209 lines
8 KiB
Text
209 lines
8 KiB
Text
-- 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: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
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-- logic.connectives.eq
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-- ====================
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-- Equality.
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import logic.prop
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-- eq
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-- --
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inductive eq {A : Type} (a : A) : A → Prop :=
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refl : eq a a
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infix `=` := eq
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definition rfl {A : Type} {a : A} := eq.refl a
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-- proof irrelevance is built in
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theorem proof_irrel {a : Prop} {H1 H2 : a} : H1 = H2 := rfl
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namespace eq
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theorem id_refl {A : Type} {a : A} (H1 : a = a) : H1 = (eq.refl a) :=
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proof_irrel
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theorem irrel {A : Type} {a b : A} (H1 H2 : a = b) : H1 = H2 :=
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proof_irrel
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theorem subst {A : Type} {a b : A} {P : A → Prop} (H1 : a = b) (H2 : P a) : P b :=
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rec H2 H1
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theorem trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b = c) : a = c :=
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subst H2 H1
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theorem symm {A : Type} {a b : A} (H : a = b) : b = a :=
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subst H (refl a)
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end eq
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calc_subst eq.subst
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calc_refl eq.refl
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calc_trans eq.trans
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namespace eq_ops
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postfix `⁻¹` := eq.symm
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reserve infixr `⬝`:75 infixr `⬝` := eq.trans
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infixr `▸` := eq.subst
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end eq_ops
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open eq_ops
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namespace eq
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-- eq_rec with arguments swapped, for transporting an element of a dependent type
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-- definition rec_on {A : Type} {a1 a2 : A} {B : A → Type} (H1 : a1 = a2) (H2 : B a1) : B a2 :=
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-- eq.rec H2 H1
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definition rec_on {A : Type} {a a' : A} {B : Πa' : A, a = a' → Type} (H1 : a = a') (H2 : B a (refl a)) : B a' H1 :=
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eq.rec (λH1 : a = a, show B a H1, from H2) H1 H1
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theorem rec_on_id {A : Type} {a : A} {B : Πa' : A, a = a' → Type} (H : a = a) (b : B a H) : rec_on H b = b :=
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refl (rec_on rfl b)
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theorem rec_on_constant {A : Type} {a a' : A} {B : Type} (H : a = a') (b : B) : rec_on H b = b :=
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rec_on H (λ(H' : a = a), rec_on_id H' b) H
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theorem rec_on_constant2 {A : Type} {a₁ a₂ a₃ a₄ : A} {B : Type} (H₁ : a₁ = a₂) (H₂ : a₃ = a₄) (b : B) : rec_on H₁ b = rec_on H₂ b :=
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rec_on_constant H₁ b ⬝ rec_on_constant H₂ b ⁻¹
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theorem rec_on_irrel {A B : Type} {a a' : A} {f : A → B} {D : B → Type} (H : a = a') (H' : f a = f a') (b : D (f a)) : rec_on H b = rec_on H' b :=
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rec_on H (λ(H : a = a) (H' : f a = f a), rec_on_id H b ⬝ rec_on_id H' b⁻¹) H H'
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theorem rec_id {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : rec b H = b :=
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id_refl H⁻¹ ▸ refl (eq.rec b (refl a))
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theorem rec_on_compose {A : Type} {a b c : A} {P : A → Type} (H1 : a = b) (H2 : b = c)
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(u : P a) :
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rec_on H2 (rec_on H1 u) = rec_on (trans H1 H2) u :=
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(show ∀(H2 : b = c), rec_on H2 (rec_on H1 u) = rec_on (trans H1 H2) u,
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from rec_on H2 (take (H2 : b = b), rec_on_id H2 _))
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H2
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end eq
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open eq
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theorem congr_fun {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) (a : A) : f a = g a :=
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H ▸ rfl
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theorem congr_arg {A : Type} {B : Type} {a b : A} (f : A → B) (H : a = b) : f a = f b :=
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H ▸ rfl
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theorem congr {A : Type} {B : Type} {f g : A → B} {a b : A} (H1 : f = g) (H2 : a = b) :
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f a = g b :=
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H1 ▸ H2 ▸ rfl
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theorem congr_arg2 {A B C : Type} {a a' : A} {b b' : B} (f : A → B → C) (Ha : a = a') (Hb : b = b') : f a b = f a' b' :=
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congr (congr_arg f Ha) Hb
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theorem congr_arg3 {A B C D : Type} {a a' : A} {b b' : B} {c c' : C} (f : A → B → C → D) (Ha : a = a') (Hb : b = b') (Hc : c = c') : f a b c = f a' b' c' :=
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congr (congr_arg2 f Ha Hb) Hc
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theorem congr_arg4 {A B C D E : Type} {a a' : A} {b b' : B} {c c' : C} {d d' : D} (f : A → B → C → D → E) (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d') : f a b c d = f a' b' c' d' :=
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congr (congr_arg3 f Ha Hb Hc) Hd
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theorem congr_arg5 {A B C D E F : Type} {a a' : A} {b b' : B} {c c' : C} {d d' : D} {e e' : E} (f : A → B → C → D → E → F) (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d') (He : e = e') : f a b c d e = f a' b' c' d' e' :=
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congr (congr_arg4 f Ha Hb Hc Hd) He
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theorem congr2 {A B C : Type} {a a' : A} {b b' : B} (f f' : A → B → C) (Hf : f = f') (Ha : a = a') (Hb : b = b') : f a b = f' a' b' :=
