refactor(builtin/Nat): use obtain-from instead of ExistsElim, and use more user-friendly argument order for Induction
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
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Leonardo de Moura
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3 changed files with 139 additions and 145 deletions
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@ -37,22 +37,20 @@ Axiom PlusSucc (a b : Nat) : a + (b + 1) = (a + b) + 1.
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Axiom MulZero (a : Nat) : a * 0 = 0.
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Axiom MulZero (a : Nat) : a * 0 = 0.
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Axiom MulSucc (a b : Nat) : a * (b + 1) = a * b + a.
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Axiom MulSucc (a b : Nat) : a * (b + 1) = a * b + a.
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Axiom LeDef (a b : Nat) : a ≤ b ⇔ ∃ c, a + c = b.
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Axiom LeDef (a b : Nat) : a ≤ b ⇔ ∃ c, a + c = b.
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Axiom Induction {P : Nat → Bool} (Hb : P 0) (iH : Π (n : Nat) (H : P n), P (n + 1)) (a : Nat) : P a.
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Axiom Induction {P : Nat → Bool} (a : Nat) (H1 : P 0) (H2 : Π (n : Nat) (iH : P n), P (n + 1)) : P a.
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Theorem ZeroNeOne : 0 ≠ 1 := Trivial.
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Theorem ZeroNeOne : 0 ≠ 1 := Trivial.
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Theorem NeZeroPred' (a : Nat) : a ≠ 0 ⇒ ∃ b, b + 1 = a
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Theorem NeZeroPred' (a : Nat) : a ≠ 0 ⇒ ∃ b, b + 1 = a
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:= Induction (show 0 ≠ 0 ⇒ ∃ b, b + 1 = 0,
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:= Induction a
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assume H : 0 ≠ 0, FalseElim (∃ b, b + 1 = 0) H)
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(assume H : 0 ≠ 0, FalseElim (∃ b, b + 1 = 0) H)
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(λ (n : Nat) (iH : n ≠ 0 ⇒ ∃ b, b + 1 = n),
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(λ (n : Nat) (iH : n ≠ 0 ⇒ ∃ b, b + 1 = n),
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assume H : n + 1 ≠ 0,
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assume H : n + 1 ≠ 0,
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DisjCases (EM (n = 0))
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DisjCases (EM (n = 0))
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(λ Heq0 : n = 0, ExistsIntro 0 (calc 0 + 1 = n + 1 : { Symm Heq0 }))
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(λ Heq0 : n = 0, ExistsIntro 0 (calc 0 + 1 = n + 1 : { Symm Heq0 }))
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(λ Hne0 : n ≠ 0,
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(λ Hne0 : n ≠ 0,
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ExistsElim (MP iH Hne0)
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obtain (w : Nat) (Hw : w + 1 = n), from (MP iH Hne0),
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(λ (w : Nat) (Hw : w + 1 = n),
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ExistsIntro (w + 1) (calc w + 1 + 1 = n + 1 : { Hw }))).
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ExistsIntro (w + 1) (calc w + 1 + 1 = n + 1 : { Hw }))))
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a.
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Theorem NeZeroPred {a : Nat} (H : a ≠ 0) : ∃ b, b + 1 = a
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Theorem NeZeroPred {a : Nat} (H : a ≠ 0) : ∃ b, b + 1 = a
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:= MP (NeZeroPred' a) H.
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:= MP (NeZeroPred' a) H.
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@ -60,107 +58,106 @@ Theorem NeZeroPred {a : Nat} (H : a ≠ 0) : ∃ b, b + 1 = a
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Theorem Destruct {B : Bool} {a : Nat} (H1: a = 0 → B) (H2 : Π n, a = n + 1 → B) : B
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Theorem Destruct {B : Bool} {a : Nat} (H1: a = 0 → B) (H2 : Π n, a = n + 1 → B) : B
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:= DisjCases (EM (a = 0))
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:= DisjCases (EM (a = 0))
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(λ Heq0 : a = 0, H1 Heq0)
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(λ Heq0 : a = 0, H1 Heq0)
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(λ Hne0 : a ≠ 0, ExistsElim (NeZeroPred Hne0)
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(λ Hne0 : a ≠ 0, obtain (w : Nat) (Hw : w + 1 = a), from (NeZeroPred Hne0),
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(λ (w : Nat) (Hw : w + 1 = a), H2 w (Symm Hw))).
