feat(frontends/lean): new semantics for "protected" declarations
closes #426
This commit is contained in:
Leonardo de Moura
parent
eceed03044
commit
a35cce38b3
77 changed files with 738 additions and 681 deletions
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@ -38,7 +38,7 @@ namespace precategory
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-- symmetric associativity proof.
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definition op_op' {ob : Type} (C : precategory ob) : opposite (opposite C) = C :=
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begin
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apply (rec_on C), intros (hom', homH', comp', ID', assoc', id_left', id_right'),
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apply (precategory.rec_on C), intros (hom', homH', comp', ID', assoc', id_left', id_right'),
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apply (ap (λassoc'', precategory.mk hom' @homH' comp' ID' assoc'' id_left' id_right')),
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repeat ( apply funext.path_pi ; intros ),
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apply ap,
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@ -16,10 +16,10 @@ namespace natural_transformation
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variables {C D : Precategory} {F G H I : functor C D}
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definition natural_map [coercion] (η : F ⟹ G) : Π (a : C), F a ⟶ G a :=
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rec (λ x y, x) η
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natural_transformation.rec (λ x y, x) η
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theorem naturality (η : F ⟹ G) : Π⦃a b : C⦄ (f : a ⟶ b), G f ∘ η a = η b ∘ F f :=
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rec (λ x y, y) η
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natural_transformation.rec (λ x y, y) η
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protected definition compose (η : G ⟹ H) (θ : F ⟹ G) : F ⟹ H :=
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natural_transformation.mk
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@ -52,9 +52,9 @@ namespace natural_transformation
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protected definition assoc (η₃ : H ⟹ I) (η₂ : G ⟹ H) (η₁ : F ⟹ G) :
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η₃ ∘n (η₂ ∘n η₁) = (η₃ ∘n η₂) ∘n η₁ :=
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begin
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apply (rec_on η₃), intros (η₃1, η₃2),
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apply (rec_on η₂), intros (η₂1, η₂2),
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apply (rec_on η₁), intros (η₁1, η₁2),
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apply (natural_transformation.rec_on η₃), intros (η₃1, η₃2),
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apply (natural_transformation.rec_on η₂), intros (η₂1, η₂2),
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apply (natural_transformation.rec_on η₁), intros (η₁1, η₁2),
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fapply natural_transformation.congr,
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apply funext.path_pi, intro a,
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apply assoc,
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@ -72,7 +72,7 @@ namespace natural_transformation
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protected definition id_left (η : F ⟹ G) : id ∘n η = η :=
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begin
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apply (rec_on η), intros (η₁, nat₁),
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apply (natural_transformation.rec_on η), intros (η₁, nat₁),
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fapply (natural_transformation.congr F G),
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apply funext.path_pi, intro a,
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apply id_left,
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@ -84,7 +84,7 @@ namespace natural_transformation
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protected definition id_right (η : F ⟹ G) : η ∘n id = η :=
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begin
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apply (rec_on η), intros (η₁, nat₁),
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apply (natural_transformation.rec_on η), intros (η₁, nat₁),
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fapply (natural_transformation.congr F G),
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apply funext.path_pi, intro a,
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apply id_right,
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@ -21,7 +21,7 @@ namespace funext
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include F
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protected definition ap [instance] (f g : Π x, P x) : is_equiv (@apD10 A P f g) :=
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rec_on F (λ(H : Π A P f g, _), !H)
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funext.rec_on F (λ(H : Π A P f g, _), !H)
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definition path_pi {f g : Π x, P x} : f ∼ g → f = g :=
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is_equiv.inv (@apD10 A P f g)
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@ -8,18 +8,18 @@ import init.datatypes init.reserved_notation
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namespace bool
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definition cond {A : Type} (b : bool) (t e : A) :=
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rec_on b e t
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bool.rec_on b e t
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definition bor (a b : bool) :=
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rec_on a (rec_on b ff tt) tt
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bool.rec_on a (bool.rec_on b ff tt) tt
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notation a || b := bor a b
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definition band (a b : bool) :=
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rec_on a ff (rec_on b ff tt)
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bool.rec_on a ff (bool.rec_on b ff tt)
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notation a && b := band a b
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definition bnot (a : bool) :=
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rec_on a tt ff
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bool.rec_on a tt ff
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end bool
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@ -145,13 +145,15 @@ namespace is_equiv
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variables {B C : Type} (f : A → B) {f' : A → B} [Hf : is_equiv f]
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include Hf
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variable (g : B → C)
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definition cancel_R (g : B → C) [Hgf : is_equiv (g ∘ f)] : (is_equiv g) :=
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have Hfinv [visible] : is_equiv (f⁻¹), from inv_closed f,
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@homotopy_closed _ _ _ _ (compose (f⁻¹) (g ∘ f)) (λb, ap g (@retr _ _ f _ b))
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@homotopy_closed _ _ _ _ (is_equiv.compose (f⁻¹) (g ∘ f)) (λb, ap g (@retr _ _ f _ b))
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definition cancel_L (g : C → A) [Hgf : is_equiv (f ∘ g)] : (is_equiv g) :=
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have Hfinv [visible] : is_equiv (f⁻¹), from inv_closed f,
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@homotopy_closed _ _ _ _ (compose (f ∘ g) (f⁻¹)) (λa, sect f (g a))
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@homotopy_closed _ _ _ _ (is_equiv.compose (f ∘ g) (f⁻¹)) (λa, sect f (g a))
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--Rewrite rules
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definition moveR_M {x : A} {y : B} (p : x = (inv f) y) : (f x = y) :=
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@ -260,8 +262,8 @@ namespace equiv
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-- calc enviroment
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-- Note: Calculating with substitutions needs univalence
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calc_trans trans
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calc_refl refl
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calc_symm symm
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calc_trans equiv.trans
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calc_refl equiv.refl
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calc_symm equiv.symm
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end equiv
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@ -46,7 +46,7 @@ namespace eq
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variables {a b c a' : A}
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definition subst {P : A → Type} (H₁ : a = b) (H₂ : P a) : P b :=
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rec H₂ H₁
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eq.rec H₂ H₁
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definition trans (H₁ : a = b) (H₂ : b = c) : a = c :=
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subst H₂ H₁
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@ -211,13 +211,13 @@ namespace decidable
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variables {p q : Type}
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definition pos_witness [C : decidable p] (H : p) : p :=
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rec_on C (λ Hp, Hp) (λ Hnp, absurd H Hnp)
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decidable.rec_on C (λ Hp, Hp) (λ Hnp, absurd H Hnp)
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definition neg_witness [C : decidable p] (H : ¬ p) : ¬ p :=
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rec_on C (λ Hp, absurd Hp H) (λ Hnp, Hnp)
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decidable.rec_on C (λ Hp, absurd Hp H) (λ Hnp, Hnp)
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definition by_cases {q : Type} [C : decidable p] (Hpq : p → q) (Hnpq : ¬p → q) : q :=
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rec_on C (assume Hp, Hpq Hp) (assume Hnp, Hnpq Hnp)
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decidable.rec_on C (assume Hp, Hpq Hp) (assume Hnp, Hnpq Hnp)
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definition em (p : Type) [H : decidable p] : sum p ¬p :=
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by_cases (λ Hp, sum.inl Hp) (λ Hnp, sum.inr Hnp)
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@ -228,7 +228,7 @@ namespace decidable
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(assume H₁ : ¬p, empty.rec (λ e, p) (H H₁))
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definition decidable_iff_equiv (Hp : decidable p) (H : p ↔ q) : decidable q :=
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rec_on Hp
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decidable.rec_on Hp
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(assume Hp : p, inl (iff.elim_left H Hp))
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(assume Hnp : ¬p, inr (iff.elim_left (iff.flip_sign H) Hnp))
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@ -23,7 +23,7 @@ namespace nat
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notation a ≤ b := le a b
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definition pred (a : nat) : nat :=
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cases_on a zero (λ a₁, a₁)
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nat.cases_on a zero (λ a₁, a₁)
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protected definition is_inhabited [instance] : inhabited nat :=
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inhabited.mk zero
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@ -31,39 +31,39 @@ namespace nat
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protected definition has_decidable_eq [instance] : decidable_eq nat :=
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λn m : nat,
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have general : ∀n, decidable (n = m), from
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rec_on m
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(λ n, cases_on n
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nat.rec_on m
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(λ n, nat.cases_on n
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(inl rfl)
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(λ m, inr (λ (e : succ m = zero), down (no_confusion e))))
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(λ (m' : nat) (ih : ∀n, decidable (n = m')) (n : nat), cases_on n
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(inr (λ h, down (no_confusion h)))
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(λ m, inr (λ (e : succ m = zero), down (nat.no_confusion e))))
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(λ (m' : nat) (ih : ∀n, decidable (n = m')) (n : nat), nat.cases_on n
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(inr (λ h, down (nat.no_confusion h)))
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(λ (n' : nat),
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decidable.rec_on (ih n')
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(assume Heq : n' = m', inl (eq.rec_on Heq rfl))
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(assume Hne : n' ≠ m',
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have H1 : succ n' ≠ succ m', from
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assume Heq, down (no_confusion Heq (λ e : n' = m', Hne e)),
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assume Heq, down (nat.no_confusion Heq (λ e : n' = m', Hne e)),
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inr H1))),
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general n
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-- less-than is well-founded
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definition lt.wf [instance] : well_founded lt :=
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well_founded.intro (λn, rec_on n
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well_founded.intro (λn, nat.rec_on n
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(acc.intro zero (λ (y : nat) (hlt : y < zero),
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have aux : ∀ {n₁}, y < n₁ → zero = n₁ → acc lt y, from
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λ n₁ hlt, lt.cases_on hlt
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(λ heq, down (no_confusion heq))
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(λ b hlt heq, down (no_confusion heq)),
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(λ heq, down (nat.no_confusion heq))
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(λ b hlt heq, down (nat.no_confusion heq)),
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aux hlt rfl))
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(λ (n : nat) (ih : acc lt n),
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acc.intro (succ n) (λ (m : nat) (hlt : m < succ n),
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have aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m, from
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λ n₁ hlt, lt.cases_on hlt
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(λ (heq : succ n = succ m),
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down (no_confusion heq (λ (e : n = m),
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down (nat.no_confusion heq (λ (e : n = m),
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eq.rec_on e ih)))
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(λ b (hlt : m < b) (heq : succ n = succ b),
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down (no_confusion heq (λ (e : n = b),
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down (nat.no_confusion heq (λ (e : n = b),
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acc.inv (eq.rec_on e ih) hlt))),
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aux hlt rfl)))
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@ -76,12 +76,12 @@ namespace nat
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definition not_lt_zero (a : nat) : ¬ a < zero :=
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have aux : ∀ {b}, a < b → b = zero → empty, from
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λ b H, lt.cases_on H
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(λ heq, down (no_confusion heq))
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(λ b h₁ heq, down (no_confusion heq)),
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(λ heq, down (nat.no_confusion heq))
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(λ b h₁ heq, down (nat.no_confusion heq)),
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λ H, aux H rfl
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definition zero_lt_succ (a : nat) : zero < succ a :=
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rec_on a
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nat.rec_on a
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(lt.base zero)
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(λ a (hlt : zero < succ a), lt.step hlt)
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@ -114,9 +114,9 @@ namespace nat
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aux
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definition lt.is_decidable_rel [instance] : decidable_rel lt :=
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λ a b, rec_on b
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λ a b, nat.rec_on b
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(λ (a : nat), inr (not_lt_zero a))
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(λ (b₁ : nat) (ih : ∀ a, decidable (a < b₁)) (a : nat), cases_on a
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(λ (b₁ : nat) (ih : ∀ a, decidable (a < b₁)) (a : nat), nat.cases_on a
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(inl !zero_lt_succ)
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(λ a, decidable.rec_on (ih a)
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(λ h_pos : a < b₁, inl (lt.succ_of_lt h_pos))
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@ -135,8 +135,8 @@ namespace nat
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definition eq_or_lt_of_le {a b : nat} (H : a ≤ b) : sum (a = b) (a < b) :=
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have aux : Π (a₁ b₁ : nat) (hlt : a₁ < b₁), a₁ = a → b₁ = (succ b) → sum (a = b) (a < b), from
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λ a₁ b₁ hlt, lt.rec_on hlt
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(λ h₁, eq.rec_on h₁ (λ h₂, down (no_confusion h₂ (λ h₃, eq.rec_on h₃ (sum.inl rfl)))))
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(λ b₁ hlt ih h₁, eq.rec_on h₁ (λ h₂, down (no_confusion h₂ (λ h₃, eq.rec_on h₃ (sum.inr hlt))))),
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(λ h₁, eq.rec_on h₁ (λ h₂, down (nat.no_confusion h₂ (λ h₃, eq.rec_on h₃ (sum.inl rfl)))))
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(λ b₁ hlt ih h₁, eq.rec_on h₁ (λ h₂, down (nat.no_confusion h₂ (λ h₃, eq.rec_on h₃ (sum.inr hlt))))),
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aux a (succ b) H rfl rfl
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definition le.of_eq_or_lt {a b : nat} (H : sum (a = b) (a < b)) : a ≤ b :=
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@ -155,7 +155,7 @@ namespace nat
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end
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definition lt.irrefl (a : nat) : ¬ a < a :=
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rec_on a
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nat.rec_on a
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!not_lt_zero
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(λ (a : nat) (ih : ¬ a < a) (h : succ a < succ a),
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ih (lt.of_succ_lt_succ h))
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@ -166,11 +166,11 @@ namespace nat
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(λ b hlt (ih : ¬ b < a) (h : succ b < a), ih (lt.of_succ_lt h))
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definition lt.trichotomy (a b : nat) : a < b ⊎ a = b ⊎ b < a :=
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rec_on b
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(λa, cases_on a
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nat.rec_on b
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(λa, nat.cases_on a
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(sum.inr (sum.inl rfl))
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(λ a₁, sum.inr (sum.inr !zero_lt_succ)))
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(λ b₁ (ih : ∀a, a < b₁ ⊎ a = b₁ ⊎ b₁ < a) (a : nat), cases_on a
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(λ b₁ (ih : ∀a, a < b₁ ⊎ a = b₁ ⊎ b₁ < a) (a : nat), nat.cases_on a
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(sum.inl !zero_lt_succ)
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(λ a, sum.rec_on (ih a)
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(λ h : a < b₁, sum.inl (lt.succ_of_lt h))
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@ -282,23 +282,23 @@ namespace nat
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notation a ≥ b := ge a b
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definition add (a b : nat) : nat :=
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rec_on b a (λ b₁ r, succ r)
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nat.rec_on b a (λ b₁ r, succ r)
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notation a + b := add a b
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definition sub (a b : nat) : nat :=
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rec_on b a (λ b₁ r, pred r)
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nat.rec_on b a (λ b₁ r, pred r)
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notation a - b := sub a b
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definition mul (a b : nat) : nat :=
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rec_on b zero (λ b₁ r, r + a)
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nat.rec_on b zero (λ b₁ r, r + a)
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notation a * b := mul a b
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local attribute sub [reducible]
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definition succ_sub_succ_eq_sub (a b : nat) : succ a - succ b = a - b :=
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rec_on b
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nat.rec_on b
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rfl
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(λ b₁ (ih : succ a - succ b₁ = a - b₁),
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eq.rec_on ih (eq.refl (pred (succ a - succ b₁))))
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@ -307,7 +307,7 @@ namespace nat
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eq.rec_on (succ_sub_succ_eq_sub a b) rfl
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definition zero_sub_eq_zero (a : nat) : zero - a = zero :=
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rec_on a
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nat.rec_on a
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rfl
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(λ a₁ (ih : zero - a₁ = zero), calc
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zero - succ a₁ = pred (zero - a₁) : rfl
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@ -333,12 +333,12 @@ namespace nat
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λ h₁ h₂, aux h₁ h₂
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definition pred_le (a : nat) : pred a ≤ a :=
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cases_on a
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nat.cases_on a
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(le.refl zero)
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(λ a₁, le.of_lt (lt.base a₁))
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definition sub_le (a b : nat) : a - b ≤ a :=
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rec_on b
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nat.rec_on b
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(le.refl a)
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(λ b₁ ih, le.trans !pred_le ih)
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@ -13,25 +13,25 @@ inhabited.mk pos_num.one
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namespace pos_num
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definition succ (a : pos_num) : pos_num :=
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rec_on a (bit0 one) (λn r, bit0 r) (λn r, bit1 n)
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pos_num.rec_on a (bit0 one) (λn r, bit0 r) (λn r, bit1 n)
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definition is_one (a : pos_num) : bool :=
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rec_on a tt (λn r, ff) (λn r, ff)
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pos_num.rec_on a tt (λn r, ff) (λn r, ff)
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definition pred (a : pos_num) : pos_num :=
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rec_on a one (λn r, bit0 n) (λn r, cond (is_one n) one (bit1 r))
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pos_num.rec_on a one (λn r, bit0 n) (λn r, cond (is_one n) one (bit1 r))
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definition size (a : pos_num) : pos_num :=
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rec_on a one (λn r, succ r) (λn r, succ r)
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pos_num.rec_on a one (λn r, succ r) (λn r, succ r)
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definition add (a b : pos_num) : pos_num :=
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rec_on a
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pos_num.rec_on a
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||||
succ
|
||||
(λn f b, rec_on b
|
||||
(λn f b, pos_num.rec_on b
|
||||
(succ (bit1 n))
|
||||
(λm r, succ (bit1 (f m)))
|
||||
(λm r, bit1 (f m)))
|
||||
(λn f b, rec_on b
|
||||
(λn f b, pos_num.rec_on b
|
||||
(bit1 n)
|
||||
(λm r, bit1 (f m))
|
||||
(λm r, bit0 (f m)))
|
||||
|
|
@ -40,7 +40,7 @@ namespace pos_num
|
|||
notation a + b := add a b
|
||||
|
||||
definition mul (a b : pos_num) : pos_num :=
|
||||
rec_on a
|
||||
pos_num.rec_on a
|
||||
b
|
||||
(λn r, bit0 r + b)
|
||||
(λn r, bit0 r)
|
||||
|
|
@ -55,19 +55,19 @@ inhabited.mk num.zero
|
|||
namespace num
|
||||
open pos_num
|
||||
definition succ (a : num) : num :=
|
||||
rec_on a (pos one) (λp, pos (succ p))
|
||||
num.rec_on a (pos one) (λp, pos (succ p))
|
||||
|
||||
definition pred (a : num) : num :=
|
||||
rec_on a zero (λp, cond (is_one p) zero (pos (pred p)))
|
||||
num.rec_on a zero (λp, cond (is_one p) zero (pos (pred p)))
|
||||
|
||||
definition size (a : num) : num :=
|
||||
rec_on a (pos one) (λp, pos (size p))
|
||||
num.rec_on a (pos one) (λp, pos (size p))
|
||||
|
||||
definition add (a b : num) : num :=
|
||||
rec_on a b (λp_a, rec_on b (pos p_a) (λp_b, pos (pos_num.add p_a p_b)))
|
||||
num.rec_on a b (λp_a, num.rec_on b (pos p_a) (λp_b, pos (pos_num.add p_a p_b)))
|
||||
|
||||
definition mul (a b : num) : num :=
|
||||
rec_on a zero (λp_a, rec_on b zero (λp_b, pos (pos_num.mul p_a p_b)))
|
||||
num.rec_on a zero (λp_a, num.rec_on b zero (λp_b, pos (pos_num.mul p_a p_b)))
|
||||
|
||||
notation a + b := add a b
|
||||
notation a * b := mul a b
|
||||
|
|
|
|||
|
|
@ -48,128 +48,128 @@ namespace eq
|
|||
|
||||
-- The identity path is a right unit.
|
||||
definition concat_p1 (p : x = y) : p ⬝ idp = p :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
-- The identity path is a right unit.
|
||||
definition concat_1p (p : x = y) : idp ⬝ p = p :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
-- Concatenation is associative.
|
||||
definition concat_p_pp (p : x = y) (q : y = z) (r : z = t) :
|
||||
p ⬝ (q ⬝ r) = (p ⬝ q) ⬝ r :=
|
||||
rec_on r (rec_on q idp)
|
||||
eq.rec_on r (eq.rec_on q idp)
|
||||
|
||||
definition concat_pp_p (p : x = y) (q : y = z) (r : z = t) :
|
||||
(p ⬝ q) ⬝ r = p ⬝ (q ⬝ r) :=
|
||||
rec_on r (rec_on q idp)
|
||||
eq.rec_on r (eq.rec_on q idp)
|
||||
|
||||
-- The left inverse law.
|
||||
definition concat_pV (p : x = y) : p ⬝ p⁻¹ = idp :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
-- The right inverse law.
|
||||
definition concat_Vp (p : x = y) : p⁻¹ ⬝ p = idp :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
-- Several auxiliary theorems about canceling inverses across associativity. These are somewhat
|
||||
-- redundant, following from earlier theorems.
|
||||
|
||||
definition concat_V_pp (p : x = y) (q : y = z) : p⁻¹ ⬝ (p ⬝ q) = q :=
|
||||
rec_on q (rec_on p idp)
|
||||
eq.rec_on q (eq.rec_on p idp)
|
||||
|
||||
definition concat_p_Vp (p : x = y) (q : x = z) : p ⬝ (p⁻¹ ⬝ q) = q :=
|
||||
rec_on q (rec_on p idp)
|
||||
eq.rec_on q (eq.rec_on p idp)
|
||||
|
||||
definition concat_pp_V (p : x = y) (q : y = z) : (p ⬝ q) ⬝ q⁻¹ = p :=
|
||||
rec_on q (rec_on p idp)
|
||||
eq.rec_on q (eq.rec_on p idp)
|
||||
|
||||
definition concat_pV_p (p : x = z) (q : y = z) : (p ⬝ q⁻¹) ⬝ q = p :=
|
||||
rec_on q (take p, rec_on p idp) p
|
||||
eq.rec_on q (take p, eq.rec_on p idp) p
|
||||
|
||||
-- Inverse distributes over concatenation
|
||||
definition inv_pp (p : x = y) (q : y = z) : (p ⬝ q)⁻¹ = q⁻¹ ⬝ p⁻¹ :=
|
||||
rec_on q (rec_on p idp)
|
||||
eq.rec_on q (eq.rec_on p idp)
|
||||
|
||||
definition inv_Vp (p : y = x) (q : y = z) : (p⁻¹ ⬝ q)⁻¹ = q⁻¹ ⬝ p :=
|
||||
rec_on q (rec_on p idp)
|
||||
eq.rec_on q (eq.rec_on p idp)
|
||||
|
||||
-- universe metavariables
|
||||
definition inv_pV (p : x = y) (q : z = y) : (p ⬝ q⁻¹)⁻¹ = q ⬝ p⁻¹ :=
|
||||
rec_on p (take q, rec_on q idp) q
|
||||
eq.rec_on p (take q, eq.rec_on q idp) q
|
||||
|
||||
definition inv_VV (p : y = x) (q : z = y) : (p⁻¹ ⬝ q⁻¹)⁻¹ = q ⬝ p :=
|
||||
rec_on p (rec_on q idp)
|
||||
eq.rec_on p (eq.rec_on q idp)
|
||||
|
||||
-- Inverse is an involution.
|
||||
definition inv_V (p : x = y) : p⁻¹⁻¹ = p :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
-- Theorems for moving things around in equations
|
||||
-- ----------------------------------------------
|
||||
|
||||
definition moveR_Mp (p : x = z) (q : y = z) (r : y = x) :
|
||||
p = (r⁻¹ ⬝ q) → (r ⬝ p) = q :=
|
||||
rec_on r (take p h, concat_1p _ ⬝ h ⬝ concat_1p _) p
|
||||
eq.rec_on r (take p h, concat_1p _ ⬝ h ⬝ concat_1p _) p
|
||||
|
||||
definition moveR_pM (p : x = z) (q : y = z) (r : y = x) :
|
||||
r = q ⬝ p⁻¹ → r ⬝ p = q :=
|
||||
rec_on p (take q h, (concat_p1 _ ⬝ h ⬝ concat_p1 _)) q
|
||||
eq.rec_on p (take q h, (concat_p1 _ ⬝ h ⬝ concat_p1 _)) q
|
||||
|
||||
definition moveR_Vp (p : x = z) (q : y = z) (r : x = y) :
|
||||
p = r ⬝ q → r⁻¹ ⬝ p = q :=
|
||||
rec_on r (take q h, concat_1p _ ⬝ h ⬝ concat_1p _) q
|
||||
eq.rec_on r (take q h, concat_1p _ ⬝ h ⬝ concat_1p _) q
|
||||
|
||||
definition moveR_pV (p : z = x) (q : y = z) (r : y = x) :
|
||||
r = q ⬝ p → r ⬝ p⁻¹ = q :=
|
||||
rec_on p (take r h, concat_p1 _ ⬝ h ⬝ concat_p1 _) r
|
||||
eq.rec_on p (take r h, concat_p1 _ ⬝ h ⬝ concat_p1 _) r
|
||||
|
||||
definition moveL_Mp (p : x = z) (q : y = z) (r : y = x) :
|
||||
r⁻¹ ⬝ q = p → q = r ⬝ p :=
|
||||
rec_on r (take p h, (concat_1p _)⁻¹ ⬝ h ⬝ (concat_1p _)⁻¹) p
|
||||
eq.rec_on r (take p h, (concat_1p _)⁻¹ ⬝ h ⬝ (concat_1p _)⁻¹) p
|
||||
|
||||
definition moveL_pM (p : x = z) (q : y = z) (r : y = x) :
|
||||
q ⬝ p⁻¹ = r → q = r ⬝ p :=
|
||||
rec_on p (take q h, (concat_p1 _)⁻¹ ⬝ h ⬝ (concat_p1 _)⁻¹) q
|
||||
eq.rec_on p (take q h, (concat_p1 _)⁻¹ ⬝ h ⬝ (concat_p1 _)⁻¹) q
|
||||
|
||||
definition moveL_Vp (p : x = z) (q : y = z) (r : x = y) :
|
||||
r ⬝ q = p → q = r⁻¹ ⬝ p :=
|
||||
rec_on r (take q h, (concat_1p _)⁻¹ ⬝ h ⬝ (concat_1p _)⁻¹) q
|
||||
eq.rec_on r (take q h, (concat_1p _)⁻¹ ⬝ h ⬝ (concat_1p _)⁻¹) q
|
||||
|
||||
definition moveL_pV (p : z = x) (q : y = z) (r : y = x) :
|
||||
q ⬝ p = r → q = r ⬝ p⁻¹ :=
|
||||
rec_on p (take r h, (concat_p1 _)⁻¹ ⬝ h ⬝ (concat_p1 _)⁻¹) r
|
||||
eq.rec_on p (take r h, (concat_p1 _)⁻¹ ⬝ h ⬝ (concat_p1 _)⁻¹) r
|
||||
|
||||
definition moveL_1M (p q : x = y) :
|
||||
p ⬝ q⁻¹ = idp → p = q :=
|
||||
rec_on q (take p h, (concat_p1 _)⁻¹ ⬝ h) p
|
||||
eq.rec_on q (take p h, (concat_p1 _)⁻¹ ⬝ h) p
|
||||
|
||||
definition moveL_M1 (p q : x = y) :
|
||||
q⁻¹ ⬝ p = idp → p = q :=
|
||||
rec_on q (take p h, (concat_1p _)⁻¹ ⬝ h) p
|
||||
eq.rec_on q (take p h, (concat_1p _)⁻¹ ⬝ h) p
|
||||
|
||||
definition moveL_1V (p : x = y) (q : y = x) :
|
||||
p ⬝ q = idp → p = q⁻¹ :=
|
||||
rec_on q (take p h, (concat_p1 _)⁻¹ ⬝ h) p
|
||||
eq.rec_on q (take p h, (concat_p1 _)⁻¹ ⬝ h) p
|
||||
|
||||
definition moveL_V1 (p : x = y) (q : y = x) :
|
||||
q ⬝ p = idp → p = q⁻¹ :=
|
||||
rec_on q (take p h, (concat_1p _)⁻¹ ⬝ h) p
|
||||
eq.rec_on q (take p h, (concat_1p _)⁻¹ ⬝ h) p
|
||||
|
||||
definition moveR_M1 (p q : x = y) :
|
||||
idp = p⁻¹ ⬝ q → p = q :=
|
||||
rec_on p (take q h, h ⬝ (concat_1p _)) q
|
||||
eq.rec_on p (take q h, h ⬝ (concat_1p _)) q
|
||||
|
||||
definition moveR_1M (p q : x = y) :
|
||||
idp = q ⬝ p⁻¹ → p = q :=
|
||||
rec_on p (take q h, h ⬝ (concat_p1 _)) q
|
||||
eq.rec_on p (take q h, h ⬝ (concat_p1 _)) q
|
||||
|
||||
definition moveR_1V (p : x = y) (q : y = x) :
|
||||
idp = q ⬝ p → p⁻¹ = q :=
|
||||
rec_on p (take q h, h ⬝ (concat_p1 _)) q
|
||||
eq.rec_on p (take q h, h ⬝ (concat_p1 _)) q
|
||||
|
||||
definition moveR_V1 (p : x = y) (q : y = x) :
|
||||
idp = p ⬝ q → p⁻¹ = q :=
|
||||
rec_on p (take q h, h ⬝ (concat_1p _)) q
|
||||
eq.rec_on p (take q h, h ⬝ (concat_1p _)) q
|
||||
|
||||
|
||||
-- Transport
|
||||
|
|
@ -197,10 +197,10 @@ namespace eq
|
|||
definition ap10 {f g : A → B} (H : f = g) : f ∼ g := apD10 H
|
||||
|
||||
definition ap11 {f g : A → B} (H : f = g) {x y : A} (p : x = y) : f x = g y :=
|
||||
rec_on H (rec_on p idp)
|
||||
eq.rec_on H (eq.rec_on p idp)
|
||||
|
||||
definition apD (f : Πa:A, P a) {x y : A} (p : x = y) : p ▹ (f x) = f y :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
|
||||
-- calc enviroment
|
||||
|
|
@ -216,19 +216,19 @@ namespace eq
|
|||
|
||||
definition moveR_transport_p (P : A → Type) {x y : A} (p : x = y) (u : P x) (v : P y) :
|
||||
u = p⁻¹ ▹ v → p ▹ u = v :=
|
||||
rec_on p (take v, id) v
|
||||
eq.rec_on p (take v, id) v
|
||||
|
||||
definition moveR_transport_V (P : A → Type) {x y : A} (p : y = x) (u : P x) (v : P y) :
|
||||
u = p ▹ v → p⁻¹ ▹ u = v :=
|
||||
rec_on p (take u, id) u
|
||||
eq.rec_on p (take u, id) u
|
||||
|
||||
definition moveL_transport_V (P : A → Type) {x y : A} (p : x = y) (u : P x) (v : P y) :
|
||||
p ▹ u = v → u = p⁻¹ ▹ v :=
|
||||
rec_on p (take v, id) v
|
||||
eq.rec_on p (take v, id) v
|
||||
|
||||
definition moveL_transport_p (P : A → Type) {x y : A} (p : y = x) (u : P x) (v : P y) :
|
||||
p⁻¹ ▹ u = v → u = p ▹ v :=
|
||||
rec_on p (take u, id) u
|
||||
eq.rec_on p (take u, id) u
|
||||
|
||||
-- Functoriality of functions
|
||||
-- --------------------------
|
||||
|
|
@ -244,111 +244,111 @@ namespace eq
|
|||
-- Functions commute with concatenation.
