 Subject: [PATCH 29/34] Polish --- _posts/2026-06-28-six-less-than-seven.md | 95 ++++++++++-------------- 1 file changed, 39 insertions(+), 56 deletions(-) diff --git a/_posts/2026-06-28-six-less-than-seven.md b/_posts/2026-06-28-six-less-than-seven.md index 58dee38..ab0d9b3 100644 --- a/_posts/2026-06-28-six-less-than-seven.md +++ b/_posts/2026-06-28-six-less-than-seven.md @@ -25,7 +25,7 @@ take 1 twice. then $(1 \le 1) \lor (1 \le 1)$ is $T \lor T$, so $T$. checking pairs by hand settles those two pairs. $\mathbb{N}^2$ is infinite, so no amount of checking gets through it, and the $\forall$ has to be discharged some other way. -right now $\le$, $+$ and $\mathbb{N}$ are all undefined, so that line is notation. the rest of the article pays the debt in order. +right now $\le$, $+$ and $\mathbb{N}$ are all undefined, so that line is only notation. the sections below define each one before proving the theorem. ## What the proof rests on @@ -96,7 +96,7 @@ $$ ### Natural numbers -$\mathbb{N} = \lbrace 0,1,2,3,\ldots \rbrace$ is a listing. the ellipsis carries the definition, which means there is no definition yet. +$\mathbb{N} = \lbrace 0,1,2,3,\ldots \rbrace$ lists the first few naturals and assumes we know how the list continues. we still need a definition. what we need instead is an *inductive* definition: a finite set of rules that generate every natural and nothing else. two rules suffice. [^peano] @@ -114,13 +114,13 @@ n = 0 n = \operatorname{succ}(d) \text{ for some } d \in \mathbb{N}. $$ -Lean's inductive definition also gives us induction. to prove something about every natural, prove it for $0$ and prove that $d$ having it forces $\operatorname{succ}(d)$ to have it. that is the `induction` tactic later, and it is the reason this definition is worth the trouble. +Lean's inductive definition also gives us induction. to prove something about every natural, prove it for $0$ and prove that $d$ having it forces $\operatorname{succ}(d)$ to have it. that is the `induction` tactic used later. the Lean artefact names this inductive type `MyNat`; i write $\mathbb{N}$ in the prose. [^peano]: Giuseppe Peano, *Arithmetices principia, nova methodo exposita*, 1889. The Latin original is on archive.org: -this is a choice, not the only option. the naturals can be built in several ways, and the constructions agree on everything we care about here. [^ordinals] +there are other constructions of the naturals. [^ordinals]
four other ways to define ℕ @@ -141,7 +141,7 @@ $$ simpler to write, but $n < m$ is no longer $n \in m$, so order has to be defined separately. [^zermelo] -**Church numerals.** a number is a function that applies another function that many times. $n$ is $\lambda f. \lambda x. f^n(x)$, so $3$ is $\lambda f. \lambda x. f(f(f(x)))$. addition composes the two iterates of $f$. this is how the naturals appear in untyped lambda calculus. [^church] +**Church numerals.** a number is a function that applies another function that many times. $n$ is $\lambda f. \lambda x. f^n(x)$, so $3$ is $\lambda f. \lambda x. f(f(f(x)))$. addition can be defined by $\lambda m.\lambda n.\lambda f.\lambda x.\,m\,f\,(n\,f\,x)$: apply $f$ $n$ times, then $m$ more times. this is how the naturals appear in untyped lambda calculus. [^church] **Peano axioms as first-order theory.** rather than constructing the naturals, state the properties they must have: $0$ is not a successor, $\operatorname{succ}$ is injective, and the induction schema is included. this leaves the objects unspecified and constrains them instead. [^pa] @@ -159,19 +159,17 @@ i am using zero and succ, which is what the Natural Number Game uses and what Le [^pa]: The first-order theory is usually attributed to Peano 1889 by way of Dedekind. Hájek and Pudlák, *Metamathematics of First-Order Arithmetic* (1998), chapter I, is the reference treatment. First-order induction is an axiom schema, with one instance per formula, and the theory has nonstandard models. the inductive type here has only the constructors stated above. -### Our zero and Lean's zero +### Constructor zero and numeral zero -the definition above introduces a constructor, written `MyNat.zero` in Lean. the character `0` is a numeral, which is what a person types. they denote the same natural number. they are not syntactically the same term, but Lean reduces them to the same term, so they are definitionally equal. +the definition above introduces the constructor `MyNat.zero`. Lean's numeral `0` elaborates to the same natural number. the two expressions are not syntactically identical, but Lean reduces them to the same term, so they are definitionally equal. -this matters mechanically. `rfl` closes a goal up to definitional equality. `zero_eq_0` makes that equality available as an equation for `rewrite`. +for the mathematics in this article, treat `MyNat.zero` and `0` as the same thing. `zero_eq_0` only handles Lean's bookkeeping: after a case split produces the constructor name, the lemma gives `rewrite` an explicit equation using the numeral. `succ n = n + 1` is different: it is not definitional here. `succ_eq_add_one` proves it from the two addition axioms. -the proof below carries `rewrite[zero_eq_0] at hc` for exactly this reason. - ### Addition -$6 + 1 = 7$ gives one sum. $\mathbb{N}^2$ has infinitely many, so we need rules for all pairs. in the artefact, `add` is an opaque function and NNG4 supplies two axioms that characterise recursion on the second argument: +$6 + 1 = 7$ gives one sum. $\mathbb{N}^2$ has infinitely many, so we need rules for all pairs. in the artefact, `add` is opaque and constrained by two axioms. both reduce the second argument: $$ \begin{aligned} @@ -218,11 +216,11 @@ $$ there is a gap, and the gap is itself a natural number. $c$ cannot be negative because its type is $\mathbb{N}$. -the gap is a natural, so it is either zero or a successor, which gives two pictures. in (I) the gap is $c = 0$ and $a = b$, the case where $\le$ holds because the two numbers are equal. in (II) the gap is nonzero and $a + c = b$ with $a$ strictly below $b$. +there are two cases. in (I), the gap is $c = 0$ and $a = b$. in (II), the gap is a successor and $a$ is strictly below $b$. ![Number line showing the gap c as zero in case I and nonzero in case II](/assets/blog/lean_numberline_two_cases.svg) -the proof below hits that split as a case distinction. once a gap `c` is in hand, `cases c` asks which of the two pictures applies, and the two branches close with different lemmas. +the proof later makes the same distinction. once it has a gap `c`, `cases c` separates zero from successor. this is the definition the Natural Number Game uses. Lean's own `Nat.le` is an inductive type instead, built from reflexivity and a successor step: [^natle] @@ -287,11 +285,11 @@ theorem zero_add (n : ℕ) : 0 + n = n := by rfl ``` -the two axioms describe addition through its second argument. `a + 0 = a` is available directly as `add_zero`, while `0 + n = n` has to be derived as `zero_add`. the two look symmetric and only one is assumed. this asymmetry is why `succ_add`, `add_comm` and `add_assoc` all need their own inductive proofs. +the two axioms describe addition through its second argument. `a + 0 = a` is available directly as `add_zero`, while `0 + n = n` has to be derived as `zero_add`. the two look symmetric and only one is assumed. this asymmetry is why `succ_add` and `add_assoc` need their own inductive proofs.
