refactor(library/logic): rename theorems
This commit is contained in:
Jeremy Avigad
parent
3e9a484851
commit
da719e6ee4
10 changed files with 99 additions and 82 deletions
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@ -48,8 +48,8 @@ let u : {x | (∃y, P y) → P x} :=
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has_property u Hex
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theorem epsilon_spec {A : Type} {P : A → Prop} (Hex : ∃y, P y) :
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P (@epsilon A (exists_imp_nonempty Hex) P) :=
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epsilon_spec_aux (exists_imp_nonempty Hex) P Hex
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P (@epsilon A (nonempty_of_exists Hex) P) :=
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epsilon_spec_aux (nonempty_of_exists Hex) P Hex
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theorem epsilon_singleton {A : Type} (a : A) : @epsilon A (nonempty.intro a) (λx, x = a) = a :=
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epsilon_spec (exists_intro a (eq.refl a))
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@ -60,7 +60,7 @@ epsilon_spec (exists_intro a (eq.refl a))
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theorem axiom_of_choice {A : Type} {B : A → Type} {R : Πx, B x → Prop} (H : ∀x, ∃y, R x y) :
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∃f, ∀x, R x (f x) :=
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let f := λx, @epsilon _ (exists_imp_nonempty (H x)) (λy, R x y),
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let f := λx, @epsilon _ (nonempty_of_exists (H x)) (λy, R x y),
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H := take x, epsilon_spec (H x)
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in exists_intro f H
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@ -1,12 +1,16 @@
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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Leonardo de Moura
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/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: cast.lean
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Author: Leonardo de Moura
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Casts and heterogeneous equality. See also init.datatypes and init.logic.
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-/
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import logic.eq logic.quantifiers
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open eq.ops
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-- cast.lean
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-- =========
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section
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universe variable u
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variables {A B : Type.{u}}
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@ -30,20 +34,24 @@ namespace heq
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definition type_eq (H : a == b) : A = B :=
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heq.rec_on H (eq.refl A)
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theorem drec_on {C : Π {B : Type} (b : B), a == b → Type} (H₁ : a == b) (H₂ : C a (refl a)) : C b H₁ :=
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theorem drec_on {C : Π {B : Type} (b : B), a == b → Type} (H₁ : a == b) (H₂ : C a (refl a)) :
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C b H₁ :=
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rec (λ H₁ : a == a, show C a H₁, from H₂) H₁ H₁
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theorem to_cast_eq (H : a == b) : cast (type_eq H) a = b :=
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drec_on H !cast_eq
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end heq
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theorem eq_rec_heq {A : Type} {P : A → Type} {a a' : A} (H : a = a') (p : P a) : eq.rec_on H p == p :=
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theorem eq_rec_heq {A : Type} {P : A → Type} {a a' : A} (H : a = a') (p : P a) :
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eq.rec_on H p == p :=
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eq.drec_on H !heq.refl
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section
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universe variables u v
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variables {A A' B C : Type.{u}} {P P' : A → Type.{v}} {a a' : A} {b : B}
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theorem hcongr_fun {f : Π x, P x} {f' : Π x, P' x} (a : A) (H₁ : f == f') (H₂ : P = P') : f a == f' a :=
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theorem hcongr_fun {f : Π x, P x} {f' : Π x, P' x} (a : A) (H₁ : f == f') (H₂ : P = P') :
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f a == f' a :=
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have aux : ∀ (f : Π x, P x) (f' : Π x, P x), f == f' → f a == f' a, from
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take f f' H, heq.to_eq H ▸ heq.refl (f a),
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(H₂ ▸ aux) f f' H₁
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@ -105,10 +113,11 @@ section
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take H : P = P, heq.refl (eq.rec_on H f a),
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(H ▸ aux) H
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theorem rec_on_pull (H : P = P') (f : Π x, P x) (a : A) : eq.rec_on H f a = eq.rec_on (congr_fun H a) (f a) :=
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theorem rec_on_pull (H : P = P') (f : Π x, P x) (a : A) :
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eq.rec_on H f a = eq.rec_on (congr_fun H a) (f a) :=