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H2 w (Symm Hw)).
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Theorem ZeroPlus (a : Nat) : 0 + a = a
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Theorem ZeroPlus (a : Nat) : 0 + a = a
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:= Induction (show 0 + 0 = 0, Trivial)
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:= Induction a
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(λ (n : Nat) (iH : 0 + n = n),
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(show 0 + 0 = 0, Trivial)
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calc 0 + (n + 1) = (0 + n) + 1 : PlusSucc 0 n
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(λ (n : Nat) (iH : 0 + n = n),
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... = n + 1 : { iH })
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calc 0 + (n + 1) = (0 + n) + 1 : PlusSucc 0 n
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a.
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... = n + 1 : { iH }).
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Theorem SuccPlus (a b : Nat) : (a + 1) + b = (a + b) + 1
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Theorem SuccPlus (a b : Nat) : (a + 1) + b = (a + b) + 1
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:= Induction (calc (a + 1) + 0 = a + 1 : PlusZero (a + 1)
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:= Induction b
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... = (a + 0) + 1 : { Symm (PlusZero a) })
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(calc (a + 1) + 0 = a + 1 : PlusZero (a + 1)
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(λ (n : Nat) (iH : (a + 1) + n = (a + n) + 1),
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... = (a + 0) + 1 : { Symm (PlusZero a) })
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calc (a + 1) + (n + 1) = ((a + 1) + n) + 1 : PlusSucc (a + 1) n
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(λ (n : Nat) (iH : (a + 1) + n = (a + n) + 1),
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... = ((a + n) + 1) + 1 : { iH }
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calc (a + 1) + (n + 1) = ((a + 1) + n) + 1 : PlusSucc (a + 1) n
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... = (a + (n + 1)) + 1 : { show (a + n) + 1 = a + (n + 1), Symm (PlusSucc a n) })
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... = ((a + n) + 1) + 1 : { iH }
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b.
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... = (a + (n + 1)) + 1 : { show (a + n) + 1 = a + (n + 1), Symm (PlusSucc a n) }).
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Theorem PlusComm (a b : Nat) : a + b = b + a
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Theorem PlusComm (a b : Nat) : a + b = b + a
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:= Induction (calc a + 0 = a : PlusZero a
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:= Induction b
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... = 0 + a : Symm (ZeroPlus a))
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(calc a + 0 = a : PlusZero a
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(λ (n : Nat) (iH : a + n = n + a),
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... = 0 + a : Symm (ZeroPlus a))
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calc a + (n + 1) = (a + n) + 1 : PlusSucc a n
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(λ (n : Nat) (iH : a + n = n + a),
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... = (n + a) + 1 : { iH }
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calc a + (n + 1) = (a + n) + 1 : PlusSucc a n
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... = (n + 1) + a : Symm (SuccPlus n a))
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... = (n + a) + 1 : { iH }
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b.
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... = (n + 1) + a : Symm (SuccPlus n a)).
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Theorem PlusAssoc (a b c : Nat) : a + (b + c) = (a + b) + c
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Theorem PlusAssoc (a b c : Nat) : a + (b + c) = (a + b) + c
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:= Induction (calc 0 + (b + c) = b + c : ZeroPlus (b + c)
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:= Induction a
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... = (0 + b) + c : { Symm (ZeroPlus b) })
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(calc 0 + (b + c) = b + c : ZeroPlus (b + c)
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(λ (n : Nat) (iH : n + (b + c) = (n + b) + c),
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... = (0 + b) + c : { Symm (ZeroPlus b) })
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calc (n + 1) + (b + c) = (n + (b + c)) + 1 : SuccPlus n (b + c)
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(λ (n : Nat) (iH : n + (b + c) = (n + b) + c),
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... = ((n + b) + c) + 1 : { iH }
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calc (n + 1) + (b + c) = (n + (b + c)) + 1 : SuccPlus n (b + c)
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... = ((n + b) + 1) + c : Symm (SuccPlus (n + b) c)
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... = ((n + b) + c) + 1 : { iH }
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... = ((n + 1) + b) + c : { show (n + b) + 1 = (n + 1) + b, Symm (SuccPlus n b) })
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... = ((n + b) + 1) + c : Symm (SuccPlus (n + b) c)
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a.
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... = ((n + 1) + b) + c : { show (n + b) + 1 = (n + 1) + b, Symm (SuccPlus n b) }).