-one theorem often admits several proofs, each a different path through the dependency graph. Ording's *99 Variations on a Proof* takes this to its conclusion with 99 proofs of a single cubic. [^ording] +one theorem often admits several proofs, each a different path through the dependency graph. Ording's *99 Variations on a Proof* gives 99 proofs that one cubic equation has two real roots. [^ording] [^ording]: Philip Ording, *99 Variations on a Proof*, Princeton University Press, 2019. @@ -320,23 +318,20 @@ theorem le_succ_self (x : ℕ) : x ≤ succ x := by -three arithmetic theorems remain: +two arithmetic theorems remain on the path to `le_total`: $$ \begin{aligned} \operatorname{succ}(a) + b &= \operatorname{succ}(a+b) && (\texttt{succ\_add}) \\ -(a+b)+c &= a+(b+c) && (\texttt{add\_assoc}) \\ -a+b &= b+a && (\texttt{add\_comm}) +(a+b)+c &= a+(b+c) && (\texttt{add\_assoc}) \end{aligned} $$ -`le_total` depends on `succ_add` and `add_assoc`. it does not depend on `add_comm`, though the artefact proves that too. - ## Tactics as state transitions a Lean proof has a state: the hypotheses you have, and the goal you owe. a tactic changes that state. the proof is the sequence of changes. -the tables below are static copies of something you can drive yourself. [the whole development runs in the Lean web editor][lean-live], and clicking a line shows its state in the panel on the right. +the tables below are static copies of Lean's goal window. the Lean documentation puts it this way: [^tactics] @@ -351,7 +346,7 @@ written as two columns, with `zero_le` as the example: | `x : ℕ` | `0 ≤ x` | {: .table-equal-2} -after unfolding the definition of $\le$: +`use x` needs an existential goal. Lean therefore unfolds the reducible `LE` instance and `MyNat.le` until it sees the definition below. this happens automatically; there is no separate `unfold` tactic in the proof. | Givens | Goal | | --- | --- | @@ -412,7 +407,7 @@ some tactics split the state in two instead of changing it. those are the ones t | `c : ℕ`, `hc : x = d + c` | `x ≤ succ d ∨ succ d ≤ x` | {: .table-equal-2} -and `right`, which picks a side of the goal and discards the other: +and `right`, which chooses the second constructor of `Or`: | Givens | Goal | | --- | --- | @@ -421,7 +416,7 @@ and `right`, which picks a side of the goal and discards the other: -`left`, `right` and `use` commit to a choice the goal did not force. the first two pick a disjunct and discard the other, `use` picks a witness and discards every other candidate. pick wrong and the remaining goal is unprovable even though the original was fine, so you undo and try again. the rewriting tactics do not have this property: they transform a goal into an equivalent one, so a provable goal stays provable. +`left` and `right` choose which constructor of `Or` to build. `use w` supplies `w` as the witness for an existential. a bad choice can leave a goal that cannot be proved, even when another branch or witness would have worked. rewriting is different: it replaces equals by equals without choosing a branch or witness. ## Tests and proofs @@ -433,9 +428,7 @@ assert comparable(1, 1) assert comparable(0, 255) ``` -green suite, three pairs, out of infinitely many. property-based testing does better. QuickCheck, Hypothesis and proptest generate values of `a` and `b` and assert the property on each, which samples more widely and still samples. [^quickcheck] - -[^quickcheck]: Koen Claessen and John Hughes, *QuickCheck: A Lightweight Tool for Random Testing of Haskell Programs*, ICFP 2000: +green suite, three pairs, out of infinitely many. testing more pairs still settles only the pairs tested. the Lean proof keeps `x` and `y` arbitrary. the parts that do correspond: @@ -455,7 +448,7 @@ that makes `sorry` the counterpart of a stubbed test: it lets you write the shap ## The last step -what follows is the final theorem only, the bottom node of the dependency graph. it uses the definition of $\mathbb{N}$, the two addition axioms, the definition of $\le$, and six earlier theorems. the artefact also proves `add_comm`, but `le_total` does not use it. +what follows is the final theorem only, the bottom node of the dependency graph. it uses the definition of $\mathbb{N}$, the two addition axioms, the definition of $\le$, and six earlier theorems. ```lean theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by @@ -491,19 +484,9 @@ theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by rfl ``` -[open the whole development in the Lean web editor][lean-live] to click through the proof states yourself. it contains the six supporting theorems and the separate proof of `add_comm`. - -the induction is on `y`, which gives two cases. - -in the zero case the goal is `x ≤ 0 ∨ 0 ≤ x`. the right disjunct is `zero_le`, proved above, so `right` followed by `exact zero_le x` closes it. - -in the successor case the goal is `x ≤ succ d ∨ succ d ≤ x`, with `hd : x ≤ d ∨ d ≤ x` available. splitting `hd` gives two branches. - -in the `inl` branch, `x ≤ d`, so there is a gap `c` with `d = x + c`. the same gap extended by one witnesses `x ≤ succ d`, which is `use c + 1`. - -in the `inr` branch, `d ≤ x`, so there is a gap `c` with `x = d + c`. this branch needs a second split, on `c` itself, and it is exactly the (I) against (II) distinction from the number line. +[open the whole development in the Lean web editor][lean-live]. clicking a line shows its hypotheses and goal. -case (I), `c` is zero, so `x = d` and `x ≤ succ d` follows from `le_succ_self`. case (II), `c` is `succ a`, so `x = succ(d + a)` and `succ d ≤ x` holds with witness `a`. +the indentation records the nesting, but the code still reads as one vertical sequence. the proof state branches: `induction y` creates two obligations, then `cases hd` and `cases c` split them again. the diagram puts that shape on the page and shows where each branch closes. ![Lean proof state overview](/assets/blog/lean_state_overview.png) @@ -511,13 +494,13 @@ every node in that diagram is one of the two-column states from earlier, and eve the labelled frames are where the state splits. `induction y` opens the `succ d` frame, `cases hd` opens `inl` and `inr` inside it, and `cases c` splits again inside `inr`. each frame starts at its own filled dot, so the nesting on the page is the nesting of the proof. -`left` and `right` discard a disjunct. the top right branch goes from `x ≤ 0 ∨ 0 ≤ x` to `0 ≤ x` under `right`, and the left half never appears again. the same happens inside the frames, where the goal is written `…∨…` while both halves are still live and collapses to a single inequality the moment `left` or `right` fires. +`left` and `right` choose a constructor of `Or`. the top right branch goes from `x ≤ 0 ∨ 0 ≤ x` to `0 ≤ x` under `right`. inside the frames, each `…∨…` goal becomes a single inequality after `left` or `right`. four goals get closed, one per leaf, and this proof uses two tactics to do it. `rfl` closes the two that end in an equation whose sides are the same term, `(x+c)+1 = (x+c)+1` and `succ(d+a) = succ(d+a)`, both rewritten until the two halves are literally identical. `exact` closes the other two by naming a result proved earlier, `zero_le x` on the far right and `le_succ_self d` in the middle. every rewrite above them exists to reach one of those two endings. the remaining circles lower down are merge points where the branches rejoin. ## In English -the same development written out as a mathematician would write it, with the tactics replaced by prose. the Lean above and the proof below are the same argument. +the same development in textbook-style prose, inspired by the proofs in Velleman and Axler. the Lean above and the proof below are the same argument.