|
||||
definition ap_pp (f : A → B) {x y z : A} (p : x = y) (q : y = z) :
|
||||
ap f (p ⬝ q) = (ap f p) ⬝ (ap f q) :=
|
||||
rec_on q (rec_on p idp)
|
||||
eq.rec_on q (eq.rec_on p idp)
|
||||
|
||||
definition ap_p_pp (f : A → B) {w x y z : A} (r : f w = f x) (p : x = y) (q : y = z) :
|
||||
r ⬝ (ap f (p ⬝ q)) = (r ⬝ ap f p) ⬝ (ap f q) :=
|
||||
rec_on q (take p, rec_on p (concat_p_pp r idp idp)) p
|
||||
eq.rec_on q (take p, eq.rec_on p (concat_p_pp r idp idp)) p
|
||||
|
||||
definition ap_pp_p (f : A → B) {w x y z : A} (p : x = y) (q : y = z) (r : f z = f w) :
|
||||
(ap f (p ⬝ q)) ⬝ r = (ap f p) ⬝ (ap f q ⬝ r) :=
|
||||
rec_on q (rec_on p (take r, concat_pp_p _ _ _)) r
|
||||
eq.rec_on q (eq.rec_on p (take r, concat_pp_p _ _ _)) r
|
||||
|
||||
-- Functions commute with path inverses.
|
||||
definition inverse_ap (f : A → B) {x y : A} (p : x = y) : (ap f p)⁻¹ = ap f (p⁻¹) :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
definition ap_V {A B : Type} (f : A → B) {x y : A} (p : x = y) : ap f (p⁻¹) = (ap f p)⁻¹ :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
-- [ap] itself is functorial in the first argument.
|
||||
|
||||
definition ap_idmap (p : x = y) : ap id p = p :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
definition ap_compose (f : A → B) (g : B → C) {x y : A} (p : x = y) :
|
||||
ap (g ∘ f) p = ap g (ap f p) :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
-- Sometimes we don't have the actual function [compose].
|
||||
definition ap_compose' (f : A → B) (g : B → C) {x y : A} (p : x = y) :
|
||||
ap (λa, g (f a)) p = ap g (ap f p) :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
-- The action of constant maps.
|
||||
definition ap_const (p : x = y) (z : B) :
|
||||
ap (λu, z) p = idp :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
-- Naturality of [ap].
|
||||
definition concat_Ap {f g : A → B} (p : Π x, f x = g x) {x y : A} (q : x = y) :
|
||||
(ap f q) ⬝ (p y) = (p x) ⬝ (ap g q) :=
|
||||
rec_on q (concat_1p _ ⬝ (concat_p1 _)⁻¹)
|
||||
eq.rec_on q (concat_1p _ ⬝ (concat_p1 _)⁻¹)
|
||||
|
||||
-- Naturality of [ap] at identity.
|
||||
definition concat_A1p {f : A → A} (p : Πx, f x = x) {x y : A} (q : x = y) :
|
||||
(ap f q) ⬝ (p y) = (p x) ⬝ q :=
|
||||
rec_on q (concat_1p _ ⬝ (concat_p1 _)⁻¹)
|
||||
eq.rec_on q (concat_1p _ ⬝ (concat_p1 _)⁻¹)
|
||||
|
||||
definition concat_pA1 {f : A → A} (p : Πx, x = f x) {x y : A} (q : x = y) :
|
||||
(p x) ⬝ (ap f q) = q ⬝ (p y) :=
|
||||
rec_on q (concat_p1 _ ⬝ (concat_1p _)⁻¹)
|
||||
eq.rec_on q (concat_p1 _ ⬝ (concat_1p _)⁻¹)
|
||||
|
||||
-- Naturality with other paths hanging around.
|
||||
|
||||
definition concat_pA_pp {f g : A → B} (p : Πx, f x = g x) {x y : A} (q : x = y)
|
||||
{w z : B} (r : w = f x) (s : g y = z) :
|
||||
(r ⬝ ap f q) ⬝ (p y ⬝ s) = (r ⬝ p x) ⬝ (ap g q ⬝ s) :=
|
||||
rec_on s (rec_on q idp)
|
||||
eq.rec_on s (eq.rec_on q idp)
|
||||
|
||||
definition concat_pA_p {f g : A → B} (p : Πx, f x = g x) {x y : A} (q : x = y)
|
||||
{w : B} (r : w = f x) :
|
||||
(r ⬝ ap f q) ⬝ p y = (r ⬝ p x) ⬝ ap g q :=
|
||||
rec_on q idp
|
||||
eq.rec_on q idp
|
||||
|
||||
-- TODO: try this using the simplifier, and compare proofs
|
||||
definition concat_A_pp {f g : A → B} (p : Πx, f x = g x) {x y : A} (q : x = y)
|
||||
{z : B} (s : g y = z) :
|
||||
(ap f q) ⬝ (p y ⬝ s) = (p x) ⬝ (ap g q ⬝ s) :=
|
||||
rec_on s (rec_on q
|
||||
eq.rec_on s (eq.rec_on q
|
||||
(calc
|
||||
(ap f idp) ⬝ (p x ⬝ idp) = idp ⬝ p x : idp
|
||||
... = p x : concat_1p _
|
||||
... = (p x) ⬝ (ap g idp ⬝ idp) : idp))
|
||||
-- This also works:
|
||||
-- rec_on s (rec_on q (concat_1p _ ▹ idp))
|
||||
-- eq.rec_on s (eq.rec_on q (concat_1p _ ▹ idp))
|
||||
|
||||
definition concat_pA1_pp {f : A → A} (p : Πx, f x = x) {x y : A} (q : x = y)
|
||||
{w z : A} (r : w = f x) (s : y = z) :
|
||||
(r ⬝ ap f q) ⬝ (p y ⬝ s) = (r ⬝ p x) ⬝ (q ⬝ s) :=
|
||||
rec_on s (rec_on q idp)
|
||||
eq.rec_on s (eq.rec_on q idp)
|
||||
|
||||
definition concat_pp_A1p {g : A → A} (p : Πx, x = g x) {x y : A} (q : x = y)
|
||||
{w z : A} (r : w = x) (s : g y = z) :
|
||||
(r ⬝ p x) ⬝ (ap g q ⬝ s) = (r ⬝ q) ⬝ (p y ⬝ s) :=
|
||||
rec_on s (rec_on q idp)
|
||||
eq.rec_on s (eq.rec_on q idp)
|
||||
|
||||
definition concat_pA1_p {f : A → A} (p : Πx, f x = x) {x y : A} (q : x = y)
|
||||
{w : A} (r : w = f x) :
|
||||
(r ⬝ ap f q) ⬝ p y = (r ⬝ p x) ⬝ q :=
|
||||
rec_on q idp
|
||||
eq.rec_on q idp
|
||||
|
||||
definition concat_A1_pp {f : A → A} (p : Πx, f x = x) {x y : A} (q : x = y)
|
||||
{z : A} (s : y = z) :
|
||||
(ap f q) ⬝ (p y ⬝ s) = (p x) ⬝ (q ⬝ s) :=
|
||||
rec_on s (rec_on q (concat_1p _ ▹ idp))
|
||||
eq.rec_on s (eq.rec_on q (concat_1p _ ▹ idp))
|
||||
|
||||
definition concat_pp_A1 {g : A → A} (p : Πx, x = g x) {x y : A} (q : x = y)
|
||||
{w : A} (r : w = x) :
|
||||
(r ⬝ p x) ⬝ ap g q = (r ⬝ q) ⬝ p y :=
|
||||
rec_on q idp
|
||||
eq.rec_on q idp
|
||||
|
||||
definition concat_p_A1p {g : A → A} (p : Πx, x = g x) {x y : A} (q : x = y)
|
||||
{z : A} (s : g y = z) :
|
||||
p x ⬝ (ap g q ⬝ s) = q ⬝ (p y ⬝ s) :=
|
||||
begin
|
||||
apply (rec_on s),
|
||||
apply (rec_on q),
|
||||
apply (eq.rec_on s),
|
||||
apply (eq.rec_on q),
|
||||
apply (concat_1p (p x) ▹ idp)
|
||||
end
|
||||
|
||||
|
|
@ -361,11 +361,11 @@ namespace eq
|
|||
|
||||
definition apD10_pp {f f' f'' : Πx, P x} (h : f = f') (h' : f' = f'') (x : A) :
|
||||
apD10 (h ⬝ h') x = apD10 h x ⬝ apD10 h' x :=
|
||||
rec_on h (take h', rec_on h' idp) h'
|
||||
eq.rec_on h (take h', eq.rec_on h' idp) h'
|
||||
|
||||
definition apD10_V {f g : Πx : A, P x} (h : f = g) (x : A) :
|
||||
apD10 (h⁻¹) x = (apD10 h x)⁻¹ :=
|
||||
rec_on h idp
|
||||
eq.rec_on h idp
|
||||
|
||||
definition ap10_1 {f : A → B} (x : A) : ap10 (refl f) x = idp := idp
|
||||
|
||||
|
|
@ -378,7 +378,7 @@ namespace eq
|
|||
-- [ap10] also behaves nicely on paths produced by [ap]
|
||||
definition ap_ap10 (f g : A → B) (h : B → C) (p : f = g) (a : A) :
|
||||
ap h (ap10 p a) = ap10 (ap (λ f', h ∘ f') p) a:=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
|
||||
-- Transport and the groupoid structure of paths
|
||||
|
|
@ -389,7 +389,7 @@ namespace eq
|
|||
|
||||
definition transport_pp (P : A → Type) {x y z : A} (p : x = y) (q : y = z) (u : P x) :
|
||||
p ⬝ q ▹ u = q ▹ p ▹ u :=
|
||||
rec_on q (rec_on p idp)
|
||||
eq.rec_on q (eq.rec_on p idp)
|
||||
|
||||
definition transport_pV (P : A → Type) {x y : A} (p : x = y) (z : P y) :
|
||||
p ▹ p⁻¹ ▹ z = z :=
|
||||
|
|
@ -405,18 +405,18 @@ namespace eq
|
|||
ap (transport P r) (transport_pp P p q u)
|
||||
= (transport_pp P p (q ⬝ r) u) ⬝ (transport_pp P q r (p ▹ u))
|
||||
:> ((p ⬝ (q ⬝ r)) ▹ u = r ▹ q ▹ p ▹ u) :=
|
||||
rec_on r (rec_on q (rec_on p idp))
|
||||
eq.rec_on r (eq.rec_on q (eq.rec_on p idp))
|
||||
|
||||
-- Here is another coherence lemma for transport.
|
||||
definition transport_pVp (P : A → Type) {x y : A} (p : x = y) (z : P x) :
|
||||
transport_pV P p (transport P p z) = ap (transport P p) (transport_Vp P p z) :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
-- Dependent transport in a doubly dependent type.
|
||||
-- should P, Q and y all be explicit here?
|
||||
definition transportD (P : A → Type) (Q : Π a : A, P a → Type)
|
||||
{a a' : A} (p : a = a') (b : P a) (z : Q a b) : Q a' (p ▹ b) :=
|
||||
rec_on p z
|
||||
eq.rec_on p z
|
||||
-- In Coq the variables B, C and y are explicit, but in Lean we can probably have them implicit using the following notation
|
||||
notation p `▹D`:65 x:64 := transportD _ _ p _ x
|
||||
|
||||
|
|
@ -431,32 +431,32 @@ namespace eq
|
|||
definition transport2_is_ap10 (Q : A → Type) {x y : A} {p q : x = y} (r : p = q)
|
||||
(z : Q x) :
|
||||
transport2 Q r z = ap10 (ap (transport Q) r) z :=
|
||||
rec_on r idp
|
||||
eq.rec_on r idp
|
||||
|
||||
definition transport2_p2p (P : A → Type) {x y : A} {p1 p2 p3 : x = y}
|
||||
(r1 : p1 = p2) (r2 : p2 = p3) (z : P x) :
|
||||
transport2 P (r1 ⬝ r2) z = transport2 P r1 z ⬝ transport2 P r2 z :=
|
||||
rec_on r1 (rec_on r2 idp)
|
||||
eq.rec_on r1 (eq.rec_on r2 idp)
|
||||
|
||||
definition transport2_V (Q : A → Type) {x y : A} {p q : x = y} (r : p = q) (z : Q x) :
|
||||
transport2 Q (r⁻¹) z = ((transport2 Q r z)⁻¹) :=
|
||||
rec_on r idp
|
||||
eq.rec_on r idp
|
||||
|
||||
definition transportD2 (B C : A → Type) (D : Π(a:A), B a → C a → Type)
|
||||
{x1 x2 : A} (p : x1 = x2) (y : B x1) (z : C x1) (w : D x1 y z) : D x2 (p ▹ y) (p ▹ z) :=
|
||||
rec_on p w
|
||||
eq.rec_on p w
|
||||
|
||||
notation p `▹D2`:65 x:64 := transportD2 _ _ _ p _ _ x
|
||||
|
||||
definition concat_AT (P : A → Type) {x y : A} {p q : x = y} {z w : P x} (r : p = q)
|
||||
(s : z = w) :
|
||||
ap (transport P p) s ⬝ transport2 P r w = transport2 P r z ⬝ ap (transport P q) s :=
|
||||
rec_on r (concat_p1 _ ⬝ (concat_1p _)⁻¹)
|
||||
eq.rec_on r (concat_p1 _ ⬝ (concat_1p _)⁻¹)
|
||||
|
||||
-- TODO (from Coq library): What should this be called?
|
||||
definition ap_transport {P Q : A → Type} {x y : A} (p : x = y) (f : Πx, P x → Q x) (z : P x) :
|
||||
f y (p ▹ z) = (p ▹ (f x z)) :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
|
||||
-- Transporting in particular fibrations
|
||||
|
|
@ -473,34 +473,34 @@ namespace eq
|
|||
|
||||
-- Transporting in a constant fibration.
|
||||
definition transport_const (p : x = y) (z : B) : transport (λx, B) p z = z :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
definition transport2_const {p q : x = y} (r : p = q) (z : B) :
|
||||
transport_const p z = transport2 (λu, B) r z ⬝ transport_const q z :=
|
||||
rec_on r (concat_1p _)⁻¹
|
||||
eq.rec_on r (concat_1p _)⁻¹
|
||||
|
||||
-- Transporting in a pulled back fibration.
|
||||
-- TODO: P can probably be implicit
|
||||
definition transport_compose (P : B → Type) (f : A → B) (p : x = y) (z : P (f x)) :
|
||||
transport (P ∘ f) p z = transport P (ap f p) z :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
definition transport_precompose (f : A → B) (g g' : B → C) (p : g = g') :
|
||||
transport (λh : B → C, g ∘ f = h ∘ f) p idp = ap (λh, h ∘ f) p :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
definition apD10_ap_precompose (f : A → B) (g g' : B → C) (p : g = g') (a : A) :
|
||||
apD10 (ap (λh : B → C, h ∘ f) p) a = apD10 p (f a) :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
definition apD10_ap_postcompose (f : B → C) (g g' : A → B) (p : g = g') (a : A) :
|
||||
apD10 (ap (λh : A → B, f ∘ h) p) a = ap f (apD10 p a) :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
-- A special case of [transport_compose] which seems to come up a lot.
|
||||
definition transport_idmap_ap (P : A → Type) x y (p : x = y) (u : P x) :
|
||||
transport P p u = transport (λz, z) (ap P p) u :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
|
||||
-- The behavior of [ap] and [apD]
|
||||
|
|
@ -509,7 +509,7 @@ namespace eq
|
|||
-- In a constant fibration, [apD] reduces to [ap], modulo [transport_const].
|
||||
definition apD_const (f : A → B) (p: x = y) :
|
||||
apD f p = transport_const p (f x) ⬝ ap f p :=
|
||||
rec_on p idp
|
||||
eq.rec_on p idp
|
||||
|
||||
|
||||
-- The 2-dimensional groupoid structure
|
||||
|
|
@ -518,13 +518,13 @@ namespace eq
|
|||
-- Horizontal composition of 2-dimensional paths.
|
||||
definition concat2 {p p' : x = y} {q q' : y = z} (h : p = p') (h' : q = q') :
|
||||
p ⬝ q = p' ⬝ q' :=
|
||||
rec_on h (rec_on h' idp)
|
||||
eq.rec_on h (eq.rec_on h' idp)
|
||||
|
||||
infixl `◾`:75 := concat2
|
||||
|
||||
-- 2-dimensional path inversion
|
||||
definition inverse2 {p q : x = y} (h : p = q) : p⁻¹ = q⁻¹ :=
|
||||
rec_on h idp
|
||||
eq.rec_on h idp
|
||||
|
||||
|
||||
-- Whiskering
|
||||
|
|
@ -539,47 +539,47 @@ namespace eq
|
|||
-- Unwhiskering, a.k.a. cancelling
|
||||
|
||||
definition cancelL {x y z : A} (p : x = y) (q r : y = z) : (p ⬝ q = p ⬝ r) → (q = r) :=
|
||||
rec_on p (take r, rec_on r (take q a, (concat_1p q)⁻¹ ⬝ a)) r q
|
||||
eq.rec_on p (take r, eq.rec_on r (take q a, (concat_1p q)⁻¹ ⬝ a)) r q
|
||||
|
||||
definition cancelR {x y z : A} (p q : x = y) (r : y = z) : (p ⬝ r = q ⬝ r) → (p = q) :=
|
||||
rec_on r (rec_on p (take q a, a ⬝ concat_p1 q)) q
|
||||
eq.rec_on r (eq.rec_on p (take q a, a ⬝ concat_p1 q)) q
|
||||
|
||||
-- Whiskering and identity paths.
|
||||
|
||||
definition whiskerR_p1 {p q : x = y} (h : p = q) :
|
||||
(concat_p1 p)⁻¹ ⬝ whiskerR h idp ⬝ concat_p1 q = h :=
|
||||
rec_on h (rec_on p idp)
|
||||
eq.rec_on h (eq.rec_on p idp)
|
||||
|
||||
definition whiskerR_1p (p : x = y) (q : y = z) :
|
||||
whiskerR idp q = idp :> (p ⬝ q = p ⬝ q) :=
|
||||
rec_on q idp
|
||||
eq.rec_on q idp
|
||||
|
||||
definition whiskerL_p1 (p : x = y) (q : y = z) :
|
||||
whiskerL p idp = idp :> (p ⬝ q = p ⬝ q) :=
|
||||
rec_on q idp
|
||||
eq.rec_on q idp
|
||||
|
||||
definition whiskerL_1p {p q : x = y} (h : p = q) :
|
||||
(concat_1p p) ⁻¹ ⬝ whiskerL idp h ⬝ concat_1p q = h :=
|
||||
rec_on h (rec_on p idp)
|
||||
eq.rec_on h (eq.rec_on p idp)
|
||||
|
||||
definition concat2_p1 {p q : x = y} (h : p = q) :
|
||||
h ◾ idp = whiskerR h idp :> (p ⬝ idp = q ⬝ idp) :=
|
||||
rec_on h idp
|
||||
eq.rec_on h idp
|
||||
|
||||
definition concat2_1p {p q : x = y} (h : p = q) :
|
||||
idp ◾ h = whiskerL idp h :> (idp ⬝ p = idp ⬝ q) :=
|
||||
rec_on h idp
|
||||
eq.rec_on h idp
|
||||
|
||||
-- TODO: note, 4 inductions
|
||||
-- The interchange law for concatenation.
|
||||
definition concat_concat2 {p p' p'' : x = y} {q q' q'' : y = z}
|
||||
(a : p = p') (b : p' = p'') (c : q = q') (d : q' = q'') :
|
||||
(a ◾ c) ⬝ (b ◾ d) = (a ⬝ b) ◾ (c ⬝ d) :=
|
||||
rec_on d (rec_on c (rec_on b (rec_on a idp)))
|
||||
eq.rec_on d (eq.rec_on c (eq.rec_on b (eq.rec_on a idp)))
|
||||
|
||||
definition concat_whisker {x y z : A} (p p' : x = y) (q q' : y = z) (a : p = p') (b : q = q') :
|
||||
(whiskerR a q) ⬝ (whiskerL p' b) = (whiskerL p b) ⬝ (whiskerR a q') :=
|
||||
rec_on b (rec_on a (concat_1p _)⁻¹)
|
||||
eq.rec_on b (eq.rec_on a (concat_1p _)⁻¹)
|
||||
|
||||
-- Structure corresponding to the coherence equations of a bicategory.
|
||||
|
||||
|
|
@ -589,12 +589,12 @@ namespace eq
|
|||
⬝ concat_p_pp p (q ⬝ r) s
|
||||
⬝ whiskerR (concat_p_pp p q r) s
|
||||
= concat_p_pp p q (r ⬝ s) ⬝ concat_p_pp (p ⬝ q) r s :=
|
||||
rec_on s (rec_on r (rec_on q (rec_on p idp)))
|
||||
eq.rec_on s (eq.rec_on r (eq.rec_on q (eq.rec_on p idp)))
|
||||
|
||||
-- The 3-cell witnessing the left unit triangle.
|
||||
definition triangulator (p : x = y) (q : y = z) :
|
||||
concat_p_pp p idp q ⬝ whiskerR (concat_p1 p) q = whiskerL p (concat_1p q) :=
|
||||
rec_on q (rec_on p idp)
|
||||
eq.rec_on q (eq.rec_on p idp)
|
||||
|
||||
definition eckmann_hilton {x:A} (p q : idp = idp :> (x = x)) : p ⬝ q = q ⬝ p :=
|
||||
(!whiskerR_p1 ◾ !whiskerL_1p)⁻¹
|
||||
|
|
@ -607,23 +607,23 @@ namespace eq
|
|||
|
||||
-- The action of functions on 2-dimensional paths
|
||||
definition ap02 (f:A → B) {x y : A} {p q : x = y} (r : p = q) : ap f p = ap f q :=
|
||||
rec_on r idp
|
||||
eq.rec_on r idp
|
||||
|
||||
definition ap02_pp (f : A → B) {x y : A} {p p' p'' : x = y} (r : p = p') (r' : p' = p'') :
|
||||
ap02 f (r ⬝ r') = ap02 f r ⬝ ap02 f r' :=
|
||||
rec_on r (rec_on r' idp)
|
||||
eq.rec_on r (eq.rec_on r' idp)
|
||||
|
||||
definition ap02_p2p (f : A → B) {x y z : A} {p p' : x = y} {q q' :y = z} (r : p = p')
|
||||
(s : q = q') :
|
||||
ap02 f (r ◾ s) = ap_pp f p q
|
||||
⬝ (ap02 f r ◾ ap02 f s)
|
||||
⬝ (ap_pp f p' q')⁻¹ :=
|
||||
rec_on r (rec_on s (rec_on q (rec_on p idp)))
|
||||
-- rec_on r (rec_on s (rec_on p (rec_on q idp)))
|
||||
eq.rec_on r (eq.rec_on s (eq.rec_on q (eq.rec_on p idp)))
|
||||
-- eq.rec_on r (eq.rec_on s (eq.rec_on p (eq.rec_on q idp)))
|
||||
|
||||
definition apD02 {p q : x = y} (f : Π x, P x) (r : p = q) :
|
||||
apD f p = transport2 P r (f x) ⬝ apD f q :=
|
||||
rec_on r (concat_1p _)⁻¹
|
||||
eq.rec_on r (concat_1p _)⁻¹
|
||||
|
||||
-- And now for a lemma whose statement is much longer than its proof.
