Natural Number Game Sol

https://adam.math.hhu.de/#/g/leanprover-community/nng4

Tutorial World

Level 1 / 8 : The rfl tactic

rfl

Level 2 / 8 : the rw tactic

rw[h]
rfl

Level 3 / 8 : Numbers

rw[two_eq_succ_one]
rw[one_eq_succ_zero]
rfl

Level 4 / 8 : rewriting backwards

rw[← one_eq_succ_zero]
rw[← two_eq_succ_one]
rfl

Level 5 / 8 : Adding zero

rw[add_zero b]
rw[add_zero c]
rfl

Level 6 / 8 : Precision rewriting

rw[add_zero b]
rw[add_zero c]
rfl

Level 7 / 8 : add_succ

rw[one_eq_succ_zero]
rw[add_succ]
rw[add_zero]
rfl

Level 8 / 8 : 2+2=4

nth_rewrite 2 [two_eq_succ_one]
rw[add_succ 2]
nth_rewrite 1 [one_eq_succ_zero]
rw[add_succ 2]
rw[add_zero 2]
rw[<- three_eq_succ_two]
rw[<- four_eq_succ_three]
rfl

Addition World

Level 1 / 5 : zero_add

induction n with d hd
rw[add_zero]
rfl
rw[add_succ 0]
rw[hd]
rfl

Level 2 / 5 : succ_add

induction b with d hd
rw[succ_eq_add_one]
rw[succ_eq_add_one]
rw[add_zero]
rw[add_zero]
rfl
rw[add_succ]
rw[add_succ]
rw[hd]
rfl

Level 3 / 5 : add_comm (level boss)

induction b with d hd
rw[add_zero]
rw[zero_add]
rfl
rw[add_succ]
rw[hd]
rw[succ_add]
rfl

Level 4 / 5 : add_assoc (associativity of addition)

induction b with d hd
rw[add_zero]
rw[zero_add]
rfl
rw[succ_add]
rw[add_succ]
rw[succ_add]
rw[add_succ]
rw[hd]
rfl

Level 5 / 5 : add_right_comm

induction b with d hd
rw[add_zero]
rw[add_zero]
rfl
rw[add_succ]
rw[succ_add]
rw[hd]
rw[add_succ]
rfl

Multiplication World

Level 1 / 9 : mul_one

rw[one_eq_succ_zero]
rw[mul_succ]
rw[mul_zero]
rw[zero_add]
rfl

Level 2 / 9 : zero_mul

induction m with d hd
rw[mul_zero]
rfl
rw[mul_succ]
rw[hd]
rw[add_zero]
rfl

Level 3 / 9 : succ_mul

induction b with d hd
rw[mul_zero]
rw[mul_zero]
rw[add_zero]
rfl
rw[mul_succ]
rw[mul_succ]
rw[hd]
rw[add_assoc]
rw[add_comm]
nth_rewrite 2 [add_assoc]
nth_rewrite 3 [add_comm]
rw[succ_eq_add_one]
rw[succ_eq_add_one]
rw[← add_assoc]
rw[← add_assoc]
rw[add_assoc]
rw[add_comm]
nth_rewrite 3 [add_comm]
rw[add_comm]
rw[← add_assoc]
rfl

Level 4 / 9 : mul_comm

induction b with d hd
rw[mul_zero]
rw[zero_mul]
rfl
rw[succ_mul]
rw[mul_succ]
rw[hd]
rfl

Level 5 / 9 : one_mul

rw[mul_comm, mul_one]
rfl

Level 6 / 9 : two_mul

rw[mul_comm]
rw[two_eq_succ_one]
rw[mul_succ]
rw[mul_one]
rfl

Level 7 / 9 : mul_add

induction a with d hd
rw[zero_mul]
rw[zero_mul]
rw[zero_mul]
rw[add_zero]
rfl
rw[succ_mul]
rw[succ_mul]
rw[succ_mul]
rw[hd]
nth_rewrite 2 [← add_assoc]
nth_rewrite 1 [← add_assoc]
nth_rewrite 2 [add_right_comm]
rfl

Level 8 / 9 : add_mul

rw[mul_comm]
rw[mul_add]
rw[mul_comm]
nth_rewrite 2 [mul_comm]
rfl

Level 9 / 9 : mul_assoc

induction c with d hd
repeat rw[mul_zero]
rfl
rw[mul_succ]
rw[mul_succ]
rw[hd]
rw[mul_add]
rfl

Implication World

Level 1 / 11 : The exact tactic

exact h1

Level 2 / 11 : exact practice.

rw[zero_add] at h
rw[zero_add] at h
exact h

Level 3 / 11 : The apply tactic.

apply h2 at h1
exact h1

Level 4 / 11 : succ_inj : the successor function is injective

rw[four_eq_succ_three] at h
rw[← succ_eq_add_one] at h
apply succ_inj at h
exact h

Level 5 / 11 : Arguing backwards

apply succ_inj
rw[succ_eq_add_one]
rw[← four_eq_succ_three]
exact h

Level 6 / 11 : intro

intro h
exact h

Level 7 / 11 : intro practice

intro h
rw[← succ_eq_add_one] at h
rw[← succ_eq_add_one] at h
apply succ_inj at h
exact h

