refactor(library): using section variables

This commit is contained in:
Leonardo de Moura 2014-10-02 18:25:00 -07:00
parent d5cad765a0
commit a52b21c92d
2 changed files with 47 additions and 30 deletions

View file

@ -113,24 +113,26 @@ infix `<->` := iff
infix `↔` := iff
namespace iff
theorem def {a b : Prop} : (a ↔ b) = ((a → b) ∧ (b → a)) :=
section
variables {a b c : Prop}
theorem def : (a ↔ b) = ((a → b) ∧ (b → a)) :=
rfl
theorem intro {a b : Prop} (H₁ : a → b) (H₂ : b → a) : a ↔ b :=
theorem intro (H₁ : a → b) (H₂ : b → a) : a ↔ b :=
and.intro H₁ H₂
theorem elim {a b c : Prop} (H₁ : (a → b) → (b → a) → c) (H₂ : a ↔ b) : c :=
theorem elim (H₁ : (a → b) → (b → a) → c) (H₂ : a ↔ b) : c :=
and.rec H₁ H₂
theorem elim_left {a b : Prop} (H : a ↔ b) : a → b :=
theorem elim_left (H : a ↔ b) : a → b :=
elim (assume H₁ H₂, H₁) H
definition mp := @elim_left
theorem elim_right {a b : Prop} (H : a ↔ b) : b → a :=
theorem elim_right (H : a ↔ b) : b → a :=
elim (assume H₁ H₂, H₂) H
theorem flip_sign {a b : Prop} (H₁ : a ↔ b) : ¬a ↔ ¬b :=
theorem flip_sign (H₁ : a ↔ b) : ¬a ↔ ¬b :=
intro
(assume Hna, mt (elim_right H₁) Hna)
(assume Hnb, mt (elim_left H₁) Hnb)
@ -141,21 +143,23 @@ namespace iff
theorem rfl {a : Prop} : a ↔ a :=
refl a
theorem trans {a b c : Prop} (H₁ : a ↔ b) (H₂ : b ↔ c) : a ↔ c :=
theorem trans (H₁ : a ↔ b) (H₂ : b ↔ c) : a ↔ c :=
intro
(assume Ha, elim_left H₂ (elim_left H₁ Ha))
(assume Hc, elim_right H₁ (elim_right H₂ Hc))
theorem symm {a b : Prop} (H : a ↔ b) : b ↔ a :=
theorem symm (H : a ↔ b) : b ↔ a :=
intro
(assume Hb, elim_right H Hb)
(assume Ha, elim_left H Ha)
theorem true_elim {a : Prop} (H : a ↔ true) : a :=
theorem true_elim (H : a ↔ true) : a :=
mp (symm H) trivial
theorem false_elim {a : Prop} (H : a ↔ false) : ¬a :=
theorem false_elim (H : a ↔ false) : ¬a :=
assume Ha : a, mp H Ha
end
end iff
calc_refl iff.refl
@ -168,12 +172,13 @@ iff.intro (λ Ha, H ▸ Ha) (λ Hb, H⁻¹ ▸ Hb)
-- comm and assoc for and / or
-- ---------------------------
namespace and
theorem comm {a b : Prop} : a ∧ b ↔ b ∧ a :=
section
variables {a b c : Prop}
theorem comm : a ∧ b ↔ b ∧ a :=
iff.intro (λH, swap H) (λH, swap H)
theorem assoc {a b c : Prop} : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
theorem assoc : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
iff.intro
(assume H, intro
(elim_left (elim_left H))
@ -181,13 +186,16 @@ namespace and
(assume H, intro
(intro (elim_left H) (elim_left (elim_right H)))
(elim_right (elim_right H)))
end
end and
namespace or
theorem comm {a b : Prop} : a b ↔ b a :=
section
variables {a b c : Prop}
theorem comm : a b ↔ b a :=
iff.intro (λH, swap H) (λH, swap H)
theorem assoc {a b c : Prop} : (a b) c ↔ a (b c) :=
theorem assoc : (a b) c ↔ a (b c) :=
iff.intro
(assume H, elim H
(assume H₁, elim H₁
@ -199,4 +207,5 @@ namespace or
(assume H₁, elim H₁
(assume Hb, inl (inr Hb))
(assume Hc, inr Hc)))
end
end or

View file

@ -22,26 +22,29 @@ definition rfl {A : Type} {a : A} := eq.refl a
theorem proof_irrel {a : Prop} {H1 H2 : a} : H1 = H2 := rfl
namespace eq
theorem id_refl {A : Type} {a : A} (H1 : a = a) : H1 = (eq.refl a) :=
section
variables {A : Type}
variables {a b c : A}
theorem id_refl (H1 : a = a) : H1 = (eq.refl a) :=
proof_irrel
theorem irrel {A : Type} {a b : A} (H1 H2 : a = b) : H1 = H2 :=
theorem irrel (H1 H2 : a = b) : H1 = H2 :=
proof_irrel
theorem subst {A : Type} {a b : A} {P : A → Prop} (H1 : a = b) (H2 : P a) : P b :=
theorem subst {P : A → Prop} (H1 : a = b) (H2 : P a) : P b :=
rec H2 H1
theorem trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b = c) : a = c :=
theorem trans (H1 : a = b) (H2 : b = c) : a = c :=
subst H2 H1
theorem symm {A : Type} {a b : A} (H : a = b) : b = a :=
theorem symm (H : a = b) : b = a :=
subst H (refl a)
namespace ops
postfix `⁻¹` := symm
infixr `⬝` := trans
infixr `▸` := subst
end ops
end
namespace ops
postfix `⁻¹` := symm
infixr `⬝` := trans
infixr `▸` := subst
end ops
end eq
calc_subst eq.subst
@ -175,17 +178,22 @@ definition ne {A : Type} (a b : A) := ¬(a = b)
infix `≠` := ne
namespace ne
theorem intro {A : Type} {a b : A} (H : a = b → false) : a ≠ b :=
section
variable {A : Type}
variables {a b : A}
theorem intro (H : a = b → false) : a ≠ b :=
H
theorem elim {A : Type} {a b : A} (H1 : a ≠ b) (H2 : a = b) : false :=
theorem elim (H1 : a ≠ b) (H2 : a = b) : false :=
H1 H2
theorem irrefl {A : Type} {a : A} (H : a ≠ a) : false :=
theorem irrefl (H : a ≠ a) : false :=
H rfl
theorem symm {A : Type} {a b : A} (H : a ≠ b) : b ≠ a :=
theorem symm (H : a ≠ b) : b ≠ a :=
assume H1 : b = a, H (H1⁻¹)
end
end ne
theorem a_neq_a_elim {A : Type} {a : A} (H : a ≠ a) : false :=