feat(builtin/Nat): flip orientation of associativity axioms for + and *
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura
parent
d1bd56b3d3
commit
97ead50a3e
9 changed files with 79 additions and 44 deletions
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@ -54,7 +54,7 @@ theorem OddPlusOne {a : Nat} (H : odd a) : even (a + 1)
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:= obtain (w : Nat) (Hw : a = 2*w + 1), from H,
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exists_intro (w + 1)
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(calc a + 1 = 2*w + 1 + 1 : { Hw }
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... = 2*w + (1 + 1) : symm (add_assoc _ _ _)
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... = 2*w + (1 + 1) : add_assoc _ _ _
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... = 2*w + 2*1 : refl _
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... = 2*(w + 1) : symm (distributer _ _ _))
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@ -83,15 +83,15 @@ theorem add_comm (a b : Nat) : a + b = b + a
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... = (n + a) + 1 : { iH }
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... = (n + 1) + a : symm (add_succl n a))
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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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theorem add_assoc (a b c : Nat) : (a + b) + c = a + (b + c)
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:= symm (induction_on a
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(calc 0 + (b + c) = b + c : add_zerol (b + c)
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... = (0 + b) + c : { symm (add_zerol b) })
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(λ (n : Nat) (iH : n + (b + c) = (n + b) + c),
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calc (n + 1) + (b + c) = (n + (b + c)) + 1 : add_succl n (b + c)
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... = ((n + b) + c) + 1 : { iH }
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... = ((n + b) + 1) + c : symm (add_succl (n + b) c)
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... = ((n + 1) + b) + c : { have (n + b) + 1 = (n + 1) + b : symm (add_succl n b) })
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... = ((n + 1) + b) + c : { have (n + b) + 1 = (n + 1) + b : symm (add_succl n b) }))
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theorem mul_zerol (a : Nat) : 0 * a = 0
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:= induction_on a
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@ -109,12 +109,12 @@ theorem mul_succl (a b : Nat) : (a + 1) * b = a * b + b
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(λ (n : Nat) (iH : (a + 1) * n = a * n + n),
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calc (a + 1) * (n + 1) = (a + 1) * n + (a + 1) : mul_succr (a + 1) n
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... = a * n + n + (a + 1) : { iH }
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... = a * n + n + a + 1 : add_assoc (a * n + n) a 1
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... = a * n + (n + a) + 1 : { have a * n + n + a = a * n + (n + a) : symm (add_assoc (a * n) n a) }
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... = a * n + n + a + 1 : symm (add_assoc (a * n + n) a 1)
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... = a * n + (n + a) + 1 : { have a * n + n + a = a * n + (n + a) : add_assoc (a * n) n a }
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... = a * n + (a + n) + 1 : { add_comm n a }
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... = a * n + a + n + 1 : { add_assoc (a * n) a n }
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... = a * n + a + n + 1 : { symm (add_assoc (a * n) a n) }
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... = a * (n + 1) + n + 1 : { symm (mul_succr a n) }
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... = a * (n + 1) + (n + 1) : symm (add_assoc (a * (n + 1)) n 1))
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... = a * (n + 1) + (n + 1) : add_assoc (a * (n + 1)) n 1)
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theorem mul_onel (a : Nat) : 1 * a = a
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:= induction_on a
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@ -148,12 +148,12 @@ theorem distributer (a b c : Nat) : a * (b + c) = a * b + a * c
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(λ (n : Nat) (iH : n * (b + c) = n * b + n * c),
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calc (n + 1) * (b + c) = n * (b + c) + (b + c) : mul_succl n (b + c)
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... = n * b + n * c + (b + c) : { iH }
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... = n * b + n * c + b + c : add_assoc (n * b + n * c) b c
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... = n * b + (n * c + b) + c : { symm (add_assoc (n * b) (n * c) b) }
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... = n * b + n * c + b + c : symm (add_assoc (n * b + n * c) b c)
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... = n * b + (n * c + b) + c : { add_assoc (n * b) (n * c) b }
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... = n * b + (b + n * c) + c : { add_comm (n * c) b }