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heq.to_eq (calc
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eq.rec_on H f a == f a : rec_on_app H f a
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... == eq.rec_on (congr_fun H a) (f a) : heq.symm (eq_rec_heq (congr_fun H a) (f a)))
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eq.rec_on H f a == f a : rec_on_app H f a
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... == eq.rec_on (congr_fun H a) (f a) : heq.symm (eq_rec_heq (congr_fun H a) (f a)))
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theorem cast_app (H : P = P') (f : Π x, P x) (a : A) : cast (pi_eq H) f a == f a :=
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have H₁ : ∀ (H : (Π x, P x) = (Π x, P x)), cast H f a == f a, from
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@ -116,7 +125,6 @@ section
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have H₂ : ∀ (H : (Π x, P x) = (Π x, P' x)), cast H f a == f a, from
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H ▸ H₁,
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H₂ (pi_eq H)
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end
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section
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@ -144,7 +152,8 @@ section
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(Hd : cast (dcongr_arg3 D Ha Hb Hc) d = d') : f a b c d = f a' b' c' d' :=
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heq.to_eq (hcongr_arg4 f Ha (eq_rec_to_heq Hb) (eq_rec_to_heq Hc) (eq_rec_to_heq Hd))
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--mixed versions (we want them for example if C a' b' is a subsingleton, like a proposition. Then proving eq is easier than proving heq)
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-- mixed versions (we want them for example if C a' b' is a subsingleton, like a proposition.
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-- Then proving eq is easier than proving heq)
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theorem hdcongr_arg3 (f : Πa b, C a b → F) (Ha : a = a') (Hb : b == b')
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(Hc : cast (heq.to_eq (hcongr_arg2 C Ha Hb)) c = c')
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: f a b c = f a' b' c' :=
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@ -5,7 +5,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Module: logic.connectives
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Authors: Jeremy Avigad, Leonardo de Moura
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The propositional connectives. See also init.datatypes.
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The propositional connectives. See also init.datatypes and init.logic.
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-/
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variables {a b c d : Prop}
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@ -1,6 +1,10 @@
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--- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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--- Released under Apache 2.0 license as described in the file LICENSE.
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--- Author: Jeremy Avigad
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/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: logic.default
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Author: Jeremy Avigad
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-/
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import logic.eq logic.connectives logic.cast logic.subsingleton
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import logic.quantifiers logic.instances logic.identities
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@ -1,16 +1,13 @@
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/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: logic.eq
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Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
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Additional declarations/theorems about equality
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Additional declarations/theorems about equality. See also init.datatypes and init.logic.
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-/
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-- logic.eq
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-- ========
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-- Equality.
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open eq.ops
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namespace eq
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@ -1,36 +1,37 @@
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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Authors: Jeremy Avigad, Leonardo de Moura
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/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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-- logic.identities
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-- ============================
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Module: logic.identities
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Authors: Jeremy Avigad, Leonardo de Moura
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-- Useful logical identities. In the absence of propositional extensionality, some of the
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-- calculations use the type class support provided by logic.instances
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Useful logical identities. Since we are not using propositional extensionality, some of the
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calculations use the type class support provided by logic.instances.
