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feat(builtin/kernel): add Not.*Elim theorems

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2014-01-01 11:35:21 -08:00
parent a8bc9fb4e0
commit c423198a69
4 changed files with 53 additions and 0 deletions
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18
src/builtin/kernel.lean
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@ -306,24 +306,36 @@ Theorem NotAnd (a b : Bool) : (¬ (a ∧ b)) == (¬ a ∨ ¬ b)
(Case (λ y, (¬ (false ∧ y)) == (¬ false ∨ ¬ y)) Trivial Trivial b)
a
Theorem NotAndElim {a b : Bool} (H : ¬ (a ∧ b)) : ¬ a ∨ ¬ b
:= EqMP (NotAnd a b) H.
Theorem NotOr (a b : Bool) : (¬ (a ∨ b)) == (¬ a ∧ ¬ b)
:= Case (λ x, (¬ (x ∨ b)) == (¬ x ∧ ¬ b))
(Case (λ y, (¬ (true ∨ y)) == (¬ true ∧ ¬ y)) Trivial Trivial b)
(Case (λ y, (¬ (false ∨ y)) == (¬ false ∧ ¬ y)) Trivial Trivial b)
a
Theorem NotOrElim {a b : Bool} (H : ¬ (a ∨ b)) : ¬ a ∧ ¬ b
:= EqMP (NotOr a b) H.
Theorem NotEq (a b : Bool) : (¬ (a == b)) == ((¬ a) == b)
:= Case (λ x, (¬ (x == b)) == ((¬ x) == b))
(Case (λ y, (¬ (true == y)) == ((¬ true) == y)) Trivial Trivial b)
(Case (λ y, (¬ (false == y)) == ((¬ false) == y)) Trivial Trivial b)
a
Theorem NotEqElim {a b : Bool} (H : ¬ (a == b)) : (¬ a) == b
:= EqMP (NotEq a b) H.
Theorem NotImp (a b : Bool) : (¬ (a ⇒ b)) == (a ∧ ¬ b)
:= Case (λ x, (¬ (x ⇒ b)) == (x ∧ ¬ b))
(Case (λ y, (¬ (true ⇒ y)) == (true ∧ ¬ y)) Trivial Trivial b)
(Case (λ y, (¬ (false ⇒ y)) == (false ∧ ¬ y)) Trivial Trivial b)
a
Theorem NotImpElim {a b : Bool} (H : ¬ (a ⇒ b)) : a ∧ ¬ b
:= EqMP (NotImp a b) H.
Theorem NotCongr {a b : Bool} (H : a == b) : (¬ a) == (¬ b)
:= Congr2 not H.
@ -339,11 +351,17 @@ Theorem NotForall (A : (Type U)) (P : A → Bool) : (¬ (∀ x : A, P x)) == (
NotCongr (ForallEqIntro (λ x : A, (Symm (DoubleNeg (P x)))))
in Trans L2 L1.
Theorem NotForallElim {A : (Type U)} {P : A → Bool} (H : ¬ (∀ x : A, P x)) : ∃ x : A, ¬ P x
:= EqMP (NotForall A P) H.
Theorem NotExists (A : (Type U)) (P : A → Bool) : (¬ ∃ x : A, P x) == (∀ x : A, ¬ P x)
:= let L1 : (¬ ∃ x : A, P x) == (¬ ¬ ∀ x : A, ¬ P x) := Refl (¬ ∃ x : A, P x),
L2 : (¬ ¬ ∀ x : A, ¬ P x) == (∀ x : A, ¬ P x) := DoubleNeg (∀ x : A, ¬ P x)
in Trans L1 L2.
Theorem NotExistsElim {A : (Type U)} {P : A → Bool} (H : ¬ ∃ x : A, P x) : ∀ x : A, ¬ P x
:= EqMP (NotExists A P) H.
Theorem UnfoldExists1 {A : TypeU} {P : A → Bool} (a : A) (H : ∃ x : A, P x) : P a ∨ (∃ x : A, x ≠ a ∧ P x)
:= ExistsElim H
(λ (w : A) (H1 : P w),

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src/builtin/obj/kernel.olean
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27
tests/lean/exists8.lean Normal file
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@ -0,0 +1,27 @@
Import int.
Variable P : Int -> Int -> Bool
Theorem T1 (R1 : not (exists x y, P x y)) : forall x y, not (P x y) :=
ForallIntro (fun a,
ForallIntro (fun b,
ForallElim (NotExistsElim (ForallElim (NotExistsElim R1) a)) b))
Axiom Ax : forall x, exists y, P x y
Theorem T2 : exists x y, P x y :=
Refute (fun R : not (exists x y, P x y),
let L1 : forall x y, not (P x y) := ForallIntro (fun a,
ForallIntro (fun b,
ForallElim (NotExistsElim (ForallElim (NotExistsElim R) a)) b)),
L2 : exists y, P 0 y := ForallElim Ax 0
in ExistsElim L2 (fun (w : Int) (H : P 0 w),
Absurd H (ForallElim (ForallElim L1 0) w))).
Theorem T3 (A : (Type U)) (P : A -> A -> Bool) (a : A) (H1 : forall x, exists y, P x y) : exists x y, P x y :=
Refute (fun R : not (exists x y, P x y),
let L1 : forall x y, not (P x y) := ForallIntro (fun a,
ForallIntro (fun b,
ForallElim (NotExistsElim (ForallElim (NotExistsElim R) a)) b)),
L2 : exists y, P a y := ForallElim H1 a
in ExistsElim L2 (fun (w : A) (H : P a w),
Absurd H (ForallElim (ForallElim L1 a) w))).

8
tests/lean/exists8.lean.expected.out Normal file
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@ -0,0 +1,8 @@
Set: pp::colors
Set: pp::unicode
Imported 'int'
Assumed: P
Proved: T1
Assumed: Ax
Proved: T2
Proved: T3