|
||||
definition apD02_pp (P : A → Type) (f : Π x:A, P x) {x y : A}
|
||||
|
|
@ -632,22 +632,22 @@ namespace eq
|
|||
⬝ whiskerL (transport2 P r1 (f x)) (apD02 f r2)
|
||||
⬝ concat_p_pp _ _ _
|
||||
⬝ (whiskerR ((transport2_p2p P r1 r2 (f x))⁻¹) (apD f p3)) :=
|
||||
rec_on r2 (rec_on r1 (rec_on p1 idp))
|
||||
eq.rec_on r2 (eq.rec_on r1 (eq.rec_on p1 idp))
|
||||
end eq
|
||||
|
||||
namespace eq
|
||||
variables {A B C D E : Type} {a a' : A} {b b' : B} {c c' : C} {d d' : D}
|
||||
|
||||
theorem congr_arg2 (f : A → B → C) (Ha : a = a') (Hb : b = b') : f a b = f a' b' :=
|
||||
rec_on Ha (rec_on Hb idp)
|
||||
eq.rec_on Ha (eq.rec_on Hb idp)
|
||||
|
||||
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' :=
|
||||
rec_on Ha (congr_arg2 (f a) Hb Hc)
|
||||
eq.rec_on Ha (congr_arg2 (f a) 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' :=
|
||||
rec_on Ha (congr_arg3 (f a) Hb Hc Hd)
|
||||
eq.rec_on Ha (congr_arg3 (f a) Hb Hc Hd)
|
||||
|
||||
end eq
|
||||
|
||||
|
|
@ -661,7 +661,7 @@ variables {a a' : A}
|
|||
|
||||
theorem dcongr_arg2 (f : Πa, B a → F) (Ha : a = a') (Hb : (Ha ▹ b) = b')
|
||||
: f a b = f a' b' :=
|
||||
rec_on Hb (rec_on Ha idp)
|
||||
eq.rec_on Hb (eq.rec_on Ha idp)
|
||||
|
||||
/- From the Coq version:
|
||||
|
||||
|
|
|
|||
|
|
@ -11,7 +11,7 @@ import ..datatypes ..logic
|
|||
namespace empty
|
||||
|
||||
protected theorem elim (A : Type) (H : empty) : A :=
|
||||
rec (λe, A) H
|
||||
empty.rec (λe, A) H
|
||||
|
||||
end empty
|
||||
|
||||
|
|
|
|||
|
|
@ -60,8 +60,8 @@ namespace prod
|
|||
(iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
|
||||
acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
|
||||
have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
|
||||
@lex.rec_on A B Ra Rb (λp₁ p₂ h, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
|
||||
p (xa, xb) lt
|
||||
@prod.lex.rec_on A B Ra Rb (λp₁ p₂ h, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
|
||||
p (xa, xb) lt
|
||||
(λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
|
||||
show acc (lex Ra Rb) (a₁, b₁), from
|
||||
have Ra₁ : Ra a₁ xa, from eq.rec_on eq₂ H,
|
||||
|
|
@ -79,7 +79,7 @@ namespace prod
|
|||
|
||||
-- Relational product is a subrelation of the lex
|
||||
definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
|
||||
λa b H, rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
|
||||
λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
|
||||
|
||||
-- The relational product of well founded relations is well-founded
|
||||
definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
|
||||
|
|
|
|||
|
|
@ -13,7 +13,7 @@ namespace acc
|
|||
variables {A : Type} {R : A → A → Type}
|
||||
|
||||
definition inv {x y : A} (H₁ : acc R x) (H₂ : R y x) : acc R y :=
|
||||
rec_on H₁ (λ x₁ ac₁ iH H₂, ac₁ y H₂) H₂
|
||||
acc.rec_on H₁ (λ x₁ ac₁ iH H₂, ac₁ y H₂) H₂
|
||||
end acc
|
||||
|
||||
inductive well_founded [class] {A : Type} (R : A → A → Type) : Type :=
|
||||
|
|
|
|||
|
|
@ -19,21 +19,21 @@ namespace natural_transformation
|
|||
variables {C D : Category} {F G H I : functor C D}
|
||||
|
||||
definition natural_map [coercion] (η : F ⟹ G) : Π(a : C), F a ⟶ G a :=
|
||||
rec (λ x y, x) η
|
||||
natural_transformation.rec (λ x y, x) η
|
||||
|
||||
theorem naturality (η : F ⟹ G) : Π⦃a b : C⦄ (f : a ⟶ b), G f ∘ η a = η b ∘ F f :=
|
||||
rec (λ x y, y) η
|
||||
natural_transformation.rec (λ x y, y) η
|
||||
|
||||
protected definition compose (η : G ⟹ H) (θ : F ⟹ G) : F ⟹ H :=
|
||||
natural_transformation.mk
|
||||
(λ a, η a ∘ θ a)
|
||||
(λ a b f,
|
||||
calc
|
||||
H f ∘ (η a ∘ θ a) = (H f ∘ η a) ∘ θ a : assoc
|
||||
... = (η b ∘ G f) ∘ θ a : naturality η f
|
||||
... = η b ∘ (G f ∘ θ a) : assoc
|
||||
... = η b ∘ (θ b ∘ F f) : naturality θ f
|
||||
... = (η b ∘ θ b) ∘ F f : assoc)
|
||||
H f ∘ (η a ∘ θ a) = (H f ∘ η a) ∘ θ a : assoc
|
||||
... = (η b ∘ G f) ∘ θ a : naturality η f
|
||||
... = η b ∘ (G f ∘ θ a) : assoc
|
||||
... = η b ∘ (θ b ∘ F f) : naturality θ f
|
||||
... = (η b ∘ θ b) ∘ F f : assoc)
|
||||
--congr_arg (λx, η b ∘ x) (naturality θ f) -- this needed to be explicit for some reason (on Oct 24)
|
||||
|
||||
infixr `∘n`:60 := compose
|
||||
|
|
@ -46,8 +46,8 @@ namespace natural_transformation
|
|||
protected definition ID {C D : Category} (F : functor C D) : natural_transformation F F := id
|
||||
|
||||
protected theorem id_left (η : F ⟹ G) : natural_transformation.compose id η = η :=
|
||||
rec (λf H, dcongr_arg2 mk (funext (take x, !id_left)) !proof_irrel) η
|
||||
natural_transformation.rec (λf H, dcongr_arg2 mk (funext (take x, !id_left)) !proof_irrel) η
|
||||
|
||||
protected theorem id_right (η : F ⟹ G) : natural_transformation.compose η id = η :=
|
||||
rec (λf H, dcongr_arg2 mk (funext (take x, !id_right)) !proof_irrel) η
|
||||
natural_transformation.rec (λf H, dcongr_arg2 mk (funext (take x, !id_right)) !proof_irrel) η
|
||||
end natural_transformation
|
||||
|
|
|
|||
|
|
@ -14,7 +14,7 @@ namespace bool
|
|||
local attribute band [reducible]
|
||||
|
||||
theorem dichotomy (b : bool) : b = ff ∨ b = tt :=
|
||||
cases_on b (or.inl rfl) (or.inr rfl)
|
||||
bool.cases_on b (or.inl rfl) (or.inr rfl)
|
||||
|
||||
theorem cond.ff {A : Type} (t e : A) : cond ff t e = e :=
|
||||
rfl
|
||||
|
|
@ -35,24 +35,24 @@ namespace bool
|
|||
notation a || b := bor a b
|
||||
|
||||
theorem bor.tt_right (a : bool) : a || tt = tt :=
|
||||
cases_on a rfl rfl
|
||||
bool.cases_on a rfl rfl
|
||||
|
||||
theorem bor.ff_left (a : bool) : ff || a = a :=
|
||||
cases_on a rfl rfl
|
||||
bool.cases_on a rfl rfl
|
||||
|
||||
theorem bor.ff_right (a : bool) : a || ff = a :=
|
||||
cases_on a rfl rfl
|
||||
bool.cases_on a rfl rfl
|
||||
|
||||
theorem bor.id (a : bool) : a || a = a :=
|
||||
cases_on a rfl rfl
|
||||
bool.cases_on a rfl rfl
|
||||
|
||||
theorem bor.comm (a b : bool) : a || b = b || a :=
|
||||
cases_on a
|
||||
(cases_on b rfl rfl)
|
||||
(cases_on b rfl rfl)
|
||||
bool.cases_on a
|
||||
(bool.cases_on b rfl rfl)
|
||||
(bool.cases_on b rfl rfl)
|
||||
|
||||
theorem bor.assoc (a b c : bool) : (a || b) || c = a || (b || c) :=
|
||||
cases_on a
|
||||
bool.cases_on a
|
||||
(calc (ff || b) || c = b || c : {!bor.ff_left}
|
||||
... = ff || (b || c) : !bor.ff_left⁻¹)
|
||||
(calc (tt || b) || c = tt || c : {!bor.tt_left}
|
||||
|
|
@ -60,7 +60,7 @@ namespace bool
|
|||
... = tt || (b || c) : !bor.tt_left⁻¹)
|
||||
|
||||
theorem bor.to_or {a b : bool} : a || b = tt → a = tt ∨ b = tt :=
|
||||
rec_on a
|
||||
bool.rec_on a
|
||||
(assume H : ff || b = tt,
|
||||
have Hb : b = tt, from !bor.ff_left ▸ H,
|
||||
or.inr Hb)
|
||||
|
|
@ -70,24 +70,24 @@ namespace bool
|
|||
rfl
|
||||
|
||||
theorem band.tt_left (a : bool) : tt && a = a :=
|
||||
cases_on a rfl rfl
|
||||
bool.cases_on a rfl rfl
|
||||
|
||||
theorem band.ff_right (a : bool) : a && ff = ff :=
|
||||
cases_on a rfl rfl
|
||||
bool.cases_on a rfl rfl
|
||||
|
||||
theorem band.tt_right (a : bool) : a && tt = a :=
|
||||
cases_on a rfl rfl
|
||||
bool.cases_on a rfl rfl
|
||||
|
||||
theorem band.id (a : bool) : a && a = a :=
|
||||
cases_on a rfl rfl
|
||||
bool.cases_on a rfl rfl
|
||||
|
||||
theorem band.comm (a b : bool) : a && b = b && a :=
|
||||
cases_on a
|
||||
(cases_on b rfl rfl)
|
||||
(cases_on b rfl rfl)
|
||||
bool.cases_on a
|
||||
(bool.cases_on b rfl rfl)
|
||||
(bool.cases_on b rfl rfl)
|
||||
|
||||
theorem band.assoc (a b c : bool) : (a && b) && c = a && (b && c) :=
|
||||
cases_on a
|
||||
bool.cases_on a
|
||||
(calc (ff && b) && c = ff && c : {!band.ff_left}
|
||||
... = ff : !band.ff_left
|
||||
... = ff && (b && c) : !band.ff_left⁻¹)
|
||||
|
|
@ -108,7 +108,7 @@ namespace bool
|
|||
band.eq_tt_elim_left (!band.comm ⬝ H)
|
||||
|
||||
theorem bnot.bnot (a : bool) : bnot (bnot a) = a :=
|
||||
cases_on a rfl rfl
|
||||
bool.cases_on a rfl rfl
|
||||
|
||||
theorem bnot.false : bnot ff = tt :=
|
||||
rfl
|
||||
|
|
@ -116,13 +116,15 @@ namespace bool
|
|||
theorem bnot.true : bnot tt = ff :=
|
||||
rfl
|
||||
|
||||
protected definition is_inhabited [instance] : inhabited bool :=
|
||||
inhabited.mk ff
|
||||
|
||||
protected definition has_decidable_eq [instance] : decidable_eq bool :=
|
||||
take a b : bool,
|
||||
rec_on a
|
||||
(rec_on b (inl rfl) (inr ff_ne_tt))
|
||||
(rec_on b (inr (ne.symm ff_ne_tt)) (inl rfl))
|
||||
|
||||
end bool
|
||||
|
||||
open bool
|
||||
|
||||
protected definition bool.inhabited [instance] : inhabited bool :=
|
||||
inhabited.mk ff
|
||||
|
||||
protected definition bool.decidable_eq [instance] : decidable_eq bool :=
|
||||
take a b : bool,
|
||||
bool.rec_on a
|
||||
(bool.rec_on b (inl rfl) (inr ff_ne_tt))
|
||||
(bool.rec_on b (inr (ne.symm ff_ne_tt)) (inl rfl))
|
||||
|
|
|
|||
|
|
@ -10,7 +10,7 @@ import logic.cast logic.subsingleton
|
|||
|
||||
namespace empty
|
||||
protected theorem elim (A : Type) (H : empty) : A :=
|
||||
rec (λe, A) H
|
||||
empty.rec (λe, A) H
|
||||
|
||||
protected theorem subsingleton [instance] : subsingleton empty :=
|
||||
subsingleton.intro (λ a b, !elim a)
|
||||
|
|
|
|||
|
|
@ -57,30 +57,30 @@ nat.cases_on (n - m)
|
|||
(take k, neg_succ_of_nat k) -- m < n, and n - m = succ k
|
||||
|
||||
definition neg (a : ℤ) : ℤ :=
|
||||
cases_on a
|
||||
int.cases_on a
|
||||
(take m, -- a = of_nat m
|
||||
nat.cases_on m 0 (take m', neg_succ_of_nat m'))
|
||||
(take m, of_nat (succ m)) -- a = neg_succ_of_nat m
|
||||
|
||||
definition add (a b : ℤ) : ℤ :=
|
||||
cases_on a
|
||||
int.cases_on a
|
||||
(take m, -- a = of_nat m
|
||||
cases_on b
|
||||
int.cases_on b
|
||||
(take n, of_nat (m + n)) -- b = of_nat n
|
||||
(take n, sub_nat_nat m (succ n))) -- b = neg_succ_of_nat n
|
||||
(take m, -- a = neg_succ_of_nat m
|
||||
cases_on b
|
||||
int.cases_on b
|
||||
(take n, sub_nat_nat n (succ m)) -- b = of_nat n
|
||||
(take n, neg_of_nat (succ m + succ n))) -- b = neg_succ_of_nat n
|
||||
|
||||
definition mul (a b : ℤ) : ℤ :=
|
||||
cases_on a
|
||||
int.cases_on a
|
||||
(take m, -- a = of_nat m
|
||||
cases_on b
|
||||
int.cases_on b
|
||||
(take n, of_nat (m * n)) -- b = of_nat n
|
||||
(take n, neg_of_nat (m * succ n))) -- b = neg_succ_of_nat n
|
||||
(take m, -- a = neg_succ_of_nat m
|
||||
cases_on b
|
||||
int.cases_on b
|
||||
(take n, neg_of_nat (succ m * n)) -- b = of_nat n
|
||||
(take n, of_nat (succ m * succ n))) -- b = neg_succ_of_nat n
|
||||
|
||||
|
|
@ -94,22 +94,22 @@ infix * := int.mul
|
|||
/- some basic functions and properties -/
|
||||
|
||||
theorem of_nat_inj {m n : ℕ} (H : of_nat m = of_nat n) : m = n :=
|
||||
no_confusion H (λe, e)
|
||||
int.no_confusion H (λe, e)
|
||||
|
||||
theorem neg_succ_of_nat_inj {m n : ℕ} (H : neg_succ_of_nat m = neg_succ_of_nat n) : m = n :=
|
||||
no_confusion H (λe, e)
|
||||
int.no_confusion H (λe, e)
|
||||
|
||||
definition has_decidable_eq [instance] : decidable_eq ℤ :=
|
||||
take a b,
|
||||
cases_on a
|
||||
int.cases_on a
|
||||
(take m,
|
||||
cases_on b
|
||||
int.cases_on b
|
||||
(take n,
|
||||
if H : m = n then inl (congr_arg of_nat H) else inr (take H1, H (of_nat_inj H1)))
|
||||
(take n', inr (assume H, no_confusion H)))
|
||||
(take n', inr (assume H, int.no_confusion H)))
|
||||
(take m',
|
||||
cases_on b
|
||||
(take n, inr (assume H, no_confusion H))
|
||||
int.cases_on b
|
||||
(take n, inr (assume H, int.no_confusion H))
|
||||
(take n',
|
||||
(if H : m' = n' then inl (congr_arg neg_succ_of_nat H) else
|
||||
inr (take H1, H (neg_succ_of_nat_inj H1)))))
|
||||
|
|
@ -137,12 +137,12 @@ calc
|
|||
... = neg_succ_of_nat (pred (n - m)) : rfl
|
||||
end
|
||||
|
||||
definition nat_abs (a : ℤ) : ℕ := cases_on a (take n, n) (take n', succ n')
|
||||
definition nat_abs (a : ℤ) : ℕ := int.cases_on a (take n, n) (take n', succ n')
|
||||
|
||||
theorem nat_abs_of_nat (n : ℕ) : nat_abs (of_nat n) = n := rfl
|
||||
|
||||
theorem nat_abs_eq_zero {a : ℤ} : nat_abs a = 0 → a = 0 :=
|
||||
cases_on a
|
||||
int.cases_on a
|
||||
(take m, assume H : nat_abs (of_nat m) = 0, congr_arg of_nat H)
|
||||
(take m', assume H : nat_abs (neg_succ_of_nat m') = 0, absurd H (succ_ne_zero _))
|
||||
|
||||
|
|
@ -206,10 +206,10 @@ theorem abstr_of_lt {p : ℕ × ℕ} (H : pr1 p < pr2 p) :
|
|||
abstr p = neg_succ_of_nat (pred (pr2 p - pr1 p)) :=
|
||||
sub_nat_nat_of_lt H
|
||||
|
||||
definition repr (a : ℤ) : ℕ × ℕ := cases_on a (take m, (m, 0)) (take m, (0, succ m))
|
||||
definition repr (a : ℤ) : ℕ × ℕ := int.cases_on a (take m, (m, 0)) (take m, (0, succ m))
|
||||
|
||||
theorem abstr_repr (a : ℤ) : abstr (repr a) = a :=
|
||||
cases_on a (take m, (sub_nat_nat_of_ge (zero_le m))) (take m, rfl)
|
||||
int.cases_on a (take m, (sub_nat_nat_of_ge (zero_le m))) (take m, rfl)
|
||||
|
||||
theorem repr_sub_nat_nat (m n : ℕ) : repr (sub_nat_nat m n) ≡ (m, n) :=
|
||||
or.elim (@le_or_gt n m)
|
||||
|
|
@ -299,7 +299,7 @@ or.elim (@le_or_gt n m)
|
|||
end
|
||||
|
||||
theorem cases_of_nat (a : ℤ) : (∃n : ℕ, a = of_nat n) ∨ (∃n : ℕ, a = - of_nat n) :=
|
||||
cases_on a
|
||||
int.cases_on a
|
||||
(take n, or.inl (exists.intro n rfl))
|
||||
(take n', or.inr (exists.intro (succ n') rfl))
|
||||
|
||||
|
|
@ -329,9 +329,9 @@ or.elim (cases_of_nat_succ a)
|
|||
definition padd (p q : ℕ × ℕ) : ℕ × ℕ := (pr1 p + pr1 q, pr2 p + pr2 q)
|
||||
|
||||
theorem repr_add (a b : ℤ) : repr (add a b) ≡ padd (repr a) (repr b) :=
|
||||
cases_on a
|
||||
int.cases_on a
|
||||
(take m,
|
||||
cases_on b
|
||||
int.cases_on b
|
||||
(take n, !equiv.refl)
|
||||
(take n',
|
||||
have H1 : equiv (repr (add (of_nat m) (neg_succ_of_nat n'))) (m, succ n'),
|
||||
|
|
@ -340,7 +340,7 @@ cases_on a
|
|||
from rfl,
|
||||
(!zero_add ▸ H2)⁻¹ ▸ H1))
|
||||
(take m',
|
||||
cases_on b
|
||||
int.cases_on b
|
||||
(take n,
|
||||
have H1 : equiv (repr (add (neg_succ_of_nat m') (of_nat n))) (n, succ m'),
|
||||
from !repr_sub_nat_nat,
|
||||
|
|
@ -394,7 +394,7 @@ begin
|
|||
apply H2
|
||||
end
|
||||
|
||||
theorem add_zero (a : ℤ) : a + 0 = a := cases_on a (take m, rfl) (take m', rfl)
|
||||
theorem add_zero (a : ℤ) : a + 0 = a := int.cases_on a (take m, rfl) (take m', rfl)
|
||||
|
||||
theorem zero_add (a : ℤ) : 0 + a = a := add.comm a 0 ▸ add_zero a
|
||||
|
||||
|
|
@ -404,7 +404,7 @@ definition pneg (p : ℕ × ℕ) : ℕ × ℕ := (pr2 p, pr1 p)
|
|||
|
||||
-- note: this is =, not just ≡
|
||||
theorem repr_neg (a : ℤ) : repr (- a) = pneg (repr a) :=
|
||||
cases_on a
|
||||
int.cases_on a
|
||||
(take m,
|
||||
nat.cases_on m rfl (take m', rfl))
|
||||
(take m', rfl)
|
||||
|
|
@ -465,13 +465,13 @@ H⁻¹ ▸ H1⁻¹ ▸ H2⁻¹ ▸ H3
|
|||
context
|
||||
attribute nat_abs [reducible]
|
||||
theorem mul_nat_abs (a b : ℤ) : nat_abs (a * b) = #nat (nat_abs a) * (nat_abs b) :=
|
||||
cases_on a
|
||||
int.cases_on a
|
||||
(take m,
|
||||
cases_on b
|
||||
int.cases_on b
|
||||
(take n, rfl)
|
||||
(take n', !nat_abs_neg ▸ rfl))
|
||||
(take m',
|
||||
cases_on b
|
||||
int.cases_on b
|
||||
(take n, !nat_abs_neg ▸ rfl)
|
||||
(take n', rfl))
|
||||
end
|
||||
|
|
@ -486,9 +486,9 @@ nat.cases_on m rfl (take m', rfl)
|
|||
|
||||
-- note: we have =, not just ≡
|
||||
theorem repr_mul (a b : ℤ) : repr (mul a b) = pmul (repr a) (repr b) :=
|
||||
cases_on a
|
||||
int.cases_on a
|
||||
(take m,
|
||||
cases_on b
|
||||
int.cases_on b
|
||||
(take n,
|
||||
(calc
|
||||
pmul (repr m) (repr n) = (m * n + 0 * 0, m * 0 + 0 * n) : rfl
|
||||
|
|
@ -500,7 +500,7 @@ cases_on a
|
|||
... = (m * 0 + 0, m * succ n' + 0 * 0) : zero_mul
|
||||
... = repr (mul m (neg_succ_of_nat n')) : repr_neg_of_nat)⁻¹))
|
||||
(take m',
|
||||
cases_on b
|
||||
int.cases_on b
|
||||
(take n,
|
||||
(calc
|
||||
pmul (repr (neg_succ_of_nat m')) (repr n) =
|
||||
|
|
|
|||
|
|
@ -16,7 +16,7 @@ open int eq.ops
|
|||
|
||||
namespace int
|
||||
|
||||
private definition nonneg (a : ℤ) : Prop := cases_on a (take n, true) (take n, false)
|
||||
private definition nonneg (a : ℤ) : Prop := int.cases_on a (take n, true) (take n, false)
|
||||
definition le (a b : ℤ) : Prop := nonneg (sub b a)
|
||||
definition lt (a b : ℤ) : Prop := le (add a 1) b
|
||||
|
||||
|
|
@ -25,15 +25,15 @@ infix <= := int.le
|
|||
infix ≤ := int.le
|
||||
infix < := int.lt
|
||||
|
||||
private definition decidable_nonneg [instance] (a : ℤ) : decidable (nonneg a) := cases_on a _ _
|
||||
private definition decidable_nonneg [instance] (a : ℤ) : decidable (nonneg a) := int.cases_on a _ _
|
||||
definition decidable_le [instance] (a b : ℤ) : decidable (a ≤ b) := decidable_nonneg _
|
||||
definition decidable_lt [instance] (a b : ℤ) : decidable (a < b) := decidable_nonneg _
|
||||
|
||||
private theorem nonneg.elim {a : ℤ} : nonneg a → ∃n : ℕ, a = n :=
|
||||
cases_on a (take n H, exists.intro n rfl) (take n' H, false.elim H)
|
||||
int.cases_on a (take n H, exists.intro n rfl) (take n' H, false.elim H)
|
||||
|
||||
private theorem nonneg_or_nonneg_neg (a : ℤ) : nonneg a ∨ nonneg (-a) :=
|
||||
cases_on a (take n, or.inl trivial) (take n, or.inr trivial)
|
||||
int.cases_on a (take n, or.inl trivial) (take n, or.inr trivial)
|
||||
|
||||
theorem le.intro {a b : ℤ} {n : ℕ} (H : a + n = b) : a ≤ b :=
|
||||
have H1 : b - a = n, from (eq_add_neg_of_add_eq (!add.comm ▸ H))⁻¹,
|
||||
|
|
|
|||
|
|
@ -137,7 +137,7 @@ head (a :: l) := a
|
|||
theorem head_cons [h : inhabited T] (a : T) (l : list T) : head (a::l) = a
|
||||
|
||||
theorem head_concat [h : inhabited T] {s : list T} (t : list T) : s ≠ nil → head (s ++ t) = head s :=
|
||||
cases_on s
|
||||
list.cases_on s
|
||||
(take H : nil ≠ nil, absurd rfl H)
|
||||
(take (x : T) (s : list T), take H : x::s ≠ nil,
|
||||
calc
|
||||
|
|
@ -154,7 +154,7 @@ theorem tail_nil : tail (@nil T) = nil
|
|||
theorem tail_cons (a : T) (l : list T) : tail (a::l) = l
|
||||
|
||||
theorem cons_head_tail [h : inhabited T] {l : list T} : l ≠ nil → (head l)::(tail l) = l :=
|
||||
cases_on l
|
||||
list.cases_on l
|
||||
(assume H : nil ≠ nil, absurd rfl H)
|
||||
(take x l, assume H : x::l ≠ nil, rfl)
|
||||
|
||||
|
|
@ -173,7 +173,7 @@ theorem mem_cons (x y : T) (l : list T) : x ∈ y::l ↔ (x = y ∨ x ∈ l) :=
|
|||
iff.rfl
|
||||
|
||||
theorem mem_concat_imp_or {x : T} {s t : list T} : x ∈ s ++ t → x ∈ s ∨ x ∈ t :=
|
||||
induction_on s or.inr
|
||||
list.induction_on s or.inr
|
||||
(take y s,
|
||||
assume IH : x ∈ s ++ t → x ∈ s ∨ x ∈ t,
|
||||
assume H1 : x ∈ y::s ++ t,
|
||||
|
|
@ -182,7 +182,7 @@ induction_on s or.inr
|
|||
iff.elim_right or.assoc H3)
|
||||
|
||||
theorem mem_or_imp_concat {x : T} {s t : list T} : x ∈ s ∨ x ∈ t → x ∈ s ++ t :=
|
||||
induction_on s
|
||||
list.induction_on s
|
||||
(take H, or.elim H false.elim (assume H, H))
|
||||
(take y s,
|
||||
assume IH : x ∈ s ∨ x ∈ t → x ∈ s ++ t,
|
||||
|
|
@ -200,7 +200,7 @@ iff.intro mem_concat_imp_or mem_or_imp_concat
|
|||
local attribute mem [reducible]
|
||||
local attribute append [reducible]
|
||||
theorem mem_split {x : T} {l : list T} : x ∈ l → ∃s t : list T, l = s ++ (x::t) :=
|
||||
induction_on l
|
||||
list.induction_on l
|
||||
(take H : x ∈ nil, false.elim (iff.elim_left !mem_nil H))
|
||||
(take y l,
|
||||
assume IH : x ∈ l → ∃s t : list T, l = s ++ (x::t),
|
||||
|
|
@ -216,7 +216,7 @@ induction_on l
|
|||
!exists.intro (!exists.intro H4)))
|
||||
|
||||
definition mem.is_decidable [instance] (H : decidable_eq T) (x : T) (l : list T) : decidable (x ∈ l) :=
|
||||
rec_on l
|
||||
list.rec_on l
|
||||
(decidable.inr (not_of_iff_false !mem_nil))
|
||||
(take (h : T) (l : list T) (iH : decidable (x ∈ l)),
|
||||
show decidable (x ∈ h::l), from
|
||||
|
|
@ -255,7 +255,7 @@ theorem find_nil (x : T) : find x nil = 0
|
|||
theorem find_cons (x y : T) (l : list T) : find x (y::l) = if x = y then 0 else succ (find x l)
|
||||
|
||||
theorem find.not_mem {l : list T} {x : T} : ¬x ∈ l → find x l = length l :=
|