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-/
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import logic.connectives logic.instances logic.quantifiers logic.cast
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open relation decidable relation.iff_ops
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theorem or_right_comm (a b c : Prop) : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b :=
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theorem or.right_comm (a b c : Prop) : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b :=
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calc
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(a ∨ b) ∨ c ↔ a ∨ (b ∨ c) : or.assoc
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... ↔ a ∨ (c ∨ b) : {or.comm}
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... ↔ (a ∨ c) ∨ b : iff.symm or.assoc
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theorem or_left_comm (a b c : Prop) : a ∨ (b ∨ c)↔ b ∨ (a ∨ c) :=
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theorem or.left_comm (a b c : Prop) : a ∨ (b ∨ c)↔ b ∨ (a ∨ c) :=
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calc
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a ∨ (b ∨ c) ↔ (a ∨ b) ∨ c : iff.symm or.assoc
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... ↔ (b ∨ a) ∨ c : {or.comm}
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... ↔ b ∨ (a ∨ c) : or.assoc
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theorem and_right_comm (a b c : Prop) : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ b :=
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theorem and.right_comm (a b c : Prop) : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ b :=
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calc
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(a ∧ b) ∧ c ↔ a ∧ (b ∧ c) : and.assoc
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... ↔ a ∧ (c ∧ b) : {and.comm}
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... ↔ (a ∧ c) ∧ b : iff.symm and.assoc
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theorem and_left_comm (a b c : Prop) : a ∧ (b ∧ c)↔ b ∧ (a ∧ c) :=
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theorem and.left_comm (a b c : Prop) : a ∧ (b ∧ c)↔ b ∧ (a ∧ c) :=
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calc
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a ∧ (b ∧ c) ↔ (a ∧ b) ∧ c : iff.symm and.assoc
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... ↔ (b ∧ a) ∧ c : {and.comm}
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@ -46,14 +47,14 @@ iff.intro
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theorem not_not_elim {a : Prop} [D : decidable a] (H : ¬¬a) : a :=
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iff.mp not_not_iff H
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theorem not_true_iff_false : (¬true) ↔ false :=
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theorem not_true_iff_false : ¬true ↔ false :=
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iff.intro (assume H, H trivial) false.elim
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theorem not_false_iff_true : (¬false) ↔ true :=
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theorem not_false_iff_true : ¬false ↔ true :=
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iff.intro (assume H, trivial) (assume H H', H')
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theorem not_or_iff_not_and_not {a b : Prop} [Da : decidable a] [Db : decidable b] :
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(¬(a ∨ b)) ↔ (¬a ∧ ¬b) :=
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¬(a ∨ b) ↔ ¬a ∧ ¬b :=
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iff.intro
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(assume H, or.elim (em a)
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(assume Ha, absurd (or.inl Ha) H)
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@ -65,7 +66,8 @@ iff.intro
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(assume Ha, absurd Ha (and.elim_left H))
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(assume Hb, absurd Hb (and.elim_right H)))
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theorem not_and {a b : Prop} [Da : decidable a] [Db : decidable b] : (¬(a ∧ b)) ↔ (¬a ∨ ¬b) :=
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theorem not_and_iff_not_or_not {a b : Prop} [Da : decidable a] [Db : decidable b] :
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¬(a ∧ b) ↔ ¬a ∨ ¬b :=
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iff.intro
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(assume H, or.elim (em a)
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(assume Ha, or.elim (em b)
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@ -77,7 +79,7 @@ iff.intro
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(assume Hna, absurd (and.elim_left N) Hna)
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(assume Hnb, absurd (and.elim_right N) Hnb))
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theorem imp_or {a b : Prop} [Da : decidable a] : (a → b) ↔ (¬a ∨ b) :=
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theorem imp_iff_not_or {a b : Prop} [Da : decidable a] : (a → b) ↔ ¬a ∨ b :=
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iff.intro
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(assume H : a → b, (or.elim (em a)
