人工智能数学验证工具LEAN4【入门介绍9】高级乘法世界：逆否策略的等效替代，提取假设的已知，tauto另类理解，更严格的归纳法假设。。。

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视频讲解：人工智能数学验证工具LEAN4【入门介绍9】高级乘法世界：逆否策略的等效替代，提取假设的已知，tauto另类理解，更严格的归纳法假设。。。_哔哩哔哩_bilibili

import Game.Levels.AdvMultiplication.L08mul_eq_zero

World "AdvMultiplication"
Level 9
Title "mul_left_cancel"

TheoremTab "*"

namespace MyNat

/-- `mul_left_cancel a b c` is a proof that if `a ≠ 0` and `a * b = a * c` then `b = c`. -/
TheoremDoc MyNat.mul_left_cancel as "mul_left_cancel" in "*"

Introduction
"
In this level we prove that if `a * b = a * c` and `a ≠ 0` then `b = c`. It is tricky, for
several reasons. One of these is that
we need to introduce a new idea: we will need to understand the concept of
mathematical induction a little better.

Starting with `induction b with d hd` is too naive, because in the inductive step
the hypothesis is `a * d = a * c → d = c` but what we know is `a * succ d = a * c`,
so the induction hypothesis does not apply!

Assume `a ≠ 0` is fixed. The actual statement we want to prove by induction on `b` is
\"for all `c`, if `a * b = a * c` then `b = c`. This *can* be proved by induction,
because we now have the flexibility to change `c`.\"
"

Statement mul_left_cancel (a b c : ℕ) (ha : a ≠ 0) (h : a * b = a * c) : b = c := by
Hint "The way to start this proof is `induction b with d hd generalizing c`."
induction b with d hd generalizing c
· Hint (hidden := true) "Use `mul_eq_zero` and remember that `tauto` will solve a goal
if there are hypotheses `a = 0` and `a ≠ 0`."
rw [mul_zero] at h
symm at h
apply mul_eq_zero at h
cases h with h1 h2
· tauto
· rw [h2]
rfl
· Hint "The inductive hypothesis `hd` is \"For all natural numbers `c`, `a * d = a * c → d = c`\".
You can `apply` it `at` any hypothesis of the form `a * d = a * ?`. "
Hint (hidden := true) "Split into cases `c = 0` and `c = succ e` with `cases c with e`."
cases c with e
· rw [mul_succ, mul_zero] at h
apply add_left_eq_zero at h
tauto
· rw [mul_succ, mul_succ] at h
apply add_right_cancel at h
apply hd at h
rw [h]
rfl