the whole thing, in prose @@ -563,13 +546,6 @@ Proof. let $a$ be arbitrary and fixed. by induction on $b$. - *Base case:* $b = 0$. both sides reduce to $\operatorname{succ}(a)$ by the zero equation. - *Inductive step:* let $d$ be arbitrary and take $b = \operatorname{succ}(d)$. inductive hypothesis: $\operatorname{succ}(a) + d = \operatorname{succ}(a + d)$. the left side becomes $\operatorname{succ}(\operatorname{succ}(a) + d)$, then $\operatorname{succ}(\operatorname{succ}(a + d))$ by the hypothesis. the right side becomes the same, by the successor equation under the outer successor. $\Box$ -**Lemma (`add_comm`).** $a + b = b + a$. - -Proof. let $a$ be arbitrary and fixed. by induction on $b$. - -- *Base case:* $b = 0$. both sides equal $a$, by the zero equation and by `zero_add`. -- *Inductive step:* let $d$ be arbitrary and take $b = \operatorname{succ}(d)$. inductive hypothesis: $a + d = d + a$. the left side is $\operatorname{succ}(a + d)$, hence $\operatorname{succ}(d + a)$ by the hypothesis. the right side is $\operatorname{succ}(d + a)$ by `succ_add`. $\Box$ - **Lemma (`add_assoc`).** $(a + b) + c = a + (b + c)$. Proof. let $a$ and $c$ be arbitrary and fixed. by induction on the middle summand $b$, which occurs under a successor on both sides. @@ -602,17 +578,22 @@ each case establishes one half of the goal, so the goal holds at $\operatorname{
-## What it cost +## From code to paper -eight theorems and eighty-three lines of tactics, for a statement that needs no defending to anyone who has counted to seven. +i have only just started learning how to prove theorems, and i am still unsure why this beautiful part of mathematics has stayed so far out of sight during my time at ZHAW. did i miss a definition? why exactly is this move allowed? Lean turns those questions into goals and errors. this was an exercise in precision, not in making a short theorem long. -`le_total` itself takes thirty lines. six theorems on its dependency path take another forty-two: `succ_eq_add_one` at four lines, `zero_add` at nine, `succ_add` at eleven, `add_assoc` at thirteen, `zero_le` at three, `le_succ_self` at two. `add_comm` takes the remaining eleven lines, though `le_total` does not use it. four of the six dependencies are about addition, and none of those four mention $\le$ at all. proving that two numbers compare turns out to be mostly a matter of proving that addition behaves. +i suspect many mathematicians learning Lean make the opposite transition, from paper proofs into a proof assistant. i am coming from computer science and using Lean on the way into proof-based maths. it felt approachable because it is also a functional programming language. i can work in Neovim with the Lean plugin, inspect the proof state, and start with mathlib documentation when i get stuck. Lean materialises the proof as something i can run and get a checkmark for. -what the artefact assumes is small and explicit: the inductive definition of `MyNat`, an opaque `add`, the axioms `add_zero` and `add_succ`, and the definition of `≤`. everything after that is derived. `sorry` is the explicit escape hatch for an unfinished proof, and Lean reports its presence. +the structure and state keeping are training for paper proofs, where there is no compiler or proof checker. next is linear algebra at the [FernUniversität in Hagen](https://www.fernuni-hagen.de/mi/studium/module/lin_alg.shtml), alongside part-time studies at ZHAW. it is my first course where proofs are the work rather than a step inside it. i am eager to see how much transfers. -the full single-file solution is in the [Lean web editor][lean-live]. it opens in the Lean 4 web editor with the whole development in it, so you can click any line and watch the givens and goal in the right-hand panel, exactly the two columns from earlier. put the cursor inside the `inr` branch and you can see the case split on `c` open up. no install, and it typechecks end to end. +reading Peano also made me wish i had learned Latin. some of the historical material would be much more approachable. -next is linear algebra at the [FernUniversität in Hagen](https://www.fernuni-hagen.de/mi/studium/module/lin_alg.shtml), alongside part-time studies at ZHAW. first course where proofs are the work rather than a step inside it, which is why i wanted this done now. +the Lean file ends with the concrete pair from the title: + +```lean +#check le_total 6 7 +-- le_total 6 7 : 6 ≤ 7 ∨ 7 ≤ 6 +``` ## Further reading @@ -629,4 +610,6 @@ next is linear algebra at the [FernUniversität in Hagen](https://www.fernuni-ha - and the list of 1000 theorems: - Kevin Buzzard, co-author of Natural Number Game 4: -[lean-live]: https://live.lean-lang.org/#codez=LTAEEEFcBcAsHsBOAuUBjARgZgCwEZQAKAEQEMA7ASwEoAoEesAFVkoGdQATeNSAWwCm5aKFKU%2BHaPFBsEAd1BwBM6AIAOHRAICOkSls6hK5UGsTx4AN2MBzRotjKknAYluh4AM1DlS0SIikADY%2B%2FBiubAA0PqSCQQCejPagDgJIAnygQQIA%2BlLQwUQAHqDxoKiAqITU5aAlgCZEpaCAFESNDUVJIKCe5plqAhTSpEWU8BLRbAWI0O5%2BqUbknJBo05bKLp7GlNPw5PZeMf6BIeRhEdFmVjNBIaSQw0GUpIhl2Xx8pBzBOzap9nKkZQohiEnBmIgAigBRYgAOg6YAAkqBIGxlEoDgFCic%2BOFEFwBBsqNtyBxuqM5r5DljTog2PYADL9Ew2WLKD6iUAYD7seGgWDQaAaZAAemFpE4sRh7zgMNgsEgMJcwoAxMKbMLshQLqtEMA0KM%2BJAifFheRyDYcLRxGokCIALJ%2BWAPDAwpikZaUNAwgASpFWVr4NqmoAdcGdrvd0y9AGEdtBAjbUQGg%2FbHeG3R6veA1GoEuBoMnbSG05QXRmozDox8BHTrUXQ07SxHMzCAHJwABKAjkblUheDDfTkc9EZg8H7IkZFBhkKCpDLw69ACFuWgJ6Ap%2BQZ3OFy2EQslsT1yuBa5KDXmxWAPJ9cgABTclj8AmPflUbgvZAKMPp7GgMJXNhPXXQcm03GcijMADVxAktdwrKtUVrQMi3Au0BG%2FcsRy7TwgnXcDZ3nS8R0A4C6wHOCfyZCCoNItdyMnajCPgkiYIYjcmJ3YivS7HsthfdjQJYr0AFUk0Eyi0Iw0gYQAZWtWh7BYZRPHga54Dkdx1k2YlPgWURhlGDg5FcNFSAAayEDwdVAYVQD4eBQQ2ARDDJPh7HRbN3UcDdPSEVFDFbVsAHEcFAIRrHMchBGEUAkViLpYkoB4nlATSZXsJFuB8eARAKSyPDkcgIlYNRrNcOZVOuIINK0gkdJGEl7AwGA4q4QY0HM8gNOyTgbGUBynM9PxGrYOEGE6flBTYEU1S2eUXX1PgNSZbVXD1A0jS2E0gtC3liHYNA53EVxUBYdgjA4bq8p8oRpl7Mp9nRXawoi8awGIAANLp%2BkOGsZCWWBRA4MwMOgMozGMaZzUiexVMQaVoZ%2BIF7PSeZ%2FC2ShVkURdjNMnK8osqyrFcOF7CXSFgoRVtQBe0AGGMRYPWxu14lbWY5EcLRaAAH1AAAvVxpFQVn2YLPm2CWNAalF2ZACTCEM2b8Wg3zcFrVFAABtHMcm67hoAAXUVsWYUltA12ukadiIXxBHKABeY2%2FFbXKrfIapCEhpBtod0A8AABkDoPqgAIgqEPQHtgA%2BJ2C1oW2azUd1lFluPjAmCg0GUVB91ged%2BMMVO0q5l8UnWW4ghEZBHdTmFBfMRT1g8TwxeKGoxeqEWlar%2B3aBSaU0CBkp0tgPvQD5k36%2BkFJo9juuhbHie%2FFNqXOUjmPa7N6XCC8VuMDoWgm6kVvCBKLuO%2Fb2Zq7Hgeh7S%2BbF7nqeUnX2nl6nx%2FN9XjBX5NreiGPrMfeil04FHIFneYExd6zAAN4mFQGLAAvjUK8LdZhFxMJzUyY9oE92bq3XYVoSRgIgWdeAsl4zuCLlg7mKQpAULcOaUwvt6GUKYYQQBIg1AHyUOkTIU8cg6ByP7GW3d57mEjqAER1dQCIFwo3AkHhiqiNbjIr%2B5spEKO8OwHIz9KigAVg%2BeAaheZSNfkwRAkAXwS1XjkV%2BAAxYISZ7CthpPiQkmMdhjVoLwrQmQdi5CEVvXRQsX6oAII7f%2B0jHYYDKHIvCviMiKDkPAQR2gcjBICS%2FGoAAmSR%2F8CAyNibI%2BRiTMhwC0IE9JwToApJSKgLA%2BTV55KKXE0pjg%2BFdHgAENJGSpZ5FgJUmoYVImr0aa0kpCSOl%2BK6FjKpfTzY5FUgEepoAACsTSNFhQmfEnx0yklASKL04JGxsarIAGybOlhsnZ7S0gzNRKscgxz%2BmHNWQAdiuaAS5ty8L2HFKCI8xjSC6DZJwQw595axwMbHEBxDM7KFAeAcFNRkWF27sXbBKQAW%2B1rgCxSQwRiZABSEiRhBSAqL8J3DkABqMxjtSAq0MsS8FCzt4UohbHalFK6WEH%2FpwaooyNHktAHSgVilOjGB0JAYI21D6KOyEQClP9Kid0dqAQAwEREGlqq6IP8GWivQEQjO4Ds4bkhNCmRgAL8lrtkQAl%2BT2EhK2YgtMQphQYPYAA6nnEQVUaoKCeGya4cwymaGlfoFyihpAXBJnMZwZ4mH7AqO9UAykOIUBwMFVk4VyCRR2DFLh5hrAuEMHbEepg5xZwQEEFwtJkC8jKQDRZQiSVZPgU26WJhHYmDpYUmJiROgpDYEgZ4vI02hrSvjNaJhil23ZBQcKRQs6IDQOwAQZNOgUypjTYg4BWxIg9a8d4HbemtuUe2%2F%2BXafCGr7ZyRIKQtB8VUJrAJLzFlTwNmPR9vYBCaxJVvT9D7uw%2Fr%2Fayj9X75G0CPRSgROLCDtqqDUERdKr3wP7WPRmh5GrXpHo%2FZ%2B0cx5Aafb%2BgRQj%2FaAZft%2B%2FioHOCkvgBRh98iUg2I0YYWAhgCPZKo8%2B%2F9UsGOyOA9R9j%2FHdlQYyMe4JcHlU1EQ6gf%2BPK15CvZYa%2FevtYkYYPB6a2P9cPMYFqEzjlHBPPtI%2Bk8jhGBPEZo3RkTxnf0kvA1xpj48O1cD5PqqOFnuP2dZQBrzdnNawAwLZqzvHzYicg9B0QrKlqZBFSq0AsnaWKbXnSilrSNNM2JGvXTLn8Oea4wF0zwiQsgYc0LUr1HYPgoi3hPTl6%2BRdoK0Z0Lvm%2BP%2Fas5JmrHWQOwHILVxSUWSUfGHcpn%2BOrEvcpU4a6WBreU%2FzpWgNVd7MtYe0%2FfOAeGDPNaIyB4r5nCutdo45lre2hY5ABQN%2Brq8TB9fXj16jYW0CVefV1zgL3f1vY%2B9Zvzh3ev9a85F8TMHzuKtPjJ6lIi2hqfvciVEtQv1FfO5diDfyovZDZRkgQQRvDg9VTUeormz7oZSCiZQeAx4CCKJGE9LbWVZPaB5HyjaTJ8j9MocIVlSA5nMN5dwcAUSJUQGJt4FKMf5EKODso%2BPUCE7KC0Mo0OMspEw1pkwZRct83y15ygNh%2BQWapzTgRir2jXdY3yDjO30DVg4Oxjbo9sl82MCEJ093skpDQDbgA5HyEI5bpawDXO7yzvXnsWZSNkTwBZg%2Fec1sEuntGAn8dJ%2FD6Wvbw8h%2Bo4ABMJou0ZGzwZPkyLNO%2FIHiQZbv3ee6Qj78v%2Fu%2BRB8rzb9A9uM9a%2B2xn3bVXztkaNrMQPHfM88bA6E97ogRD9%2BDxHgk0fJ%2Bx8D4XlIhvlhZFyME1EOOuCt9cxSwzMeAtPd7%2BPhvMfdf68n4P398%2BB9k9EAP2PX27%2BQeVIPAQnUV95FdiES57zFIgljkAA +## Notes + +[lean-live]: https://live.lean-lang.org/#url=https%3A%2F%2Fgist.githubusercontent.com%2Fcb341%2F824784c269bf2a8465ba1d8715103008%2Fraw%2F28f6fd2095de270d72a43dbd7601acad51c0ce89%2Flean_le_total From feffe3e5e58c59a82ddc883af16566fdb1f6b751 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 21:59:09 +0200 Subject: [PATCH 30/34] Final Polish? --- _posts/2026-06-28-six-less-than-seven.md | 22 ++++++++++------------ 1 file changed, 10 insertions(+), 12 deletions(-) diff --git a/_posts/2026-06-28-six-less-than-seven.md b/_posts/2026-06-28-six-less-than-seven.md index ab0d9b3..816dd04 