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(assume Ha : a, or.inr (H Ha))
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@ -85,27 +87,29 @@ iff.intro
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(assume (H : ¬a ∨ b) (Ha : a),
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or_resolve_right H (not_not_iff⁻¹ ▸ Ha))
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theorem not_implies {a b : Prop} [Da : decidable a] [Db : decidable b] : (¬(a → b)) ↔ (a ∧ ¬b) :=
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calc (¬(a → b)) ↔ (¬(¬a ∨ b)) : {imp_or}
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... ↔ (¬¬a ∧ ¬b) : not_or_iff_not_and_not
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... ↔ (a ∧ ¬b) : {not_not_iff}
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theorem not_implies_iff_and_not {a b : Prop} [Da : decidable a] [Db : decidable b] :
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¬(a → b) ↔ a ∧ ¬b :=
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calc
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¬(a → b) ↔ ¬(¬a ∨ b) : {imp_iff_not_or}
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... ↔ ¬¬a ∧ ¬b : not_or_iff_not_and_not
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... ↔ a ∧ ¬b : {not_not_iff}
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theorem peirce {a b : Prop} [D : decidable a] : ((a → b) → a) → a :=
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assume H, by_contradiction (assume Hna : ¬a,
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have Hnna : ¬¬a, from not_not_of_not_implies (mt H Hna),
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absurd (not_not_elim Hnna) Hna)
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theorem not_exists_forall {A : Type} {P : A → Prop} [D : ∀x, decidable (P x)]
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theorem forall_not_of_not_exists {A : Type} {P : A → Prop} [D : ∀x, decidable (P x)]
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(H : ¬∃x, P x) : ∀x, ¬P x :=
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take x, or.elim (em (P x))
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(assume Hp : P x, absurd (exists_intro x Hp) H)
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(assume Hn : ¬P x, Hn)
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theorem not_forall_exists {A : Type} {P : A → Prop} [D : ∀x, decidable (P x)]
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theorem exists_not_of_not_forall {A : Type} {P : A → Prop} [D : ∀x, decidable (P x)]
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[D' : decidable (∃x, ¬P x)] (H : ¬∀x, P x) :
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∃x, ¬P x :=
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@by_contradiction _ D' (assume H1 : ¬∃x, ¬P x,
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have H2 : ∀x, ¬¬P x, from @not_exists_forall _ _ (take x, not.decidable (D x)) H1,
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have H2 : ∀x, ¬¬P x, from @forall_not_of_not_exists _ _ (take x, not.decidable (D x)) H1,
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have H3 : ∀x, P x, from take x, @not_not_elim _ (D x) (H2 x),
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absurd H3 H)
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@ -119,32 +123,32 @@ iff.intro
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(assume H1 : a, absurd H1 H)
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(assume H2 : false, false.elim H2)
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theorem a_neq_a {A : Type} (a : A) : (a ≠ a) ↔ false :=
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theorem ne_self_iff_false {A : Type} (a : A) : (a ≠ a) ↔ false :=
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iff.intro
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(assume H, false.of_ne H)
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(assume H, false.elim H)
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theorem eq_id {A : Type} (a : A) : (a = a) ↔ true :=
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theorem eq_self_iff_true {A : Type} (a : A) : (a = a) ↔ true :=
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iff_true_intro rfl
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theorem heq_id {A : Type} (a : A) : (a == a) ↔ true :=
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theorem heq_self_iff_true {A : Type} (a : A) : (a == a) ↔ true :=
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iff_true_intro (heq.refl a)
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theorem a_iff_not_a (a : Prop) : (a ↔ ¬a) ↔ false :=
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theorem iff_not_self (a : Prop) : (a ↔ ¬a) ↔ false :=
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iff.intro
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(assume H,
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have H' : ¬a, from assume Ha, (H ▸ Ha) Ha,
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H' (H⁻¹ ▸ H'))
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(assume H, false.elim H)
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theorem true_eq_false : (true ↔ false) ↔ false :=
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not_true_iff_false ▸ (a_iff_not_a true)
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theorem true_iff_false : (true ↔ false) ↔ false :=
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not_true_iff_false ▸ (iff_not_self true)