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... = n * b + b + n * c + c : { add_assoc (n * b) b (n * c) }
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... = n * b + b + n * c + c : { symm (add_assoc (n * b) b (n * c)) }
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... = (n + 1) * b + n * c + c : { symm (mul_succl n b) }
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... = (n + 1) * b + (n * c + c) : symm (add_assoc ((n + 1) * b) (n * c) c)
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... = (n + 1) * b + (n * c + c) : add_assoc ((n + 1) * b) (n * c) c
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... = (n + 1) * b + (n + 1) * c : { symm (mul_succl n c) })
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theorem distributel (a b c : Nat) : (a + b) * c = a * c + b * c
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@ -162,8 +162,8 @@ theorem distributel (a b c : Nat) : (a + b) * c = a * c + b * c
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... = a * c + c * b : { mul_comm c a }
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... = a * c + b * c : { mul_comm c b }
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theorem mul_assoc (a b c : Nat) : a * (b * c) = a * b * c
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:= induction_on a
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theorem mul_assoc (a b c : Nat) : (a * b) * c = a * (b * c)
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:= symm (induction_on a
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(calc 0 * (b * c) = 0 : mul_zerol (b * c)
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... = 0 * c : symm (mul_zerol c)
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... = (0 * b) * c : { symm (mul_zerol b) })
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@ -171,7 +171,13 @@ theorem mul_assoc (a b c : Nat) : a * (b * c) = a * b * c
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calc (n + 1) * (b * c) = n * (b * c) + (b * c) : mul_succl n (b * c)
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... = n * b * c + (b * c) : { iH }
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... = (n * b + b) * c : symm (distributel (n * b) b c)
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... = (n + 1) * b * c : { symm (mul_succl n b) })
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... = (n + 1) * b * c : { symm (mul_succl n b) }))
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theorem add_left_comm (a b c : Nat) : a + (b + c) = b + (a + c)
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:= left_comm add_comm add_assoc a b c
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theorem mul_left_comm (a b c : Nat) : a * (b * c) = b * (a * c)
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:= left_comm mul_comm mul_assoc a b c
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theorem add_injr {a b c : Nat} : a + b = a + c → b = c
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:= induction_on a
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@ -181,13 +187,13 @@ theorem add_injr {a b c : Nat} : a + b = a + c → b = c
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... = c : add_zerol c)
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(λ (n : Nat) (iH : n + b = n + c → b = c) (H : n + 1 + b = n + 1 + c),
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let L1 : n + b + 1 = n + c + 1
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:= (calc n + b + 1 = n + (b + 1) : symm (add_assoc n b 1)
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:= (calc n + b + 1 = n + (b + 1) : add_assoc n b 1
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... = n + (1 + b) : { add_comm b 1 }
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... = n + 1 + b : add_assoc n 1 b
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... = n + 1 + b : symm (add_assoc n 1 b)
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... = n + 1 + c : H
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... = n + (1 + c) : symm (add_assoc n 1 c)
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... = n + (1 + c) : add_assoc n 1 c
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... = n + (c + 1) : { add_comm 1 c }
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... = n + c + 1 : add_assoc n c 1),
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... = n + c + 1 : symm (add_assoc n c 1)),
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L2 : n + b = n + c := succ_inj L1
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in iH L2)
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@ -202,9 +208,9 @@ theorem add_eqz {a b : Nat} (H : a + b = 0) : a = 0
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(λ (n : Nat) (H1 : a = n + 1),
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absurd_elim (a = 0)
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H (calc a + b = n + 1 + b : { H1 }
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... = n + (1 + b) : symm (add_assoc n 1 b)
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... = n + (1 + b) : add_assoc n 1 b
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... = n + (b + 1) : { add_comm 1 b }
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... = n + b + 1 : add_assoc n b 1