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Theorem ZeroMul (a : Nat) : 0 * a = 0
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Theorem ZeroMul (a : Nat) : 0 * a = 0
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:= Induction (show 0 * 0 = 0, Trivial)
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:= Induction a
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(λ (n : Nat) (iH : 0 * n = 0),
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(show 0 * 0 = 0, Trivial)
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calc 0 * (n + 1) = (0 * n) + 0 : MulSucc 0 n
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(λ (n : Nat) (iH : 0 * n = 0),
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... = 0 + 0 : { iH }
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calc 0 * (n + 1) = (0 * n) + 0 : MulSucc 0 n
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... = 0 : Trivial)
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... = 0 + 0 : { iH }
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a.
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... = 0 : Trivial).
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Theorem SuccMul (a b : Nat) : (a + 1) * b = a * b + b
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Theorem SuccMul (a b : Nat) : (a + 1) * b = a * b + b
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:= Induction (calc (a + 1) * 0 = 0 : MulZero (a + 1)
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:= Induction b
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... = a * 0 : Symm (MulZero a)
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(calc (a + 1) * 0 = 0 : MulZero (a + 1)
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... = a * 0 + 0 : Symm (PlusZero (a * 0)))
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... = a * 0 : Symm (MulZero a)
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(λ (n : Nat) (iH : (a + 1) * n = a * n + n),
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... = a * 0 + 0 : Symm (PlusZero (a * 0)))
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calc (a + 1) * (n + 1) = (a + 1) * n + (a + 1) : MulSucc (a + 1) n
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(λ (n : Nat) (iH : (a + 1) * n = a * n + n),
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... = a * n + n + (a + 1) : { iH }
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calc (a + 1) * (n + 1) = (a + 1) * n + (a + 1) : MulSucc (a + 1) n
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... = a * n + n + a + 1 : PlusAssoc (a * n + n) a 1
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... = a * n + n + (a + 1) : { iH }
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... = a * n + (n + a) + 1 : { show a * n + n + a = a * n + (n + a), Symm (PlusAssoc (a * n) n a) }
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... = a * n + n + a + 1 : PlusAssoc (a * n + n) a 1
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... = a * n + (a + n) + 1 : { PlusComm n a }
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... = a * n + (n + a) + 1 : { show a * n + n + a = a * n + (n + a), Symm (PlusAssoc (a * n) n a) }
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... = a * n + a + n + 1 : { PlusAssoc (a * n) a n }
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... = a * n + (a + n) + 1 : { PlusComm n a }
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... = a * (n + 1) + n + 1 : { Symm (MulSucc a n) }
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... = a * n + a + n + 1 : { PlusAssoc (a * n) a n }
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... = a * (n + 1) + (n + 1) : Symm (PlusAssoc (a * (n + 1)) n 1))
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... = a * (n + 1) + n + 1 : { Symm (MulSucc a n) }
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b.
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... = a * (n + 1) + (n + 1) : Symm (PlusAssoc (a * (n + 1)) n 1)).
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Theorem OneMul (a : Nat) : 1 * a = a
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Theorem OneMul (a : Nat) : 1 * a = a
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:= Induction (show 1 * 0 = 0, Trivial)
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:= Induction a
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(λ (n : Nat) (iH : 1 * n = n),
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(show 1 * 0 = 0, Trivial)
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calc 1 * (n + 1) = 1 * n + 1 : MulSucc 1 n
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(λ (n : Nat) (iH : 1 * n = n),
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... = n + 1 : { iH })
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calc 1 * (n + 1) = 1 * n + 1 : MulSucc 1 n
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a.
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... = n + 1 : { iH }).
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Theorem MulOne (a : Nat) : a * 1 = a
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Theorem MulOne (a : Nat) : a * 1 = a
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:= Induction (show 0 * 1 = 0, Trivial)
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:= Induction a
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(λ (n : Nat) (iH : n * 1 = n),
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(show 0 * 1 = 0, Trivial)
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calc (n + 1) * 1 = n * 1 + 1 : SuccMul n 1
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(λ (n : Nat) (iH : n * 1 = n),
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... = n + 1 : { iH })
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calc (n + 1) * 1 = n * 1 + 1 : SuccMul n 1
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a.
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... = n + 1 : { iH }).