100644 --- a/_posts/2026-06-28-six-less-than-seven.md +++ b/_posts/2026-06-28-six-less-than-seven.md @@ -1,5 +1,5 @@ --- -title: "Proving comparability of 6 and 7, the hard way" +title: "Proving comparability of 6 and 7, from first principles" date: 2026-09-05 description: "Proving that any two natural numbers compare, from an inductive definition of the naturals and two axioms for addition, in Lean." tags: ["theoretical mathematics", "first principles"] @@ -20,6 +20,8 @@ $$ any two natural numbers compare. one of them is at most the other. +we are proving a little more than comparability of 6 and 7. `le_total` proves that every pair of natural numbers is comparable, with 6 and 7 as one instance. its name describes the property: `le` is $\le$, and `total` means that the relation compares any two elements. this is the comparability condition of a total order. here `total` does not mean that a function is defined for every input, and this theorem alone does not establish the other order laws. + take 6 and 7. then $(6 \le 7) \lor (7 \le 6)$ is $T \lor F$, so $T$. take 1 twice. then $(1 \le 1) \lor (1 \le 1)$ is $T \lor T$, so $T$. @@ -448,7 +450,7 @@ that makes `sorry` the counterpart of a stubbed test: it lets you write the shap ## The last step -what follows is the final theorem only, the bottom node of the dependency graph. it uses the definition of $\mathbb{N}$, the two addition axioms, the definition of $\le$, and six earlier theorems. +what follows is the final theorem, the bottom node of the dependency graph. it uses the definition of $\mathbb{N}$, the two addition axioms, the definition of $\le$, and six earlier theorems. the last command checks the concrete pair from the title. ```lean theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by @@ -482,6 +484,9 @@ theorem le_total (x y : ℕ) : x ≤ y ∨ y ≤ x := by use a rewrite[succ_add] rfl + +#check le_total 6 7 +-- le_total 6 7 : 6 ≤ 7 ∨ 7 ≤ 6 ``` [open the whole development in the Lean web editor][lean-live]. clicking a line shows its hypotheses and goal. @@ -490,7 +495,7 @@ the indentation records the nesting, but the code still reads as one vertical se ![Lean proof state overview](/assets/blog/lean_state_overview.png) -every node in that diagram is one of the two-column states from earlier, and every edge is a tactic. three things it shows that the linear listing hides. +every node in the graphviz diagram is one of the two-column states from earlier, and every edge is a tactic. three things it shows that the linear listing hides. the labelled frames are where the state splits. `induction y` opens the `succ d` frame, `cases hd` opens `inl` and `inr` inside it, and `cases c` splits again inside `inr`. each frame starts at its own filled dot, so the nesting on the page is the nesting of the proof. @@ -580,21 +585,14 @@ each case establishes one half of the goal, so the goal holds at $\operatorname{ ## From code to paper -i have only just started learning how to prove theorems, and i am still unsure why this beautiful part of mathematics has stayed so far out of sight during my time at ZHAW. did i miss a definition? why exactly is this move allowed? Lean turns those questions into goals and errors. this was an exercise in precision, not in making a short theorem long. +there are two parts to this for me. the first is the mathematics itself. i have only just started learning how to prove theorems, and i am still unsure why this beautiful part of mathematics has stayed so far out of sight during my time at ZHAW. definitions are beautiful because they are precise. they say exactly what the objects and relations mean, and the proof shows what follows from them. working through `le_total` was an exercise in that precision. -i suspect many mathematicians learning Lean make the opposite transition, from paper proofs into a proof assistant. i am coming from computer science and using Lean on the way into proof-based maths. it felt approachable because it is also a functional programming language. i can work in Neovim with the Lean plugin, inspect the proof state, and start with mathlib documentation when i get stuck. Lean materialises the proof as something i can run and get a checkmark for. +the second is the transition from computer science into proof-based maths. i suspect many mathematicians learning Lean move in the opposite direction, from paper proofs into a proof assistant. Lean felt approachable to me because it is also a functional programming language. i can work in Neovim with the Lean plugin, inspect the proof state, and start with mathlib documentation when i get stuck. Lean materialises the proof as something i can run and get a checkmark for. the structure and state keeping are training for paper proofs, where there is no compiler or proof checker. next is linear algebra at the [FernUniversität in Hagen](https://www.fernuni-hagen.de/mi/studium/module/lin_alg.shtml), alongside part-time studies at ZHAW. it is my first course where proofs are the work rather than a step inside it. i am eager to see how much transfers. reading Peano also made me wish i had learned Latin. some of the historical material would be much more approachable. -the Lean file ends with the concrete pair from the title: - -```lean -#check le_total 6 7 --- le_total 6 7 : 6 ≤ 7 ∨ 7 ≤ 6 -``` - ## Further reading - Natural Number Game 4: From 57990501a5897e9c1850566bb5a4490e7e380b8b Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 22:00:43 +0200 Subject: [PATCH 31/34] Drop duplicate asset --- assets/blog/dependency_graph.svg | 2 -- 1 file changed, 2 deletions(-) delete mode 100644 assets/blog/dependency_graph.svg diff --git a/assets/blog/dependency_graph.svg b/assets/blog/dependency_graph.svg deleted file mode 100644 index 9efe3e4..0000000 --- a/assets/blog/dependency_graph.svg +++ /dev/null @@ -1,2 +0,0 @@ - -a + succ(d) = succ(a + d)a + 0 = a1 = succ(0)succ(n) = n + 10 + n = nsucc(a) + b = succ(a + b)a + b = b + aa + b + c = a + (b + c)0 ≤ xx ≤ succ(x)x ≤ y ∨ y ≤ x \ No newline at end of file From f5fb1962a51d1eb6a13e0332878728784b44448b Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 22:01:11 +0200 Subject: [PATCH 32/34] Drop dot --- proof.dot | 109 ------------------------------------------------------ 1 file changed, 109 deletions(-) delete mode 100644 proof.dot diff --git a/proof.dot b/proof.dot deleted file mode 100644 index df3560c..0000000 --- a/proof.dot +++ /dev/null @@ -1,109 +0,0 @@ -digraph proof { - compound=true; - rankdir=TB; - node [shape=box, style="rounded,filled", fillcolor="#f0f0f0", - fontname="Helvetica", fontsize=11]; - edge [fontname="Helvetica", fontsize=10]; - - // ── pseudo-states ── - start_top [shape=circle, width=.2, style=filled, fillcolor=black, label=""]; - end_top [shape=doublecircle, width=.15, style=filled, fillcolor=black, label=""]; - - // ── top level ── - G0 [label="x ≤ y ∨ y ≤ x"]; - Z0 [label="x ≤ 0 ∨ 0 ≤ x"]; - Z1 [label="0 ≤ x"]; - - start_top -> G0; - G0 -> Z0 [label="induction y (0)"]; - G0 -> start_succ [label="induction y (succ d, hd)", lhead=cluster_succ]; - Z0 -> Z1 [label="right"]; - Z1 -> end_top [label="exact zero_le x"]; - - // ── succ branch ── - subgraph cluster_succ { - label="succ d (hd : x≤d ∨ d≤x)"; - style=rounded; fontname="Helvetica"; - start_succ [shape=circle, width=.2, style=filled, fillcolor=black, label=""]; - end_succ [shape=doublecircle, width=.15, style=filled, fillcolor=black, label=""]; - - S0 [label="x ≤ succ d ∨ succ d ≤ x"]; - - start_succ -> S0; - S0 -> start_inl [label="cases hd (inl hl)", lhead=cluster_inl]; - S0 -> start_inr [label="cases hd (inr hr)", lhead=cluster_inr]; - - // inl sub-scope - subgraph cluster_inl { - label="inl"; - style=rounded; fontname="Helvetica"; - start_inl [shape=circle, width=.2, style=filled, fillcolor=black, label=""]; - end_inl [shape=doublecircle, width=.15, style=filled, fillcolor=black, label=""]; - - IL0 [label="…∨…\n(hl : x≤d)"]; - IL1 [label="…∨…\n(hc : d = x+c)"]; - IL2 [label="x ≤ succ(x+c) ∨ succ(x+c) ≤ x"]; - IL3 [label="x ≤ succ(x+c)"]; - IL4 [label="x ≤ (x+c)+1"]; - IL5 [label="(x+c)+1 = x+(c+1)"]; - IL6 [label="(x+c)+1 = (x+c)+1"]; - - start_inl -> IL0; - IL0 -> IL1 [label="cases' hl with c hc"]; - IL1 -> IL2 [label="rewrite[hc]"]; - IL2 -> IL3 [label="left"]; - IL3 -> IL4 [label="rewrite[succ_eq_add_one]"]; - IL4 -> IL5 [label="use c+1"]; - IL5 -> IL6 [label="rewrite[← add_assoc]"]; - IL6 -> end_inl [label="rfl"]; - } - - end_inl -> end_succ [ltail=cluster_inl]; - - // inr sub-scope - subgraph cluster_inr { - label="inr"; - style=rounded; fontname="Helvetica"; - start_inr [shape=circle, width=.2, style=filled, fillcolor=black, label=""]; - end_inr [shape=doublecircle, width=.15, style=filled, fillcolor=black, label=""]; - - IR0 [label="…∨…\n(hr : d≤x)"]; - IR1 [label="…∨…\n(hc : x = d+c)"]; - - // c = 0 - RZ0 [label="…∨…\n(hc : x = d+0)"]; - RZ2 [label="…∨…\n(hc : x = d)"]; - RZ3 [label="x ≤ succ d"]; - RZ4 [label="d ≤ succ d"]; - - // c = succ a - RS0 [label="…∨…\n(hc : x = d + succ a)"]; - RS1 [label="…∨…\n(hc : x = succ(d+a))"]; - RS2 [label="succ d ≤ x"]; - RS3 [label="succ d ≤ succ(d+a)"]; - RS4 [label="succ(d+a) = succ d + a"]; - RS5 [label="succ(d+a) = succ(d+a)"]; - - start_inr -> IR0; - IR0 -> IR1 [label="cases' hr with c hc"]; - IR1 -> RZ0 [label="cases c (0)"]; - IR1 -> RS0 [label="cases c (succ a)"]; - - RZ0 -> RZ2 [label="rewrite[add_zero d] at hc"]; - RZ2 -> RZ3 [label="left"]; - RZ3 -> RZ4 [label="rewrite[hc]"]; - RZ4 -> end_inr [label="exact le_succ_self d"]; - - RS0 -> RS1 [label="rewrite[add_succ] at hc"]; - RS1 -> RS2 [label="right"]; - RS2 -> RS3 [label="rewrite[hc]"]; - RS3 -> RS4 [label="use a"]; - RS4 -> RS5 [label="rewrite[succ_add]"]; - RS5 -> end_inr [label="rfl"]; - } - - end_inr -> end_succ [ltail=cluster_inr]; - } - - end_succ -> end_top [ltail=cluster_succ]; -} From 39b8e9aa2ef2d5ac4a2ecbd777069ee7a8e624b5 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 22:02:38 +0200 Subject: [PATCH 33/34] Readd unused assets (follow up) --- assets/blog/flowers.webp | Bin 0 -> 10340 bytes .../chess-widget/lichess-widget-embed.webp | Bin 0 -> 96368 bytes webp-gallery/dani_with_cool_glasses.webp | Bin 0 -> 53824 bytes ...ge_stating_career_as_software_developer.webp | Bin 0 -> 47448 bytes webp-gallery/thumbs/dani_with_cool_glasses.webp | Bin 0 -> 14384 bytes ...ge_stating_career_as_software_developer.webp | Bin 0 -> 11802 bytes 6 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 assets/blog/flowers.webp create mode 100644 assets/projects/chess-widget/lichess-widget-embed.webp create mode 100644 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zU_;H7hO#V}LubSPCHlMZ*}VRqw7;cY3_&v~M_D0ZA#hSppZ@1af2sM(3JXYpfjv?C_v-$ocD!=jo%|3k ur>g&6OJ~FTi~E0s=uGX`rhM9T60|>Fb;k`SQt(A8izCf{_EMI`0saqG2%T^M literal 0 HcmV?d00001 From 9abea44d046031bda2cc6bcde7fd4d76fbd5dbdc Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 22:04:10 +0200 Subject: [PATCH 34/34] SLug --- ...en.md => 2026-06-28-comparing-six-and-seven-from-scratch.md} | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) rename _posts/{2026-06-28-six-less-than-seven.md => 2026-06-28-comparing-six-and-seven-from-scratch.md} (99%) diff --git a/_posts/2026-06-28-six-less-than-seven.md b/_posts/2026-06-28-comparing-six-and-seven-from-scratch.md similarity index 99% rename from _posts/2026-06-28-six-less-than-seven.md rename to _posts/2026-06-28-comparing-six-and-seven-from-scratch.md index 816dd04..12ac9a7 100644 --- a/_posts/2026-06-28-six-less-than-seven.md +++ b/_posts/2026-06-28-comparing-six-and-seven-from-scratch.md @@ -1,5 +1,5 @@ --- -title: "Proving comparability of 6 and 7, from first principles" +title: "Comparing 6 and 7 from Scratch" date: 2026-09-05 description: "Proving that any two natural numbers compare, from an inductive definition of the naturals and two axioms for addition, in Lean." tags: ["theoretical mathematics", "first principles"]
-Prerequisites +refresher: ∧, ¬, →, ↔, ⊤, ⊥, contrapositive, De Morgan -$P \land Q$ holds when both sides hold. +each connective is fixed by its truth table. -$\lnot P$ holds when $P$ does not. +| $P$ | $Q$ | $P \land Q$ | $P \to Q$ | $P \leftrightarrow Q$ | $\lnot P$ | +| --- | --- | --- | --- | --- | --- | +| $F$ | $F$ | $F$ | $T$ | $T$ | $T$ | +| $F$ | $T$ | $F$ | $T$ | $F$ | $T$ | +| $T$ | $F$ | $F$ | $F$ | $F$ | $F$ | +| $T$ | $T$ | $T$ | $T$ | $T$ | $F$ | -$P \to Q$ holds unless $P$ holds and $Q$ fails. the two rows where $P$ is false both come out true, which is worth its own article. +the two rows of $\to$ where $P$ is false both come out true, which is worth its own article. -$P \leftrightarrow Q$ is $P \to Q$ together with $Q \to P$. it is the $:\Leftrightarrow$ used to define $\le$ below. +$P \leftrightarrow Q$ is $(P \to Q) \land (Q \to P)$. it is the $:\Leftrightarrow$ used to define $\le$ below. $\top$ is the statement that always holds, $\bot$ the statement that never does. -the converse of $P \to Q$ is $Q \to P$, a different statement. the contrapositive is $\lnot Q \to \lnot P$, the same statement. +the converse of $P \to Q$ is $Q \to P$, a different statement. the contrapositive is $\lnot Q \to \lnot P$, the same statement: + +$$ +(P \to Q) \Leftrightarrow (\lnot Q \to \lnot P) +$$ the standard equivalences are the De Morgan, commutative, associative, idempotent, distributive and absorption laws, plus double negation. the two De Morgan laws: @@ -86,7 +93,15 @@ $$ $\mathbb{N} = \lbrace 0,1,2,3,\ldots \rbrace$ is a listing. the ellipsis carries the definition, which means there is no definition yet. -a natural number is either zero or the successor of a natural number. [^peano] +what we need instead is an *inductive* definition: a finite set of rules that generate every natural and nothing else. two rules suffice. [^peano] + +$$ +\frac{}{\;0 \in \mathbb{N}\;} +\qquad +\frac{d \in \mathbb{N}}{\;\operatorname{succ}(d) \in \mathbb{N}\;} +$$ + +zero is a natural. the successor of a natural is a natural. every natural is reached by applying the second rule to the first some finite number of times, and nothing else is in $\mathbb{N}$. written as cases: $$ n \in \mathbb{N} \stackrel{\mathrm{def}}{=} @@ -96,26 +111,36 @@ n \in \mathbb{N} \stackrel{\mathrm{def}}{=} \end{cases} $$ +the same shape gives us induction for free. to prove something about every natural, prove it for $0$ and prove that $d$ having it forces $\operatorname{succ}(d)$ to have it. that is the `induction` tactic later, and it is the reason this definition is worth the trouble. + [^peano]: Giuseppe Peano, *Arithmetices principia, nova methodo exposita*, 1889. The Latin original is on archive.org: -there is more than one way to build the naturals out of sets. [^ordinals] +this is a choice, not the only option. the naturals can be built in several ways, and the constructions agree on everything we care about here. [^ordinals]
-Other encodings of the same numbers +four other ways to define ℕ -Zermelo: +**von Neumann ordinals.** each number is the set of all smaller numbers. $$ -0=\emptyset,\quad 1=\lbrace\emptyset\rbrace,\quad 2=\lbrace\lbrace\emptyset\rbrace\rbrace,\quad\ldots +0=\varnothing,\quad 1=\lbrace\varnothing\rbrace,\quad 2=\lbrace\varnothing,\lbrace\varnothing\rbrace\rbrace,\quad\ldots $$ -von Neumann: +$n < m$ becomes $n \in m$, so order comes for free. this is the standard construction in set theory. + +**Zermelo ordinals.** each number is the singleton of the previous one. $$ -0=\emptyset,\quad 1=\lbrace\emptyset\rbrace,\quad 2=\lbrace\emptyset,\lbrace\emptyset\rbrace\rbrace,\quad\ldots +0=\varnothing,\quad 1=\lbrace\varnothing\rbrace,\quad 2=\lbrace\lbrace\varnothing\rbrace\rbrace,\quad\ldots $$ -the von Neumann version has the property that each number is the set of all smaller numbers, so $n < m$ becomes $n \in m$. order comes for free. i am using zero and succ instead, which is what the Natural Number Game uses. +simpler to write, but $n < m$ is no longer $n \in m$, so order has to be defined separately. + +**Church numerals.** a number is a function that applies another function that many times. $n$ is $\lambda f. \lambda x. f^n(x)$, so $3$ is $\lambda f. \lambda x. f(f(f(x)))$. addition is function composition. this is how the naturals appear in untyped lambda calculus. + +**Peano axioms as first-order theory.** rather than constructing the naturals, state the properties they must have: $0$ is not a successor, $\operatorname{succ}$ is injective, and induction holds. this leaves the objects unspecified and constrains them instead. + +i am using zero and succ, which is what the Natural Number Game uses and what Lean's own `Nat` is.