Level 8 / 11 : ≠

apply h2 at h1
exact h1

Level 9 / 11 : zero_ne_succ

rw[one_eq_succ_zero]
apply zero_ne_succ 

Level 10 / 11 : 1 ≠ 0

intro h
symm at h
apply zero_ne_one at h
exact h

Level 11 / 11 : 2 + 2 ≠ 5

rw[add_succ]
rw[add_succ]
rw[add_zero]
intro h
apply succ_inj at h
apply succ_inj at h
apply succ_inj at h
apply succ_inj at h
apply zero_ne_succ at h
exact h

Power World

Level 1 / 10 : zero_pow_zero

rw[pow_zero]
rfl

Level 2 / 10 : zero_pow_succ

rw[pow_succ]
rw[mul_zero]
rfl

Level 3 / 10 : pow_one

rw[one_eq_succ_zero]
rw[pow_succ]
rw[pow_zero]
rw[one_mul]
rfl

Level 4 / 10 : one_pow

induction m with d hd
rw[pow_zero]
rfl
rw[pow_succ]
rw[hd]
rw[mul_one]
rfl

Level 5 / 10 : pow_two

rw[two_eq_succ_one]
rw[pow_succ]
rw[pow_one]
rfl

Level 6 / 10 : pow_add

induction n with d hd
rw[add_zero, pow_zero, mul_one]
rfl
rw[pow_succ,succ_eq_add_one,← add_assoc, ← succ_eq_add_one]
rw[pow_succ, hd, mul_assoc]
rfl

Level 7 / 10 : mul_pow

induction n with d hd
rw[pow_zero, pow_zero, pow_zero, mul_one]
rfl
rw[pow_succ, pow_succ, pow_succ, hd]
nth_rewrite 3 [mul_comm]
nth_rewrite 2 [mul_assoc]
nth_rewrite 5 [mul_comm]
nth_rewrite 2 [← mul_assoc]
nth_rewrite 2 [← mul_assoc]
nth_rewrite 1 [← mul_assoc]
rfl

Level 8 / 10 : pow_pow

induction n with d hd
rw[pow_zero]
rw[mul_zero]
rw[pow_zero]
rfl
rw[pow_succ]
rw[hd]
rw[mul_succ]
rw[pow_add]
rfl

Level 9 / 10 : add_sq

rw[pow_two]
rw[mul_add]
rw[add_mul]
rw[add_mul]
rw[← pow_two]
rw[← pow_two]
rw[← add_right_comm]
nth_rewrite 3 [add_comm]
rw[add_assoc]
nth_rewrite 2 [mul_comm]
rw[add_assoc]
rw[← add_mul]
rw[← two_mul]
rw[← add_assoc]
rfl

Level 10 / 10 : Fermat's Last Theorem

不会.

Advanced Addition World

Level 1 / 6 : add_right_cancel

induction n with d hd
rw[add_zero]
rw[add_zero]
intro h
exact h
rw[add_succ]
rw[add_succ]
intro hh
apply succ_inj at hh
apply hd at hh
exact hh

Level 2 / 6 : add_left_cancel

induction n with d hd
rw[zero_add]
rw[zero_add]
intro h
exact h
rw[succ_eq_add_one]
rw[add_right_comm]
rw[add_right_comm d 1 b]
rw[← succ_eq_add_one]
rw[← succ_eq_add_one]
intro h
apply succ_inj at h
apply hd at h
exact h

Level 3 / 6 : add_left_eq_self

induction y
rw[add_zero]
intro h
exact h
rw[add_succ]
intro h
apply succ_inj at h
apply n_ih at h
exact h

Level 4 / 6 : add_right_eq_self

induction x
rw[zero_add]
intro h
exact h
rw[succ_add]
intro h
apply succ_inj at h
apply n_ih at h
exact h

Level 5 / 6 : add_right_eq_zero

cases b with d
rw[add_zero]
intro h
exact h
intro h
induction a with e he
rw[zero_add] at h
symm at h
apply zero_ne_succ at h
cases h
rw[succ_add] at h
symm at h
apply zero_ne_succ at h
cases h

Level 6 / 6 : add_left_eq_zero

rw[add_comm]
apply add_right_eq_zero

Algorithm World

Level 1 / 9 : add_left_comm

rw[add_comm, ← add_assoc, add_assoc]
nth_rewrite 2 [add_comm]
rw[add_assoc]
rfl

Level 2 / 9 : making life easier

repeat rw[add_assoc]
nth_rewrite 2 [add_comm]
rw[add_assoc]
rfl

Level 3 / 9 : making life simple

simp only [add_assoc, add_left_comm, add_comm]