||||
rec_on l
|
||||
list.rec_on l
|
||||
(assume P₁ : ¬x ∈ nil, _)
|
||||
(take y l,
|
||||
assume iH : ¬x ∈ l → find x l = length l,
|
||||
|
|
|
|||
|
|
@ -47,7 +47,7 @@ nat.induction_on x
|
|||
/- successor and predecessor -/
|
||||
|
||||
theorem succ_ne_zero (n : ℕ) : succ n ≠ 0 :=
|
||||
assume H, no_confusion H
|
||||
assume H, nat.no_confusion H
|
||||
|
||||
-- add_rewrite succ_ne_zero
|
||||
|
||||
|
|
@ -58,7 +58,7 @@ theorem pred_succ (n : ℕ) : pred (succ n) = n :=
|
|||
rfl
|
||||
|
||||
theorem eq_zero_or_eq_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(or.inl rfl)
|
||||
(take m IH, or.inr
|
||||
(show succ m = succ (pred (succ m)), from congr_arg succ !pred_succ⁻¹))
|
||||
|
|
@ -67,23 +67,23 @@ theorem exists_eq_succ_of_ne_zero {n : ℕ} (H : n ≠ 0) : ∃k : ℕ, n = succ
|
|||
exists.intro _ (or_resolve_right !eq_zero_or_eq_succ_pred H)
|
||||
|
||||
theorem succ.inj {n m : ℕ} (H : succ n = succ m) : n = m :=
|
||||
no_confusion H (λe, e)
|
||||
nat.no_confusion H (λe, e)
|
||||
|
||||
theorem succ.ne_self {n : ℕ} : succ n ≠ n :=
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(take H : 1 = 0,
|
||||
have ne : 1 ≠ 0, from !succ_ne_zero,
|
||||
absurd H ne)
|
||||
(take k IH H, IH (succ.inj H))
|
||||
|
||||
theorem discriminate {B : Prop} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B :=
|
||||
have H : n = n → B, from cases_on n H1 H2,
|
||||
have H : n = n → B, from nat.cases_on n H1 H2,
|
||||
H rfl
|
||||
|
||||
theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1)
|
||||
(H3 : ∀ (n : ℕ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a :=
|
||||
have stronger : P a ∧ P (succ a), from
|
||||
induction_on a
|
||||
nat.induction_on a
|
||||
(and.intro H1 H2)
|
||||
(take k IH,
|
||||
have IH1 : P k, from and.elim_left IH,
|
||||
|
|
@ -93,12 +93,12 @@ have stronger : P a ∧ P (succ a), from
|
|||
|
||||
theorem sub_induction {P : ℕ → ℕ → Prop} (n m : ℕ) (H1 : ∀m, P 0 m)
|
||||
(H2 : ∀n, P (succ n) 0) (H3 : ∀n m, P n m → P (succ n) (succ m)) : P n m :=
|
||||
have general : ∀m, P n m, from induction_on n
|
||||
have general : ∀m, P n m, from nat.induction_on n
|
||||
(take m : ℕ, H1 m)
|
||||
(take k : ℕ,
|
||||
assume IH : ∀m, P k m,
|
||||
take m : ℕ,
|
||||
cases_on m (H2 k) (take l, (H3 k l (IH l)))),
|
||||
nat.cases_on m (H2 k) (take l, (H3 k l (IH l)))),
|
||||
general m
|
||||
|
||||
/- addition -/
|
||||
|
|
@ -110,7 +110,7 @@ theorem add_succ (n m : ℕ) : n + succ m = succ (n + m) :=
|
|||
rfl
|
||||
|
||||
theorem zero_add (n : ℕ) : 0 + n = n :=
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
!add_zero
|
||||
(take m IH, show 0 + succ m = succ m, from
|
||||
calc
|
||||
|
|
@ -118,7 +118,7 @@ induction_on n
|
|||
... = succ m : IH)
|
||||
|
||||
theorem add.succ_left (n m : ℕ) : (succ n) + m = succ (n + m) :=
|
||||
induction_on m
|
||||
nat.induction_on m
|
||||
(!add_zero ▸ !add_zero)
|
||||
(take k IH, calc
|
||||
succ n + succ k = succ (succ n + k) : add_succ
|
||||
|
|
@ -126,7 +126,7 @@ induction_on m
|
|||
... = succ (n + succ k) : add_succ)
|
||||
|
||||
theorem add.comm (n m : ℕ) : n + m = m + n :=
|
||||
induction_on m
|
||||
nat.induction_on m
|
||||
(!add_zero ⬝ !zero_add⁻¹)
|
||||
(take k IH, calc
|
||||
n + succ k = succ (n+k) : add_succ
|
||||
|
|
@ -137,7 +137,7 @@ theorem succ_add_eq_add_succ (n m : ℕ) : succ n + m = n + succ m :=
|
|||
!add.succ_left ⬝ !add_succ⁻¹
|
||||
|
||||
theorem add.assoc (n m k : ℕ) : (n + m) + k = n + (m + k) :=
|
||||
induction_on k
|
||||
nat.induction_on k
|
||||
(!add_zero ▸ !add_zero)
|
||||
(take l IH,
|
||||
calc
|
||||
|
|
@ -153,7 +153,7 @@ theorem add.right_comm (n m k : ℕ) : n + m + k = n + k + m :=
|
|||
right_comm add.comm add.assoc n m k
|
||||
|
||||
theorem add.cancel_left {n m k : ℕ} : n + m = n + k → m = k :=
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(take H : 0 + m = 0 + k,
|
||||
!zero_add⁻¹ ⬝ H ⬝ !zero_add)
|
||||
(take (n : ℕ) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k),
|
||||
|
|
@ -170,7 +170,7 @@ have H2 : m + n = m + k, from !add.comm ⬝ H ⬝ !add.comm,
|
|||
add.cancel_left H2
|
||||
|
||||
theorem eq_zero_of_add_eq_zero_right {n m : ℕ} : n + m = 0 → n = 0 :=
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(take (H : 0 + m = 0), rfl)
|
||||
(take k IH,
|
||||
assume H : succ k + m = 0,
|
||||
|
|
@ -203,12 +203,12 @@ rfl
|
|||
-- commutativity, distributivity, associativity, identity
|
||||
|
||||
theorem zero_mul (n : ℕ) : 0 * n = 0 :=
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
!mul_zero
|
||||
(take m IH, !mul_succ ⬝ !add_zero ⬝ IH)
|
||||
|
||||
theorem succ_mul (n m : ℕ) : (succ n) * m = (n * m) + m :=
|
||||
induction_on m
|
||||
nat.induction_on m
|
||||
(!mul_zero ⬝ !mul_zero⁻¹ ⬝ !add_zero⁻¹)
|
||||
(take k IH, calc
|
||||
succ n * succ k = succ n * k + succ n : mul_succ
|
||||
|
|
@ -220,7 +220,7 @@ induction_on m
|
|||
... = n * succ k + succ k : mul_succ)
|
||||
|
||||
theorem mul.comm (n m : ℕ) : n * m = m * n :=
|
||||
induction_on m
|
||||
nat.induction_on m
|
||||
(!mul_zero ⬝ !zero_mul⁻¹)
|
||||
(take k IH, calc
|
||||
n * succ k = n * k + n : mul_succ
|
||||
|
|
@ -228,7 +228,7 @@ induction_on m
|
|||
... = (succ k) * n : succ_mul)
|
||||
|
||||
theorem mul.right_distrib (n m k : ℕ) : (n + m) * k = n * k + m * k :=
|
||||
induction_on k
|
||||
nat.induction_on k
|
||||
(calc
|
||||
(n + m) * 0 = 0 : mul_zero
|
||||
... = 0 + 0 : add_zero
|
||||
|
|
@ -251,7 +251,7 @@ calc
|
|||
... = n * m + n * k : mul.comm
|
||||
|
||||
theorem mul.assoc (n m k : ℕ) : (n * m) * k = n * (m * k) :=
|
||||
induction_on k
|
||||
nat.induction_on k
|
||||
(calc
|
||||
(n * m) * 0 = n * (m * 0) : mul_zero)
|
||||
(take l IH,
|
||||
|
|
@ -273,10 +273,10 @@ calc
|
|||
... = n : mul_one
|
||||
|
||||
theorem eq_zero_or_eq_zero_of_mul_eq_zero {n m : ℕ} : n * m = 0 → n = 0 ∨ m = 0 :=
|
||||
cases_on n
|
||||
nat.cases_on n
|
||||
(assume H, or.inl rfl)
|
||||
(take n',
|
||||
cases_on m
|
||||
nat.cases_on m
|
||||
(assume H, or.inr rfl)
|
||||
(take m',
|
||||
assume H : succ n' * succ m' = 0,
|
||||
|
|
|
|||
|
|
@ -52,7 +52,7 @@ theorem add_div_self_left {x : ℕ} (z : ℕ) (H : x > 0) : (x + z) div x = succ
|
|||
!add.comm ▸ !add_div_self_right H
|
||||
|
||||
theorem add_mul_div_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) div z = x div z + y :=
|
||||
induction_on y
|
||||
nat.induction_on y
|
||||
(calc (x + zero * z) div z = (x + zero) div z : zero_mul
|
||||
... = x div z : add_zero
|
||||
... = x div z + zero : add_zero)
|
||||
|
|
@ -107,7 +107,7 @@ theorem add_mod_right {x z : ℕ} (H : x > 0) : (x + z) mod x = z mod x :=
|
|||
!add.comm ▸ add_mod_left H
|
||||
|
||||
theorem add_mul_mod_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) mod z = x mod z :=
|
||||
induction_on y
|
||||
nat.induction_on y
|
||||
(calc (x + zero * z) mod z = (x + zero) mod z : zero_mul
|
||||
... = x mod z : add_zero)
|
||||
(take y,
|
||||
|
|
@ -131,7 +131,7 @@ theorem mul_mod_right {m n : ℕ} : (m * n) mod m = 0 :=
|
|||
!mul.comm ▸ !mul_mod_left
|
||||
|
||||
theorem mod_lt {x y : ℕ} (H : y > 0) : x mod y < y :=
|
||||
case_strong_induction_on x
|
||||
nat.case_strong_induction_on x
|
||||
(show 0 mod y < y, from !zero_mod⁻¹ ▸ H)
|
||||
(take x,
|
||||
assume IH : ∀x', x' ≤ x → x' mod y < y,
|
||||
|
|
@ -160,7 +160,7 @@ by_cases_zero_pos y
|
|||
(take y,
|
||||
assume H : y > 0,
|
||||
show x = x div y * y + x mod y, from
|
||||
case_strong_induction_on x
|
||||
nat.case_strong_induction_on x
|
||||
(show 0 = (0 div y) * y + 0 mod y, by simp)
|
||||
(take x,
|
||||
assume IH : ∀x', x' ≤ x → x' = x' div y * y + x' mod y,
|
||||
|
|
@ -254,7 +254,7 @@ have H1 : n mod 1 < 1, from mod_lt !succ_pos,
|
|||
eq_zero_of_le_zero (le_of_lt_succ H1)
|
||||
|
||||
theorem mod_self (n : ℕ) : n mod n = 0 :=
|
||||
cases_on n (by simp)
|
||||
nat.cases_on n (by simp)
|
||||
(take n,
|
||||
have H : (succ n * 1) mod (succ n * 1) = succ n * (1 mod 1),
|
||||
from !mul_mod_mul_left,
|
||||
|
|
@ -411,7 +411,7 @@ private definition gcd.lt.dec (x y₁ : nat) : (succ y₁, x mod succ y₁) ≺
|
|||
mod_lt (succ_pos y₁)
|
||||
|
||||
definition gcd.F (p₁ : nat × nat) : (Π p₂ : nat × nat, p₂ ≺ p₁ → nat) → nat :=
|
||||
prod.cases_on p₁ (λx y, cases_on y
|
||||
prod.cases_on p₁ (λx y, nat.cases_on y
|
||||
(λ f, x)
|
||||
(λ y₁ (f : Πp₂, p₂ ≺ (x, succ y₁) → nat), f (succ y₁, x mod succ y₁) !gcd.lt.dec))
|
||||
|
||||
|
|
@ -429,7 +429,7 @@ calc gcd n 1 = gcd 1 (n mod 1) : gcd_succ n zero
|
|||
... = 1 : gcd_zero_right
|
||||
|
||||
theorem gcd_def (x y : ℕ) : gcd x y = if y = 0 then x else gcd y (x mod y) :=
|
||||
cases_on y
|
||||
nat.cases_on y
|
||||
(calc gcd x 0 = x : gcd_zero_right x
|
||||
... = if 0 = 0 then x else gcd zero (x mod zero) : (if_pos rfl)⁻¹)
|
||||
(λy₁, calc
|
||||
|
|
@ -437,7 +437,7 @@ cases_on y
|
|||
... = if succ y₁ = 0 then x else gcd (succ y₁) (x mod succ y₁) : (if_neg (succ_ne_zero y₁))⁻¹)
|
||||
|
||||
theorem gcd_self (n : ℕ) : gcd n n = n :=
|
||||
cases_on n
|
||||
nat.cases_on n
|
||||
rfl
|
||||
(λn₁, calc
|
||||
gcd (succ n₁) (succ n₁) = gcd (succ n₁) (succ n₁ mod succ n₁) : gcd_succ (succ n₁) n₁
|
||||
|
|
@ -445,7 +445,7 @@ cases_on n
|
|||
... = succ n₁ : gcd_zero_right)
|
||||
|
||||
theorem gcd_zero_left (n : nat) : gcd 0 n = n :=
|
||||
cases_on n
|
||||
nat.cases_on n
|
||||
rfl
|
||||
(λ n₁, calc
|
||||
gcd 0 (succ n₁) = gcd (succ n₁) (0 mod succ n₁) : gcd_succ
|
||||
|
|
@ -471,7 +471,7 @@ theorem gcd.induction {P : ℕ → ℕ → Prop}
|
|||
let Q : nat × nat → Prop := λ p : nat × nat, P (pr₁ p) (pr₂ p) in
|
||||
have aux : Q (m, n), from
|
||||
well_founded.induction (m, n) (λp, prod.cases_on p
|
||||
(λm n, cases_on n
|
||||
(λm n, nat.cases_on n
|
||||
(λ ih, show P (pr₁ (m, 0)) (pr₂ (m, 0)), from H0 m)
|
||||
(λ n₁ (ih : ∀p₂, p₂ ≺ (m, succ n₁) → P (pr₁ p₂) (pr₂ p₂)),
|
||||
have hlt₁ : 0 < succ n₁, from succ_pos n₁,
|
||||
|
|
|
|||
|
|
@ -43,7 +43,7 @@ iff.intro
|
|||
(take H, lt_of_le_and_ne (and.elim_left H) (and.elim_right H))
|
||||
|
||||
theorem le_add_right (n k : ℕ) : n ≤ n + k :=
|
||||
induction_on k
|
||||
nat.induction_on k
|
||||
(calc n ≤ n : le.refl n
|
||||
... = n + zero : add_zero)
|
||||
(λ k (ih : n ≤ n + k), calc
|
||||
|
|
@ -340,7 +340,7 @@ or_of_or_of_imp_left (succ_le_or_eq_of_le H)
|
|||
(take H2 : succ n ≤ succ m, show n ≤ m, from le_of_succ_le_succ H2)
|
||||
|
||||
theorem le_pred_self (n : ℕ) : pred n ≤ n :=
|
||||
cases_on n
|
||||
nat.cases_on n
|
||||
(pred_zero⁻¹ ▸ !le.refl)
|
||||
(take k : ℕ, (!pred_succ)⁻¹ ▸ !self_le_succ)
|
||||
|
||||
|
|
@ -367,7 +367,7 @@ protected theorem strong_induction_on {P : nat → Prop} (n : ℕ) (H : ∀n, (
|
|||
P n :=
|
||||
have H1 : ∀ {n m : nat}, m < n → P m, from
|
||||
take n,
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(show ∀m, m < 0 → P m, from take m H, absurd H !not_lt_zero)
|
||||
(take n',
|
||||
assume IH : ∀ {m : nat}, m < n' → P m,
|
||||
|
|
@ -385,7 +385,7 @@ protected theorem case_strong_induction_on {P : nat → Prop} (a : nat) (H0 : P
|
|||
strong_induction_on a (
|
||||
take n,
|
||||
show (∀m, m < n → P m) → P n, from
|
||||
cases_on n
|
||||
nat.cases_on n
|
||||
(assume H : (∀m, m < 0 → P m), show P 0, from H0)
|
||||
(take n,
|
||||
assume H : (∀m, m < succ n → P m),
|
||||
|
|
@ -395,7 +395,7 @@ strong_induction_on a (
|
|||
/- pos -/
|
||||
|
||||
theorem by_cases_zero_pos {P : ℕ → Prop} (y : ℕ) (H0 : P 0) (H1 : ∀ {y : nat}, y > 0 → P y) : P y :=
|
||||
cases_on y H0 (take y, H1 !succ_pos)
|
||||
nat.cases_on y H0 (take y, H1 !succ_pos)
|
||||
|
||||
theorem eq_zero_or_pos (n : ℕ) : n = 0 ∨ n > 0 :=
|
||||
or_of_or_of_imp_left
|
||||
|
|
|
|||
|
|
@ -23,7 +23,7 @@ theorem sub_succ (n m : ℕ) : n - succ m = pred (n - m) :=
|
|||
rfl
|
||||
|
||||
theorem zero_sub (n : ℕ) : 0 - n = 0 :=
|
||||
induction_on n !sub_zero
|
||||
nat.induction_on n !sub_zero
|
||||
(take k : nat,
|
||||
assume IH : 0 - k = 0,
|
||||
calc
|
||||
|
|
@ -35,10 +35,10 @@ theorem succ_sub_succ (n m : ℕ) : succ n - succ m = n - m :=
|
|||
succ_sub_succ_eq_sub n m
|
||||
|
||||
theorem sub_self (n : ℕ) : n - n = 0 :=
|
||||
induction_on n !sub_zero (take k IH, !succ_sub_succ ⬝ IH)
|
||||
nat.induction_on n !sub_zero (take k IH, !succ_sub_succ ⬝ IH)
|
||||
|
||||
theorem add_sub_add_right (n k m : ℕ) : (n + k) - (m + k) = n - m :=
|
||||
induction_on k
|
||||
nat.induction_on k
|
||||
(calc
|
||||
(n + 0) - (m + 0) = n - (m + 0) : {!add_zero}
|
||||
... = n - m : {!add_zero})
|
||||
|
|
@ -54,7 +54,7 @@ theorem add_sub_add_left (k n m : ℕ) : (k + n) - (k + m) = n - m :=
|
|||
!add.comm ▸ !add.comm ▸ !add_sub_add_right
|
||||
|
||||
theorem add_sub_cancel (n m : ℕ) : n + m - m = n :=
|
||||
induction_on m
|
||||
nat.induction_on m
|
||||
(!add_zero⁻¹ ▸ !sub_zero)
|
||||
(take k : ℕ,
|
||||
assume IH : n + k - k = n,
|
||||
|
|
@ -67,7 +67,7 @@ theorem add_sub_cancel_left (n m : ℕ) : n + m - n = m :=
|
|||
!add.comm ▸ !add_sub_cancel
|
||||
|
||||
theorem sub_sub (n m k : ℕ) : n - m - k = n - (m + k) :=
|
||||
induction_on k
|
||||
nat.induction_on k
|
||||
(calc
|
||||
n - m - 0 = n - m : sub_zero
|
||||
... = n - (m + 0) : add_zero)
|
||||
|
|
@ -107,7 +107,7 @@ rfl
|
|||
/- interaction with multiplication -/
|
||||
|
||||
theorem mul_pred_left (n m : ℕ) : pred n * m = n * m - m :=
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(calc
|
||||
pred 0 * m = 0 * m : pred_zero
|
||||
... = 0 : zero_mul
|
||||
|
|
@ -127,7 +127,7 @@ calc
|
|||
... = n * m - n : mul.comm
|
||||
|
||||
theorem mul_sub_right_distrib (n m k : ℕ) : (n - m) * k = n * k - m * k :=
|
||||
induction_on m
|
||||
nat.induction_on m
|
||||
(calc
|
||||
(n - 0) * k = n * k : sub_zero
|
||||
... = n * k - 0 : sub_zero
|
||||
|
|
|
|||
|
|
@ -11,10 +11,10 @@ open bool
|
|||
|
||||
namespace pos_num
|
||||
theorem succ_not_is_one (a : pos_num) : is_one (succ a) = ff :=
|
||||
induction_on a rfl (take n iH, rfl) (take n iH, rfl)
|
||||
pos_num.induction_on a rfl (take n iH, rfl) (take n iH, rfl)
|
||||
|
||||
theorem pred.succ (a : pos_num) : pred (succ a) = a :=
|
||||
rec_on a
|
||||
pos_num.rec_on a
|
||||
rfl
|
||||
(take (n : pos_num) (iH : pred (succ n) = n),
|
||||
calc
|
||||
|
|
@ -59,7 +59,7 @@ namespace pos_num
|
|||
rfl
|
||||
|
||||
theorem mul.one_right (a : pos_num) : a * one = a :=
|
||||
induction_on a
|
||||
pos_num.induction_on a
|
||||
rfl
|
||||
(take (n : pos_num) (iH : n * one = n),
|
||||
calc bit1 n * one = bit0 (n * one) + one : rfl
|
||||
|
|
|
|||
|
|
@ -11,7 +11,7 @@ open eq.ops decidable
|
|||
|
||||
namespace option
|
||||
definition is_none {A : Type} (o : option A) : Prop :=
|
||||
rec true (λ a, false) o
|
||||
option.rec true (λ a, false) o
|
||||
|
||||
theorem is_none_none {A : Type} : is_none (@none A) :=
|
||||
trivial
|
||||
|
|
@ -20,19 +20,19 @@ namespace option
|
|||
not_false
|
||||
|
||||
theorem none_ne_some {A : Type} (a : A) : none ≠ some a :=
|
||||
assume H, no_confusion H
|
||||
assume H, option.no_confusion H
|
||||
|
||||
theorem some.inj {A : Type} {a₁ a₂ : A} (H : some a₁ = some a₂) : a₁ = a₂ :=
|
||||
no_confusion H (λe, e)
|
||||
option.no_confusion H (λe, e)
|
||||
|
||||
protected definition is_inhabited [instance] (A : Type) : inhabited (option A) :=
|
||||
inhabited.mk none
|
||||
|
||||
protected definition has_decidable_eq [instance] {A : Type} (H : decidable_eq A) : decidable_eq (option A) :=
|
||||
take o₁ o₂ : option A,
|
||||
rec_on o₁
|
||||
(rec_on o₂ (inl rfl) (take a₂, (inr (none_ne_some a₂))))
|
||||
(take a₁ : A, rec_on o₂
|
||||
option.rec_on o₁
|
||||
(option.rec_on o₂ (inl rfl) (take a₂, (inr (none_ne_some a₂))))
|
||||
(take a₁ : A, option.rec_on o₂
|
||||
(inr (ne.symm (none_ne_some a₁)))
|
||||
(take a₂ : A, decidable.rec_on (H a₁ a₂)
|
||||
(assume Heq : a₁ = a₂, inl (Heq ▸ rfl))
|
||||
|
|
|
|||
|
|
@ -22,16 +22,16 @@ namespace sum
|
|||
variables {A B : Type}
|
||||
|
||||
definition inl_ne_inr (a : A) (b : B) : inl a ≠ inr b :=
|
||||
assume H, no_confusion H
|
||||
assume H, sum.no_confusion H
|
||||
|
||||
definition inr_ne_inl (b : B) (a : A) : inr b ≠ inl a :=
|
||||
assume H, no_confusion H
|
||||
assume H, sum.no_confusion H
|
||||
|
||||
definition inl_inj {a₁ a₂ : A} : intro_left B a₁ = intro_left B a₂ → a₁ = a₂ :=
|
||||
assume H, no_confusion H (λe, e)
|
||||
assume H, sum.no_confusion H (λe, e)
|
||||
|
||||
definition inr_inj {b₁ b₂ : B} : intro_right A b₁ = intro_right A b₂ → b₁ = b₂ :=
|
||||
assume H, no_confusion H (λe, e)
|
||||
assume H, sum.no_confusion H (λe, e)
|
||||
|
||||
protected definition is_inhabited_left [instance] : inhabited A → inhabited (A + B) :=
|
||||
assume H : inhabited A, inhabited.mk (inl (default A))
|
||||
|
|
|
|||
|
|
@ -12,10 +12,10 @@ namespace unit
|
|||
notation `⋆` := star
|
||||
|
||||
protected theorem equal (a b : unit) : a = b :=
|
||||
rec_on a (rec_on b rfl)
|
||||
unit.rec_on a (unit.rec_on b rfl)
|
||||
|
||||
theorem eq_star (a : unit) : a = star :=
|
||||
equal a star
|
||||
unit.equal a star
|
||||
|
||||
protected theorem subsingleton [instance] : subsingleton unit :=
|
||||
subsingleton.intro (λ a b, equal a b)
|
||||
|
|
|
|||
|
|
@ -6,18 +6,18 @@ import init.datatypes init.reserved_notation
|
|||
|
||||
namespace bool
|
||||
definition cond {A : Type} (b : bool) (t e : A) :=
|
||||
rec_on b e t
|
||||
bool.rec_on b e t
|
||||
|
||||
definition bor (a b : bool) :=
|
||||
rec_on a (rec_on b ff tt) tt
|
||||
bool.rec_on a (bool.rec_on b ff tt) tt
|
||||
|
||||
notation a || b := bor a b
|
||||
|
||||
definition band (a b : bool) :=
|
||||
rec_on a ff (rec_on b ff tt)
|
||||
bool.rec_on a ff (bool.rec_on b ff tt)
|
||||
|
||||
notation a && b := band a b
|
||||
|
||||
definition bnot (a : bool) :=
|
||||
rec_on a tt ff
|
||||
bool.rec_on a tt ff
|
||||
end bool
|
||||
|
|
|
|||
|
|
@ -82,7 +82,7 @@ bit0 : pos_num → pos_num
|
|||
|
||||
namespace pos_num
|
||||
definition succ (a : pos_num) : pos_num :=
|
||||
rec_on a (bit0 one) (λn r, bit0 r) (λn r, bit1 n)
|
||||
pos_num.rec_on a (bit0 one) (λn r, bit0 r) (λn r, bit1 n)
|
||||
end pos_num
|
||||
|
||||
inductive num : Type :=
|
||||
|
|
@ -92,7 +92,7 @@ pos : pos_num → num
|
|||
namespace num
|
||||
open pos_num
|
||||
definition succ (a : num) : num :=
|
||||
rec_on a (pos one) (λp, pos (succ p))
|
||||
num.rec_on a (pos one) (λp, pos (succ p))
|
||||
end num
|
||||
|
||||
inductive bool : Type :=
|
||||
|
|
|
|||
|
|
@ -37,13 +37,13 @@ namespace eq
|
|||
variables {a b c a': A}
|
||||
|
||||
theorem subst {P : A → Prop} (H₁ : a = b) (H₂ : P a) : P b :=
|
||||
rec H₂ H₁
|
||||
eq.rec H₂ H₁
|
||||
|
||||
theorem trans (H₁ : a = b) (H₂ : b = c) : a = c :=
|
||||
subst H₂ H₁
|
||||
|
||||
definition symm (H : a = b) : b = a :=
|
||||
rec (refl a) H
|
||||
eq.rec (refl a) H
|
||||
|
||||
namespace ops
|
||||
notation H `⁻¹` := symm H --input with \sy or \-1 or \inv
|
||||
|
|
@ -124,10 +124,10 @@ namespace heq
|
|||
eq.rec_on (to_eq H₁) H₂
|
||||
|
||||
theorem subst {P : ∀T : Type, T → Prop} (H₁ : a == b) (H₂ : P A a) : P B b :=
|
||||
rec_on H₁ H₂
|
||||
heq.rec_on H₁ H₂
|
||||
|
||||
theorem symm (H : a == b) : b == a :=
|
||||
rec_on H (refl a)
|
||||
heq.rec_on H (refl a)
|
||||
|
||||
theorem of_eq (H : a = a') : a == a' :=
|
||||
eq.subst H (refl a)
|
||||
|
|
@ -167,7 +167,7 @@ notation a ∨ b := or a b
|
|||
|
||||
namespace or
|
||||
theorem elim (H₁ : a ∨ b) (H₂ : a → c) (H₃ : b → c) : c :=
|
||||
rec H₂ H₃ H₁
|
||||
or.rec H₂ H₃ H₁
|
||||
end or
|
||||
|
||||
/- iff -/
|
||||
|
|
@ -254,15 +254,15 @@ namespace decidable
|
|||
variables {p q : Prop}
|
||||
|
||||
definition rec_on_true [H : decidable p] {H1 : p → Type} {H2 : ¬p → Type} (H3 : p) (H4 : H1 H3)
|
||||
: rec_on H H1 H2 :=
|
||||
rec_on H (λh, H4) (λh, !false.rec (h H3))
|
||||
: decidable.rec_on H H1 H2 :=
|
||||