@@ -146,26 +171,26 @@ $$ worked on $1 + 2 = 3$. the right column names the axiom used and which direction it was applied in. $(\rightarrow)$ unfolds, replacing a name by its definition. $(\leftarrow)$ folds, recognising a definition and naming it. $$ -\begin{array}{ll} -& \textbf{Axioms} \\ -\colorbox{#fff3cd}{i.} & 1 \stackrel{\mathrm{def}}{=} \operatorname{succ}(0) \\ -\colorbox{#cfe2ff}{ii.} & 2 \stackrel{\mathrm{def}}{=} \operatorname{succ}(1) \\ -\colorbox{#e2d9f3}{iii.} & 3 \stackrel{\mathrm{def}}{=} \operatorname{succ}(2) \\ -\colorbox{#f8d7da}{iv.} & a+0 \stackrel{\mathrm{def}}{=} a \\ -\colorbox{#d1e7dd}{v.} & a+\operatorname{succ}(b) \stackrel{\mathrm{def}}{=} \operatorname{succ}(a+b) +\begin{array}{rl} +& \textbf{Axioms} \\[2pt] +\colorbox{#fff3cd}{$\vphantom{Ag}\text{i.}$} & 1 \stackrel{\mathrm{def}}{=} \operatorname{succ}(0) \\[2pt] +\colorbox{#cfe2ff}{$\vphantom{Ag}\text{ii.}$} & 2 \stackrel{\mathrm{def}}{=} \operatorname{succ}(1) \\[2pt] +\colorbox{#e2d9f3}{$\vphantom{Ag}\text{iii.}$} & 3 \stackrel{\mathrm{def}}{=} \operatorname{succ}(2) \\[2pt] +\colorbox{#f8d7da}{$\vphantom{Ag}\text{iv.}$} & a+0 \stackrel{\mathrm{def}}{=} a \\[2pt] +\colorbox{#d1e7dd}{$\vphantom{Ag}\text{v.}$} & a+\operatorname{succ}(b) \stackrel{\mathrm{def}}{=} \operatorname{succ}(a+b) \end{array} \qquad \begin{array}{l|l} \textbf{Statement} & \textbf{Reason} \\ \hline -1 + 2 & \text{given} \\ -1 + \colorbox{#cfe2ff}{$\operatorname{succ}(1)$} & \colorbox{#cfe2ff}{$\text{ii.}$} \; (\rightarrow) \\ -\colorbox{#d1e7dd}{$\operatorname{succ}(1 + 1)$} & \colorbox{#d1e7dd}{$\text{v.}$} \; (\rightarrow) \\ -\operatorname{succ}(1 + \colorbox{#fff3cd}{$\operatorname{succ}(0)$}) & \colorbox{#fff3cd}{$\text{i.}$} \; (\rightarrow) \\ -\operatorname{succ}(\operatorname{succ}(\colorbox{#d1e7dd}{$1 + 0$})) & \colorbox{#d1e7dd}{$\text{v.}$} \; (\rightarrow) \\ -\operatorname{succ}(\operatorname{succ}(\colorbox{#f8d7da}{$1$})) & \colorbox{#f8d7da}{$\text{iv.}$} \; (\rightarrow) \\ -\operatorname{succ}(\colorbox{#cfe2ff}{$2$}) & \colorbox{#cfe2ff}{$\text{ii.}$} \; (\leftarrow) \\ -\colorbox{#e2d9f3}{$3$} & \colorbox{#e2d9f3}{$\text{iii.}$} \; (\leftarrow) +1 + 2 & \text{given} \\[2pt] +1 + \colorbox{#cfe2ff}{$\vphantom{Ag}\operatorname{succ}(1)$} & \colorbox{#cfe2ff}{$\vphantom{Ag}\text{ii.}$} \; (\rightarrow) \\[2pt] +\colorbox{#d1e7dd}{$\vphantom{Ag}\operatorname{succ}(1 + 1)$} & \colorbox{#d1e7dd}{$\vphantom{Ag}\text{v.}$} \; (\rightarrow) \\[2pt] +\operatorname{succ}(1 + \colorbox{#fff3cd}{$\vphantom{Ag}\operatorname{succ}(0)$}) & \colorbox{#fff3cd}{$\vphantom{Ag}\text{i.}$} \; (\rightarrow) \\[2pt] +\operatorname{succ}(\operatorname{succ}(\colorbox{#d1e7dd}{$\vphantom{Ag}1 + 0$})) & \colorbox{#d1e7dd}{$\vphantom{Ag}\text{v.}$} \; (\rightarrow) \\[2pt] +\operatorname{succ}(\operatorname{succ}(\colorbox{#f8d7da}{$\vphantom{Ag}1$})) & \colorbox{#f8d7da}{$\vphantom{Ag}\text{iv.}$} \; (\rightarrow) \\[2pt] +\operatorname{succ}(\colorbox{#cfe2ff}{$\vphantom{Ag}2$}) & \colorbox{#cfe2ff}{$\vphantom{Ag}\text{ii.}$} \; (\leftarrow) \\[2pt] +\colorbox{#e2d9f3}{$\vphantom{Ag}3$} & \colorbox{#e2d9f3}{$\vphantom{Ag}\text{iii.}$} \; (\leftarrow) \end{array} $$ @@ -179,9 +204,23 @@ $$ there is a gap, and the gap is itself a natural number. the second half carries the content, since there are no negative naturals available to serve as gaps. -![Number line showing the gap c in both cases](/assets/blog/lean_numberline_two_cases.svg) +the gap is a natural, so it is either zero or a successor, which gives two pictures. in (I) the gap is $c = 0$ and $a = b$, the case where $\le$ holds because the two numbers are equal. in (II) the gap is nonzero and $a + c = b$ with $a$ strictly below $b$. + +![Number line showing the gap c as zero in case I and nonzero in case II](/assets/blog/lean_numberline_two_cases.svg) + +that split is not decoration. the proof below hits it as a real case distinction: once a gap `c` is in hand, `cases c` asks which of the two pictures applies, and the two branches close with different lemmas. + +this is the definition the Natural Number Game uses. Lean's own `Nat.le` is an inductive type instead, built from reflexivity and a successor step: [^natle] + +```lean +protected inductive Nat.le (n : Nat) : Nat → Prop + | refl : Nat.le n n + | step {m} : Nat.le n m → Nat.le n (succ m) +``` -this is the definition the Natural Number Game uses. [^nng] mathlib defines $\le$ differently, so code pasted from here into a mathlib project will not typecheck unchanged. +both say the same thing about the same numbers. the existential version hands you a gap to compute with, the inductive version hands you a chain of steps to recurse on. code pasted from here into a mathlib project will not typecheck unchanged. + +[^natle]: `Init/Prelude.lean`, line 2022 in the Lean 4 source: . mathlib inherits this definition rather than replacing it. [^nng]: Natural Number Game 4, by Kevin Buzzard and Mohammad Pedramfar: @@ -203,15 +242,11 @@ a Lean proof is written as a list of tactics. the ones in this article: `rw` is the short form of `rewrite`. `zero_eq_0`, `succ_eq_add_one` and `cases'` are Natural Number Game spellings, so pasting this into a fresh mathlib project gets you errors on the names before anything interesting. -## One lemma, three ways +## The smaller lemmas before the theorem, a smaller one: $0 \le x$ for every natural $x$. it sits in the dependency graph as `zero_le`, and the theorem's base case consumes it. -unfolding the definition, $0 \le x$ means $\exists c, x = 0 + c$. take $c := x$. the goal becomes $x = 0 + x$, which is `zero_add`. - -by induction on $x$: the base case is $0 \le 0$ with $c := 0$, and the step goes from $0 \le d$ to $0 \le \operatorname{succ}(d)$. longer, and it pulls more lemmas into the graph. - -in Lean, the first route is three lines. +unfolding the definition, $0 \le x$ means $\exists c, x = 0 + c$. take $c := x$. the goal becomes $x = 0 + x$, which is `zero_add`. in Lean that is three lines. ```lean theorem zero_le (x : ℕ) : 0 ≤ x := by @@ -220,14 +255,51 @@ theorem zero_le (x : ℕ) : 0 ≤ x := by rfl ``` +
+the same lemma by induction on x + +the base case is $0 \le 0$ with $c := 0$. the step goes from $0 \le d$ to $0 \le \operatorname{succ}(d)$, which means turning a gap `c` into `succ c`. + +longer than the first route, and it pulls `add_succ` into the graph on top of `zero_add`. the first route is shorter because it picks the witness in one move instead of building it up one successor at a time. + +
+ one theorem, several proofs, each a different path through the dependency graph. Ording's *99 Variations on a Proof* takes this to its conclusion with 99 proofs of a single cubic. [^ording] [^ording]: Philip Ording, *99 Variations on a Proof*, Princeton University Press, 2019. +the main proof calls two more results by name. both are proved the same way, from the definitions above. + +
+succ_eq_add_one and le_succ_self + +`succ_eq_add_one` connects the constructor to the numeral, the split from earlier: + +```lean +theorem succ_eq_add_one (n : ℕ) : succ n = n + 1 := by + rw [one_eq_succ_zero] + rw [add_succ] + rw [add_zero] + rfl +``` + +`le_succ_self` says every number is below its own successor. the gap is one: + +```lean +theorem le_succ_self (x : ℕ) : x ≤ succ x := by + use 1 + rw [succ_eq_add_one] + rfl +``` + +
+ ## Tactics as state transitions a Lean proof has a state: the hypotheses you have, and the goal you owe. a tactic changes that state. the proof is the sequence of changes. +the tables below are static copies of something you can drive yourself. [the whole development runs in the Lean web editor](https://tinyurl.com/4tr5uc7c), and clicking a line shows its state in the panel on the right. + the Lean documentation puts it this way: [^tactics] > A proof term is a representation of a mathematical proof; tactics are commands, or instructions, that describe how to build such a proof. @@ -266,7 +338,12 @@ after `rfl`, no goals remain. the left column only grows and the right column only shrinks. the proof is finished when the right column is empty, which is a condition rather than a judgement call. -the same shape, for the tactics that split the state in two. `induction` on `y`: +some tactics split the state in two instead of changing it. those are the ones that make a proof branch. + +
