165 lines
7.3 KiB
Text
165 lines
7.3 KiB
Text
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(*
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Nat library full of "holes".
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We provide only the proof skeletons, and let Lean infer the rest.
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*)
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Import kernel.
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Variable Nat : Type.
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Alias ℕ : Nat.
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Namespace Nat.
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Builtin numeral.
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Builtin add : Nat → Nat → Nat.
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Infixl 65 + : add.
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Builtin mul : Nat → Nat → Nat.
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Infixl 70 * : mul.
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Builtin le : Nat → Nat → Bool.
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Infix 50 <= : le.
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Infix 50 ≤ : le.
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Definition ge (a b : Nat) := b ≤ a.
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Infix 50 >= : ge.
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Infix 50 ≥ : ge.
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Definition lt (a b : Nat) := ¬ (a ≥ b).
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Infix 50 < : lt.
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Definition gt (a b : Nat) := ¬ (a ≤ b).
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Infix 50 > : gt.
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Definition id (a : Nat) := a.
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Notation 55 | _ | : id.
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Axiom SuccInj {a b : Nat} (H : a + 1 = b + 1) : a = b
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Axiom PlusZero (a : Nat) : a + 0 = a.
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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 MulSucc (a b : Nat) : a * (b + 1) = a * b + a.
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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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Theorem ZeroNeOne : 0 ≠ 1 := Trivial.
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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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(λ (n : Nat) (iH : 0 + n = n),
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calc 0 + (n + 1) = (0 + n) + 1 : PlusSucc _ _
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... = n + 1 : { iH })
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a.
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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 _
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... = (a + 0) + 1 : { Symm (PlusZero _) })
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(λ (n : Nat) (iH : (a + 1) + n = (a + n) + 1),
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calc (a + 1) + (n + 1) = ((a + 1) + n) + 1 : PlusSucc _ _
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... = ((a + n) + 1) + 1 : { iH }
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... = (a + (n + 1)) + 1 : { Symm (PlusSucc _ _) })
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b.
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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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... = 0 + a : Symm (ZeroPlus a))
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(λ (n : Nat) (iH : a + n = n + a),
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calc a + (n + 1) = (a + n) + 1 : PlusSucc _ _
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... = (n + a) + 1 : { iH }
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... = (n + 1) + a : Symm (SuccPlus _ _))
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b.
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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 _
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... = (0 + b) + c : { Symm (ZeroPlus _) })
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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 : SuccPlus _ _
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... = ((n + b) + c) + 1 : { iH }
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... = ((n + b) + 1) + c : Symm (SuccPlus _ _)
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... = ((n + 1) + b) + c : { Symm (SuccPlus _ _) })
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a.
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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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(λ (n : Nat) (iH : 0 * n = 0),
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calc 0 * (n + 1) = (0 * n) + 0 : MulSucc _ _
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... = 0 + 0 : { iH }
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... = 0 : Trivial)
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a.
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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 _
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... = a * 0 : Symm (MulZero _)
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... = a * 0 + 0 : Symm (PlusZero _))
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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) : MulSucc _ _
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... = a * n + n + (a + 1) : { iH }
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... = a * n + n + a + 1 : PlusAssoc _ _ _
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... = a * n + (n + a) + 1 : { Symm (PlusAssoc _ _ _) }
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... = a * n + (a + n) + 1 : { PlusComm _ _ }
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... = a * n + a + n + 1 : { PlusAssoc _ _ _ }
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... = a * (n + 1) + n + 1 : { Symm (MulSucc _ _) }
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... = a * (n + 1) + (n + 1) : Symm (PlusAssoc _ _ _))
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b.
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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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(λ (n : Nat) (iH : 1 * n = n),
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calc 1 * (n + 1) = 1 * n + 1 : MulSucc _ _
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... = n + 1 : { iH })
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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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(λ (n : Nat) (iH : n * 1 = n),
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calc (n + 1) * 1 = n * 1 + 1 : SuccMul _ _
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... = n + 1 : { iH })
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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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... = 0 * a : Symm (ZeroMul a))
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(λ (n : Nat) (iH : a * n = n * a),
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calc a * (n + 1) = a * n + a : MulSucc _ _
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... = n * a + a : { iH }
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... = (n + 1) * a : Symm (SuccMul _ _))
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b.
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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 _
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... = 0 + 0 : Trivial
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... = 0 * b + 0 : { Symm (ZeroMul _) }
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... = 0 * b + 0 * c : { Symm (ZeroMul _) })
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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) : SuccMul _ _
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... = n * b + n * c + (b + c) : { iH }
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... = n * b + n * c + b + c : PlusAssoc _ _ _
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... = n * b + (n * c + b) + c : { Symm (PlusAssoc _ _ _) }
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... = n * b + (b + n * c) + c : { PlusComm _ _ }
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... = n * b + b + n * c + c : { PlusAssoc _ _ _ }
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... = (n + 1) * b + n * c + c : { Symm (SuccMul _ _) }
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... = (n + 1) * b + (n * c + c) : Symm (PlusAssoc _ _ _)
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... = (n + 1) * b + (n + 1) * c : { Symm (SuccMul _ _) })
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a.
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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 _ _
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... = c * a + c * b : Distribute _ _ _
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... = a * c + c * b : { MulComm _ _ }
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... = a * c + b * c : { MulComm _ _}.
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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 _
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... = 0 * c : Symm (ZeroMul _)
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... = (0 * b) * c : { Symm (ZeroMul _) })
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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) + (b * c) : SuccMul _ _
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... = n * b * c + (b * c) : { iH }
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... = (n * b + b) * c : Symm (Distribute2 _ _ _)
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... = (n + 1) * b * c : { Symm (SuccMul _ _) })
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a.
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SetOpaque ge true.
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SetOpaque lt true.
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SetOpaque gt true.
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SetOpaque id true.
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EndNamespace.
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