lean2/library/logic/eq.lean

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
import general_notation logic.prop data.unit.decl
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-- logic.eq
-- ====================
-- Equality.
-- eq
-- --
inductive eq {A : Type} (a : A) : A → Prop :=
refl : eq a a
notation a = b := eq a b
definition rfl {A : Type} {a : A} := eq.refl a
-- proof irrelevance is built in
theorem proof_irrel {a : Prop} (H₁ H₂ : a) : H₁ = H₂ :=
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rfl
namespace eq
variables {A : Type}
variables {a b c : A}
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theorem id_refl (H₁ : a = a) : H₁ = (eq.refl a) :=
rfl
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theorem irrel (H₁ H₂ : a = b) : H₁ = H₂ :=
!proof_irrel
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theorem subst {P : A → Prop} (H₁ : a = b) (H₂ : P a) : P b :=
rec H₂ H₁
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theorem trans (H₁ : a = b) (H₂ : b = c) : a = c :=
subst H₂ H₁
theorem symm (H : a = b) : b = a :=
subst H (refl a)
namespace ops
notation H `⁻¹` := symm H
notation H1 ⬝ H2 := trans H1 H2
notation H1 ▸ H2 := subst H1 H2
end ops
end eq
calc_subst eq.subst
calc_refl eq.refl
calc_trans eq.trans
calc_symm eq.symm
open eq.ops
namespace eq
variables {A B : Type} {a a' a₁ a₂ a₃ a₄ : A}
definition drec_on {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
--can we remove the theorems about drec_on and only have the rec_on versions?
-- theorem drec_on_id {B : Πa' : A, a = a' → Type} (H : a = a) (b : B a H) : drec_on H b = b :=
-- rfl
-- theorem drec_on_constant (H : a = a') {B : Type} (b : B) : drec_on H b = b :=
-- drec_on H rfl
-- theorem drec_on_constant2 (H₁ : a₁ = a₂) (H₂ : a₃ = a₄) (b : B) : drec_on H₁ b = drec_on H₂ b :=
-- drec_on_constant H₁ b ⬝ (drec_on_constant H₂ b)⁻¹
-- theorem drec_on_irrel_arg {f : A → B} {D : B → Type} (H : a = a') (H' : f a = f a')
-- (b : D (f a)) : drec_on H b = drec_on H' b :=
-- drec_on H (λ(H' : f a = f a), !drec_on_id⁻¹) H'
-- theorem drec_on_irrel {D : A → Type} (H H' : a = a') (b : D a) :
-- drec_on H b = drec_on H' b :=
-- !drec_on_irrel_arg
-- theorem drec_on_compose {a b c : A} {P : A → Type} (H₁ : a = b) (H₂ : b = c)
-- (u : P a) : drec_on H₂ (drec_on H₁ u) = drec_on (trans H₁ H₂) u :=
-- (show ∀ H₂ : b = c, drec_on H₂ (drec_on H₁ u) = drec_on (trans H₁ H₂) u,
-- from drec_on H₂ (take (H₂ : b = b), drec_on_id H₂ _))
-- H₂
theorem rec_on_id {B : A → Type} (H : a = a) (b : B a) : rec_on H b = b :=
rfl
theorem rec_on_constant (H : a = a') {B : Type} (b : B) : rec_on H b = b :=
drec_on H rfl
theorem rec_on_constant2 (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
theorem rec_on_irrel_arg {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 :=
drec_on H (λ(H' : f a = f a), !rec_on_id⁻¹) H'
theorem rec_on_irrel {a a' : A} {D : A → Type} (H H' : a = a') (b : D a) :
drec_on H b = drec_on H' b :=
!rec_on_irrel_arg
--do we need the following?