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... = n + b + 1 : symm (add_assoc n b 1)
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... ≠ 0 : succ_nz (n + b)))
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theorem le_intro {a b c : Nat} (H : a + c = b) : a ≤ b
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@ -221,22 +227,22 @@ theorem le_zero (a : Nat) : 0 ≤ a := le_intro (add_zerol a)
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theorem le_trans {a b c : Nat} (H1 : a ≤ b) (H2 : b ≤ c) : a ≤ c
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:= obtain (w1 : Nat) (Hw1 : a + w1 = b), from (le_elim H1),
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obtain (w2 : Nat) (Hw2 : b + w2 = c), from (le_elim H2),
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le_intro (calc a + (w1 + w2) = a + w1 + w2 : add_assoc a w1 w2
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le_intro (calc a + (w1 + w2) = a + w1 + w2 : symm (add_assoc a w1 w2)
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... = b + w2 : { Hw1 }
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... = c : Hw2)
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theorem le_add {a b : Nat} (H : a ≤ b) (c : Nat) : a + c ≤ b + c
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:= obtain (w : Nat) (Hw : a + w = b), from (le_elim H),
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le_intro (calc a + c + w = a + (c + w) : symm (add_assoc a c w)
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le_intro (calc a + c + w = a + (c + w) : add_assoc a c w
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... = a + (w + c) : { add_comm c w }
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... = a + w + c : add_assoc a w c
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... = a + w + c : symm (add_assoc a w c)
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... = b + c : { Hw })
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theorem le_antisym {a b : Nat} (H1 : a ≤ b) (H2 : b ≤ a) : a = b
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:= obtain (w1 : Nat) (Hw1 : a + w1 = b), from (le_elim H1),
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obtain (w2 : Nat) (Hw2 : b + w2 = a), from (le_elim H2),
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let L1 : w1 + w2 = 0
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:= add_injr (calc a + (w1 + w2) = a + w1 + w2 : { add_assoc a w1 w2 }
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:= add_injr (calc a + (w1 + w2) = a + w1 + w2 : { symm (add_assoc a w1 w2) }
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... = b + w2 : { Hw1 }
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... = a : Hw2
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... = a + 0 : symm (add_zeror a)),
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@ -249,9 +255,9 @@ theorem not_lt_0 (a : Nat) : ¬ a < 0
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:= not_intro (λ H : a + 1 ≤ 0,
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obtain (w : Nat) (Hw1 : a + 1 + w = 0), from (le_elim H),
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absurd
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(calc a + w + 1 = a + (w + 1) : symm (add_assoc _ _ _)
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(calc a + w + 1 = a + (w + 1) : add_assoc _ _ _
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... = a + (1 + w) : { add_comm _ _ }
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... = a + 1 + w : add_assoc _ _ _
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... = a + 1 + w : symm (add_assoc _ _ _)
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... = 0 : Hw1)
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(succ_nz (a + w)))
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@ -263,7 +269,7 @@ theorem lt_elim {a b : Nat} (H : a < b) : ∃ x, a + 1 + x = b
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theorem lt_le {a b : Nat} (H : a < b) : a ≤ b
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:= obtain (w : Nat) (Hw : a + 1 + w = b), from (le_elim H),
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le_intro (calc a + (1 + w) = a + 1 + w : add_assoc _ _ _
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le_intro (calc a + (1 + w) = a + 1 + w : symm (add_assoc _ _ _)
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... = b : Hw)
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theorem lt_ne {a b : Nat} (H : a < b) : a ≠ b
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@ -271,7 +277,7 @@ theorem lt_ne {a b : Nat} (H : a < b) : a ≠ b
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obtain (w : Nat) (Hw : a + 1 + w = b), from (lt_elim H),
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absurd (calc w + 1 = 1 + w : add_comm _ _
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... = 0 :
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add_injr (calc b + (1 + w) = b + 1 + w : add_assoc b 1 w
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add_injr (calc b + (1 + w) = b + 1 + w : symm (add_assoc b 1 w)
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... = a + 1 + w : { symm H1 }
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... = b : Hw
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... = b + 0 : symm (add_zeror b)))
