59 lines
No EOL
1.6 KiB
Text
59 lines
No EOL
1.6 KiB
Text
import tactic
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using Nat
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rewrite_set basic
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add_rewrite add_zerol add_succl eq_id : basic
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theorem add_assoc (a b c : Nat) : (a + b) + c = a + (b + c)
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:= induction_on a
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(have (0 + b) + c = 0 + (b + c) :
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by simp basic)
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(λ (n : Nat) (iH : (n + b) + c = n + (b + c)),
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have ((n + 1) + b) + c = (n + 1) + (b + c) :
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by simp basic)
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check add_zerol
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check add_succl
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check @eq_id
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-- print environment 1
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add_rewrite add_assoc add_comm mul_zeror mul_zerol mul_succr : basic
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theorem mul_succl_1 (a b : Nat) : (a + 1) * b = a * b + b
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:= induction_on b
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(have (a + 1) * 0 = a * 0 + 0 :
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by simp basic)
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(λ (n : Nat) (iH : (a + 1) * n = a * n + n),
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have (a + 1) * (n + 1) = a * (n + 1) + (n + 1) :
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by simp basic)
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exit
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rewrite_set first_pass
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add_rewrite mul_succr eq_id : first_pass
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rewrite_set sort_add
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add_rewrite add_assoc add_comm add_left_comm eq_id : sort_add
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theorem mul_succl_2 (a b : Nat) : (a + 1) * b = a * b + b
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:= induction_on b
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(have (a + 1) * 0 = a * 0 + 0 :
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by simp basic)
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(λ (n : Nat) (iH : (a + 1) * n = a * n + n),
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have (a + 1) * (n + 1) = a * (n + 1) + (n + 1) :
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by Then (simp first_pass) (simp sort_add))
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exit
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theorem mul_succl_3 (a b : Nat) : (a + 1) * b = a * b + b
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:= induction_on b
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(have (a + 1) * 0 = a * 0 + 0 :
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by simp basic)
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(λ (n : Nat) (iH : (a + 1) * n = a * n + n),
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calc (a + 1) * (n + 1) = (a * n + n) + (a + 1) : by simp first_pass
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... = (a * n + a) + (n + 1) : by simp sort_add
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... = a * (n + 1) + (n + 1) : by simp first_pass) |