-- theorem rec_on_irrel_fun {B : A → Type} {a : A} {f f' : Π x, B x} {D : Π a, B a → Type} (H : f = f') (H' : f a = f' a) (b : D a (f a)) :
-- rec_on H b = rec_on H' b :=
-- sorry
-- the
-- theorem rec_on_comm_ap {B : A → Type} {C : Πa, B a → Type} {a a' : A} {f : Π x, C a x}
-- (H : a = a') (H' : a = a') (b : B a) : rec_on H f b = rec_on H' (f b) :=
-- sorry
theorem rec_on_compose {a b c : A} {P : A → Type} (H₁ : a = b) (H₂ : b = c)
(u : P a) : rec_on H₂ (rec_on H₁ u) = rec_on (trans H₁ H₂) u :=
(show ∀ H₂ : b = c, rec_on H₂ (rec_on H₁ u) = rec_on (trans H₁ H₂) u,
from drec_on H₂ (take (H₂ : b = b), rec_on_id H₂ _))
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H₂
end eq
open eq
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section
variables {A B C D E F : Type}
variables {a a' : A} {b b' : B} {c c' : C} {d d' : D} {e e' : E}
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theorem congr_fun {B : A → Type} {f g : Π x, B x} (H : f = g) (a : A) : f a = g a :=
H ▸ rfl
theorem congr_arg (f : A → B) (H : a = a') : f a = f a' :=
H ▸ rfl
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theorem congr {f g : A → B} (H₁ : f = g) (H₂ : a = a') : f a = g a' :=
H₁ ▸ H₂ ▸ rfl
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theorem congr_arg2 (f : A → B → C) (Ha : a = a') (Hb : b = b') : f a b = f a' b' :=
congr (congr_arg f Ha) Hb
theorem congr_arg3 (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
theorem congr_arg4 (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
theorem congr_arg5 (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 (f f' : A → B → C) (Hf : f = f') (Ha : a = a') (Hb : b = b') : f a b = f' a' b' :=
Hf ▸ congr_arg2 f Ha Hb
theorem congr3 (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
theorem congr4 (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
theorem congr5 (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
end
theorem equal_f {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) : ∀x, f x = g x :=
take x, congr_fun H x
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section
variables {a b c : Prop}
theorem eqmp (H₁ : a = b) (H₂ : a) : b :=
H₁ ▸ H₂
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theorem eqmpr (H₁ : a = b) (H₂ : b) : a :=
H₁⁻¹ ▸ H₂
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theorem eq_true_elim (H : a = true) : a :=
H⁻¹ ▸ trivial
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theorem eq_false_elim (H : a = false) : ¬a :=
assume Ha : a, H ▸ Ha
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theorem imp_trans (H₁ : a → b) (H₂ : b → c) : a → c :=
assume Ha, H₂ (H₁ Ha)
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theorem imp_eq_trans (H₁ : a → b) (H₂ : b = c) : a → c :=
assume Ha, H₂ ▸ (H₁ Ha)
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theorem eq_imp_trans (H₁ : a = b) (H₂ : b → c) : a → c :=
assume Ha, H₂ (H₁ ▸ Ha)
end
-- ne
-- --
definition ne {A : Type} (a b : A) := ¬(a = b)
notation a ≠ b := ne a b
namespace ne
variable {A : Type}
variables {a b : A}
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theorem intro : (a = b → false) → a ≠ b :=
assume H, H
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theorem elim : a ≠ b → a = b → false :=
assume H₁ H₂, H₁ H₂
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theorem irrefl : a ≠ a → false :=
assume H, H rfl
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theorem symm : a ≠ b → b ≠ a :=
assume (H : a ≠ b) (H₁ : b = a), H (H₁⁻¹)
end ne
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section
variables {A : Type} {a b c : A}
theorem a_neq_a_elim : a ≠ a → false :=
assume H, H rfl
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theorem eq_ne_trans : a = b → b ≠ c → a ≠ c :=
assume H₁ H₂, H₁⁻¹ ▸ H₂
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theorem ne_eq_trans : a ≠ b → b = c → a ≠ c :=
assume H₁ H₂, H₂ ▸ H₁
end
calc_trans eq_ne_trans
calc_trans ne_eq_trans
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section
variables {p : Prop}
theorem p_ne_false : p → p ≠ false :=
assume (Hp : p) (Heq : p = false), Heq ▸ Hp
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theorem p_ne_true : ¬p → p ≠ true :=
assume (Hnp : ¬p) (Heq : p = true), absurd trivial (Heq ▸ Hnp)
end
theorem true_ne_false : ¬true = false :=
assume H : true = false,
H ▸ trivial
inductive subsingleton [class] (A : Type) : Prop :=
intro : (∀ a b : A, a = b) -> subsingleton A
namespace subsingleton
definition elim {A : Type} {H : subsingleton A} : ∀(a b : A), a = b := rec (fun p, p) H
end subsingleton
protected definition prop.subsingleton [instance] (P : Prop) : subsingleton P :=
subsingleton.intro (λa b, !proof_irrel)