Level 4 / 9 : the simplest approach

simp_add

Level 5 / 9 : pred

rw[← pred_succ a]
rw[← pred_succ b]
rw[h]
rfl

Level 6 / 9 : is_zero

intro h
symm at h
apply zero_ne_succ at h
exact h

Level 7 / 9 : An algorithm for equality

contrapose! h
apply succ_inj at h
exact h

Level 8 / 9 : decide

decide

Level 9 / 9 : decide again

decide

≤ World

Level 1 / 11 : The use tactic

use 0
rw[add_zero]
rfl

Level 2 / 11 : 0 ≤ x

induction x with d hd
apply le_refl
rw[succ_eq_add_one]
use d+1
rw[zero_add]
rfl

Level 3 / 11 : x ≤ succ x

rw[succ_eq_add_one]
use 1
rfl

Level 4 / 11 : x ≤ y and y ≤ z implies x ≤ z

cases hxy with a ha
cases hyz with b hb
use a+b
rw[← add_assoc]
rw[← ha]
rw[← hb]
rfl

Level 5 / 11 : x ≤ 0 → x = 0

cases hx with c hc
symm at hc
apply add_right_eq_zero at hc
exact hc

Level 6 / 11 : x ≤ y and y ≤ x implies x = y

cases hxy with c hc
cases hyx with d hd
rw[hd] at hc
symm at hc
rw[add_assoc] at hc
apply add_right_eq_self at hc
apply add_right_eq_zero at hc
rw[hc] at hd
rw[add_zero] at hd
exact hd

Level 7 / 11 : Dealing with or

cases h with hx hy
right
exact hx
left
exact hy

Level 8 / 11 : x ≤ y or y ≤ x

induction y with d hd
right
apply zero_le
cases hd
left
cases h with c hc
rw[succ_eq_add_one]
rw[hc]
rw[add_assoc]
use c+1
rfl
cases h with c hc
cases c
rw[add_zero] at hc
rw[hc]
left
use 1
rw[succ_eq_add_one]
rfl
right
rw[add_succ] at hc
rw[hc]
use a
rw[succ_add]
rfl

Level 9 / 11 : succ x ≤ succ y → x ≤ y

cases hx with c hc
use c
rw[succ_add] at hc
apply succ_inj at hc
exact hc

Level 10 / 11 : x ≤ 1

cases x
left
rfl
rw[one_eq_succ_zero] at hx
apply succ_le_succ at hx
apply le_zero at hx
right
rw[one_eq_succ_zero]
rw[hx]
rfl

Level 11 / 11 : le_two

cases x
left
rfl
cases a
rw[one_eq_succ_zero]
right
left
rfl
right
right
rw[two_eq_succ_one] at hx
apply succ_le_succ at hx
rw[one_eq_succ_zero] at hx
apply succ_le_succ at hx
apply le_zero at hx
rw[hx]
rw[← one_eq_succ_zero]
rw[← two_eq_succ_one]
rfl

Advanced Multiplication World

Level 1 / 10 : mul_le_mul_right

cases h with d hd
use d*t
rw[← add_mul, hd]
rfl

Level 2 / 10 : mul_left_ne_zero

contrapose! h
rw[h,mul_zero]
rfl

Level 3 / 10 : eq_succ_of_ne_zero

cases a
tauto
use a_1
rfl

Level 4 / 10 : one_le_of_ne_zero

cases a
tauto
rw[one_eq_succ_zero]
use a_1
rw[← one_eq_succ_zero]
rw[succ_eq_add_one]
simp_add

Level 5 / 10 : le_mul_right

cases b
rw[mul_zero] at h
tauto
rw[succ_eq_add_one]
rw[mul_add]
rw[mul_one]
use a*a_1
simp_add

Level 6 / 10 : mul_right_eq_one

have h2 : x * y ≠ 0
rw[h]
tauto
apply le_mul_right at h2
rw[h] at h2
apply le_one at h2
cases h2
rw[h_1] at h
rw[zero_mul] at h
tauto
exact h_1

Level 7 / 10 : mul_ne_zero

apply eq_succ_of_ne_zero at ha
apply eq_succ_of_ne_zero at hb
cases a
contrapose! ha
apply zero_ne_succ
cases b
contrapose! hb
apply zero_ne_succ
rw[succ_eq_add_one]
rw[succ_eq_add_one]
rw[mul_add]
rw[add_mul]
rw[add_mul]
rw[mul_one]
rw[mul_one]
rw[one_mul]
rw[<- add_assoc]
rw[← succ_eq_add_one]
apply succ_ne_zero

Level 8 / 10 : mul_eq_zero

have h2 := mul_ne_zero a b
tauto

Level 9 / 10 : mul_left_cancel

induction b with d hd generalizing c
rw[mul_zero] at h
symm at h
apply mul_eq_zero at h
tauto
cases c with e
rw[mul_zero] at h
apply mul_eq_zero at h
tauto
rw[succ_eq_add_one] at h
rw[succ_eq_add_one] at h
rw[mul_add] at h
rw[mul_add] at h
rw[mul_one] at h
apply add_right_cancel at h
apply hd at h
rw[h]
rfl

Level 10 / 10 : mul_right_eq_self

nth_rewrite 2 [<- mul_one a] at h
apply mul_left_cancel at h
exact h
exact ha