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theorem false_eq_true : (false ↔ true) ↔ false :=
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not_false_iff_true ▸ (a_iff_not_a false)
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theorem false_iff_true : (false ↔ true) ↔ false :=
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not_false_iff_true ▸ (iff_not_self false)
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theorem a_eq_true (a : Prop) : (a ↔ true) ↔ a :=
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theorem iff_true_iff (a : Prop) : (a ↔ true) ↔ a :=
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iff.intro (assume H, of_iff_true H) (assume H, iff_true_intro H)
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theorem a_eq_false (a : Prop) : (a ↔ false) ↔ ¬a :=
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theorem iff_false_iff_not (a : Prop) : (a ↔ false) ↔ ¬a :=
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iff.intro (assume H, not_of_iff_false H) (assume H, iff_false_intro H)
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@ -20,7 +20,6 @@ relation.is_equivalence.mk (@eq.refl T) (@eq.symm T) (@eq.trans T)
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theorem is_equivalence_iff [instance] : relation.is_equivalence iff :=
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relation.is_equivalence.mk @iff.refl @iff.symm @iff.trans
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/- congruences for logic operations -/
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theorem is_congruence_not : is_congruence iff iff not :=
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@ -62,14 +61,12 @@ is_congruence2.mk
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(assume H3 : a1 ↔ a2, iff.trans (iff.symm H1) (iff.trans H3 H2))
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(assume H3 : b1 ↔ b2, iff.trans H1 (iff.trans H3 (iff.symm H2))))
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-- theorem is_congruence_const_iff [instance] := is_congruence.const iff iff.refl
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definition is_congruence_not_compose [instance] := is_congruence.compose is_congruence_not
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definition is_congruence_and_compose [instance] := is_congruence.compose21 is_congruence_and
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definition is_congruence_or_compose [instance] := is_congruence.compose21 is_congruence_or
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definition is_congruence_implies_compose [instance] := is_congruence.compose21 is_congruence_imp
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definition is_congruence_iff_compose [instance] := is_congruence.compose21 is_congruence_iff
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/- a general substitution operation with respect to an arbitrary congruence -/
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namespace general_subst
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@ -77,13 +74,11 @@ namespace general_subst
|
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{a b : T} (H : R a b) (H1 : P a) : P b := iff.elim_left (is_congruence.app C H) H1
|
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end general_subst
|
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/- iff can be coerced to implication -/
|
||||
|
||||
definition mp_like_iff [instance] : relation.mp_like iff :=
|
||||
relation.mp_like.mk (λa b (H : a ↔ b), iff.elim_left H)
|
||||
|
||||
|
||||
/- support for calculations with iff -/
|
||||
|
||||
namespace iff
|
||||
|
|
|
|||
|
|
@ -1,22 +1,22 @@
|
|||
logic
|
||||
=====
|
||||
|
||||
Additional constructions and axioms. By default, `import logic` does not
|
||||
import any axioms.
|
||||
Logical constructions and theorems, beyond what has already been
|
||||
declared in init.datatypes and init.logic.
|
||||
|
||||
Logical operations and connectives.
|
||||
The subfolder logic.axioms declares additional axioms. The command
|
||||
`import logic` does not import any axioms by default.
|
||||
|
||||
* [connectives](connectives.lean) : the propositional connectives
|
||||
* [eq](eq.lean) : additional theorems for equality and disequality
|
||||
* [cast](cast.lean) : casts and heterogeneous equality
|
||||
* [quantifiers](quantifiers.lean) : existential and universal quantifiers
|
||||
* [identities](identities.lean) : some useful identities
|
||||
* [instances](instances.lean) : class instances for eq and iff
|
||||
* [subsingleton](subsingleton.lean)
|
||||
* [default](default.lean)
|
||||
|
||||
Constructively, inhabited types have a witness, while nonempty types
|
||||
are "proof irrelevant". Classically (assuming the axioms in
|
||||
logic.axioms.hilbert) the two are equivalent. Type class inferences
|
||||
are set up to use "inhabited" however, so users should use that to
|
||||
declare that types have an element. Use "nonempty" in the hypothesis
|
||||
of a theorem when the theorem does not depend on the witness chosen.
|
||||
Subfolders:
|
||||
|
||||
* [axioms](axioms/axioms.md) : additional axioms
|
||||
* [examples](examples/examples.md)
|
||||
|
|
@ -4,16 +4,18 @@ Released under Apache 2.0 license as described in the file LICENSE.
|
|||
|
||||
Module: logic.quantifiers
|
||||
Authors: Leonardo de Moura, Jeremy Avigad
|
||||
|
||||
Universal and existential quantifiers. See also init.logic.