decidable.rec_on H (λh, H4) (λh, !false.rec (h H3))
|
||||
|
||||
definition rec_on_false [H : decidable p] {H1 : p → Type} {H2 : ¬p → Type} (H3 : ¬p) (H4 : H2 H3)
|
||||
: rec_on H H1 H2 :=
|
||||
rec_on H (λh, false.rec _ (H3 h)) (λh, H4)
|
||||
: decidable.rec_on H H1 H2 :=
|
||||
decidable.rec_on H (λh, false.rec _ (H3 h)) (λh, H4)
|
||||
|
||||
definition by_cases {q : Type} [C : decidable p] (Hpq : p → q) (Hnpq : ¬p → q) : q :=
|
||||
rec_on C (assume Hp, Hpq Hp) (assume Hnp, Hnpq Hnp)
|
||||
decidable.rec_on C (assume Hp, Hpq Hp) (assume Hnp, Hnpq Hnp)
|
||||
|
||||
theorem em (p : Prop) [H : decidable p] : p ∨ ¬p :=
|
||||
by_cases (λ Hp, or.inl Hp) (λ Hnp, or.inr Hnp)
|
||||
|
|
|
|||
|
|
@ -24,7 +24,7 @@ namespace nat
|
|||
notation a ≤ b := le a b
|
||||
|
||||
definition pred (a : nat) : nat :=
|
||||
cases_on a zero (λ a₁, a₁)
|
||||
nat.cases_on a zero (λ a₁, a₁)
|
||||
|
||||
protected definition is_inhabited [instance] : inhabited nat :=
|
||||
inhabited.mk zero
|
||||
|
|
@ -40,17 +40,17 @@ namespace nat
|
|||
|
||||
-- less-than is well-founded
|
||||
definition lt.wf [instance] : well_founded lt :=
|
||||
well_founded.intro (λn, rec_on n
|
||||
well_founded.intro (λn, nat.rec_on n
|
||||
(acc.intro zero (λ (y : nat) (hlt : y < zero),
|
||||
have aux : ∀ {n₁}, y < n₁ → zero = n₁ → acc lt y, from
|
||||
λ n₁ hlt, lt.cases_on hlt
|
||||
(λ heq, no_confusion heq)
|
||||
(λ b hlt heq, no_confusion heq),
|
||||
λ n₁ hlt, nat.lt.cases_on hlt
|
||||
(λ heq, nat.no_confusion heq)
|
||||
(λ b hlt heq, nat.no_confusion heq),
|
||||
aux hlt rfl))
|
||||
(λ (n : nat) (ih : acc lt n),
|
||||
acc.intro (succ n) (λ (m : nat) (hlt : m < succ n),
|
||||
have aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m, from
|
||||
λ n₁ hlt, lt.cases_on hlt
|
||||
λ n₁ hlt, nat.lt.cases_on hlt
|
||||
(λ (heq : succ n = succ m),
|
||||
nat.no_confusion heq (λ (e : n = m),
|
||||
eq.rec_on e ih))
|
||||
|
|
@ -73,7 +73,7 @@ namespace nat
|
|||
λ H, aux H rfl
|
||||
|
||||
definition zero_lt_succ (a : nat) : zero < succ a :=
|
||||
rec_on a
|
||||
nat.rec_on a
|
||||
(lt.base zero)
|
||||
(λ a (hlt : zero < succ a), lt.step hlt)
|
||||
|
||||
|
|
@ -106,9 +106,9 @@ namespace nat
|
|||
aux
|
||||
|
||||
definition lt.is_decidable_rel [instance] : decidable_rel lt :=
|
||||
λ a b, rec_on b
|
||||
λ a b, nat.rec_on b
|
||||
(λ (a : nat), inr (not_lt_zero a))
|
||||
(λ (b₁ : nat) (ih : ∀ a, decidable (a < b₁)) (a : nat), cases_on a
|
||||
(λ (b₁ : nat) (ih : ∀ a, decidable (a < b₁)) (a : nat), nat.cases_on a
|
||||
(inl !zero_lt_succ)
|
||||
(λ a, decidable.rec_on (ih a)
|
||||
(λ h_pos : a < b₁, inl (succ_lt_succ h_pos))
|
||||
|
|
@ -147,7 +147,7 @@ namespace nat
|
|||
end
|
||||
|
||||
definition lt.irrefl (a : nat) : ¬ a < a :=
|
||||
rec_on a
|
||||
nat.rec_on a
|
||||
!not_lt_zero
|
||||
(λ (a : nat) (ih : ¬ a < a) (h : succ a < succ a),
|
||||
ih (lt_of_succ_lt_succ h))
|
||||
|
|
@ -158,11 +158,11 @@ namespace nat
|
|||
(λ b hlt (ih : ¬ b < a) (h : succ b < a), ih (lt_of_succ_lt h))
|
||||
|
||||
definition lt.trichotomy (a b : nat) : a < b ∨ a = b ∨ b < a :=
|
||||
rec_on b
|
||||
(λa, cases_on a
|
||||
nat.rec_on b
|
||||
(λa, nat.cases_on a
|
||||
(or.inr (or.inl rfl))
|
||||
(λ a₁, or.inr (or.inr !zero_lt_succ)))
|
||||
(λ b₁ (ih : ∀a, a < b₁ ∨ a = b₁ ∨ b₁ < a) (a : nat), cases_on a
|
||||
(λ b₁ (ih : ∀a, a < b₁ ∨ a = b₁ ∨ b₁ < a) (a : nat), nat.cases_on a
|
||||
(or.inl !zero_lt_succ)
|
||||
(λ a, or.rec_on (ih a)
|
||||
(λ h : a < b₁, or.inl (succ_lt_succ h))
|
||||
|
|
@ -274,24 +274,24 @@ namespace nat
|
|||
notation a ≥ b := ge a b
|
||||
|
||||
definition add (a b : nat) : nat :=
|
||||
rec_on b a (λ b₁ r, succ r)
|
||||
nat.rec_on b a (λ b₁ r, succ r)
|
||||
|
||||
notation a + b := add a b
|
||||
|
||||
definition sub (a b : nat) : nat :=
|
||||
rec_on b a (λ b₁ r, pred r)
|
||||
nat.rec_on b a (λ b₁ r, pred r)
|
||||
|
||||
notation a - b := sub a b
|
||||
|
||||
definition mul (a b : nat) : nat :=
|
||||
rec_on b zero (λ b₁ r, r + a)
|
||||
nat.rec_on b zero (λ b₁ r, r + a)
|
||||
|
||||
notation a * b := mul a b
|
||||
|
||||
context
|
||||
attribute sub [reducible]
|
||||
definition succ_sub_succ_eq_sub (a b : nat) : succ a - succ b = a - b :=
|
||||
induction_on b
|
||||
nat.induction_on b
|
||||
rfl
|
||||
(λ b₁ (ih : succ a - succ b₁ = a - b₁),
|
||||
eq.rec_on ih (eq.refl (pred (succ a - succ b₁))))
|
||||
|
|
@ -301,7 +301,7 @@ namespace nat
|
|||
eq.rec_on (succ_sub_succ_eq_sub a b) rfl
|
||||
|
||||
definition zero_sub_eq_zero (a : nat) : zero - a = zero :=
|
||||
induction_on a
|
||||
nat.induction_on a
|
||||
rfl
|
||||
(λ a₁ (ih : zero - a₁ = zero),
|
||||
eq.rec_on ih (eq.refl (pred (zero - a₁))))
|
||||
|
|
@ -325,12 +325,12 @@ namespace nat
|
|||
λ h₁ h₂, aux h₁ h₂
|
||||
|
||||
definition pred_le (a : nat) : pred a ≤ a :=
|
||||
cases_on a
|
||||
nat.cases_on a
|
||||
(le.refl zero)
|
||||
(λ a₁, le_of_lt (lt.base a₁))
|
||||
|
||||
definition sub_le (a b : nat) : a - b ≤ a :=
|
||||
induction_on b
|
||||
nat.induction_on b
|
||||
(le.refl a)
|
||||
(λ b₁ ih, le.trans !pred_le ih)
|
||||
|
||||
|
|
|
|||
|
|
@ -14,22 +14,22 @@ inhabited.mk pos_num.one
|
|||
|
||||
namespace pos_num
|
||||
definition is_one (a : pos_num) : bool :=
|
||||
rec_on a tt (λn r, ff) (λn r, ff)
|
||||
pos_num.rec_on a tt (λn r, ff) (λn r, ff)
|
||||
|
||||
definition pred (a : pos_num) : pos_num :=
|
||||
rec_on a one (λn r, bit0 n) (λn r, cond (is_one n) one (bit1 r))
|
||||
pos_num.rec_on a one (λn r, bit0 n) (λn r, cond (is_one n) one (bit1 r))
|
||||
|
||||
definition size (a : pos_num) : pos_num :=
|
||||
rec_on a one (λn r, succ r) (λn r, succ r)
|
||||
pos_num.rec_on a one (λn r, succ r) (λn r, succ r)
|
||||
|
||||
definition add (a b : pos_num) : pos_num :=
|
||||
rec_on a
|
||||
pos_num.rec_on a
|
||||
succ
|
||||
(λn f b, rec_on b
|
||||
(λn f b, pos_num.rec_on b
|
||||
(succ (bit1 n))
|
||||
(λm r, succ (bit1 (f m)))
|
||||
(λm r, bit1 (f m)))
|
||||
(λn f b, rec_on b
|
||||
(λn f b, pos_num.rec_on b
|
||||
(bit1 n)
|
||||
(λm r, bit1 (f m))
|
||||
(λm r, bit0 (f m)))
|
||||
|
|
@ -38,7 +38,7 @@ namespace pos_num
|
|||
notation a + b := add a b
|
||||
|
||||
definition mul (a b : pos_num) : pos_num :=
|
||||
rec_on a
|
||||
pos_num.rec_on a
|
||||
b
|
||||
(λn r, bit0 r + b)
|
||||
(λn r, bit0 r)
|
||||
|
|
@ -54,16 +54,16 @@ namespace num
|
|||
open pos_num
|
||||
|
||||
definition pred (a : num) : num :=
|
||||
rec_on a zero (λp, cond (is_one p) zero (pos (pred p)))
|
||||
num.rec_on a zero (λp, cond (is_one p) zero (pos (pred p)))
|
||||
|
||||
definition size (a : num) : num :=
|
||||
rec_on a (pos one) (λp, pos (size p))
|
||||
num.rec_on a (pos one) (λp, pos (size p))
|
||||
|
||||
definition add (a b : num) : num :=
|
||||
rec_on a b (λp_a, rec_on b (pos p_a) (λp_b, pos (pos_num.add p_a p_b)))
|
||||
num.rec_on a b (λp_a, num.rec_on b (pos p_a) (λp_b, pos (pos_num.add p_a p_b)))
|
||||
|
||||
definition mul (a b : num) : num :=
|
||||
rec_on a zero (λp_a, rec_on b zero (λp_b, pos (pos_num.mul p_a p_b)))
|
||||
num.rec_on a zero (λp_a, num.rec_on b zero (λp_b, pos (pos_num.mul p_a p_b)))
|
||||
|
||||
notation a + b := add a b
|
||||
notation a * b := mul a b
|
||||
|
|
|
|||
|
|
@ -59,8 +59,8 @@ namespace prod
|
|||
(iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
|
||||
acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
|
||||
have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
|
||||
@lex.rec_on A B Ra Rb (λp₁ p₂, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
|
||||
p (xa, xb) lt
|
||||
@prod.lex.rec_on A B Ra Rb (λp₁ p₂, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
|
||||
p (xa, xb) lt
|
||||
(λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
|
||||
show acc (lex Ra Rb) (a₁, b₁), from
|
||||
have Ra₁ : Ra a₁ xa, from eq.rec_on eq₂ H,
|
||||
|
|
@ -78,7 +78,7 @@ namespace prod
|
|||
|
||||
-- Relational product is a subrelation of the lex
|
||||
definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
|
||||
λa b H, rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
|
||||
λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
|
||||
|
||||
-- The relational product of well founded relations is well-founded
|
||||
definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
|
||||
|
|
|
|||
|
|
@ -48,8 +48,8 @@ namespace sigma
|
|||
(iHb : ∀ (y : B xa), Rb xa y xb → acc (lex Ra Rb) ⟨xa, y⟩),
|
||||
acc.intro ⟨xa, xb⟩ (λp (lt : p ≺ ⟨xa, xb⟩),
|
||||
have aux : xa = xa → xb == xb → acc (lex Ra Rb) p, from
|
||||
@lex.rec_on A B Ra Rb (λp₁ p₂, p₂.1 = xa → p₂.2 == xb → acc (lex Ra Rb) p₁)
|
||||
p ⟨xa, xb⟩ lt
|
||||
@sigma.lex.rec_on A B Ra Rb (λp₁ p₂, p₂.1 = xa → p₂.2 == xb → acc (lex Ra Rb) p₁)
|
||||
p ⟨xa, xb⟩ lt
|
||||
(λ (a₁ : A) (b₁ : B a₁) (a₂ : A) (b₂ : B a₂) (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ == xb),
|
||||
begin cases eq₂, exact (iHa a₁ H b₁) end)
|
||||
(λ (a : A) (b₁ b₂ : B a) (H : Rb a b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ == xb),
|
||||
|
|
|
|||
|
|
@ -36,7 +36,7 @@ namespace heq
|
|||
|
||||
theorem drec_on {C : Π {B : Type} (b : B), a == b → Type} (H₁ : a == b) (H₂ : C a (refl a)) :
|
||||
C b H₁ :=
|
||||
rec (λ H₁ : a == a, show C a H₁, from H₂) H₁ H₁
|
||||
heq.rec (λ H₁ : a == a, show C a H₁, from H₂) H₁ H₁
|
||||
|
||||
theorem to_cast_eq (H : a == b) : cast (type_eq H) a = b :=
|
||||
drec_on H !cast_eq
|
||||
|
|
|
|||
|
|
@ -22,17 +22,17 @@ namespace eq
|
|||
definition drec_on {B : Πa' : A, a = a' → Type} (H₁ : a = a') (H₂ : B a (refl a)) : B a' H₁ :=
|
||||
eq.rec (λH₁ : a = a, show B a H₁, from H₂) H₁ H₁
|
||||
|
||||
theorem rec_on_id {B : A → Type} (H : a = a) (b : B a) : rec_on H b = b :=
|
||||
theorem rec_on_id {B : A → Type} (H : a = a) (b : B a) : eq.rec_on H b = b :=
|
||||
rfl
|
||||
|
||||
theorem rec_on_constant (H : a = a') {B : Type} (b : B) : rec_on H b = b :=
|
||||
theorem rec_on_constant (H : a = a') {B : Type} (b : B) : eq.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 :=
|
||||
theorem rec_on_constant2 (H₁ : a₁ = a₂) (H₂ : a₃ = a₄) (b : B) : eq.rec_on H₁ b = eq.rec_on H₂ b :=
|
||||
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 :=
|
||||
eq.rec_on H b = eq.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) :
|
||||
|
|
@ -40,8 +40,8 @@ namespace eq
|
|||
proof_irrel H H' ▸ rfl
|
||||
|
||||
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,
|
||||
(u : P a) : eq.rec_on H₂ (eq.rec_on H₁ u) = eq.rec_on (trans H₁ H₂) u :=
|
||||
(show ∀ H₂ : b = c, eq.rec_on H₂ (eq.rec_on H₁ u) = eq.rec_on (trans H₁ H₂) u,
|
||||
from drec_on H₂ (take (H₂ : b = b), rec_on_id H₂ _))
|
||||
H₂
|
||||
end eq
|
||||
|
|
|
|||
|
|
@ -12,7 +12,7 @@ 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
|
||||
definition elim {A : Type} {H : subsingleton A} : ∀(a b : A), a = b := subsingleton.rec (fun p, p) H
|
||||
end subsingleton
|
||||
|
||||
protected definition prop.subsingleton [instance] (P : Prop) : subsingleton P :=
|
||||
|
|
|
|||
|
|
@ -22,6 +22,7 @@ Author: Leonardo de Moura
|
|||
#include "library/parser_nested_exception.h"
|
||||
#include "library/aliases.h"
|
||||
#include "library/private.h"
|
||||
#include "library/protected.h"
|
||||
#include "library/choice.h"
|
||||
#include "library/placeholder.h"
|
||||
#include "library/deep_copy.h"
|
||||
|
|
@ -1086,7 +1087,8 @@ expr parser::id_to_expr(name const & id, pos_info const & p) {
|
|||
|
||||
for (name const & ns : get_namespaces(m_env)) {
|
||||
auto new_id = ns + id;
|
||||
if (!ns.is_anonymous() && m_env.find(new_id)) {
|
||||
if (!ns.is_anonymous() && m_env.find(new_id) &&
|
||||
(!id.is_atomic() || !is_protected(m_env, new_id))) {
|
||||
auto r = save_pos(mk_constant(new_id, ls), p);
|
||||
save_type_info(r);
|
||||
add_ref_index(new_id, p);
|
||||
|
|
|
|||
|
|
@ -18,6 +18,7 @@ Author: Leonardo de Moura
|
|||
#include "library/expr_pair.h"
|
||||
#include "library/placeholder.h"
|
||||
#include "library/private.h"
|
||||
#include "library/protected.h"
|
||||
#include "library/explicit.h"
|
||||
#include "library/typed_expr.h"
|
||||
#include "library/num.h"
|
||||
|
|
@ -418,7 +419,9 @@ auto pretty_fn::pp_const(expr const & e) -> result {
|
|||
for (name const & ns : get_namespaces(m_env)) {
|
||||
if (!ns.is_anonymous()) {
|
||||
name new_n = n.replace_prefix(ns, name());
|
||||
if (new_n != n && !new_n.is_anonymous()) {
|
||||
if (new_n != n &&
|
||||
!new_n.is_anonymous() &&
|
||||
(!new_n.is_atomic() || !is_protected(m_env, n))) {
|
||||
n = new_n;
|
||||
break;
|
||||
}
|
||||
|
|
|
|||
|
|
@ -17,6 +17,7 @@ Author: Leonardo de Moura
|
|||
#include "library/aliases.h"
|
||||
#include "library/type_util.h"
|
||||
#include "library/unifier.h"
|
||||
#include "library/protected.h"
|
||||
#include "library/reducible.h"
|
||||
#include "library/projection.h"
|
||||
#include "library/scoped_ext.h"
|
||||
|
|
@ -571,7 +572,7 @@ optional<name> is_essentially_atomic(environment const & env, name const & n) {
|
|||
for (name const & ns : ns_list) {
|
||||
if (is_prefix_of(ns, n)) {
|
||||
auto n_prime = n.replace_prefix(ns, name());
|
||||
if (n_prime.is_atomic())
|
||||
if (n_prime.is_atomic() && !is_protected(env, n))
|
||||
return optional<name>(n_prime);
|
||||
break;
|
||||
}
|
||||
|
|
@ -586,8 +587,11 @@ void server::display_decl(name const & d, environment const & env, options const
|
|||
// using namespace override resolution rule
|
||||
list<name> const & ns_list = get_namespaces(env);
|
||||
for (name const & ns : ns_list) {
|
||||
if (is_prefix_of(ns, d) && ns != d) {
|
||||
display_decl(d.replace_prefix(ns, name()), d, env, o);
|
||||
name new_d = d.replace_prefix(ns, name());
|
||||
if (new_d != d &&
|
||||
!new_d.is_anonymous() &&
|
||||
(!new_d.is_atomic() || !is_protected(env, d))) {
|
||||
display_decl(new_d, d, env, o);
|
||||
return;
|
||||
}
|
||||
}
|
||||
|
|
|
|||
|
|
@ -10,6 +10,6 @@ namespace foo
|
|||
{A₂ : Type} {B₂ : A₂ → Type} {a₂ : A₂} {b₂ : B₂ a₂}
|
||||
(H : foo.mk A₁ B₁ a₁ b₁ = foo.mk A₂ B₂ a₂ b₂)
|
||||
: A₁ = A₂
|
||||
:= lift.down (no_confusion H (λ e₁ e₂ e₃ e₄, e₁))
|
||||
:= lift.down (foo.no_confusion H (λ e₁ e₂ e₃ e₄, e₁))
|
||||
|
||||
end foo
|
||||
|
|
|
|||
|
|
@ -6,12 +6,12 @@ namespace list
|
|||
open lift
|
||||
|
||||
theorem nil_ne_cons {A : Type} (a : A) (v : list A) : nil ≠ cons a v :=
|
||||
λ H, down (no_confusion H)
|
||||
λ H, down (list.no_confusion H)
|
||||
|
||||
theorem cons.inj₁ {A : Type} (a₁ a₂ : A) (v₁ v₂ : list A) : cons a₁ v₁ = cons a₂ v₂ → a₁ = a₂ :=
|
||||
λ H, down (no_confusion H (λ (h₁ : a₁ = a₂) (h₂ : v₁ = v₂), h₁))
|
||||
λ H, down (list.no_confusion H (λ (h₁ : a₁ = a₂) (h₂ : v₁ = v₂), h₁))
|
||||
|
||||
theorem cons.inj₂ {A : Type} (a₁ a₂ : A) (v₁ v₂ : list A) : cons a₁ v₁ = cons a₂ v₂ → v₁ = v₂ :=
|
||||
λ H, down (no_confusion H (λ (h₁ : a₁ = a₂) (h₂ : v₁ = v₂), h₂))
|
||||
λ H, down (list.no_confusion H (λ (h₁ : a₁ = a₂) (h₂ : v₁ = v₂), h₂))
|
||||
|
||||
end list
|
||||
|
|
|
|||
|
|
@ -6,7 +6,7 @@ namespace sigma
|
|||
variables {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂}
|
||||
|
||||
definition dpair.inj (H : ⟨a₁, b₁⟩ = ⟨a₂, b₂⟩) : Σ (e₁ : a₁ = a₂), eq.rec b₁ e₁ = b₂ :=
|
||||
down (no_confusion H (λ e₁ e₂, ⟨e₁, e₂⟩))
|
||||
down (sigma.no_confusion H (λ e₁ e₂, ⟨e₁, e₂⟩))
|
||||
|
||||
definition dpair.inj₁ (H : ⟨a₁, b₁⟩ = ⟨a₂, b₂⟩) : a₁ = a₂ :=
|
||||
(dpair.inj H).1
|
||||
|
|
@ -29,7 +29,7 @@ namespace foo
|
|||
|
||||
definition foo.inj (H : mk A₁ B₁ a₁ b₁ = mk A₂ B₂ a₂ b₂) :
|
||||
Σ (e₁ : A₁ = A₂) (e₂ : eq.rec B₁ e₁ = B₂) (e₃ : eq.rec a₁ e₁ = a₂), eq.rec (eq.rec (eq.rec b₁ e₁) e₂) e₃ = b₂ :=
|
||||
down (no_confusion H (λ e₁ e₂ e₃ e₄, ⟨e₁, e₂, e₃, e₄⟩))
|
||||
down (foo.no_confusion H (λ e₁ e₂ e₃ e₄, ⟨e₁, e₂, e₃, e₄⟩))
|
||||
|
||||
definition foo.inj₁ (H : mk A₁ B₁ a₁ b₁ = mk A₂ B₂ a₂ b₂) : A₁ = A₂ :=
|
||||
(foo.inj H).1
|
||||
|
|
|
|||
15
tests/lean/interactive/auto_comp.input
Normal file
15
tests/lean/interactive/auto_comp.input
Normal file
|
|
@ -0,0 +1,15 @@
|
|||
VISIT auto_comp.lean
|
||||
SYNC 8
|
||||
prelude
|
||||
inductive nat := zero : nat, succ : nat → nat
|
||||
inductive bool := ff, tt
|
||||
namespace nat
|
||||
inductive le : nat → nat → Type.{0} := refl : Π a, le a a
|
||||
|
||||
end nat
|
||||
|
||||
WAIT
|
||||
FINDP 6
|
||||
rec_on
|
||||
FINDP 8
|
||||
rec_on
|
||||
12
tests/lean/interactive/auto_comp.input.expected.out
Normal file
12
tests/lean/interactive/auto_comp.input.expected.out
Normal file
|
|
@ -0,0 +1,12 @@
|
|||
-- BEGINWAIT
|
||||
-- ENDWAIT
|
||||
-- BEGINFINDP
|
||||
le.rec_on|le ?a ?a → (Π (a : nat), ?C a a) → ?C ?a ?a
|
||||
nat.rec_on|Π (n : nat), ?C zero → (Π (a : nat), ?C a → ?C (succ a)) → ?C n
|
||||
bool.rec_on|Π (n : bool), ?C bool.ff → ?C bool.tt → ?C n
|
||||
-- ENDFINDP
|
||||
-- BEGINFINDP
|
||||
nat.le.rec_on|nat.le ?a ?a → (Π (a : nat), ?C a a) → ?C ?a ?a
|
||||
nat.rec_on|Π (n : nat), ?C nat.zero → (Π (a : nat), ?C a → ?C (nat.succ a)) → ?C n
|
||||
bool.rec_on|Π (n : bool), ?C bool.ff → ?C bool.tt → ?C n
|
||||
-- ENDFINDP
|
||||
|
|
@ -1,7 +1,7 @@
|
|||
VISIT eq2.lean
|
||||
WAIT
|
||||
REPLACE 134
|
||||
(λ (b₂ : B a₁) (H₂ : b₁ = b₂) (c₂ : C a₁ b₂) (H₃ : (rec_on (congr_arg2_dep C (refl a₁) H₂) c₁) = c₂),
|
||||
(λ (b₂ : B a₁) (H₂ : b₁ = b₂) (c₂ : C a₁ b₂) (H₃ : (eq.rec_on (congr_arg2_dep C (refl a₁) H₂) c₁) = c₂),
|
||||
WAIT
|
||||
REPLACE 134
|
||||
(λ (b₂ : B a₁) (H₂ : b₁ = b₂) (c₂ : C a₁ b₂) (H₃ : _ = c₂),
|
||||
|
|
|
|||
|
|
@ -28,7 +28,7 @@ namespace eq
|
|||
proof_irrel
|
||||
|
||||
theorem subst {A : Type} {a b : A} {P : A → Prop} (H1 : a = b) (H2 : P a) : P b :=
|
||||
rec H2 H1
|
||||
eq.rec H2 H1
|
||||
|
||||
theorem trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b = c) : a = c :=
|
||||
subst H2 H1
|
||||
|
|
@ -69,7 +69,7 @@ namespace eq
|
|||
theorem drec_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)) : drec_on H b = drec_on H' b :=
|
||||
drec_on H (λ(H : a = a) (H' : f a = f a), drec_on_id H b ⬝ drec_on_id H' b⁻¹) H H'
|
||||
|
||||
theorem rec_id {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : rec b H = b :=
|
||||
theorem rec_id {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : eq.rec b H = b :=
|
||||
id_refl H⁻¹ ▸ refl (eq.rec b (refl a))
|
||||
|
||||
theorem drec_on_compose {A : Type} {a b c : A} {P : A → Type} (H1 : a = b) (H2 : b = c)
|
||||
|
|
|
|||
|
|
@ -2,26 +2,26 @@ VISIT extra_type.lean
|
|||
SET
|
||||
pp.notation false
|
||||
SYNC 20
|
||||
import general_notation
|
||||
import logic.connectives logic.decidable logic.inhabited
|
||||
|
||||
open eq eq.ops decidable
|
||||
|
||||
prelude
|
||||
definition Prop := Type.{0}
|
||||
inductive eq {A : Type} (a : A) : A → Prop := refl : eq a a
|
||||
infix `=`:50 := eq
|
||||
definition rfl {A : Type} {a : A} := eq.refl a
|
||||
inductive or (A B : Prop) : Prop := inl {} : A → or A B, inr {} : B → or A B
|
||||
infix `∨`:35 := or
|
||||
inductive bool : Type :=
|
||||
ff : bool,
|
||||
tt : bool
|
||||
namespace bool
|
||||
protected definition rec_on {C : bool → Type} (b : bool) (H₁ : C ff) (H₂ : C tt) : C b :=
|
||||
rec H₁ H₂ b
|
||||
|
||||
protected definition cases_on {p : bool → Prop} (b : bool) (H₁ : p ff) (H₂ : p tt) : p b :=
|
||||
rec H₁ H₂ b
|
||||
|
||||
protected definition rec_on2 {C : bool → Type} (b : bool) (H₁ : C ff) (H₂ : C tt) : C b :=
|
||||
bool.rec H₁ H₂ b
|
||||
protected definition cases_on2 {p : bool → Prop} (b : bool) (H₁ : p ff) (H₂ : p tt) : p b :=
|
||||
bool.rec H₁ H₂ b
|
||||
definition cond {A : Type} (b : bool) (t e : A) :=
|
||||
rec_on b e t
|
||||
bool.rec_on b e t
|
||||
|
||||
theorem dichotomy (b : bool) : b = ff ∨ b = tt :=
|
||||
cases_on b (or.inl rfl) (or.inr rfl)
|
||||
bool.cases_on2 b (or.inl rfl) (or.inr rfl)
|
||||
end bool
|
||||
WAIT
|
||||
INFO 20 13
|
||||
INFO 20 19
|
||||