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Theorem MulComm (a b : Nat) : a * b = b * a
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Theorem MulComm (a b : Nat) : a * b = b * a
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:= Induction (calc a * 0 = 0 : MulZero a
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:= Induction b
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... = 0 * a : Symm (ZeroMul a))
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(calc a * 0 = 0 : MulZero a
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(λ (n : Nat) (iH : a * n = n * a),
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... = 0 * a : Symm (ZeroMul a))
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calc a * (n + 1) = a * n + a : MulSucc a n
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(λ (n : Nat) (iH : a * n = n * a),
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... = n * a + a : { iH }
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calc a * (n + 1) = a * n + a : MulSucc a n
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... = (n + 1) * a : Symm (SuccMul n a))
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... = n * a + a : { iH }
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b.
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... = (n + 1) * a : Symm (SuccMul n a)).
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Theorem Distribute (a b c : Nat) : a * (b + c) = a * b + a * c
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Theorem Distribute (a b c : Nat) : a * (b + c) = a * b + a * c
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:= Induction (calc 0 * (b + c) = 0 : ZeroMul (b + c)
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:= Induction a
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... = 0 + 0 : Trivial
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(calc 0 * (b + c) = 0 : ZeroMul (b + c)
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... = 0 * b + 0 : { Symm (ZeroMul b) }
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... = 0 + 0 : Trivial
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... = 0 * b + 0 * c : { Symm (ZeroMul c) })
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... = 0 * b + 0 : { Symm (ZeroMul b) }
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(λ (n : Nat) (iH : n * (b + c) = n * b + n * c),
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... = 0 * b + 0 * c : { Symm (ZeroMul c) })
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calc (n + 1) * (b + c) = n * (b + c) + (b + c) : SuccMul n (b + c)
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(λ (n : Nat) (iH : n * (b + c) = n * b + n * c),
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... = n * b + n * c + (b + c) : { iH }
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calc (n + 1) * (b + c) = n * (b + c) + (b + c) : SuccMul n (b + c)
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... = n * b + n * c + b + c : PlusAssoc (n * b + n * c) b c
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... = n * b + n * c + (b + c) : { iH }
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... = n * b + (n * c + b) + c : { Symm (PlusAssoc (n * b) (n * c) b) }
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... = n * b + n * c + b + c : PlusAssoc (n * b + n * c) b c
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... = n * b + (b + n * c) + c : { PlusComm (n * c) b }
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... = n * b + (n * c + b) + c : { Symm (PlusAssoc (n * b) (n * c) b) }
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... = n * b + b + n * c + c : { PlusAssoc (n * b) b (n * c) }
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... = n * b + (b + n * c) + c : { PlusComm (n * c) b }
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... = (n + 1) * b + n * c + c : { Symm (SuccMul n b) }
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... = n * b + b + n * c + c : { PlusAssoc (n * b) b (n * c) }
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... = (n + 1) * b + (n * c + c) : Symm (PlusAssoc ((n + 1) * b) (n * c) c)
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... = (n + 1) * b + n * c + c : { Symm (SuccMul n b) }
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... = (n + 1) * b + (n + 1) * c : { Symm (SuccMul n c) })
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... = (n + 1) * b + (n * c + c) : Symm (PlusAssoc ((n + 1) * b) (n * c) c)
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a.
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... = (n + 1) * b + (n + 1) * c : { Symm (SuccMul n c) }).
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Theorem Distribute2 (a b c : Nat) : (a + b) * c = a * c + b * c
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Theorem Distribute2 (a b c : Nat) : (a + b) * c = a * c + b * c
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:= calc (a + b) * c = c * (a + b) : MulComm (a + b) c
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:= calc (a + b) * c = c * (a + b) : MulComm (a + b) c
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@ -169,34 +166,34 @@ Theorem Distribute2 (a b c : Nat) : (a + b) * c = a * c + b * c
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... = a * c + b * c : { MulComm c b }.
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... = a * c + b * c : { MulComm c b }.