|
||||
-/
|
||||
|
||||
open inhabited nonempty
|
||||
|
||||
theorem exists_not_forall {A : Type} {p : A → Prop} (H : ∃x, p x) : ¬∀x, ¬p x :=
|
||||
theorem not_forall_not_of_exists {A : Type} {p : A → Prop} (H : ∃x, p x) : ¬∀x, ¬p x :=
|
||||
assume H1 : ∀x, ¬p x,
|
||||
obtain (w : A) (Hw : p w), from H,
|
||||
absurd Hw (H1 w)
|
||||
|
||||
theorem forall_not_exists {A : Type} {p : A → Prop} (H2 : ∀x, p x) : ¬∃x, ¬p x :=
|
||||
theorem not_exists_not_of_forall {A : Type} {p : A → Prop} (H2 : ∀x, p x) : ¬∃x, ¬p x :=
|
||||
assume H1 : ∃x, ¬p x,
|
||||
obtain (w : A) (Hw : ¬p w), from H1,
|
||||
absurd (H2 w) Hw
|
||||
|
|
@ -41,12 +43,14 @@ iff.intro
|
|||
(assume Hl, obtain a Hp, from Hl, Hp)
|
||||
(assume Hr, inhabited.destruct H (take a, exists_intro a Hr))
|
||||
|
||||
theorem forall_and_distribute {A : Type} (φ ψ : A → Prop) : (∀x, φ x ∧ ψ x) ↔ (∀x, φ x) ∧ (∀x, ψ x) :=
|
||||
theorem forall_and_distribute {A : Type} (φ ψ : A → Prop) :
|
||||
(∀x, φ x ∧ ψ x) ↔ (∀x, φ x) ∧ (∀x, ψ x) :=
|
||||
iff.intro
|
||||
(assume H, and.intro (take x, and.elim_left (H x)) (take x, and.elim_right (H x)))
|
||||
(assume H, take x, and.intro (and.elim_left H x) (and.elim_right H x))
|
||||
|
||||
theorem exists_or_distribute {A : Type} (φ ψ : A → Prop) : (∃x, φ x ∨ ψ x) ↔ (∃x, φ x) ∨ (∃x, ψ x) :=
|
||||
theorem exists_or_distribute {A : Type} (φ ψ : A → Prop) :
|
||||
(∃x, φ x ∨ ψ x) ↔ (∃x, φ x) ∨ (∃x, ψ x) :=
|
||||
iff.intro
|
||||
(assume H, obtain (w : A) (Hw : φ w ∨ ψ w), from H,
|
||||
or.elim Hw
|
||||
|
|
@ -58,7 +62,7 @@ iff.intro
|
|||
(assume H2, obtain (w : A) (Hw : ψ w), from H2,
|
||||
exists_intro w (or.inr Hw)))
|
||||
|
||||
theorem exists_imp_nonempty {A : Type} {P : A → Prop} (H : ∃x, P x) : nonempty A :=
|
||||
theorem nonempty_of_exists {A : Type} {P : A → Prop} (H : ∃x, P x) : nonempty A :=
|
||||
obtain w Hw, from H, nonempty.intro w
|
||||
|
||||
section
|
||||
|
|
|
|||
|
|
@ -1,8 +1,11 @@
|
|||
/-
|
||||
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Floris van Doorn
|
||||
|
||||
Module: logic.subsingleton
|
||||
Author: Floris van Doorn
|
||||
-/
|
||||
|
||||
import logic.eq
|
||||
|
||||
inductive subsingleton [class] (A : Type) : Prop :=
|
||||
|
|
@ -28,7 +31,8 @@ subsingleton.intro (fun d1 d2,
|
|||
d2)
|
||||
d1)
|
||||
|
||||
protected theorem rec_subsingleton [instance] {p : Prop} [H : decidable p] {H1 : p → Type} {H2 : ¬p → Type}
|
||||
protected theorem rec_subsingleton [instance] {p : Prop} [H : decidable p]
|
||||
{H1 : p → Type} {H2 : ¬p → Type}
|
||||
(H3 : Π(h : p), subsingleton (H1 h)) (H4 : Π(h : ¬p), subsingleton (H2 h))
|
||||
: subsingleton (decidable.rec_on H H1 H2) :=
|
||||
: subsingleton (decidable.rec_on H H1 H2) :=
|
||||
decidable.rec_on H (λh, H3 h) (λh, H4 h) --this can be proven using dependent version of "by_cases"
|
||||
|
|
|
|||
Loading…
Reference in a new issue