|
|
|
|||
|
|
@ -4,12 +4,12 @@
|
|||
-- BEGINWAIT
|
||||
-- ENDWAIT
|
||||
-- BEGININFO
|
||||
-- EXTRA_TYPE|20|13
|
||||
-- EXTRA_TYPE|20|19
|
||||
or.inl rfl
|
||||
--
|
||||
or (eq ff ff) (eq ff tt)
|
||||
-- ACK
|
||||
-- SYMBOL|20|13
|
||||
-- SYMBOL|20|19
|
||||
(
|
||||
-- ACK
|
||||
-- ENDINFO
|
||||
|
|
|
|||
|
|
@ -7,7 +7,6 @@ pos_num.size|pos_num → pos_num
|
|||
pos_num.bit0|pos_num → pos_num
|
||||
pos_num.is_inhabited|inhabited pos_num
|
||||
pos_num.is_one|pos_num → bool
|
||||
pos_num.inc|pos_num → pos_num
|
||||
pos_num.ibelow|pos_num → Prop
|
||||
pos_num.binduction_on|Π (n : pos_num), (Π (n : pos_num), pos_num.ibelow n → ?C n) → ?C n
|
||||
pos_num.induction_on|Π (n : pos_num), ?C pos_num.one → (Π (a : pos_num), ?C a → ?C (pos_num.bit1 a)) → (Π (a : pos_num), ?C a → ?C (pos_num.bit0 a)) → ?C n
|
||||
|
|
@ -20,7 +19,6 @@ pos_num.cases_on|Π (n : pos_num), ?C pos_num.one → (Π (a : pos_num), ?C (pos
|
|||
pos_num.pred|pos_num → pos_num
|
||||
pos_num.mul|pos_num → pos_num → pos_num
|
||||
pos_num.no_confusion_type|Type → pos_num → pos_num → Type
|
||||
pos_num.num_bits|pos_num → pos_num
|
||||
pos_num.no_confusion|eq ?v1 ?v2 → pos_num.no_confusion_type ?P ?v1 ?v2
|
||||
pos_num.rec_on|Π (n : pos_num), ?C pos_num.one → (Π (a : pos_num), ?C a → ?C (pos_num.bit1 a)) → (Π (a : pos_num), ?C a → ?C (pos_num.bit0 a)) → ?C n
|
||||
pos_num.brec_on|Π (n : pos_num), (Π (n : pos_num), pos_num.below n → ?C n) → ?C n
|
||||
|
|
@ -36,7 +34,6 @@ pos_num.size|pos_num → pos_num
|
|||
pos_num.bit0|pos_num → pos_num
|
||||
pos_num.is_inhabited|inhabited pos_num
|
||||
pos_num.is_one|pos_num → bool
|
||||
pos_num.inc|pos_num → pos_num
|
||||
pos_num.ibelow|pos_num → Prop
|
||||
pos_num.binduction_on|Π (n : pos_num), (Π (n : pos_num), pos_num.ibelow n → ?C n) → ?C n
|
||||
pos_num.induction_on|Π (n : pos_num), ?C pos_num.one → (Π (a : pos_num), ?C a → ?C (pos_num.bit1 a)) → (Π (a : pos_num), ?C a → ?C (pos_num.bit0 a)) → ?C n
|
||||
|
|
@ -60,7 +57,6 @@ pos_num.size|pos_num → pos_num
|
|||
pos_num.bit0|pos_num → pos_num
|
||||
pos_num.is_inhabited|inhabited pos_num
|
||||
pos_num.is_one|pos_num → bool
|
||||
pos_num.inc|pos_num → pos_num
|
||||
pos_num.ibelow|pos_num → Prop
|
||||
pos_num.binduction_on|Π (n : pos_num), (Π (n : pos_num), pos_num.ibelow n → ?C n) → ?C n
|
||||
pos_num.induction_on|Π (n : pos_num), ?C pos_num.one → (Π (a : pos_num), ?C a → ?C (pos_num.bit1 a)) → (Π (a : pos_num), ?C a → ?C (pos_num.bit0 a)) → ?C n
|
||||
|
|
|
|||
11
tests/lean/protected_test.lean
Normal file
11
tests/lean/protected_test.lean
Normal file
|
|
@ -0,0 +1,11 @@
|
|||
namespace nat
|
||||
check induction_on -- ERROR
|
||||
check rec_on -- ERROR
|
||||
check nat.induction_on
|
||||
check lt.rec_on -- OK
|
||||
check nat.lt.rec_on
|
||||
namespace lt
|
||||
check rec_on -- ERROR
|
||||
check lt.rec_on
|
||||
end lt
|
||||
end nat
|
||||
7
tests/lean/protected_test.lean.expected.out
Normal file
7
tests/lean/protected_test.lean.expected.out
Normal file
|
|
@ -0,0 +1,7 @@
|
|||
protected_test.lean:2:8: error: unknown identifier 'induction_on'
|
||||
protected_test.lean:3:8: error: unknown identifier 'rec_on'
|
||||
nat.induction_on : ∀ (n : ℕ), ?C 0 → (∀ (a : ℕ), ?C a → ?C (succ a)) → ?C n
|
||||
lt.rec_on : ?a < ?a_1 → ?C (succ ?a) → (∀ {b : ℕ}, ?a < b → ?C b → ?C (succ b)) → ?C ?a_1
|
||||
lt.rec_on : ?a < ?a_1 → ?C (succ ?a) → (∀ {b : ℕ}, ?a < b → ?C b → ?C (succ b)) → ?C ?a_1
|
||||
protected_test.lean:8:10: error: unknown identifier 'rec_on'
|
||||
lt.rec_on : ?a < ?a_1 → ?C (succ ?a) → (∀ {b : ℕ}, ?a < b → ?C b → ?C (succ b)) → ?C ?a_1
|
||||
|
|
@ -18,8 +18,8 @@ context sec_cat
|
|||
|
||||
attribute foo [class]
|
||||
parameters {ob : Type} {mor : ob → ob → Type} {Cat : category ob mor}
|
||||
definition compose := rec (λ comp id assoc idr idl, comp) Cat
|
||||
definition id := rec (λ comp id assoc idr idl, id) Cat
|
||||
definition compose := category.rec (λ comp id assoc idr idl, comp) Cat
|
||||
definition id := category.rec (λ comp id assoc idr idl, id) Cat
|
||||
infixr ∘ := compose
|
||||
inductive is_section {A B : ob} (f : mor A B) : Type :=
|
||||
mk : ∀g, g ∘ f = id → is_section f
|
||||
|
|
|
|||
|
|
@ -18,8 +18,8 @@ section sec_cat
|
|||
|
||||
attribute foo [class]
|
||||
variables {ob : Type} {mor : ob → ob → Type} {Cat : category ob mor}
|
||||
definition compose := rec (λ comp id assoc idr idl, comp) Cat
|
||||
definition id := rec (λ comp id assoc idr idl, id) Cat
|
||||
definition compose := category.rec (λ comp id assoc idr idl, comp) Cat
|
||||
definition id := category.rec (λ comp id assoc idr idl, id) Cat
|
||||
infixr ∘ := compose
|
||||
end sec_cat
|
||||
end category
|
||||
|
|
|
|||
|
|
@ -44,7 +44,7 @@ if_neg H
|
|||
|
||||
definition rec_measure_aux {dom codom : Type} (default : codom) (measure : dom → ℕ)
|
||||
(rec_val : dom → (dom → codom) → codom) : ℕ → dom → codom :=
|
||||
rec (λx, default) (λm g x, if measure x < succ m then rec_val x g else default)
|
||||
nat.rec (λx, default) (λm g x, if measure x < succ m then rec_val x g else default)
|
||||
|
||||
definition rec_measure {dom codom : Type} (default : codom) (measure : dom → ℕ)
|
||||
(rec_val : dom → (dom → codom) → codom) (x : dom) : codom :=
|
||||
|
|
@ -62,7 +62,7 @@ theorem rec_measure_aux_spec {dom codom : Type} (default : codom) (measure : dom
|
|||
∀x, f' m x = restrict default measure f m x :=
|
||||
let f' := rec_measure_aux default measure rec_val in
|
||||
let f := rec_measure default measure rec_val in
|
||||
case_strong_induction_on m
|
||||
nat.case_strong_induction_on m
|
||||
(take x,
|
||||
have H1 : f' 0 x = default, from rfl,
|
||||
have H2 : ¬ measure x < 0, from !not_lt_zero,
|
||||
|
|
|
|||
|
|
@ -8,9 +8,9 @@ two : Three
|
|||
namespace Three
|
||||
|
||||
theorem disj (a : Three) : a = zero ∨ a = one ∨ a = two :=
|
||||
rec (or.inl rfl) (or.inr (or.inl rfl)) (or.inr (or.inr rfl)) a
|
||||
Three.rec (or.inl rfl) (or.inr (or.inl rfl)) (or.inr (or.inr rfl)) a
|
||||
|
||||
example (a : Three) : a ≠ zero → a ≠ one → a = two :=
|
||||
rec (λ h₁ h₂, absurd rfl h₁) (λ h₁ h₂, absurd rfl h₂) (λ h₁ h₂, rfl) a
|
||||
Three.rec (λ h₁ h₂, absurd rfl h₁) (λ h₁ h₂, absurd rfl h₂) (λ h₁ h₂, rfl) a
|
||||
|
||||
end Three
|
||||
|
|
|
|||
|
|
@ -6,7 +6,7 @@ monday, tuesday, wednesday, thursday, friday, saturday, sunday
|
|||
namespace day
|
||||
|
||||
definition next_weekday d :=
|
||||
rec_on d tuesday wednesday thursday friday monday monday monday
|
||||
day.rec_on d tuesday wednesday thursday friday monday monday monday
|
||||
|
||||
example : next_weekday (next_weekday saturday) = tuesday :=
|
||||
rfl
|
||||
|
|
|
|||
|
|
@ -3,12 +3,12 @@ open prod
|
|||
|
||||
namespace nat
|
||||
namespace manual
|
||||
definition brec_on {C : nat → Type} (n : nat) (F : Π (n : nat), @below C n → C n) : C n :=
|
||||
have general : C n × @below C n, from
|
||||
rec_on n
|
||||
definition brec_on {C : nat → Type} (n : nat) (F : Π (n : nat), @nat.below C n → C n) : C n :=
|
||||
have general : C n × @nat.below C n, from
|
||||
nat.rec_on n
|
||||
(pair (F zero unit.star) unit.star)
|
||||
(λ (n₁ : nat) (r₁ : C n₁ × @below C n₁),
|
||||
have b : @below C (succ n₁), from
|
||||
(λ (n₁ : nat) (r₁ : C n₁ × @nat.below C n₁),
|
||||
have b : @nat.below C (succ n₁), from
|
||||
r₁,
|
||||
have c : C (succ n₁), from
|
||||
F (succ n₁) b,
|
||||
|
|
@ -17,12 +17,12 @@ namespace nat
|
|||
end manual
|
||||
|
||||
definition fib (n : nat) :=
|
||||
brec_on n (λ (n : nat),
|
||||
cases_on n
|
||||
(λ (b₀ : below zero), succ zero)
|
||||
(λ (n₁ : nat), cases_on n₁
|
||||
(λ b₁ : below (succ zero), succ zero)
|
||||
(λ (n₂ : nat) (b₂ : below (succ (succ n₂))), pr₁ b₂ + pr₁ (pr₂ b₂))))
|
||||
nat.brec_on n (λ (n : nat),
|
||||
nat.cases_on n
|
||||
(λ (b₀ : nat.below zero), succ zero)
|
||||
(λ (n₁ : nat), nat.cases_on n₁
|
||||
(λ b₁ : nat.below (succ zero), succ zero)
|
||||
(λ (n₂ : nat) (b₂ : nat.below (succ (succ n₂))), pr₁ b₂ + pr₁ (pr₂ b₂))))
|
||||
|
||||
theorem fib_0 : fib 0 = 1 :=
|
||||
rfl
|
||||
|
|
|
|||
|
|
@ -12,7 +12,7 @@ namespace ftree
|
|||
namespace manual
|
||||
definition below.{l l₁ l₂} {A : Type.{l₁}} {B : Type.{l₂}} (C : ftree A B → Type.{l+1}) (t : ftree A B)
|
||||
: Type.{max l₁ l₂ (l+1)} :=
|
||||
@rec.{(max l₁ l₂ (l+1))+1 l₁ l₂}
|
||||
@ftree.rec.{(max l₁ l₂ (l+1))+1 l₁ l₂}
|
||||
A
|
||||
B
|
||||
(λ t : ftree A B, Type.{max l₁ l₂ (l+1)})
|
||||
|
|
@ -32,7 +32,7 @@ definition below.{l l₁ l₂} {A : Type.{l₁}} {B : Type.{l₂}} (C : ftree A
|
|||
|
||||
definition pbelow.{l₁ l₂} {A : Type.{l₁}} {B : Type.{l₂}} (C : ftree A B → Prop) (t : ftree A B)
|
||||
: Prop :=
|
||||
@rec.{1 l₁ l₂}
|
||||
@ftree.rec.{1 l₁ l₂}
|
||||
A
|
||||
B
|
||||
(λ t : ftree A B, Prop)
|
||||
|
|
@ -55,7 +55,7 @@ definition brec_on.{l l₁ l₂} {A : Type.{l₁}} {B : Type.{l₂}} {C : ftree
|
|||
(F : Π (t : ftree A B), @below A B C t → C t)
|
||||
: C t :=
|
||||
have gen : prod.{(l+1) (max l₁ l₂ (l+1))} (C t) (@below A B C t), from
|
||||
@rec.{(max l₁ l₂ (l+1)) l₁ l₂}
|
||||
@ftree.rec.{(max l₁ l₂ (l+1)) l₁ l₂}
|
||||
A
|
||||
B
|
||||
(λ t : ftree A B, prod.{(l+1) (max l₁ l₂ (l+1))} (C t) (@below A B C t))
|
||||
|
|
@ -82,27 +82,27 @@ prod.pr1 gen
|
|||
|
||||
definition binduction_on.{l₁ l₂} {A : Type.{l₁}} {B : Type.{l₂}} {C : ftree A B → Prop}
|
||||
(t : ftree A B)
|
||||
(F : Π (t : ftree A B), @ibelow A B C t → C t)
|
||||
(F : Π (t : ftree A B), @ftree.ibelow A B C t → C t)
|
||||
: C t :=
|
||||
have gen : C t ∧ @ibelow A B C t, from
|
||||
@rec.{0 l₁ l₂}
|
||||
have gen : C t ∧ @ftree.ibelow A B C t, from
|
||||
@ftree.rec.{0 l₁ l₂}
|
||||
A
|
||||
B
|
||||
(λ t : ftree A B, C t ∧ @ibelow A B C t)
|
||||
(have b : @ibelow A B C (leafa A B), from
|
||||
(λ t : ftree A B, C t ∧ @ftree.ibelow A B C t)
|
||||
(have b : @ftree.ibelow A B C (leafa A B), from
|
||||
true.intro,
|
||||
have c : C (leafa A B), from
|
||||
F (leafa A B) b,
|
||||
and.intro c b)
|
||||
(λ (f₁ : A → B → ftree A B)
|
||||
(f₂ : B → ftree A B)
|
||||
(r₁ : ∀ (a : A) (b : B), C (f₁ a b) ∧ @ibelow A B C (f₁ a b))
|
||||
(r₂ : ∀ (b : B), C (f₂ b) ∧ @ibelow A B C (f₂ b)),
|
||||
(r₁ : ∀ (a : A) (b : B), C (f₁ a b) ∧ @ftree.ibelow A B C (f₁ a b))
|
||||
(r₂ : ∀ (b : B), C (f₂ b) ∧ @ftree.ibelow A B C (f₂ b)),
|
||||
let fc₁ : ∀ (a : A) (b : B), C (f₁ a b) := λ (a : A) (b : B), and.elim_left (r₁ a b) in
|
||||
let fr₁ : ∀ (a : A) (b : B), @ibelow A B C (f₁ a b) := λ (a : A) (b : B), and.elim_right (r₁ a b) in
|
||||
let fr₁ : ∀ (a : A) (b : B), @ftree.ibelow A B C (f₁ a b) := λ (a : A) (b : B), and.elim_right (r₁ a b) in
|
||||
let fc₂ : ∀ (b : B), C (f₂ b) := λ (b : B), and.elim_left (r₂ b) in
|
||||
let fr₂ : ∀ (b : B), @ibelow A B C (f₂ b) := λ (b : B), and.elim_right (r₂ b) in
|
||||
have b : @ibelow A B C (node f₁ f₂), from
|
||||
let fr₂ : ∀ (b : B), @ftree.ibelow A B C (f₂ b) := λ (b : B), and.elim_right (r₂ b) in
|
||||
have b : @ftree.ibelow A B C (node f₁ f₂), from
|
||||
and.intro (and.intro fc₁ fr₁) (and.intro fc₂ fr₂),
|
||||
have c : C (node f₁ f₂), from
|
||||
F (node f₁ f₂) b,
|
||||
|
|
|
|||
|
|
@ -24,7 +24,7 @@ namespace inftree
|
|||
|
||||
definition dsub.wf (A : Type) : well_founded (@dsub A) :=
|
||||
well_founded.intro (λ (t : inftree A),
|
||||
rec_on t
|
||||
inftree.rec_on t
|
||||
(λ a, dsub.leaf.acc a)
|
||||
(λ f (ih :∀a, acc dsub (f a)), dsub.node.acc f ih))
|
||||
|
||||
|
|
|
|||
|
|
@ -21,7 +21,7 @@ namespace inftree
|
|||
|
||||
definition dsub.wf (A : Type) : well_founded (@dsub A) :=
|
||||
well_founded.intro (λ (t : inftree A),
|
||||
rec_on t
|
||||
inftree.rec_on t
|
||||
(λ a, dsub.leaf.acc a)
|
||||
(λ f t (ihf :∀a, acc dsub (f a)) (iht : acc dsub t), dsub.node.acc f ihf t iht))
|
||||
|
||||
|
|
|
|||
|
|
@ -6,14 +6,14 @@ up : A → lift A
|
|||
|
||||
namespace lift
|
||||
definition down {A : Type} (a : lift A) : A :=
|
||||
rec (λ a, a) a
|
||||
lift.rec (λ a, a) a
|
||||
|
||||
theorem down_up {A : Type} (a : A) : down (up a) = a :=
|
||||
rfl
|
||||
|
||||
|
||||
theorem up_down {A : Type} (a' : lift A) : up (down a') = a' :=
|
||||
induction_on a' (λ a, rfl)
|
||||
lift.induction_on a' (λ a, rfl)
|
||||
|
||||
end lift
|
||||
|
||||
|
|
|
|||
|
|
@ -13,13 +13,13 @@ theorem pred_zero : pred zero = zero := refl _
|
|||
theorem pred_succ (n : nat) : pred (succ n) = n := refl _
|
||||
|
||||
theorem zero_or_succ (n : nat) : n = zero ∨ n = succ (pred n)
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(or.intro_left _ (refl zero))
|
||||
(take m IH, or.intro_right _
|
||||
(show succ m = succ (pred (succ m)), from congr_arg succ (symm (pred_succ m))))
|
||||
|
||||
theorem zero_or_succ2 (n : nat) : n = zero ∨ n = succ (pred n)
|
||||
:= @induction_on _ n
|
||||
:= @nat.induction_on _ n
|
||||
(or.intro_left _ (refl zero))
|
||||
(take m IH, or.intro_right _
|
||||
(show succ m = succ (pred (succ m)), from congr_arg succ (symm (pred_succ m))))
|
||||
|
|
|
|||
|
|
@ -19,7 +19,7 @@ axiom add_assoc (n m k : nat) : (n + m) + k = n + (m + k)
|
|||
axiom add_right_comm (n m k : nat) : n + m + k = n + k + m
|
||||
set_option unifier.max_steps 50000
|
||||
theorem mul_add_distr_left (n m k : nat) : (n + m) * k = n * k + m * k
|
||||
:= induction_on k
|
||||
:= nat.induction_on k
|
||||
(calc
|
||||
(n + m) * zero = zero : eq.refl _
|
||||
... = n * zero + m * zero : eq.refl _)
|
||||
|
|
|
|||
|
|
@ -6,14 +6,14 @@ vnil : vector A zero,
|
|||
vcons : Π {n : nat}, A → vector A n → vector A (succ n)
|
||||
|
||||
namespace vector
|
||||
definition no_confusion2 {A : Type} {n : nat} {P : Type} {v₁ v₂ : vector A n} : v₁ = v₂ → no_confusion_type P v₁ v₂ :=
|
||||
definition no_confusion2 {A : Type} {n : nat} {P : Type} {v₁ v₂ : vector A n} : v₁ = v₂ → vector.no_confusion_type P v₁ v₂ :=
|
||||
assume H₁₂ : v₁ = v₂,
|
||||
begin
|
||||
show no_confusion_type P v₁ v₂, from
|
||||
have aux : v₁ = v₁ → no_confusion_type P v₁ v₁, from
|
||||
show vector.no_confusion_type P v₁ v₂, from
|
||||
have aux : v₁ = v₁ → vector.no_confusion_type P v₁ v₁, from
|
||||
take H₁₁,
|
||||
begin
|
||||
apply (cases_on v₁),
|
||||
apply (vector.cases_on v₁),
|
||||
exact (assume h : P, h),
|
||||
|
||||
intros (n, a, v, h),
|
||||
|
|
@ -26,11 +26,11 @@ namespace vector
|
|||
|
||||
theorem vcons.inj₁ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → a₁ = a₂ :=
|
||||
begin
|
||||
intro h, apply (no_confusion h), intros, assumption
|
||||
intro h, apply (vector.no_confusion h), intros, assumption
|
||||
end
|
||||
|
||||
theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ == v₂ :=
|
||||
begin
|
||||
intro h, apply (no_confusion h), intros, eassumption
|
||||
intro h, apply (vector.no_confusion h), intros, eassumption
|
||||
end
|
||||
end vector
|
||||
|
|
|
|||
|
|
@ -3,15 +3,15 @@ import data.sigma
|
|||
namespace sigma
|
||||
namespace manual
|
||||
definition no_confusion_type {A : Type} {B : A → Type} (P : Type) (v₁ v₂ : sigma B) : Type :=
|
||||
rec_on v₁
|
||||
(λ (a₁ : A) (b₁ : B a₁), rec_on v₂
|
||||
sigma.rec_on v₁
|
||||
(λ (a₁ : A) (b₁ : B a₁), sigma.rec_on v₂
|
||||
(λ (a₂ : A) (b₂ : B a₂),
|
||||
(Π (eq₁ : a₁ = a₂), eq.rec_on eq₁ b₁ = b₂ → P) → P))
|
||||
|
||||
definition no_confusion {A : Type} {B : A → Type} {P : Type} {v₁ v₂ : sigma B} : v₁ = v₂ → no_confusion_type P v₁ v₂ :=
|
||||
assume H₁₂ : v₁ = v₂,
|
||||
have aux : v₁ = v₁ → no_confusion_type P v₁ v₁, from
|
||||
assume H₁₁, rec_on v₁
|
||||
assume H₁₁, sigma.rec_on v₁
|
||||
(λ (a₁ : A) (b₁ : B a₁) (h : Π (eq₁ : a₁ = a₁), eq.rec_on eq₁ b₁ = b₁ → P),
|
||||
h rfl rfl),
|
||||
eq.rec_on H₁₂ aux H₁₂
|
||||
|
|
@ -19,11 +19,11 @@ namespace sigma
|
|||
|
||||
theorem sigma.mk.inj_1 {A : Type} {B : A → Type} {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} (Heq : mk a₁ b₁ = mk a₂ b₂) : a₁ = a₂ :=
|
||||
begin
|
||||
apply (no_confusion Heq), intros, assumption
|
||||
apply (sigma.no_confusion Heq), intros, assumption
|
||||
end
|
||||
|
||||
theorem sigma.mk.inj_2 {A : Type} {B : A → Type} (a₁ a₂ : A) (b₁ : B a₁) (b₂ : B a₂) (Heq : mk a₁ b₁ = mk a₂ b₂) : b₁ == b₂ :=
|
||||
begin
|
||||
apply (no_confusion Heq), intros, eassumption
|
||||
apply (sigma.no_confusion Heq), intros, eassumption
|
||||
end
|
||||
end sigma
|
||||
|
|
|
|||
|
|
@ -16,19 +16,19 @@ infixr `+` := sum
|
|||
open eq
|
||||
|
||||
theorem inl_inj {A B : Type} {a1 a2 : A} (H : inl B a1 = inl B a2) : a1 = a2 :=
|
||||
let f := λs, rec_on s (λa, a1 = a) (λb, false) in
|
||||
let f := λs, sum.rec_on s (λa, a1 = a) (λb, false) in
|
||||
have H1 : f (inl B a1), from rfl,
|
||||
have H2 : f (inl B a2), from subst H H1,
|
||||
H2
|
||||
|
||||
theorem inl_neq_inr {A B : Type} {a : A} {b : B} (H : inl B a = inr A b) : false :=
|
||||
let f := λs, rec_on s (λa', a = a') (λb, false) in
|
||||
let f := λs, sum.rec_on s (λa', a = a') (λb, false) in
|
||||
have H1 : f (inl B a), from rfl,
|
||||
have H2 : f (inr A b), from subst H H1,
|
||||
H2
|
||||
|
||||
theorem inr_inj {A B : Type} {b1 b2 : B} (H : inr A b1 = inr A b2) : b1 = b2 :=
|
||||
let f := λs, rec_on s (λa, false) (λb, b1 = b) in
|
||||
let f := λs, sum.rec_on s (λa, false) (λb, b1 = b) in
|
||||
have H1 : f (inr A b1), from rfl,
|
||||
have H2 : f (inr A b2), from subst H H1,
|
||||
H2
|
||||
|
|
@ -42,16 +42,16 @@ inhabited.mk (inr A (default B))
|
|||
theorem sum_eq_decidable [instance] {A B : Type} (s1 s2 : A + B)
|
||||
(H1 : ∀a1 a2 : A, decidable (inl B a1 = inl B a2))
|
||||
(H2 : ∀b1 b2 : B, decidable (inr A b1 = inr A b2)) : decidable (s1 = s2) :=
|
||||
rec_on s1
|
||||
sum.rec_on s1
|
||||
(take a1, show decidable (inl B a1 = s2), from
|
||||
rec_on s2
|
||||
sum.rec_on s2
|
||||
(take a2, show decidable (inl B a1 = inl B a2), from H1 a1 a2)
|
||||
(take b2,
|
||||
have H3 : (inl B a1 = inr A b2) ↔ false,
|
||||
from iff.intro inl_neq_inr (assume H4, false.elim H4),
|
||||
show decidable (inl B a1 = inr A b2), from decidable_of_decidable_of_iff _ (iff.symm H3)))
|
||||
(take b1, show decidable (inr A b1 = s2), from
|
||||
rec_on s2
|
||||
sum.rec_on s2
|
||||
(take a2,
|
||||
have H3 : (inr A b1 = inl B a2) ↔ false,
|
||||
from iff.intro (assume H4, inl_neq_inr (symm H4)) (assume H4, false.elim H4),
|
||||
|
|
|
|||
|
|
@ -17,7 +17,7 @@ namespace manual
|
|||
variable {A : Type.{l₁}}
|
||||
variable (C : tree A → Type.{l₂})
|
||||
definition below (t : tree A) : Type :=
|
||||
rec_on t (λ a, one.{l₂}) (λ t₁ t₂ r₁ r₂, C t₁ × C t₂ × r₁ × r₂)
|
||||
tree.rec_on t (λ a, one.{l₂}) (λ t₁ t₂ r₁ r₂, C t₁ × C t₂ × r₁ × r₂)
|
||||
end
|
||||
|
||||
section
|
||||
|
|
@ -26,7 +26,7 @@ namespace manual
|
|||
variable {C : tree A → Type.{l₂}}
|
||||
definition below_rec_on (t : tree A) (H : Π (n : tree A), below C n → C n) : C t
|
||||
:= have general : C t × below C t, from
|
||||
rec_on t
|
||||
tree.rec_on t
|
||||
(λa, (H (leaf a) one.star, one.star))
|
||||
(λ (l r : tree A) (Hl : C l × below C l) (Hr : C r × below C r),
|
||||
have b : below C (node l r), from
|
||||
|
|
@ -41,12 +41,12 @@ end manual
|
|||
universe variables l₁ l₂
|
||||
variable {A : Type.{l₁}}
|
||||
variable {C : tree A → Type.{l₂+1}}
|
||||
definition below_rec_on (t : tree A) (H : Π (n : tree A), @below A C n → C n) : C t
|
||||
:= have general : C t × @below A C t, from
|
||||
rec_on t
|
||||
definition below_rec_on (t : tree A) (H : Π (n : tree A), @tree.below A C n → C n) : C t
|
||||
:= have general : C t × @tree.below A C t, from
|
||||
tree.rec_on t
|
||||
(λa, (H (leaf a) unit.star, unit.star))
|
||||
(λ (l r : tree A) (Hl : C l × @below A C l) (Hr : C r × @below A C r),
|
||||
have b : @below A C (node l r), from
|
||||
(λ (l r : tree A) (Hl : C l × @tree.below A C l) (Hr : C r × @tree.below A C r),
|
||||
have b : @tree.below A C (node l r), from
|
||||
((pr₁ Hl, pr₂ Hl), (pr₁ Hr, pr₂ Hr)),
|
||||
have c : C (node l r), from
|
||||
H (node l r) b,
|
