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feat(builtin/Nat): cleanup and add PlusAssoc

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2014-01-01 17:19:12 -08:00
parent aa009b6b05
commit d72b165db4
1 changed files with 30 additions and 17 deletions
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47
src/builtin/Nat.lean
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@ -31,25 +31,24 @@ Notation 55 | _ | : id.
Axiom PlusZero (a : Nat) : a + 0 = a.
Axiom PlusSucc (a b : Nat) : a + (b + 1) = (a + b) + 1.
Axiom Induction {P : Nat -> Bool} (Hb : P 0) (Hi : Pi (n : Nat) (H : P n), P (n + 1)) (a : Nat) : P a.
Axiom SuccInj {a b : Nat} (H : a + 1 = b + 1) : a = b
Axiom Induction {P : Nat → Bool} (Hb : P 0) (Hi : Π (n : Nat) (H : P n), P (n + 1)) (a : Nat) : P a.
Theorem ZeroPlus (a : Nat) : 0 + a = a
:= Induction (show 0 + 0 = 0, Trivial)
(fun (n : Nat) (H : 0 + n = n),
(show 0 + (n + 1) = n + 1,
let L1 : 0 + (n + 1) = (0 + n) + 1 := PlusSucc 0 n
in Subst L1 H))
(λ (n : Nat) (Hi : 0 + n = n),
let L1 : 0 + (n + 1) = (0 + n) + 1 := PlusSucc 0 n
in show 0 + (n + 1) = n + 1, Subst L1 Hi)
a.
Theorem SuccPlus (a b : Nat) : (a + 1) + b = (a + b) + 1
:= Induction (show (a + 1) + 0 = (a + 0) + 1,
(Subst (PlusZero (a + 1)) (Symm (PlusZero a))))
(fun (n : Nat) (H : (a + 1) + n = (a + n) + 1),
(show (a + 1) + (n + 1) = (a + (n + 1)) + 1,
let L1 : (a + 1) + (n + 1) = ((a + 1) + n) + 1 := PlusSucc (a + 1) n,
L2 : (a + 1) + (n + 1) = ((a + n) + 1) + 1 := Subst L1 H,
L3 : (a + n) + 1 = a + (n + 1) := Symm (PlusSucc a n)
in Subst L2 L3))
(λ (n : Nat) (Hi : (a + 1) + n = (a + n) + 1),
let L1 : (a + 1) + (n + 1) = ((a + 1) + n) + 1 := PlusSucc (a + 1) n,
L2 : (a + 1) + (n + 1) = ((a + n) + 1) + 1 := Subst L1 Hi,
L3 : (a + n) + 1 = a + (n + 1) := Symm (PlusSucc a n)
in show (a + 1) + (n + 1) = (a + (n + 1)) + 1, Subst L2 L3)
b.
Theorem PlusComm (a b : Nat) : a + b = b + a
@ -57,14 +56,28 @@ Theorem PlusComm (a b : Nat) : a + b = b + a
let L1 : a + 0 = a := PlusZero a,
L2 : a = 0 + a := Symm (ZeroPlus a)
in Trans L1 L2)
(fun (n : Nat) (H : a + n = n + a),
(show a + (n + 1) = (n + 1) + a,
let L1 : a + (n + 1) = (a + n) + 1 := PlusSucc a n,
L2 : a + (n + 1) = (n + a) + 1 := Subst L1 H,
L3 : (n + a) + 1 = (n + 1) + a := Symm (SuccPlus n a)
in Trans L2 L3))
(λ (n : Nat) (Hi : a + n = n + a),
let L1 : a + (n + 1) = (a + n) + 1 := PlusSucc a n,
L2 : a + (n + 1) = (n + a) + 1 := Subst L1 Hi,
L3 : (n + a) + 1 = (n + 1) + a := Symm (SuccPlus n a)
in show a + (n + 1) = (n + 1) + a, Trans L2 L3)
b.
Theorem PlusAssoc (a b c : Nat) : a + (b + c) = (a + b) + c
:= Induction (show 0 + (b + c) = (0 + b) + c,
Subst (ZeroPlus (b + c)) (Symm (ZeroPlus b)))
(λ (n : Nat) (Hi : n + (b + c) = (n + b) + c),
let L1 : (n + 1) + (b + c) = (n + (b + c)) + 1 := SuccPlus n (b + c),
L2 : (n + 1) + (b + c) = ((n + b) + c) + 1 := Subst L1 Hi,
L3 : ((n + b) + 1) + c = ((n + b) + c) + 1 := SuccPlus (n + b) c,
L4 : (n + b) + 1 = (n + 1) + b := Symm (SuccPlus n b),
L5 : ((n + 1) + b) + c = ((n + b) + c) + 1 := Subst L3 L4,
L6 : ((n + b) + c) + 1 = ((n + 1) + b) + c := Symm L5
in show (n + 1) + (b + c) = ((n + 1) + b) + c, Trans L2 L6)
a.
Theorem ZeroNeOne : 0 ≠ 1 := Trivial.
SetOpaque ge true.
SetOpaque lt true.
SetOpaque gt true.