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@ -284,40 +290,40 @@ theorem lt_nrefl (a : Nat) : ¬ a < a
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theorem lt_trans {a b c : Nat} (H1 : a < b) (H2 : b < c) : a < c
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:= obtain (w1 : Nat) (Hw1 : a + 1 + w1 = b), from (lt_elim H1),
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obtain (w2 : Nat) (Hw2 : b + 1 + w2 = c), from (lt_elim H2),
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lt_intro (calc a + 1 + (w1 + 1 + w2) = a + 1 + (w1 + (1 + w2)) : { symm (add_assoc w1 1 w2) }
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... = (a + 1 + w1) + (1 + w2) : add_assoc _ _ _
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lt_intro (calc a + 1 + (w1 + 1 + w2) = a + 1 + (w1 + (1 + w2)) : { add_assoc w1 1 w2 }
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... = (a + 1 + w1) + (1 + w2) : symm (add_assoc _ _ _)
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... = b + (1 + w2) : { Hw1 }
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... = b + 1 + w2 : add_assoc _ _ _
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... = b + 1 + w2 : symm (add_assoc _ _ _)
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... = c : Hw2)
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theorem lt_le_trans {a b c : Nat} (H1 : a < b) (H2 : b ≤ c) : a < c
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:= obtain (w1 : Nat) (Hw1 : a + 1 + w1 = b), from (lt_elim H1),
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obtain (w2 : Nat) (Hw2 : b + w2 = c), from (le_elim H2),
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lt_intro (calc a + 1 + (w1 + w2) = a + 1 + w1 + w2 : add_assoc _ _ _
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lt_intro (calc a + 1 + (w1 + w2) = a + 1 + w1 + w2 : symm (add_assoc _ _ _)
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... = b + w2 : { Hw1 }
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... = c : Hw2)
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theorem le_lt_trans {a b c : Nat} (H1 : a ≤ b) (H2 : b < c) : a < c
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:= obtain (w1 : Nat) (Hw1 : a + w1 = b), from (le_elim H1),
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obtain (w2 : Nat) (Hw2 : b + 1 + w2 = c), from (lt_elim H2),
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lt_intro (calc a + 1 + (w1 + w2) = a + 1 + w1 + w2 : add_assoc _ _ _
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... = a + (1 + w1) + w2 : { symm (add_assoc a 1 w1) }
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lt_intro (calc a + 1 + (w1 + w2) = a + 1 + w1 + w2 : symm (add_assoc _ _ _)
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... = a + (1 + w1) + w2 : { add_assoc a 1 w1 }
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... = a + (w1 + 1) + w2 : { add_comm 1 w1 }
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... = a + w1 + 1 + w2 : { add_assoc a w1 1 }
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... = a + w1 + 1 + w2 : { symm (add_assoc a w1 1) }
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... = b + 1 + w2 : { Hw1 }
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... = c : Hw2)
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theorem ne_lt_succ {a b : Nat} (H1 : a ≠ b) (H2 : a < b + 1) : a < b
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:= obtain (w : Nat) (Hw : a + 1 + w = b + 1), from (lt_elim H2),
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let L : a + w = b := add_injl (calc a + w + 1 = a + (w + 1) : symm (add_assoc _ _ _)
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let L : a + w = b := add_injl (calc a + w + 1 = a + (w + 1) : add_assoc _ _ _
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... = a + (1 + w) : { add_comm _ _ }
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... = a + 1 + w : add_assoc _ _ _
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... = a + 1 + w : symm (add_assoc _ _ _)
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... = b + 1 : Hw)
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in discriminate (λ Hz : w = 0, absurd_elim (a < b) (calc a = a + 0 : symm (add_zeror _)
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... = a + w : { symm Hz }
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... = b : L)
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H1)
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(λ (p : Nat) (Hp : w = p + 1), lt_intro (calc a + 1 + p = a + (1 + p) : symm (add_assoc _ _ _)
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(λ (p : Nat) (Hp : w = p + 1), lt_intro (calc a + 1 + p = a + (1 + p) : add_assoc _ _ _
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... = a + (p + 1) : { add_comm _ _ }
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... = a + w : { symm Hp }
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... = b : L))
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Binary file not shown.
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@ -34,6 +34,8 @@ MK_CONSTANT(Nat_mul_comm_fn, name({"Nat", "mul_comm"}));
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MK_CONSTANT(Nat_distributer_fn, name({"Nat", "distributer"}));