||||
|
|
@ -58,5 +58,5 @@ end manual
|
|||
|
||||
theorem leaf_ne_tree {A : Type} (a : A) (l r : tree A) : leaf a ≠ node l r :=
|
||||
assume h : leaf a = node l r,
|
||||
no_confusion h
|
||||
tree.no_confusion h
|
||||
end tree
|
||||
|
|
|
|||
|
|
@ -17,7 +17,7 @@ namespace tree
|
|||
variable {A : Type.{l₁}}
|
||||
variable (C : tree A → Type.{l₂})
|
||||
definition below (t : tree A) : Type :=
|
||||
rec_on t (λ a, one.{l₂}) (λ t₁ t₂ r₁ r₂, C t₁ × C t₂ × r₁ × r₂)
|
||||
tree.rec_on t (λ a, one.{l₂}) (λ t₁ t₂ r₁ r₂, C t₁ × C t₂ × r₁ × r₂)
|
||||
end
|
||||
|
||||
section
|
||||
|
|
@ -26,7 +26,7 @@ namespace tree
|
|||
variable {C : tree A → Type.{l₂}}
|
||||
definition below_rec_on (t : tree A) (H : Π (n : tree A), below C n → C n) : C t
|
||||
:= have general : C t × below C t, from
|
||||
rec_on t
|
||||
tree.rec_on t
|
||||
(λa, (H (leaf a) one.star, one.star))
|
||||
(λ (l r : tree A) (Hl : C l × below C l) (Hr : C r × below C r),
|
||||
have b : below C (node l r), from
|
||||
|
|
@ -38,29 +38,29 @@ namespace tree
|
|||
end
|
||||
end manual
|
||||
|
||||
check no_confusion
|
||||
check tree.no_confusion
|
||||
|
||||
theorem leaf_ne_tree {A : Type} (a : A) (l r : tree A) : leaf a ≠ node l r :=
|
||||
assume h : leaf a = node l r,
|
||||
no_confusion h
|
||||
tree.no_confusion h
|
||||
|
||||
constant A : Type₁
|
||||
constants l₁ l₂ r₁ r₂ : tree A
|
||||
axiom node_eq : node l₁ r₁ = node l₂ r₂
|
||||
check no_confusion node_eq
|
||||
check tree.no_confusion node_eq
|
||||
|
||||
definition tst : (l₁ = l₂ → r₁ = r₂ → l₁ = l₂) → l₁ = l₂ := no_confusion node_eq
|
||||
definition tst : (l₁ = l₂ → r₁ = r₂ → l₁ = l₂) → l₁ = l₂ := tree.no_confusion node_eq
|
||||
check tst (λ e₁ e₂, e₁)
|
||||
|
||||
theorem node.inj1 {A : Type} (l₁ l₂ r₁ r₂ : tree A) : node l₁ r₁ = node l₂ r₂ → l₁ = l₂ :=
|
||||
assume h,
|
||||
have trivial : (l₁ = l₂ → r₁ = r₂ → l₁ = l₂) → l₁ = l₂, from no_confusion h,
|
||||
have trivial : (l₁ = l₂ → r₁ = r₂ → l₁ = l₂) → l₁ = l₂, from tree.no_confusion h,
|
||||
trivial (λ e₁ e₂, e₁)
|
||||
|
||||
theorem node.inj2 {A : Type} (l₁ l₂ r₁ r₂ : tree A) : node l₁ r₁ = node l₂ r₂ → l₁ = l₂ :=
|
||||
begin
|
||||
intro h,
|
||||
apply (no_confusion h),
|
||||
apply (tree.no_confusion h),
|
||||
intros, assumption
|
||||
end
|
||||
end tree
|
||||
|
|
|
|||
|
|
@ -17,7 +17,7 @@ namespace tree
|
|||
variable {A : Type.{l₁}}
|
||||
variable (C : tree A → Type.{l₂})
|
||||
definition below (t : tree A) : Type :=
|
||||
rec_on t (λ a, one.{l₂}) (λ t₁ t₂ r₁ r₂, C t₁ × C t₂ × r₁ × r₂)
|
||||
tree.rec_on t (λ a, one.{l₂}) (λ t₁ t₂ r₁ r₂, C t₁ × C t₂ × r₁ × r₂)
|
||||
end
|
||||
|
||||
section
|
||||
|
|
@ -26,7 +26,7 @@ namespace tree
|
|||
variable {C : tree A → Type.{l₂}}
|
||||
definition below_rec_on (t : tree A) (H : Π (n : tree A), below C n → C n) : C t
|
||||
:= have general : C t × below C t, from
|
||||
rec_on t
|
||||
tree.rec_on t
|
||||
(λa, (H (leaf a) one.star, one.star))
|
||||
(λ (l r : tree A) (Hl : C l × below C l) (Hr : C r × below C r),
|
||||
have b : below C (node l r), from
|
||||
|
|
@ -37,7 +37,7 @@ namespace tree
|
|||
pr₁ general
|
||||
end
|
||||
end manual
|
||||
|
||||
local abbreviation no_confusion := @tree.no_confusion
|
||||
check no_confusion
|
||||
|
||||
theorem leaf_ne_tree {A : Type} (a : A) (l r : tree A) : leaf a ≠ node l r :=
|
||||
|
|
|
|||
|
|
@ -8,7 +8,7 @@ node : tree A → tree A → tree A
|
|||
namespace tree
|
||||
|
||||
definition height {A : Type} (t : tree A) : nat :=
|
||||
rec_on t
|
||||
tree.rec_on t
|
||||
(λ a, zero)
|
||||
(λ t₁ t₂ h₁ h₂, succ (max h₁ h₂))
|
||||
|
||||
|
|
|
|||
|
|
@ -6,7 +6,7 @@ vnil {} : vector A zero,
|
|||
vcons : Π {n : nat}, A → vector A n → vector A (succ n)
|
||||
|
||||
namespace vector
|
||||
print definition no_confusion
|
||||
print definition vector.no_confusion
|
||||
infixr `::` := vcons
|
||||
|
||||
namespace play
|
||||
|
|
@ -14,12 +14,12 @@ namespace vector
|
|||
universe variables l₁ l₂
|
||||
variable {A : Type.{l₁}}
|
||||
variable {C : Π (n : nat), vector A n → Type.{l₂+1}}
|
||||
definition brec_on {n : nat} (v : vector A n) (H : Π (n : nat) (v : vector A n), @below A C n v → C n v) : C n v :=
|
||||
have general : C n v × @below A C n v, from
|
||||
rec_on v
|
||||
definition brec_on {n : nat} (v : vector A n) (H : Π (n : nat) (v : vector A n), @vector.below A C n v → C n v) : C n v :=
|
||||
have general : C n v × @vector.below A C n v, from
|
||||
vector.rec_on v
|
||||
(pair (H zero vnil unit.star) unit.star)
|
||||
(λ (n₁ : nat) (a₁ : A) (v₁ : vector A n₁) (r₁ : C n₁ v₁ × @below A C n₁ v₁),
|
||||
have b : @below A C _ (vcons a₁ v₁), from
|
||||
(λ (n₁ : nat) (a₁ : A) (v₁ : vector A n₁) (r₁ : C n₁ v₁ × @vector.below A C n₁ v₁),
|
||||
have b : @vector.below A C _ (vcons a₁ v₁), from
|
||||
r₁,
|
||||
have c : C (succ n₁) (vcons a₁ v₁), from
|
||||
H (succ n₁) (vcons a₁ v₁) b,
|
||||
|
|
@ -30,19 +30,19 @@ namespace vector
|
|||
|
||||
print "====================="
|
||||
definition append {A : Type} {n m : nat} (w : vector A m) (v : vector A n) : vector A (n + m) :=
|
||||
brec_on w (λ (n : nat) (w : vector A n),
|
||||
cases_on w
|
||||
(λ (B : below vnil), v)
|
||||
(λ (n₁ : nat) (a₁ : A) (v₁ : vector A n₁) (B : below (vcons a₁ v₁)),
|
||||
vector.brec_on w (λ (n : nat) (w : vector A n),
|
||||
vector.cases_on w
|
||||
(λ (B : vector.below vnil), v)
|
||||
(λ (n₁ : nat) (a₁ : A) (v₁ : vector A n₁) (B : vector.below (vcons a₁ v₁)),
|
||||
vcons a₁ (pr₁ B)))
|
||||
|
||||
exit
|
||||
check brec_on
|
||||
definition bw := @below
|
||||
check vector.brec_on
|
||||
definition bw := @vector.below
|
||||
|
||||
definition sum {n : nat} (v : vector nat n) : nat :=
|
||||
brec_on v (λ (n : nat) (v : vector nat n),
|
||||
cases_on v
|
||||
vector.brec_on v (λ (n : nat) (v : vector nat n),
|
||||
vector.cases_on v
|
||||
(λ (B : bw vnil), zero)
|
||||
(λ (n₁ : nat) (a : nat) (v₁ : vector nat n₁) (B : bw (vcons a v₁)),
|
||||
a + pr₁ B))
|
||||
|
|
@ -51,8 +51,8 @@ namespace vector
|
|||
rfl
|
||||
|
||||
definition addk {n : nat} (v : vector nat n) (k : nat) : vector nat n :=
|
||||
brec_on v (λ (n : nat) (v : vector nat n),
|
||||
cases_on v
|
||||
vector.brec_on v (λ (n : nat) (v : vector nat n),
|
||||
vector.cases_on v
|
||||
(λ (B : bw vnil), vnil)
|
||||
(λ (n₁ : nat) (a₁ : nat) (v₁ : vector nat n₁) (B : bw (vcons a₁ v₁)),
|
||||
vcons (a₁+k) (pr₁ B)))
|
||||
|
|
@ -64,22 +64,22 @@ namespace vector
|
|||
rfl
|
||||
|
||||
definition head {A : Type} {n : nat} (v : vector A (succ n)) : A :=
|
||||
cases_on v
|
||||
nat.cases_on v
|
||||
(λ H : succ n = 0, nat.no_confusion H)
|
||||
(λn' h t (H : succ n = succ n'), h)
|
||||
rfl
|
||||
|
||||
definition tail {A : Type} {n : nat} (v : vector A (succ n)) : vector A n :=
|
||||
@cases_on A (λn' v, succ n = n' → vector A (pred n')) (succ n) v
|
||||
@nat.cases_on A (λn' v, succ n = n' → vector A (pred n')) (succ n) v
|
||||
(λ H : succ n = 0, nat.no_confusion H)
|
||||
(λ (n' : nat) (h : A) (t : vector A n') (H : succ n = succ n'),
|
||||
t)
|
||||
rfl
|
||||
|
||||
definition add {n : nat} (w v : vector nat n) : vector nat n :=
|
||||
@brec_on nat (λ (n : nat) (v : vector nat n), vector nat n → vector nat n) n w
|
||||
@nat.brec_on nat (λ (n : nat) (v : vector nat n), vector nat n → vector nat n) n w
|
||||
(λ (n : nat) (w : vector nat n),
|
||||
cases_on w
|
||||
vector.cases_on w
|
||||
(λ (B : bw vnil) (w : vector nat zero), vnil)
|
||||
(λ (n₁ : nat) (a₁ : nat) (v₁ : vector nat n₁) (B : bw (vcons a₁ v₁)) (v : vector nat (succ n₁)),
|
||||
vcons (a₁ + head v) (pr₁ B (tail v)))) v
|
||||
|
|
|
|||
|
|
@ -9,6 +9,9 @@ namespace vector
|
|||
-- print definition no_confusion
|
||||
infixr `::` := vcons
|
||||
|
||||
local abbreviation no_confusion := @vector.no_confusion
|
||||
local abbreviation below := @vector.below
|
||||
|
||||
theorem vcons.inj₁ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → a₁ = a₂ :=
|
||||
begin
|
||||
intro h, apply (no_confusion h), intros, assumption
|
||||
|
|
@ -27,12 +30,12 @@ namespace vector
|
|||
universe variables l₁ l₂
|
||||
variable {A : Type.{l₁}}
|
||||
variable {C : Π (n : nat), vector A n → Type.{l₂}}
|
||||
definition brec_on {n : nat} (v : vector A n) (H : Π (n : nat) (v : vector A n), @below A C n v → C n v) : C n v :=
|
||||
have general : C n v × @below A C n v, from
|
||||
rec_on v
|
||||
definition brec_on {n : nat} (v : vector A n) (H : Π (n : nat) (v : vector A n), @vector.below A C n v → C n v) : C n v :=
|
||||
have general : C n v × @vector.below A C n v, from
|
||||
vector.rec_on v
|
||||
(pair (H zero vnil unit.star) unit.star)
|
||||
(λ (n₁ : nat) (a₁ : A) (v₁ : vector A n₁) (r₁ : C n₁ v₁ × @below A C n₁ v₁),
|
||||
have b : @below A C _ (vcons a₁ v₁), from
|
||||
(λ (n₁ : nat) (a₁ : A) (v₁ : vector A n₁) (r₁ : C n₁ v₁ × @vector.below A C n₁ v₁),
|
||||
have b : @vector.below A C _ (vcons a₁ v₁), from
|
||||
r₁,
|
||||
have c : C (succ n₁) (vcons a₁ v₁), from
|
||||
H (succ n₁) (vcons a₁ v₁) b,
|
||||
|
|
@ -41,13 +44,13 @@ namespace vector
|
|||
end
|
||||
end manual
|
||||
|
||||
-- check brec_on
|
||||
-- check vector.brec_on
|
||||
|
||||
definition bw := @below
|
||||
|
||||
definition sum {n : nat} (v : vector nat n) : nat :=
|
||||
brec_on v (λ (n : nat) (v : vector nat n),
|
||||
cases_on v
|
||||
vector.brec_on v (λ (n : nat) (v : vector nat n),
|
||||
vector.cases_on v
|
||||
(λ (B : bw vnil), zero)
|
||||
(λ (n₁ : nat) (a : nat) (v₁ : vector nat n₁) (B : bw (vcons a v₁)),
|
||||
a + pr₁ B))
|
||||
|
|
@ -56,8 +59,8 @@ namespace vector
|
|||
rfl
|
||||
|
||||
definition addk {n : nat} (v : vector nat n) (k : nat) : vector nat n :=
|
||||
brec_on v (λ (n : nat) (v : vector nat n),
|
||||
cases_on v
|
||||
vector.brec_on v (λ (n : nat) (v : vector nat n),
|
||||
vector.cases_on v
|
||||
(λ (B : bw vnil), vnil)
|
||||
(λ (n₁ : nat) (a₁ : nat) (v₁ : vector nat n₁) (B : bw (vcons a₁ v₁)),
|
||||
vcons (a₁+k) (pr₁ B)))
|
||||
|
|
@ -66,8 +69,8 @@ namespace vector
|
|||
rfl
|
||||
|
||||
definition append.{l} {A : Type.{l+1}} {n m : nat} (w : vector A m) (v : vector A n) : vector A (n + m) :=
|
||||
brec_on w (λ (n : nat) (w : vector A n),
|
||||
cases_on w
|
||||
vector.brec_on w (λ (n : nat) (w : vector A n),
|
||||
vector.cases_on w
|
||||
(λ (B : bw vnil), v)
|
||||
(λ (n₁ : nat) (a₁ : A) (v₁ : vector A n₁) (B : bw (vcons a₁ v₁)),
|
||||
vcons a₁ (pr₁ B)))
|
||||
|
|
@ -76,22 +79,22 @@ namespace vector
|
|||
rfl
|
||||
|
||||
definition head {A : Type} {n : nat} (v : vector A (succ n)) : A :=
|
||||
cases_on v
|
||||
vector.cases_on v
|
||||
(λ H : succ n = 0, nat.no_confusion H)
|
||||
(λn' h t (H : succ n = succ n'), h)
|
||||
rfl
|
||||
|
||||
definition tail {A : Type} {n : nat} (v : vector A (succ n)) : vector A n :=
|
||||
@cases_on A (λn' v, succ n = n' → vector A (pred n')) (succ n) v
|
||||
@vector.cases_on A (λn' v, succ n = n' → vector A (pred n')) (succ n) v
|
||||
(λ H : succ n = 0, nat.no_confusion H)
|
||||
(λ (n' : nat) (h : A) (t : vector A n') (H : succ n = succ n'),
|
||||
t)
|
||||
rfl
|
||||
|
||||
definition add {n : nat} (w v : vector nat n) : vector nat n :=
|
||||
@brec_on nat (λ (n : nat) (v : vector nat n), vector nat n → vector nat n) n w
|
||||
@vector.brec_on nat (λ (n : nat) (v : vector nat n), vector nat n → vector nat n) n w
|
||||
(λ (n : nat) (w : vector nat n),
|
||||
cases_on w
|
||||
vector.cases_on w
|
||||
(λ (B : bw vnil) (w : vector nat zero), vnil)
|
||||
(λ (n₁ : nat) (a₁ : nat) (v₁ : vector nat n₁) (B : bw (vcons a₁ v₁)) (v : vector nat (succ n₁)),
|
||||
vcons (a₁ + head v) (pr₁ B (tail v)))) v
|
||||
|
|
@ -101,7 +104,7 @@ namespace vector
|
|||
|
||||
definition map {A B C : Type} {n : nat} (f : A → B → C) (w : vector A n) (v : vector B n) : vector C n :=
|
||||
let P := λ (n : nat) (v : vector A n), vector B n → vector C n in
|
||||
@brec_on A P n w
|
||||
@vector.brec_on A P n w
|
||||
(λ (n : nat) (w : vector A n),
|
||||
begin
|
||||
cases w with (n₁, h₁, t₁),
|
||||
|
|
|
|||
|
|
@ -7,6 +7,6 @@ vcons : Π {n : nat}, A → vector A n → vector A (succ n)
|
|||
|
||||
namespace vector
|
||||
theorem vcons.inj₁ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → a₁ = a₂ :=
|
||||
assume h, no_confusion h (λ n h t, h)
|
||||
assume h, vector.no_confusion h (λ n h t, h)
|
||||
|
||||
end vector
|
||||
|
|
|
|||
|
|
@ -7,8 +7,8 @@ vcons : Π {n : nat}, A → vector A n → vector A (succ n)
|
|||
|
||||
namespace vector
|
||||
theorem vcons.inj₁ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → a₁ = a₂ :=
|
||||
assume h, no_confusion h (λ n h t, h)
|
||||
assume h, vector.no_confusion h (λ n h t, h)
|
||||
|
||||
theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ == v₂ :=
|
||||
assume h, no_confusion h (λ n h t, t)
|
||||
assume h, vector.no_confusion h (λ n h t, t)
|
||||
end vector
|
||||
|
|
|
|||
|
|
@ -54,7 +54,7 @@ theorem pred_zero : pred 0 = 0
|
|||
theorem pred_succ (n : ℕ) : pred (succ n) = n
|
||||
|
||||
theorem zero_or_succ (n : ℕ) : n = 0 ∨ n = succ (pred n)
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(or.intro_left _ (refl 0))
|
||||
(take m IH, or.intro_right _
|
||||
(show succ m = succ (pred (succ m)), from congr_arg succ ((pred_succ m)⁻¹)))
|
||||
|
|
@ -63,7 +63,7 @@ theorem zero_or_succ2 (n : ℕ) : n = 0 ∨ ∃k, n = succ k
|
|||
:= or_of_or_of_imp_of_imp (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists.intro (pred n) H)
|
||||
|
||||
theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n
|
||||
:= induction_on n H1 (take m IH, H2 m)
|
||||
:= nat.induction_on n H1 (take m IH, H2 m)
|
||||
|
||||
theorem discriminate {B : Prop} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B
|
||||
:= or.elim (zero_or_succ n)
|
||||
|
|
@ -77,7 +77,7 @@ theorem succ_inj {n m : ℕ} (H : succ n = succ m) : n = m
|
|||
... = m : pred_succ m
|
||||
|
||||
theorem succ_ne_self (n : ℕ) : succ n ≠ n
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(take H : 1 = 0,
|
||||
have ne : 1 ≠ 0, from succ_ne_zero 0,
|
||||
absurd H ne)
|
||||
|
|
@ -85,13 +85,13 @@ theorem succ_ne_self (n : ℕ) : succ n ≠ n
|
|||
|
||||
theorem decidable_eq [instance] (n m : ℕ) : decidable (n = m)
|
||||
:= have general : ∀n, decidable (n = m), from
|
||||
rec_on m
|
||||
nat.rec_on m
|
||||
(take n,
|
||||
rec_on n
|
||||
nat.rec_on n
|
||||
(inl (refl 0))
|
||||
(λ m iH, inr (succ_ne_zero m)))
|
||||
(λ (m' : ℕ) (iH1 : ∀n, decidable (n = m')),
|
||||
take n, rec_on n
|
||||
take n, nat.rec_on n
|
||||
(inr (ne.symm (succ_ne_zero m')))
|
||||
(λ (n' : ℕ) (iH2 : decidable (n' = succ m')),
|
||||
have d1 : decidable (n' = m'), from iH1 n',
|
||||
|
|
@ -106,7 +106,7 @@ theorem decidable_eq [instance] (n m : ℕ) : decidable (n = m)
|
|||
theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1)
|
||||
(H3 : ∀ (n : ℕ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a
|
||||
:= have stronger : P a ∧ P (succ a), from
|
||||
induction_on a
|
||||
nat.induction_on a
|
||||
(and_intro H1 H2)
|
||||
(take k IH,
|
||||
have IH1 : P k, from and.elim_left IH,
|
||||
|
|
@ -116,7 +116,7 @@ theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1)
|
|||
|
||||
theorem sub_induction {P : ℕ → ℕ → Prop} (n m : ℕ) (H1 : ∀m, P 0 m)
|
||||
(H2 : ∀n, P (succ n) 0) (H3 : ∀n m, P n m → P (succ n) (succ m)) : P n m
|
||||
:= have general : ∀m, P n m, from induction_on n
|
||||
:= have general : ∀m, P n m, from nat.induction_on n
|
||||
(take m : ℕ, H1 m)
|
||||
(take k : ℕ,
|
||||
assume IH : ∀m, P k m,
|
||||
|
|
@ -137,7 +137,7 @@ theorem add_succ (n m : ℕ) : n + succ m = succ (n + m)
|
|||
---------- comm, assoc
|
||||
|
||||
theorem zero_add (n : ℕ) : 0 + n = n
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(add_zero 0)
|
||||
(take m IH, show 0 + succ m = succ m, from
|
||||
calc
|
||||
|
|
@ -145,7 +145,7 @@ theorem zero_add (n : ℕ) : 0 + n = n
|
|||
... = succ m : {IH})
|
||||
|
||||
theorem succ_add (n m : ℕ) : (succ n) + m = succ (n + m)
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(calc
|
||||
succ n + 0 = succ n : add_zero (succ n)
|
||||
... = succ (n + 0) : {symm (add_zero n)})
|
||||
|
|
@ -156,7 +156,7 @@ theorem succ_add (n m : ℕ) : (succ n) + m = succ (n + m)
|
|||
... = succ (n + succ k) : {symm (add_succ _ _)})
|
||||
|
||||
theorem add_comm (n m : ℕ) : n + m = m + n
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(trans (add_zero _) (symm (zero_add _)))
|
||||
(take k IH,
|
||||
calc
|
||||
|
|
@ -175,7 +175,7 @@ theorem add_comm_succ (n m : ℕ) : n + succ m = m + succ n
|
|||
... = m + succ n : add_comm (succ n) m
|
||||
|
||||
theorem add_assoc (n m k : ℕ) : (n + m) + k = n + (m + k)
|
||||
:= induction_on k
|
||||
:= nat.induction_on k
|
||||
(calc
|
||||
(n + m) + 0 = n + m : add_zero _
|
||||
... = n + (m + 0) : {symm (add_zero m)})
|
||||
|
|
@ -197,7 +197,7 @@ theorem add_right_comm (n m k : ℕ) : n + m + k = n + k + m
|
|||
|
||||
theorem add_cancel_left {n m k : ℕ} : n + m = n + k → m = k
|
||||
:=
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(take H : 0 + m = 0 + k,
|
||||
calc
|
||||
m = 0 + m : symm (zero_add m)
|
||||
|
|
@ -224,7 +224,7 @@ theorem add_cancel_right {n m k : ℕ} (H : n + m = k + m) : n = k
|
|||
|
||||
theorem eq_zero_of_add_eq_zero_right {n m : ℕ} : n + m = 0 → n = 0
|
||||
:=
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(take (H : 0 + m = 0), refl 0)
|
||||
(take k IH,
|
||||
assume (H : succ k + m = 0),
|
||||
|
|
@ -273,7 +273,7 @@ set_option unifier.max_steps 100000
|
|||
---------- comm, distr, assoc, identity
|
||||
|
||||
theorem mul_zero_left (n:ℕ) : 0 * n = 0
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(mul_zero_right 0)
|
||||
(take m IH,
|
||||
calc
|
||||
|
|
@ -282,7 +282,7 @@ theorem mul_zero_left (n:ℕ) : 0 * n = 0
|
|||
... = 0 : IH)
|
||||
|
||||
theorem mul_succ_left (n m:ℕ) : (succ n) * m = (n * m) + m
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(calc
|
||||
succ n * 0 = 0 : mul_zero_right _
|
||||
... = n * 0 : symm (mul_zero_right _)
|
||||
|
|
@ -297,7 +297,7 @@ theorem mul_succ_left (n m:ℕ) : (succ n) * m = (n * m) + m
|
|||
... = (n * succ k) + succ k : {symm (mul_succ_right n k)})
|
||||
|
||||
theorem mul_comm (n m:ℕ) : n * m = m * n
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(trans (mul_zero_right _) (symm (mul_zero_left _)))
|
||||
(take k IH,
|
||||
calc
|
||||
|
|
@ -306,7 +306,7 @@ theorem mul_comm (n m:ℕ) : n * m = m * n
|
|||
... = (succ k) * n : symm (mul_succ_left _ _))
|
||||
|
||||
theorem mul_add_distr_left (n m k : ℕ) : (n + m) * k = n * k + m * k
|
||||
:= induction_on k
|
||||
:= nat.induction_on k
|
||||
(calc
|
||||
(n + m) * 0 = 0 : mul_zero_right _
|
||||
... = 0 + 0 : symm (add_zero _)
|
||||
|
|
@ -329,7 +329,7 @@ theorem mul_add_distr_right (n m k : ℕ) : n * (m + k) = n * m + n * k
|
|||
... = n * m + n * k : {mul_comm _ _}
|
||||
|
||||
theorem mul_assoc (n m k:ℕ) : (n * m) * k = n * (m * k)
|
||||
:= induction_on k
|
||||
:= nat.induction_on k
|
||||
(calc
|
||||
(n * m) * 0 = 0 : mul_zero_right _
|
||||
... = n * 0 : symm (mul_zero_right _)
|
||||
|
|
@ -629,7 +629,7 @@ theorem pred_le_imp_le_or_eq {n m : ℕ} (H : pred n ≤ m) : n ≤ m ∨ n = su
|
|||
theorem mul_le_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k * n ≤ k * m
|
||||
:=
|
||||
obtain (l : ℕ) (Hl : n + l = m), from (le_elim H),
|
||||
induction_on k
|
||||
nat.induction_on k
|
||||
(have H2 : 0 * n = 0 * m,
|
||||
from calc
|
||||
0 * n = 0 : mul_zero_left n
|
||||
|
|
@ -779,7 +779,7 @@ theorem succ_lt_right {n m : ℕ} (H : n < m) : n < succ m
|
|||
---------- totality of lt and le
|
||||
|
||||
theorem le_or_lt (n m : ℕ) : n ≤ m ∨ m < n
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(or_intro_left _ (zero_le m))
|
||||
(take (k : ℕ),
|
||||
assume IH : k ≤ m ∨ m < k,
|
||||
|
|