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Hf ▸ congr_arg2 f Ha Hb
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theorem congr3 {A B C D : Type} {a a' : A} {b b' : B} {c c' : C} (f f' : A → B → C → D) (Hf : f = f') (Ha : a = a') (Hb : b = b') (Hc : c = c') : f a b c = f' a' b' c' :=
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Hf ▸ congr_arg3 f Ha Hb Hc
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theorem congr4 {A B C D E : Type} {a a' : A} {b b' : B} {c c' : C} {d d' : D} (f f' : A → B → C → D → E) (Hf : f = f') (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d') : f a b c d = f' a' b' c' d' :=
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Hf ▸ congr_arg4 f Ha Hb Hc Hd
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theorem congr5 {A B C D E F : Type} {a a' : A} {b b' : B} {c c' : C} {d d' : D} {e e' : E} (f f' : A → B → C → D → E → F) (Hf : f = f') (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d') (He : e = e') : f a b c d e = f' a' b' c' d' e' :=
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Hf ▸ congr_arg5 f Ha Hb Hc Hd He
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theorem congr_arg2_dep {A : Type} {B : A → Type} {C : Type} {a₁ a₂ : A}
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{b₁ : B a₁} {b₂ : B a₂} (f : Πa, B a → C) (H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) :
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f a₁ b₁ = f a₂ b₂ :=
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eq.rec_on H₁
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(λ (b₂ : B a₁) (H₁ : a₁ = a₁) (H₂ : eq.rec_on H₁ b₁ = b₂),
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calc
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f a₁ b₁ = f a₁ (eq.rec_on H₁ b₁) : {(eq.rec_on_id H₁ b₁)⁻¹}
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... = f a₁ b₂ : {H₂})
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b₂ H₁ H₂
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theorem congr_arg3_dep {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Type} {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} {c₁ : C a₁ b₁} {c₂ : C a₂ b₂} (f : Πa b, C a b → D)
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(H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) (H₃ : eq.rec_on (congr_arg2_dep C H₁ H₂) c₁ = c₂) :
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f a₁ b₁ c₁ = f a₂ b₂ c₂ :=
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eq.rec_on H₁
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(λ (b₂ : B a₁) (H₂ : b₁ = b₂) (c₂ : C a₁ b₂) (H₃ : _ = c₂),
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have H₃' : eq.rec_on H₂ c₁ = c₂,
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from (rec_on_irrel H₂ (congr_arg2_dep C (refl a₁) H₂) c₁⁻¹) ▸ H₃,
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congr_arg2_dep (f a₁) H₂ H₃')
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b₂ H₂ c₂ H₃
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theorem congr_arg3_ndep_dep {A B : Type} {C : A → B → Type} {D : Type} {a₁ a₂ : A} {b₁ b₂ : B} {c₁ : C a₁ b₁} {c₂ : C a₂ b₂} (f : Πa b, C a b → D)
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(H₁ : a₁ = a₂) (H₂ : b₁ = b₂) (H₃ : eq.rec_on (congr_arg2 C H₁ H₂) c₁ = c₂) :
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f a₁ b₁ c₁ = f a₂ b₂ c₂ :=
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congr_arg3_dep f H₁ (rec_on_constant H₁ b₁ ⬝ H₂) H₃
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theorem equal_f {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) : ∀x, f x = g x :=
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take x, congr_fun H x
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theorem eqmp {a b : Prop} (H1 : a = b) (H2 : a) : b :=
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H1 ▸ H2
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theorem eqmpr {a b : Prop} (H1 : a = b) (H2 : b) : a :=
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H1⁻¹ ▸ H2
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theorem eq_true_elim {a : Prop} (H : a = true) : a :=
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H⁻¹ ▸ trivial
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theorem eq_false_elim {a : Prop} (H : a = false) : ¬a :=
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assume Ha : a, H ▸ Ha
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theorem imp_trans {a b c : Prop} (H1 : a → b) (H2 : b → c) : a → c :=
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assume Ha, H2 (H1 Ha)
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theorem imp_eq_trans {a b c : Prop} (H1 : a → b) (H2 : b = c) : a → c :=
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assume Ha, H2 ▸ (H1 Ha)
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theorem eq_imp_trans {a b c : Prop} (H1 : a = b) (H2 : b → c) : a → c :=
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assume Ha, H2 (H1 ▸ Ha)
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-- ne
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-- --
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definition ne {A : Type} (a b : A) := ¬(a = b)
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infix `≠` := ne
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namespace ne
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theorem intro {A : Type} {a b : A} (H : a = b → false) : a ≠ b :=
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H
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theorem elim {A : Type} {a b : A} (H1 : a ≠ b) (H2 : a = b) : false :=
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H1 H2
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theorem irrefl {A : Type} {a : A} (H : a ≠ a) : false :=
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H rfl
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theorem symm {A : Type} {a b : A} (H : a ≠ b) : b ≠ a :=
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assume H1 : b = a, H (H1⁻¹)
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end ne
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theorem a_neq_a_elim {A : Type} {a : A} (H : a ≠ a) : false :=
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H rfl
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theorem eq_ne_trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b ≠ c) : a ≠ c :=
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H1⁻¹ ▸ H2
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theorem ne_eq_trans {A : Type} {a b c : A} (H1 : a ≠ b) (H2 : b = c) : a ≠ c :=
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H2 ▸ H1
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calc_trans eq_ne_trans
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calc_trans ne_eq_trans
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theorem p_ne_false {p : Prop} (Hp : p) : p ≠ false :=
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assume Heq : p = false, Heq ▸ Hp
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theorem p_ne_true {p : Prop} (Hnp : ¬p) : p ≠ true :=
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assume Heq : p = true, absurd trivial (Heq ▸ Hnp)
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theorem true_ne_false : ¬true = false :=
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assume H : true = false,
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H ▸ trivial
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