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MK_CONSTANT(Nat_distributel_fn, name({"Nat", "distributel"}));
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MK_CONSTANT(Nat_mul_assoc_fn, name({"Nat", "mul_assoc"}));
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MK_CONSTANT(Nat_add_left_comm_fn, name({"Nat", "add_left_comm"}));
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MK_CONSTANT(Nat_mul_left_comm_fn, name({"Nat", "mul_left_comm"}));
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MK_CONSTANT(Nat_add_injr_fn, name({"Nat", "add_injr"}));
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MK_CONSTANT(Nat_add_injl_fn, name({"Nat", "add_injl"}));
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MK_CONSTANT(Nat_add_eqz_fn, name({"Nat", "add_eqz"}));
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@ -92,6 +92,12 @@ inline expr mk_Nat_distributel_th(expr const & e1, expr const & e2, expr const &
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expr mk_Nat_mul_assoc_fn();
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bool is_Nat_mul_assoc_fn(expr const & e);
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inline expr mk_Nat_mul_assoc_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_mul_assoc_fn(), e1, e2, e3}); }
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expr mk_Nat_add_left_comm_fn();
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bool is_Nat_add_left_comm_fn(expr const & e);
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inline expr mk_Nat_add_left_comm_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_add_left_comm_fn(), e1, e2, e3}); }
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expr mk_Nat_mul_left_comm_fn();
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bool is_Nat_mul_left_comm_fn(expr const & e);
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inline expr mk_Nat_mul_left_comm_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_mul_left_comm_fn(), e1, e2, e3}); }
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expr mk_Nat_add_injr_fn();
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bool is_Nat_add_injr_fn(expr const & e);
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inline expr mk_Nat_add_injr_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_Nat_add_injr_fn(), e1, e2, e3, e4}); }
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@ -6,16 +6,16 @@
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9
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⊥
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2 + 2 + (2 + 2) + 1 ≥ 3
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3 ≤ 2 * 2 + 2 * 2 + 2 * 2 + 2 * 2 + 1
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3 ≤ 2 * 2 + (2 * 2 + (2 * 2 + (2 * 2 + 1)))
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Assumed: a
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Assumed: b
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Assumed: c
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Assumed: d
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Imported 'if_then_else'
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a * c + a * d + b * c + b * d
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a * c + (a * d + (b * c + b * d))
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trans (Nat::distributel a b (c + d))
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(trans (congr (congr2 Nat::add (Nat::distributer a c d)) (Nat::distributer b c d))
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(Nat::add_assoc (a * c + a * d) (b * c) (b * d)))
|
||||
(Nat::add_assoc (a * c) (a * d) (b * c + b * d)))
|
||||
Proved: congr2_congr1
|
||||
Proved: congr2_congr2
|
||||
Proved: congr1_congr2
|
||||
|
|
@ -28,7 +28,7 @@ trans (congr (congr2 eq
|
|||
let κ::1 := congr2 (λ x : ℕ, eq ((λ x : ℕ, x + 10) x))
|
||||
(trans (congr2 (ite (a > 0) b) (Nat::add_zeror b)) (if_a_a (a > 0) b))
|
||||
in trans (congr κ::1 (congr2 (λ x : ℕ, x + 10) (if_a_a (a > 0) b))) (eq_id (b + 10))
|
||||
a * a + a * b + b * a + b * b
|
||||
a * a + (a * b + (b * a + b * b))
|
||||
⊤ → ⊥ refl (⊤ → ⊥)
|
||||
⊤ → ⊤ refl (⊤ → ⊤)
|
||||
⊥ → ⊥ refl (⊥ → ⊥)
|
||||
|
|
|
|||
10
tests/lean/simp8.lean
Normal file
10
tests/lean/simp8.lean
Normal file
|
|
@ -0,0 +1,10 @@
|
|||
variables a b c d e f : Nat
|
||||
rewrite_set simple
|
||||
add_rewrite Nat::add_assoc Nat::add_comm Nat::add_left_comm Nat::distributer Nat::distributel : simple
|
||||
(*
|
||||
local t = parse_lean("f + (c + f + d) + (e * (a + c) + (d + a))")
|
||||
local t2, pr = simplify(t, "simple")
|
||||
print(t)
|
||||
print("====>")
|
||||
print(t2)
|
||||
*)
|
||||
11
tests/lean/simp8.lean.expected.out
Normal file
11
tests/lean/simp8.lean.expected.out
Normal file
|
|
@ -0,0 +1,11 @@
|
|||
Set: pp::colors
|
||||
Set: pp::unicode
|
||||
Assumed: a
|
||||
Assumed: b
|
||||
Assumed: c
|
||||
Assumed: d
|
||||
Assumed: e
|
||||
Assumed: f
|
||||
f + (c + f + d) + (e * (a + c) + (d + a))
|
||||
====>
|
||||
a + (c + (d + (d + (f + (f + (e * a + e * c))))))
|
||||
|
|
@ -2,7 +2,7 @@
|
|||
Set: pp::unicode
|
||||
Using: Nat
|
||||
0 + 1
|
||||
Nat::add_assoc : ∀ a b c : ℕ, a + (b + c) = a + b + c
|
||||
Nat::add_assoc : ∀ a b c : ℕ, a + b + c = a + (b + c)
|
||||
using.lean:7:6: error: unknown identifier 'add'
|
||||
Using: Nat
|
||||
0 + 1
|
||||
|
|
|
|||
Loading…
Reference in a new issue