+the branching tactics, as before and after states + +`induction` on `y` replaces one state with two, and hands the second an induction hypothesis: | Givens | Goal | | --- | --- | @@ -278,7 +355,7 @@ the same shape, for the tactics that split the state in two. `induction` on `y`: | `x d : ℕ`, `hd : x ≤ d ∨ d ≤ x` | `x ≤ succ d ∨ succ d ≤ x` | {: .table-equal-2} -`cases hd` on a disjunction hypothesis, again two states: +`cases hd` on a disjunction hypothesis, again two states, one per disjunct: | Givens | Goal | | --- | --- | @@ -290,7 +367,7 @@ the same shape, for the tactics that split the state in two. `induction` on `y`: | `hr : d ≤ x` | `x ≤ succ d ∨ succ d ≤ x` | {: .table-equal-2} -`cases' hr with c hc` on an existential hypothesis, which names the witness: +`cases' hr with c hc` on an existential hypothesis, which names the witness and keeps one state: | Givens | Goal | | --- | --- | @@ -304,6 +381,8 @@ and `right`, which picks a side of the goal and discards the other: | `c : ℕ`, `hc : x = d + c` | `succ d ≤ x` | {: .table-equal-2} +
+ `left` and `right` are the only tactics here that can lose you the proof. everything else preserves provability, and those two commit to a disjunct before you have checked it holds. ## Tests and proofs @@ -380,9 +459,9 @@ in the successor case the goal is `x ≤ succ d ∨ succ d ≤ x`, with `hd : x in the `inl` branch, `x ≤ d`, so there is a gap `c` with `d = x + c`. the same gap extended by one witnesses `x ≤ succ d`, which is `use c + 1`. -in the `inr` branch, `d ≤ x`, so there is a gap `c` with `x = d + c`. this branch needs a second split, on `c` itself, because the two cases close differently. +in the `inr` branch, `d ≤ x`, so there is a gap `c` with `x = d + c`. this branch needs a second split, on `c` itself, and it is exactly the (I) against (II) distinction from the number line. -when `c` is zero, `x = d`, so `x ≤ succ d` follows from `le_succ_self`. when `c` is `succ a`, then `x = succ(d + a)` and `succ d ≤ x` holds with witness `a`. +case (I), `c` is zero, so `x = d` and `x ≤ succ d` follows from `le_succ_self`. case (II), `c` is `succ a`, so `x = succ(d + a)` and `succ d ≤ x` holds with witness `a`. ![Lean proof state overview](/assets/blog/lean_state_overview.png) @@ -396,9 +475,9 @@ TODO eleven supporting results and twenty-eight lines of tactics, for a statement that needs no defending to anyone who has counted to seven. -the gaps are visible instead of assumed. that is what the exercise buys. +this article shows three of those eleven. the other eight are commutativity, associativity, `zero_add`, `succ_add`, `add_succ`, `add_zero`, `one_eq_succ_zero` and `zero_eq_0`, and every one of them is proved in the artefact from the two clauses of addition and nothing else. no step is assumed, quoted from a library, or left to the reader. that is the part i could not have got from a paper proof: `sorry` is the only way to skip a step, and it shows up as a warning every time you compile. -the full single-file solution, with every supporting theorem derived from the definitions above, is at . +the full single-file solution is at . it opens in the Lean 4 web editor with the whole development in it, so you can click any line and watch the givens and goal in the right-hand panel, exactly the two columns from earlier. put the cursor inside the `inr` branch and you can see the case split on `c` open up. no install, and it typechecks end to end. next is linear algebra at the [FernUniversität in Hagen](https://www.fernuni-hagen.de/mi/studium/module/lin_alg.shtml), alongside part-time studies at ZHAW. first course where proofs are the work rather than a step inside it, which is why i wanted this done now. @@ -416,3 +495,5 @@ next is linear algebra at the [FernUniversität in Hagen](https://www.fernuni-ha - Peano axioms, Trinity College Dublin: - and the list of 1000 theorems: - Kevin Buzzard, whose version of this proof mine follows: + +## Notes From bc0f377b8ccd3a09b29923f9818a4ad6d14e43a2 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 14:22:57 +0200 Subject: [PATCH 19/34] Roman --- assets/blog/lean_numberline_two_cases.svg | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/assets/blog/lean_numberline_two_cases.svg b/assets/blog/lean_numberline_two_cases.svg index 444f3e8..e0fdbc8 100644 --- a/assets/blog/lean_numberline_two_cases.svg +++ b/assets/blog/lean_numberline_two_cases.svg @@ -16,8 +16,8 @@ - - (a) + + (I) @@ -36,9 +36,10 @@ a = b + c = 0 - - (b) + + (II) From fb0a7dbd63e5297ec1c5c647e30df544345e3638 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 17:16:40 +0200 Subject: [PATCH 20/34] Nuke unused assets --- assets/blog/flowers.webp | Bin 10340 -> 0 bytes assets/blog/lean_graph.png | Bin 99704 -> 0 bytes assets/blog/lean_graph.svg | 272 ------------------ assets/blog/lean_thing.png | Bin 152532 -> 0 bytes assets/blog/numberline.png | Bin 51213 -> 0 bytes .../chess-widget/lichess-widget-embed.webp | Bin 96368 -> 0 bytes webp-gallery/dani_with_cool_glasses.webp | Bin 53824 -> 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zU_;H7hO#V}LubSPCHlMZ*}VRqw7;cY3_&v~M_D0ZA#hSppZ@1af2sM(3JXYpfjv?C_v-$ocD!=jo%|3k ur>g&6OJ~FTi~E0s=uGX`rhM9T60|>Fb;k`SQt(A8izCf{_EMI`0saqG2%T^M From 515d3ec75767efc6bf9bcb9974434d846f3a638c Mon Sep 17 00:00:00 2001 From: cb341 Date: Sat, 5 Sep 2026 17:16:54 +0200 Subject: [PATCH 21/34] Drop draft posts, improve lean --- ...-lean_real.md => 2026-06-28-lean-intro.md} | 116 +++--- _posts/2026-06-28-lean-todos.md | 226 ------------ _posts/2026-06-28-lean.md | 287 --------------- _posts/2026-06-28-todo.md | 343 ------------------ 4 files changed, 75 insertions(+), 897 deletions(-) rename _posts/{2026-06-28-lean_real.md => 2026-06-28-lean-intro.md} (71%) delete mode 100644 _posts/2026-06-28-lean-todos.md delete mode 100644 _posts/2026-06-28-lean.md delete mode 100644 _posts/2026-06-28-todo.md diff --git a/_posts/2026-06-28-lean_real.md b/_posts/2026-06-28-lean-intro.md similarity index 71% rename from _posts/2026-06-28-lean_real.md rename to _posts/2026-06-28-lean-intro.md index 915f7fd..12ecc8c 100644 --- a/_posts/2026-06-28-lean_real.md +++ b/_posts/2026-06-28-lean-intro.md @@ -1,6 +1,6 @@ --- -title: "Mathematics but not handwavey ?" -date: 2026-06-28 +title: "Proving 6 ≤ 7 the hard way" +date: 2026-09-05 description: "Proving that any two natural numbers compare, from the definition of a natural number upwards, in Lean." tags: ["theoretical mathematics", "first principles"] math: true @@ -8,7 +8,9 @@ math: true paper proofs have no compiler. i went looking for one and found Lean, by way of the [Natural Number Game](https://adam.math.hhu.de/#/g/leanprover-community/nng4), which builds the naturals from nothing and makes you prove your way back out. [^nng] i finished it end of june and wanted to write up what the last level actually took. -no AI was used for the Lean here, and none for the maths either: the definitions, the axioms i picked, the proof strategy and the dependency graph are mine. AI was used for phrasing. +no AI was used for the Lean here, and none for the maths either: the proofs, the proof strategy and the dependency graph are mine. AI was used for phrasing. the definitions of `MyNat`, `+` and `≤` are taken from the Natural Number Game, which is Apache licensed, and modified. [^nng4src] + +[^nng4src]: . the artefact carries the same attribution in its header. the seven supporting theorems and `le_total` are proved by me from those definitions. ## The theorem @@ -126,7 +128,7 @@ $$ 0=\varnothing,\quad 1=\lbrace\varnothing\rbrace,\quad 2=\lbrace\varnothing,\lbrace\varnothing\rbrace\rbrace,\quad\ldots $$ -$n < m$ becomes $n \in m$, so order comes for free. this is the standard construction in set theory. +$n < m$ becomes $n \in m$, so order comes for free. this is the standard construction in set theory. [^vonneumann] **Zermelo ordinals.** each number is the singleton of the previous one. @@ -134,11 +136,11 @@ $$ 0=\varnothing,\quad 1=\lbrace\varnothing\rbrace,\quad 2=\lbrace\lbrace\varnothing\rbrace\rbrace,\quad\ldots $$ -simpler to write, but $n < m$ is no longer $n \in m$, so order has to be defined separately. +simpler to write, but $n < m$ is no longer $n \in m$, so order has to be defined separately. [^zermelo] -**Church numerals.** a number is a function that applies another function that many times. $n$ is $\lambda f. \lambda x. f^n(x)$, so $3$ is $\lambda f. \lambda x. f(f(f(x)))$. addition is function composition. this is how the naturals appear in untyped lambda calculus. +**Church numerals.** a number is a function that applies another function that many times. $n$ is $\lambda f. \lambda x. f^n(x)$, so $3$ is $\lambda f. \lambda x. f(f(f(x)))$. addition is function composition. this is how the naturals appear in untyped lambda calculus. [^church] -**Peano axioms as first-order theory.** rather than constructing the naturals, state the properties they must have: $0$ is not a successor, $\operatorname{succ}$ is injective, and induction holds. this leaves the objects unspecified and constrains them instead. +**Peano axioms as first-order theory.