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Theorem MulAssoc (a b c : Nat) : a * (b * c) = a * b * c
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Theorem MulAssoc (a b c : Nat) : a * (b * c) = a * b * c
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:= Induction (calc 0 * (b * c) = 0 : ZeroMul (b * c)
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:= Induction a
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... = 0 * c : Symm (ZeroMul c)
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(calc 0 * (b * c) = 0 : ZeroMul (b * c)
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... = (0 * b) * c : { Symm (ZeroMul b) })
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... = 0 * c : Symm (ZeroMul c)
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(λ (n : Nat) (iH : n * (b * c) = n * b * c),
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... = (0 * b) * c : { Symm (ZeroMul b) })
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calc (n + 1) * (b * c) = n * (b * c) + (b * c) : SuccMul n (b * c)
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(λ (n : Nat) (iH : n * (b * c) = n * b * c),
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... = n * b * c + (b * c) : { iH }
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calc (n + 1) * (b * c) = n * (b * c) + (b * c) : SuccMul n (b * c)
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... = (n * b + b) * c : Symm (Distribute2 (n * b) b c)
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... = n * b * c + (b * c) : { iH }
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... = (n + 1) * b * c : { Symm (SuccMul n b) })
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... = (n * b + b) * c : Symm (Distribute2 (n * b) b c)
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a.
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... = (n + 1) * b * c : { Symm (SuccMul n b) }).
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Theorem PlusInj' (a b c : Nat) : a + b = a + c ⇒ b = c
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Theorem PlusInj' (a b c : Nat) : a + b = a + c ⇒ b = c
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:= Induction (assume H : 0 + b = 0 + c,
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:= Induction a
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calc b = 0 + b : Symm (ZeroPlus b)
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(assume H : 0 + b = 0 + c,
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... = 0 + c : H
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calc b = 0 + b : Symm (ZeroPlus b)
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... = c : ZeroPlus c)
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... = 0 + c : H
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(λ (n : Nat) (iH : n + b = n + c ⇒ b = c),
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... = c : ZeroPlus c)
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assume H : n + 1 + b = n + 1 + c,
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(λ (n : Nat) (iH : n + b = n + c ⇒ b = c),
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let L1 : n + b + 1 = n + c + 1
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assume H : n + 1 + b = n + 1 + c,
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:= (calc n + b + 1 = n + (b + 1) : Symm (PlusAssoc n b 1)
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let L1 : n + b + 1 = n + c + 1
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... = n + (1 + b) : { PlusComm b 1 }
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:= (calc n + b + 1 = n + (b + 1) : Symm (PlusAssoc n b 1)
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... = n + 1 + b : PlusAssoc n 1 b
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... = n + (1 + b) : { PlusComm b 1 }
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... = n + 1 + c : H
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... = n + 1 + b : PlusAssoc n 1 b
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... = n + (1 + c) : Symm (PlusAssoc n 1 c)
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... = n + 1 + c : H
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... = n + (c + 1) : { PlusComm 1 c }
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... = n + (1 + c) : Symm (PlusAssoc n 1 c)
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... = n + c + 1 : PlusAssoc n c 1),
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... = n + (c + 1) : { PlusComm 1 c }
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L2 : n + b = n + c := SuccInj L1
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... = n + c + 1 : PlusAssoc n c 1),
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in MP iH L2)
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L2 : n + b = n + c := SuccInj L1
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a.
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in MP iH L2).
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Theorem PlusInj {a b c : Nat} (H : a + b = a + c) : b = c
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Theorem PlusInj {a b c : Nat} (H : a + b = a + c) : b = c
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:= MP (PlusInj' a b c) H.
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:= MP (PlusInj' a b c) H.
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@ -222,36 +219,31 @@ Theorem LeRefl (a : Nat) : a ≤ a := LeIntro (PlusZero a).
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Theorem LeZero (a : Nat) : 0 ≤ a := LeIntro (ZeroPlus a).
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Theorem LeZero (a : Nat) : 0 ≤ a := LeIntro (ZeroPlus a).
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Theorem LeTrans {a b c : Nat} (H1 : a ≤ b) (H2 : b ≤ c) : a ≤ c
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Theorem LeTrans {a b c : Nat} (H1 : a ≤ b) (H2 : b ≤ c) : a ≤ c
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:= ExistsElim (LeElim H1)
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:= obtain (w1 : Nat) (Hw1 : a + w1 = b), from (LeElim H1),
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(λ (w1 : Nat) (Hw1 : a + w1 = b),
|
obtain (w2 : Nat) (Hw2 : b + w2 = c), from (LeElim H2),
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ExistsElim (LeElim H2)
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LeIntro (calc a + (w1 + w2) = a + w1 + w2 : PlusAssoc a w1 w2
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(λ (w2 : Nat) (Hw2 : b + w2 = c),
|
... = b + w2 : { Hw1 }
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LeIntro (calc a + (w1 + w2) = a + w1 + w2 : PlusAssoc a w1 w2
|
... = c : Hw2).
