59 lines
1.6 KiB
Text
59 lines
1.6 KiB
Text
|
import tactic
|
||
|
|
using Nat
|
||
|
|
|
||
|
|
rewrite_set basic
|
||
|
|
add_rewrite add_zerol add_succl eq_id : basic
|
||
|
|
|
||
|
|
theorem add_assoc (a b c : Nat) : (a + b) + c = a + (b + c)
|
||
|
|
:= induction_on a
|
||
|
|
(have (0 + b) + c = 0 + (b + c) :
|
||
|
|
by simp basic)
|
||
|
|
(λ (n : Nat) (iH : (n + b) + c = n + (b + c)),
|
||
|
|
have ((n + 1) + b) + c = (n + 1) + (b + c) :
|
||
|
|
by simp basic)
|
||
|
|
|
||
|
|
check add_zerol
|
||
|
|
check add_succl
|
||
|
|
check @eq_id
|
||
|
|
|
||
|
|
-- print environment 1
|
||
|
|
|
||
|
|
add_rewrite add_assoc add_comm mul_zeror mul_zerol mul_succr : basic
|
||
|
|
|
||
|
|
theorem mul_succl_1 (a b : Nat) : (a + 1) * b = a * b + b
|
||
|
|
:= induction_on b
|
||
|
|
(have (a + 1) * 0 = a * 0 + 0 :
|
||
|
|
by simp basic)
|
||
|
|
(λ (n : Nat) (iH : (a + 1) * n = a * n + n),
|
||
|
|
have (a + 1) * (n + 1) = a * (n + 1) + (n + 1) :
|
||
|
|
by simp basic)
|
||
|
|
|
||
|
|
exit
|
||
|
|
|
||
|
|
rewrite_set first_pass
|
||
|
|
add_rewrite mul_succr eq_id : first_pass
|
||
|
|
|
||
|
|
rewrite_set sort_add
|
||
|
|
add_rewrite add_assoc add_comm add_left_comm eq_id : sort_add
|
||
|
|
|
||
|
|
theorem mul_succl_2 (a b : Nat) : (a + 1) * b = a * b + b
|
||
|
|
:= induction_on b
|
||
|
|
(have (a + 1) * 0 = a * 0 + 0 :
|
||
|
|
by simp basic)
|
||
|
|
(λ (n : Nat) (iH : (a + 1) * n = a * n + n),
|
||
|
|
have (a + 1) * (n + 1) = a * (n + 1) + (n + 1) :
|
||
|
|
by Then (simp first_pass) (simp sort_add))
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
exit
|
||
|
|
|
||
|
|
|
||
|
|
theorem mul_succl_3 (a b : Nat) : (a + 1) * b = a * b + b
|
||
|
|
:= induction_on b
|
||
|
|
(have (a + 1) * 0 = a * 0 + 0 :
|
||
|
|
by simp basic)
|
||
|
|
(λ (n : Nat) (iH : (a + 1) * n = a * n + n),
|
||
|
|
calc (a + 1) * (n + 1) = (a * n + n) + (a + 1) : by simp first_pass
|
||
|
|
... = (a * n + a) + (n + 1) : by simp sort_add
|
||
|
|
... = a * (n + 1) + (n + 1) : by simp first_pass)
|