** rather than constructing the naturals, state the properties they must have: $0$ is not a successor, $\operatorname{succ}$ is injective, and induction holds. this leaves the objects unspecified and constrains them instead. [^pa] i am using zero and succ, which is what the Natural Number Game uses and what Lean's own `Nat` is. @@ -146,6 +148,14 @@ i am using zero and succ, which is what the Natural Number Game uses and what Le [^ordinals]: +[^vonneumann]: John von Neumann, *Zur Einführung der transfiniten Zahlen*, Acta Litt. Acad. Sc. Szeged 1 (1923), 199–208. Standard modern treatment in Kunen, *Set Theory*, chapter I. + +[^zermelo]: Ernst Zermelo, *Untersuchungen über die Grundlagen der Mengenlehre I*, Mathematische Annalen 65 (1908), 261–281. + +[^church]: Alonzo Church, *An Unsolvable Problem of Elementary Number Theory*, American Journal of Mathematics 58 (1936), 345–363. Barendregt, *The Lambda Calculus*, section 6.4 gives the arithmetic. + +[^pa]: The first-order theory is usually attributed to Peano 1889 by way of Dedekind. Hájek and Pudlák, *Metamathematics of First-Order Arithmetic* (1998), chapter I, is the reference treatment. The distinction that matters here: the first-order schema quantifies over formulas, so it does not pin down $\mathbb{N}$ up to isomorphism, whereas the inductive definition above does. + ### Our zero and Lean's zero the definition above introduces a constructor, written `MyNat.zero` in Lean. the character `0` is a numeral, which is what a person types. they denote the same natural number and they are not the same term. @@ -168,33 +178,33 @@ a & b = 0 \\ \end{cases} $$ -worked on $1 + 2 = 3$. the right column names the axiom used and which direction it was applied in. $(\rightarrow)$ unfolds, replacing a name by its definition. $(\leftarrow)$ folds, recognising a definition and naming it. +worked on $1 + 2 = 3$. the reason column gives the direction, then the definition used. $(\rightarrow)$ unfolds, replacing a name by its definition. $(\leftarrow)$ folds, recognising a definition and naming it. $$ \begin{array}{rl} -& \textbf{Axioms} \\[2pt] +& \textbf{Definitions} \\[2pt] \colorbox{#fff3cd}{$\vphantom{Ag}\text{i.}$} & 1 \stackrel{\mathrm{def}}{=} \operatorname{succ}(0) \\[2pt] \colorbox{#cfe2ff}{$\vphantom{Ag}\text{ii.}$} & 2 \stackrel{\mathrm{def}}{=} \operatorname{succ}(1) \\[2pt] \colorbox{#e2d9f3}{$\vphantom{Ag}\text{iii.}$} & 3 \stackrel{\mathrm{def}}{=} \operatorname{succ}(2) \\[2pt] \colorbox{#f8d7da}{$\vphantom{Ag}\text{iv.}$} & a+0 \stackrel{\mathrm{def}}{=} a \\[2pt] \colorbox{#d1e7dd}{$\vphantom{Ag}\text{v.}$} & a+\operatorname{succ}(b) \stackrel{\mathrm{def}}{=} \operatorname{succ}(a+b) \end{array} -\qquad +\quad \begin{array}{l|l} \textbf{Statement} & \textbf{Reason} \\ \hline 1 + 2 & \text{given} \\[2pt] -1 + \colorbox{#cfe2ff}{$\vphantom{Ag}\operatorname{succ}(1)$} & \colorbox{#cfe2ff}{$\vphantom{Ag}\text{ii.}$} \; (\rightarrow) \\[2pt] -\colorbox{#d1e7dd}{$\vphantom{Ag}\operatorname{succ}(1 + 1)$} & \colorbox{#d1e7dd}{$\vphantom{Ag}\text{v.}$} \; (\rightarrow) \\[2pt] -\operatorname{succ}(1 + \colorbox{#fff3cd}{$\vphantom{Ag}\operatorname{succ}(0)$}) & \colorbox{#fff3cd}{$\vphantom{Ag}\text{i.}$} \; (\rightarrow) \\[2pt] -\operatorname{succ}(\operatorname{succ}(\colorbox{#d1e7dd}{$\vphantom{Ag}1 + 0$})) & \colorbox{#d1e7dd}{$\vphantom{Ag}\text{v.}$} \; (\rightarrow) \\[2pt] -\operatorname{succ}(\operatorname{succ}(\colorbox{#f8d7da}{$\vphantom{Ag}1$})) & \colorbox{#f8d7da}{$\vphantom{Ag}\text{iv.}$} \; (\rightarrow) \\[2pt] -\operatorname{succ}(\colorbox{#cfe2ff}{$\vphantom{Ag}2$}) & \colorbox{#cfe2ff}{$\vphantom{Ag}\text{ii.}$} \; (\leftarrow) \\[2pt] -\colorbox{#e2d9f3}{$\vphantom{Ag}3$} & \colorbox{#e2d9f3}{$\vphantom{Ag}\text{iii.}$} \; (\leftarrow) +1 + \colorbox{#cfe2ff}{$\vphantom{Ag}\operatorname{succ}(1)$} & (\rightarrow)\; \colorbox{#cfe2ff}{$\vphantom{Ag}\text{ii.}$} \\[2pt] +\colorbox{#d1e7dd}{$\vphantom{Ag}\operatorname{succ}(1 + 1)$} & (\rightarrow)\; \colorbox{#d1e7dd}{$\vphantom{Ag}\text{v.}$} \\[2pt] +\operatorname{succ}(1 + \colorbox{#fff3cd}{$\vphantom{Ag}\operatorname{succ}(0)$}) & (\rightarrow)\; \colorbox{#fff3cd}{$\vphantom{Ag}\text{i.}$} \\[2pt] +\operatorname{succ}(\operatorname{succ}(\colorbox{#d1e7dd}{$\vphantom{Ag}1 + 0$})) & (\rightarrow)\; \colorbox{#d1e7dd}{$\vphantom{Ag}\text{v.}$} \\[2pt] +\operatorname{succ}(\operatorname{succ}(\colorbox{#f8d7da}{$\vphantom{Ag}1$})) & (\rightarrow)\; \colorbox{#f8d7da}{$\vphantom{Ag}\text{iv.}$} \\[2pt] +\operatorname{succ}(\colorbox{#cfe2ff}{$\vphantom{Ag}2$}) & (\leftarrow)\; \colorbox{#cfe2ff}{$\vphantom{Ag}\text{ii.}$} \\[2pt] +\colorbox{#e2d9f3}{$\vphantom{Ag}3$} & (\leftarrow)\; \colorbox{#e2d9f3}{$\vphantom{Ag}\text{iii.}$} \end{array} $$ -the same axiom gets used in both directions. which direction you pick is a choice, and picking wrong is how a rewrite fails to terminate. +the same definition gets used in both directions. which direction you pick is a choice, and picking wrong is how a rewrite fails to terminate. ### Less than or equal @@ -208,7 +218,7 @@ the gap is a natural, so it is either zero or a successor, which gives two pictu ![Number line showing the gap c as zero in case I and nonzero in case II](/assets/blog/lean_numberline_two_cases.svg) -that split is not decoration. the proof below hits it as a real case distinction: once a gap `c` is in hand, `cases c` asks which of the two pictures applies, and the two branches close with different lemmas. +the proof below hits that split as a case distinction. once a gap `c` is in hand, `cases c` asks which of the two pictures applies, and the two branches close with different lemmas. this is the definition the Natural Number Game uses. Lean's own `Nat.le` is an inductive type instead, built from reflexivity and a successor step: [^natle] @@ -240,7 +250,7 @@ a Lean proof is written as a list of tactics. the ones in this article: | `rfl` | closes a goal whose two sides are the same term | | `exact` | closes a goal with something already proved | -`rw` is the short form of `rewrite`. `zero_eq_0`, `succ_eq_add_one` and `cases'` are Natural Number Game spellings, so pasting this into a fresh mathlib project gets you errors on the names before anything interesting. +`rw` is the usual short form of `rewrite`, though the artefact spells it out everywhere. `zero_eq_0`, `succ_eq_add_one` and `cases'` are Natural Number Game spellings, so pasting this into a fresh mathlib project gets you errors on the names before anything interesting. ## The smaller lemmas @@ -251,20 +261,33 @@ unfolding the definition, $0 \le x$ means $\exists c, x = 0 + c$. take $c := x$. ```lean theorem zero_le (x : ℕ) : 0 ≤ x := by use x - rw [zero_add] + rewrite[zero_add] rfl ``` +the `zero_add` it leans on is where the work actually happens, and that one goes by induction: +
-the same lemma by induction on x +zero_add, by induction on n -the base case is $0 \le 0$ with $c := 0$. the step goes from $0 \le d$ to $0 \le \operatorname{succ}(d)$, which means turning a gap `c` into `succ c`. +```lean +theorem zero_add (n : ℕ) : 0 + n = n := by + induction n with + | zero => + rewrite[zero_eq_0] + rewrite[add_zero] + rfl + | succ d hd => + rewrite[add_succ] + rewrite[hd] + rfl +``` -longer than the first route, and it pulls `add_succ` into the graph on top of `zero_add`. the first route is shorter because it picks the witness in one move instead of building it up one successor at a time. +addition recurses on its second argument, so `a + 0 = a` is true by definition and `0 + n = n` is not. the two look symmetric and only one of them is free. this asymmetry is why `succ_add`, `add_comm` and `add_assoc` all need their own inductive proofs.
-one theorem, several proofs, each a different path through the dependency graph. Ording's *99 Variations on a Proof* takes this to its conclusion with 99 proofs of a single cubic. [^ording] +one theorem often admits several proofs, each a different path through the dependency graph. Ording's *99 Variations on a Proof* takes this to its conclusion with 99 proofs of a single cubic. [^ording] [^ording]: Philip Ording, *99 Variations on a Proof*, Princeton University Press, 2019. @@ -276,20 +299,19 @@ the main proof calls two more results by name. both are proved the same way, fro `succ_eq_add_one` connects the constructor to the numeral, the split from earlier: ```lean -theorem succ_eq_add_one (n : ℕ) : succ n = n + 1 := by - rw [one_eq_succ_zero] - rw [add_succ] - rw [add_zero] +theorem succ_eq_add_one n : succ n = n + 1 := by + rewrite[one_eq_succ_zero] + rewrite[add_succ] + rewrite[add_zero] rfl ``` -`le_succ_self` says every number is below its own successor. the gap is one: +`le_succ_self` says every number is below its own successor. the gap is one, and once the witness is supplied the goal is exactly the previous theorem: ```lean theorem le_succ_self (x : ℕ) : x ≤ succ x := by use 1 - rw [succ_eq_add_one] - rfl + exact succ_eq_add_one x ```