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... = b + w2 : { Hw1 }
|
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||||||
... = c : Hw2))).
|
|
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|
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Theorem LeInj {a b : Nat} (H : a ≤ b) (c : Nat) : a + c ≤ b + c
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Theorem LeInj {a b : Nat} (H : a ≤ b) (c : Nat) : a + c ≤ b + c
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||||||
:= ExistsElim (LeElim H)
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:= obtain (w : Nat) (Hw : a + w = b), from (LeElim H),
|
||||||
(λ (w : Nat) (Hw : a + w = b),
|
LeIntro (calc a + c + w = a + (c + w) : Symm (PlusAssoc a c w)
|
||||||
LeIntro (calc a + c + w = a + (c + w) : Symm (PlusAssoc a c w)
|
... = a + (w + c) : { PlusComm c w }
|
||||||
... = a + (w + c) : { PlusComm c w }
|
... = a + w + c : PlusAssoc a w c
|
||||||
... = a + w + c : PlusAssoc a w c
|
... = b + c : { Hw }).
|
||||||
... = b + c : { Hw })).
|
|
||||||
|
|
||||||
Theorem LeAntiSymm {a b : Nat} (H1 : a ≤ b) (H2 : b ≤ a) : a = b
|
Theorem LeAntiSymm {a b : Nat} (H1 : a ≤ b) (H2 : b ≤ a) : a = b
|
||||||
:= ExistsElim (LeElim H1)
|
:= obtain (w1 : Nat) (Hw1 : a + w1 = b), from (LeElim H1),
|
||||||
(λ (w1 : Nat) (Hw1 : a + w1 = b),
|
obtain (w2 : Nat) (Hw2 : b + w2 = a), from (LeElim H2),
|
||||||
ExistsElim (LeElim H2)
|
let L1 : w1 + w2 = 0
|
||||||
(λ (w2 : Nat) (Hw2 : b + w2 = a),
|
:= PlusInj (calc a + (w1 + w2) = a + w1 + w2 : { PlusAssoc a w1 w2 }
|
||||||
let L1 : w1 + w2 = 0
|
... = b + w2 : { Hw1 }
|
||||||
:= PlusInj (calc a + (w1 + w2) = a + w1 + w2 : { PlusAssoc a w1 w2 }
|
... = a : Hw2
|
||||||
... = b + w2 : { Hw1 }
|
... = a + 0 : Symm (PlusZero a)),
|
||||||
... = a : Hw2
|
L2 : w1 = 0 := PlusEq0 L1
|
||||||
... = a + 0 : Symm (PlusZero a)),
|
in calc a = a + 0 : Symm (PlusZero a)
|
||||||
L2 : w1 = 0 := PlusEq0 L1
|
... = a + w1 : { Symm L2 }
|
||||||
in calc a = a + 0 : Symm (PlusZero a)
|
... = b : Hw1.
|
||||||
... = a + w1 : { Symm L2 }
|
|
||||||
... = b : Hw1)).
|
|
||||||
|
|
||||||
SetOpaque ge true.
|
SetOpaque ge true.
|
||||||
SetOpaque lt true.
|
SetOpaque lt true.
|
||||||
|
|
|
||||||
|
|
@ -106,6 +106,8 @@ Theorem Absurd {a : Bool} (H1 : a) (H2 : ¬ a) : false
|
||||||
Theorem EqMP {a b : Bool} (H1 : a == b) (H2 : a) : b
|
Theorem EqMP {a b : Bool} (H1 : a == b) (H2 : a) : b
|
||||||
:= Subst H2 H1.
|
:= Subst H2 H1.
|
||||||
|
|
||||||
|
(* assume is a 'macro' that expands into a Discharge *)
|
||||||
|
|
||||||
Theorem ImpTrans {a b c : Bool} (H1 : a ⇒ b) (H2 : b ⇒ c) : a ⇒ c
|
Theorem ImpTrans {a b c : Bool} (H1 : a ⇒ b) (H2 : b ⇒ c) : a ⇒ c
|
||||||
:= assume Ha, MP H2 (MP H1 Ha).
|
:= assume Ha, MP H2 (MP H1 Ha).
|
||||||
|
|
||||||
|
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|
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Reference in a new issue