@ -818,7 +818,7 @@ theorem le_total (n m : ℕ) : n ≤ m ∨ m ≤ n
|
|||
|
||||
theorem strong_induction_on {P : ℕ → Prop} (n : ℕ) (IH : ∀n, (∀m, m < n → P m) → P n) : P n
|
||||
:= have stronger : ∀k, k ≤ n → P k, from
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(take (k : ℕ),
|
||||
assume H : k ≤ 0,
|
||||
have H2 : k = 0, from le_zero_inv H,
|
||||
|
|
@ -946,7 +946,7 @@ theorem mul_positive_inv_right {n m : ℕ} (H : n * m > 0) : m > 0
|
|||
theorem mul_left_inj {n m k : ℕ} (Hn : n > 0) (H : n * m = n * k) : m = k
|
||||
:=
|
||||
have general : ∀m, n * m = n * k → m = k, from
|
||||
induction_on k
|
||||
nat.induction_on k
|
||||
(take m:ℕ,
|
||||
assume H : n * m = n * 0,
|
||||
have H2 : n * m = 0,
|
||||
|
|
@ -1006,7 +1006,7 @@ theorem mul_lt {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n * m < k * l
|
|||
theorem mul_lt_left_inv {n m k : ℕ} (H : k * n < k * m) : n < m
|
||||
:=
|
||||
have general : ∀ m, k * n < k * m → n < m, from
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(take m : ℕ,
|
||||
assume H2 : k * 0 < k * m,
|
||||
have H3 : 0 < k * m, from mul_zero_right k ▸ H2,
|
||||
|
|
@ -1067,7 +1067,7 @@ theorem sub_zero_right (n : ℕ) : n - 0 = n
|
|||
theorem sub_succ_right (n m : ℕ) : n - succ m = pred (n - m)
|
||||
|
||||
theorem sub_zero_left (n : ℕ) : 0 - n = 0
|
||||
:= induction_on n (sub_zero_right 0)
|
||||
:= nat.induction_on n (sub_zero_right 0)
|
||||
(take k : ℕ,
|
||||
assume IH : 0 - k = 0,
|
||||
calc
|
||||
|
|
@ -1076,7 +1076,7 @@ theorem sub_zero_left (n : ℕ) : 0 - n = 0
|
|||
... = 0 : pred_zero)
|
||||
|
||||
theorem sub_succ_succ (n m : ℕ) : succ n - succ m = n - m
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(calc
|
||||
succ n - 1 = pred (succ n - 0) : sub_succ_right (succ n) 0
|
||||
... = pred (succ n) : {sub_zero_right (succ n)}
|
||||
|
|
@ -1095,10 +1095,10 @@ theorem sub_one (n : ℕ) : n - 1 = pred n
|
|||
... = pred n : {sub_zero_right n}
|
||||
|
||||
theorem sub_self (n : ℕ) : n - n = 0
|
||||
:= induction_on n (sub_zero_right 0) (take k IH, trans (sub_succ_succ k k) IH)
|
||||
:= nat.induction_on n (sub_zero_right 0) (take k IH, trans (sub_succ_succ k k) IH)
|
||||
|
||||
theorem sub_add_add_right (n m k : ℕ) : (n + k) - (m + k) = n - m
|
||||
:= induction_on k
|
||||
:= nat.induction_on k
|
||||
(calc
|
||||
(n + 0) - (m + 0) = n - (m + 0) : {add_zero _}
|
||||
... = n - m : {add_zero _})
|
||||
|
|
@ -1114,7 +1114,7 @@ theorem sub_add_add_left (n m k : ℕ) : (k + n) - (k + m) = n - m
|
|||
:= subst (add_comm m k) (subst (add_comm n k) (sub_add_add_right n m k))
|
||||
|
||||
theorem sub_add_left (n m : ℕ) : n + m - m = n
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(subst (symm (add_zero n)) (sub_zero_right n))
|
||||
(take k : ℕ,
|
||||
assume IH : n + k - k = n,
|
||||
|
|
@ -1124,7 +1124,7 @@ theorem sub_add_left (n m : ℕ) : n + m - m = n
|
|||
... = n : IH)
|
||||
|
||||
theorem sub_sub (n m k : ℕ) : n - m - k = n - (m + k)
|
||||
:= induction_on k
|
||||
:= nat.induction_on k
|
||||
(calc
|
||||
n - m - 0 = n - m : sub_zero_right _
|
||||
... = n - (m + 0) : {symm (add_zero m)})
|
||||
|
|
@ -1161,7 +1161,7 @@ theorem succ_sub_one (n : ℕ) : succ n - 1 = n
|
|||
---------- mul
|
||||
|
||||
theorem mul_pred_left (n m : ℕ) : pred n * m = n * m - m
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(calc
|
||||
pred 0 * m = 0 * m : {pred_zero}
|
||||
... = 0 : mul_zero_left _
|
||||
|
|
@ -1180,7 +1180,7 @@ theorem mul_pred_right (n m : ℕ) : n * pred m = n * m - n
|
|||
... = n * m - n : {mul_comm m n}
|
||||
|
||||
theorem mul_sub_distr_left (n m k : ℕ) : (n - m) * k = n * k - m * k
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(calc
|
||||
(n - 0) * k = n * k : {sub_zero_right n}
|
||||
... = n * k - 0 : symm (sub_zero_right _)
|
||||
|
|
|
|||
|
|
@ -48,7 +48,7 @@ theorem pred_zero : pred 0 = 0
|
|||
theorem pred_succ (n : ℕ) : pred (succ n) = n
|
||||
|
||||
theorem zero_or_succ (n : ℕ) : n = 0 ∨ n = succ (pred n)
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(or.intro_left _ (eq.refl 0))
|
||||
(take m IH, or.intro_right _
|
||||
(show succ m = succ (pred (succ m)), from congr_arg succ ((pred_succ m)⁻¹)))
|
||||
|
|
@ -57,7 +57,7 @@ theorem zero_or_succ2 (n : ℕ) : n = 0 ∨ ∃k, n = succ k
|
|||
:= or_of_or_of_imp_of_imp (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists.intro (pred n) H)
|
||||
|
||||
theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n
|
||||
:= induction_on n H1 (take m IH, H2 m)
|
||||
:= nat.induction_on n H1 (take m IH, H2 m)
|
||||
|
||||
theorem discriminate {B : Prop} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B
|
||||
:= or.elim (zero_or_succ n)
|
||||
|
|
@ -71,7 +71,7 @@ theorem succ_inj {n m : ℕ} (H : succ n = succ m) : n = m
|
|||
... = m : pred_succ m
|
||||
|
||||
theorem succ_ne_self (n : ℕ) : succ n ≠ n
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(take H : 1 = 0,
|
||||
have ne : 1 ≠ 0, from succ_ne_zero 0,
|
||||
absurd H ne)
|
||||
|
|
@ -79,13 +79,13 @@ theorem succ_ne_self (n : ℕ) : succ n ≠ n
|
|||
|
||||
theorem decidable_eq [instance] (n m : ℕ) : decidable (n = m)
|
||||
:= have general : ∀n, decidable (n = m), from
|
||||
rec_on m
|
||||
nat.rec_on m
|
||||
(take n,
|
||||
rec_on n
|
||||
nat.rec_on n
|
||||
(inl (eq.refl 0))
|
||||
(λ m iH, inr (succ_ne_zero m)))
|
||||
(λ (m' : ℕ) (iH1 : ∀n, decidable (n = m')),
|
||||
take n, rec_on n
|
||||
take n, nat.rec_on n
|
||||
(inr (ne.symm (succ_ne_zero m')))
|
||||
(λ (n' : ℕ) (iH2 : decidable (n' = succ m')),
|
||||
have d1 : decidable (n' = m'), from iH1 n',
|
||||
|
|
@ -100,7 +100,7 @@ theorem decidable_eq [instance] (n m : ℕ) : decidable (n = m)
|
|||
theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1)
|
||||
(H3 : ∀ (n : ℕ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a
|
||||
:= have stronger : P a ∧ P (succ a), from
|
||||
induction_on a
|
||||
nat.induction_on a
|
||||
(and.intro H1 H2)
|
||||
(take k IH,
|
||||
have IH1 : P k, from and.elim_left IH,
|
||||
|
|
@ -110,7 +110,7 @@ theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1)
|
|||
|
||||
theorem sub_induction {P : ℕ → ℕ → Prop} (n m : ℕ) (H1 : ∀m, P 0 m)
|
||||
(H2 : ∀n, P (succ n) 0) (H3 : ∀n m, P n m → P (succ n) (succ m)) : P n m
|
||||
:= have general : ∀m, P n m, from induction_on n
|
||||
:= have general : ∀m, P n m, from nat.induction_on n
|
||||
(take m : ℕ, H1 m)
|
||||
(take k : ℕ,
|
||||
assume IH : ∀m, P k m,
|
||||
|
|
@ -131,7 +131,7 @@ theorem add_succ (n m : ℕ) : n + succ m = succ (n + m)
|
|||
---------- comm, assoc
|
||||
|
||||
theorem zero_add (n : ℕ) : 0 + n = n
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(add_zero 0)
|
||||
(take m IH, show 0 + succ m = succ m, from
|
||||
calc
|
||||
|
|
@ -139,7 +139,7 @@ theorem zero_add (n : ℕ) : 0 + n = n
|
|||
... = succ m : {IH})
|
||||
|
||||
theorem succ_add (n m : ℕ) : (succ n) + m = succ (n + m)
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(calc
|
||||
succ n + 0 = succ n : add_zero (succ n)
|
||||
... = succ (n + 0) : {symm (add_zero n)})
|
||||
|
|
@ -150,7 +150,7 @@ theorem succ_add (n m : ℕ) : (succ n) + m = succ (n + m)
|
|||
... = succ (n + succ k) : {symm (add_succ _ _)})
|
||||
|
||||
theorem add_comm (n m : ℕ) : n + m = m + n
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(trans (add_zero _) (symm (zero_add _)))
|
||||
(take k IH,
|
||||
calc
|
||||
|
|
@ -169,7 +169,7 @@ theorem add_comm_succ (n m : ℕ) : n + succ m = m + succ n
|
|||
... = m + succ n : add_comm (succ n) m
|
||||
|
||||
theorem add_assoc (n m k : ℕ) : (n + m) + k = n + (m + k)
|
||||
:= induction_on k
|
||||
:= nat.induction_on k
|
||||
(calc
|
||||
(n + m) + 0 = n + m : add_zero _
|
||||
... = n + (m + 0) : {symm (add_zero m)})
|
||||
|
|
@ -191,7 +191,7 @@ theorem add_right_comm (n m k : ℕ) : n + m + k = n + k + m
|
|||
|
||||
theorem add_cancel_left {n m k : ℕ} : n + m = n + k → m = k
|
||||
:=
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(take H : 0 + m = 0 + k,
|
||||
calc
|
||||
m = 0 + m : symm (zero_add m)
|
||||
|
|
@ -218,7 +218,7 @@ theorem add_cancel_right {n m k : ℕ} (H : n + m = k + m) : n = k
|
|||
|
||||
theorem eq_zero_of_add_eq_zero_right {n m : ℕ} : n + m = 0 → n = 0
|
||||
:=
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(take (H : 0 + m = 0), eq.refl 0)
|
||||
(take k IH,
|
||||
assume (H : succ k + m = 0),
|
||||
|
|
@ -267,7 +267,7 @@ set_option unifier.max_steps 100000
|
|||
---------- comm, distr, assoc, identity
|
||||
|
||||
theorem mul_zero_left (n:ℕ) : 0 * n = 0
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(mul_zero_right 0)
|
||||
(take m IH,
|
||||
calc
|
||||
|
|
@ -276,7 +276,7 @@ theorem mul_zero_left (n:ℕ) : 0 * n = 0
|
|||
... = 0 : IH)
|
||||
|
||||
theorem mul_succ_left (n m:ℕ) : (succ n) * m = (n * m) + m
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(calc
|
||||
succ n * 0 = 0 : mul_zero_right _
|
||||
... = n * 0 : symm (mul_zero_right _)
|
||||
|
|
@ -291,7 +291,7 @@ theorem mul_succ_left (n m:ℕ) : (succ n) * m = (n * m) + m
|
|||
... = (n * succ k) + succ k : {symm (mul_succ_right n k)})
|
||||
|
||||
theorem mul_comm (n m:ℕ) : n * m = m * n
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(trans (mul_zero_right _) (symm (mul_zero_left _)))
|
||||
(take k IH,
|
||||
calc
|
||||
|
|
@ -300,7 +300,7 @@ theorem mul_comm (n m:ℕ) : n * m = m * n
|
|||
... = (succ k) * n : symm (mul_succ_left _ _))
|
||||
|
||||
theorem mul_add_distr_left (n m k : ℕ) : (n + m) * k = n * k + m * k
|
||||
:= induction_on k
|
||||
:= nat.induction_on k
|
||||
(calc
|
||||
(n + m) * 0 = 0 : mul_zero_right _
|
||||
... = 0 + 0 : symm (add_zero _)
|
||||
|
|
@ -323,7 +323,7 @@ theorem mul_add_distr_right (n m k : ℕ) : n * (m + k) = n * m + n * k
|
|||
... = n * m + n * k : {mul_comm _ _}
|
||||
|
||||
theorem mul_assoc (n m k:ℕ) : (n * m) * k = n * (m * k)
|
||||
:= induction_on k
|
||||
:= nat.induction_on k
|
||||
(calc
|
||||
(n * m) * 0 = 0 : mul_zero_right _
|
||||
... = n * 0 : symm (mul_zero_right _)
|
||||
|
|
@ -627,7 +627,7 @@ theorem pred_le_imp_le_or_eq {n m : ℕ} (H : pred n ≤ m) : n ≤ m ∨ n = su
|
|||
theorem mul_le_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k * n ≤ k * m
|
||||
:=
|
||||
obtain (l : ℕ) (Hl : n + l = m), from (le_elim H),
|
||||
induction_on k
|
||||
nat.induction_on k
|
||||
(have H2 : 0 * n = 0 * m,
|
||||
from calc
|
||||
0 * n = 0 : mul_zero_left n
|
||||
|
|
@ -777,7 +777,7 @@ theorem succ_lt_right {n m : ℕ} (H : n < m) : n < succ m
|
|||
---------- totality of lt and le
|
||||
|
||||
theorem le_or_lt (n m : ℕ) : n ≤ m ∨ m < n
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(or.intro_left _ (zero_le m))
|
||||
(take (k : ℕ),
|
||||
assume IH : k ≤ m ∨ m < k,
|
||||
|
|
@ -816,7 +816,7 @@ theorem le_total (n m : ℕ) : n ≤ m ∨ m ≤ n
|
|||
|
||||
theorem strong_induction_on {P : ℕ → Prop} (n : ℕ) (IH : ∀n, (∀m, m < n → P m) → P n) : P n
|
||||
:= have stronger : ∀k, k ≤ n → P k, from
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(take (k : ℕ),
|
||||
assume H : k ≤ 0,
|
||||
have H2 : k = 0, from le_zero_inv H,
|
||||
|
|
@ -950,7 +950,7 @@ theorem mul_positive_inv_right {n m : ℕ} (H : n * m > 0) : m > 0
|
|||
theorem mul_left_inj {n m k : ℕ} (Hn : n > 0) (H : n * m = n * k) : m = k
|
||||
:=
|
||||
have general : ∀m, n * m = n * k → m = k, from
|
||||
induction_on k
|
||||
nat.induction_on k
|
||||
(take m:ℕ,
|
||||
assume H : n * m = n * 0,
|
||||
have H2 : n * m = 0,
|
||||
|
|
@ -1010,7 +1010,7 @@ theorem mul_lt {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n * m < k * l
|
|||
theorem mul_lt_left_inv {n m k : ℕ} (H : k * n < k * m) : n < m
|
||||
:=
|
||||
have general : ∀ m, k * n < k * m → n < m, from
|
||||
induction_on n
|
||||
nat.induction_on n
|
||||
(take m : ℕ,
|
||||
assume H2 : k * 0 < k * m,
|
||||
have H3 : 0 < k * m, from mul_zero_right k ▸ H2,
|
||||
|
|
@ -1071,7 +1071,7 @@ theorem sub_zero_right (n : ℕ) : n - 0 = n
|
|||
theorem sub_succ_right (n m : ℕ) : n - succ m = pred (n - m)
|
||||
|
||||
theorem sub_zero_left (n : ℕ) : 0 - n = 0
|
||||
:= induction_on n (sub_zero_right 0)
|
||||
:= nat.induction_on n (sub_zero_right 0)
|
||||
(take k : ℕ,
|
||||
assume IH : 0 - k = 0,
|
||||
calc
|
||||
|
|
@ -1080,7 +1080,7 @@ theorem sub_zero_left (n : ℕ) : 0 - n = 0
|
|||
... = 0 : pred_zero)
|
||||
|
||||
theorem sub_succ_succ (n m : ℕ) : succ n - succ m = n - m
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(calc
|
||||
succ n - 1 = pred (succ n - 0) : sub_succ_right (succ n) 0
|
||||
... = pred (succ n) : {sub_zero_right (succ n)}
|
||||
|
|
@ -1099,10 +1099,10 @@ theorem sub_one (n : ℕ) : n - 1 = pred n
|
|||
... = pred n : {sub_zero_right n}
|
||||
|
||||
theorem sub_self (n : ℕ) : n - n = 0
|
||||
:= induction_on n (sub_zero_right 0) (take k IH, trans (sub_succ_succ k k) IH)
|
||||
:= nat.induction_on n (sub_zero_right 0) (take k IH, trans (sub_succ_succ k k) IH)
|
||||
|
||||
theorem sub_add_add_right (n m k : ℕ) : (n + k) - (m + k) = n - m
|
||||
:= induction_on k
|
||||
:= nat.induction_on k
|
||||
(calc
|
||||
(n + 0) - (m + 0) = n - (m + 0) : {add_zero _}
|
||||
... = n - m : {add_zero _})
|
||||
|
|
@ -1118,7 +1118,7 @@ theorem sub_add_add_left (n m k : ℕ) : (k + n) - (k + m) = n - m
|
|||
:= subst (add_comm m k) (subst (add_comm n k) (sub_add_add_right n m k))
|
||||
|
||||
theorem sub_add_left (n m : ℕ) : n + m - m = n
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(subst (symm (add_zero n)) (sub_zero_right n))
|
||||
(take k : ℕ,
|
||||
assume IH : n + k - k = n,
|
||||
|
|
@ -1128,7 +1128,7 @@ theorem sub_add_left (n m : ℕ) : n + m - m = n
|
|||
... = n : IH)
|
||||
|
||||
theorem sub_sub (n m k : ℕ) : n - m - k = n - (m + k)
|
||||
:= induction_on k
|
||||
:= nat.induction_on k
|
||||
(calc
|
||||
n - m - 0 = n - m : sub_zero_right _
|
||||
... = n - (m + 0) : {symm (add_zero m)})
|
||||
|
|
@ -1165,7 +1165,7 @@ theorem succ_sub_one (n : ℕ) : succ n - 1 = n
|
|||
---------- mul
|
||||
|
||||
theorem mul_pred_left (n m : ℕ) : pred n * m = n * m - m
|
||||
:= induction_on n
|
||||
:= nat.induction_on n
|
||||
(calc
|
||||
pred 0 * m = 0 * m : {pred_zero}
|
||||
... = 0 : mul_zero_left _
|
||||
|
|
@ -1184,7 +1184,7 @@ theorem mul_pred_right (n m : ℕ) : n * pred m = n * m - n
|
|||
... = n * m - n : {mul_comm m n}
|
||||
|
||||
theorem mul_sub_distr_left (n m k : ℕ) : (n - m) * k = n * k - m * k
|
||||
:= induction_on m
|
||||
:= nat.induction_on m
|
||||
(calc
|
||||
(n - 0) * k = n * k : {sub_zero_right n}
|
||||
... = n * k - 0 : symm (sub_zero_right _)
|
||||
|
|
|
|||
|
|
@ -42,62 +42,62 @@ definition concat_11 {A : Type} (x : A) : idpath x ⬝ idpath x ≈ idpath x :=
|
|||
|
||||
-- The identity path is a right unit.
|
||||
definition concat_p1 {A : Type} {x y : A} (p : x ≈ y) : p ⬝ idp ≈ p :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
-- The identity path is a right unit.
|
||||
definition concat_1p {A : Type} {x y : A} (p : x ≈ y) : idp ⬝ p ≈ p :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
-- Concatenation is associative.
|
||||
definition concat_p_pp {A : Type} {x y z t : A} (p : x ≈ y) (q : y ≈ z) (r : z ≈ t) :
|
||||
p ⬝ (q ⬝ r) ≈ (p ⬝ q) ⬝ r :=
|
||||
rec_on r (rec_on q idp)
|
||||
path.rec_on r (path.rec_on q idp)
|
||||
|
||||
definition concat_pp_p {A : Type} {x y z t : A} (p : x ≈ y) (q : y ≈ z) (r : z ≈ t) :
|
||||
(p ⬝ q) ⬝ r ≈ p ⬝ (q ⬝ r) :=
|
||||
rec_on r (rec_on q idp)
|
||||
path.rec_on r (path.rec_on q idp)
|
||||
|
||||
-- The left inverse law.
|
||||
definition concat_pV {A : Type} {x y : A} (p : x ≈ y) : p ⬝ p^ ≈ idp :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
-- The right inverse law.
|
||||
definition concat_Vp {A : Type} {x y : A} (p : x ≈ y) : p^ ⬝ p ≈ idp :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
|
||||
-- Several auxiliary theorems about canceling inverses across associativity. These are somewhat
|
||||
-- redundant, following from earlier theorems.
|
||||
|
||||
definition concat_V_pp {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : p^ ⬝ (p ⬝ q) ≈ q :=
|
||||
rec_on q (rec_on p idp)
|
||||
path.rec_on q (path.rec_on p idp)
|
||||
|
||||
definition concat_p_Vp {A : Type} {x y z : A} (p : x ≈ y) (q : x ≈ z) : p ⬝ (p^ ⬝ q) ≈ q :=
|
||||
rec_on q (rec_on p idp)
|
||||
path.rec_on q (path.rec_on p idp)
|
||||
|
||||
definition concat_pp_V {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : (p ⬝ q) ⬝ q^ ≈ p :=
|
||||
rec_on q (rec_on p idp)
|
||||
path.rec_on q (path.rec_on p idp)
|
||||
|
||||
definition concat_pV_p {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) : (p ⬝ q^) ⬝ q ≈ p :=
|
||||
rec_on q (take p, rec_on p idp) p
|
||||
path.rec_on q (take p, path.rec_on p idp) p
|
||||
|
||||
-- Inverse distributes over concatenation
|
||||
definition inv_pp {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : (p ⬝ q)^ ≈ q^ ⬝ p^ :=
|
||||
rec_on q (rec_on p idp)
|
||||
path.rec_on q (path.rec_on p idp)
|
||||
|
||||
definition inv_Vp {A : Type} {x y z : A} (p : y ≈ x) (q : y ≈ z) : (p^ ⬝ q)^ ≈ q^ ⬝ p :=
|
||||
rec_on q (rec_on p idp)
|
||||
path.rec_on q (path.rec_on p idp)
|
||||
|
||||
-- universe metavariables
|
||||
definition inv_pV {A : Type} {x y z : A} (p : x ≈ y) (q : z ≈ y) : (p ⬝ q^)^ ≈ q ⬝ p^ :=
|
||||
rec_on p (λq, rec_on q idp) q
|
||||
path.rec_on p (λq, path.rec_on q idp) q
|
||||
|
||||
definition inv_VV {A : Type} {x y z : A} (p : y ≈ x) (q : z ≈ y) : (p^ ⬝ q^)^ ≈ q ⬝ p :=
|
||||
rec_on p (rec_on q idp)
|
||||
path.rec_on p (path.rec_on q idp)
|
||||
|
||||
-- Inverse is an involution.
|
||||
definition inv_V {A : Type} {x y : A} (p : x ≈ y) : p^^ ≈ p :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
|
||||
-- Theorems for moving things around in equations
|
||||
|
|
@ -106,7 +106,7 @@ rec_on p idp
|
|||
definition moveR_Mp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
|
||||
p ≈ (r^ ⬝ q) → (r ⬝ p) ≈ q :=
|
||||
have gen : Πp q, p ≈ (r^ ⬝ q) → (r ⬝ p) ≈ q, from
|
||||
rec_on r
|
||||
path.rec_on r
|
||||
(take p q,
|
||||
assume h : p ≈ idp^ ⬝ q,
|
||||
show idp ⬝ p ≈ q, from concat_1p _ ⬝ h ⬝ concat_1p _),
|
||||
|
|
@ -114,63 +114,63 @@ gen p q
|
|||
|
||||
definition moveR_pM {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
|
||||
r ≈ q ⬝ p^ → r ⬝ p ≈ q :=
|
||||
rec_on p (take q r h, (concat_p1 _ ⬝ h ⬝ concat_p1 _)) q r
|
||||
path.rec_on p (take q r h, (concat_p1 _ ⬝ h ⬝ concat_p1 _)) q r
|
||||
|
||||
definition moveR_Vp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : x ≈ y) :
|
||||
p ≈ r ⬝ q → r^ ⬝ p ≈ q :=
|
||||
rec_on r (take p q h, concat_1p _ ⬝ h ⬝ concat_1p _) p q
|
||||
path.rec_on r (take p q h, concat_1p _ ⬝ h ⬝ concat_1p _) p q
|
||||
|
||||
definition moveR_pV {A : Type} {x y z : A} (p : z ≈ x) (q : y ≈ z) (r : y ≈ x) :
|
||||
r ≈ q ⬝ p → r ⬝ p^ ≈ q :=
|
||||
rec_on p (take q r h, concat_p1 _ ⬝ h ⬝ concat_p1 _) q r
|
||||
path.rec_on p (take q r h, concat_p1 _ ⬝ h ⬝ concat_p1 _) q r
|
||||
|
||||
definition moveL_Mp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
|
||||
r^ ⬝ q ≈ p → q ≈ r ⬝ p :=
|
||||
rec_on r (take p q h, (concat_1p _)^ ⬝ h ⬝ (concat_1p _)^) p q
|
||||
path.rec_on r (take p q h, (concat_1p _)^ ⬝ h ⬝ (concat_1p _)^) p q
|
||||
|
||||
definition moveL_pM {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
|
||||
q ⬝ p^ ≈ r → q ≈ r ⬝ p :=
|
||||
rec_on p (take q r h, (concat_p1 _)^ ⬝ h ⬝ (concat_p1 _)^) q r
|
||||
path.rec_on p (take q r h, (concat_p1 _)^ ⬝ h ⬝ (concat_p1 _)^) q r
|
||||
|
||||
definition moveL_Vp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : x ≈ y) :
|
||||
r ⬝ q ≈ p → q ≈ r^ ⬝ p :=
|
||||
rec_on r (take p q h, (concat_1p _)^ ⬝ h ⬝ (concat_1p _)^) p q
|
||||
path.rec_on r (take p q h, (concat_1p _)^ ⬝ h ⬝ (concat_1p _)^) p q
|
||||
|
||||
definition moveL_pV {A : Type} {x y z : A} (p : z ≈ x) (q : y ≈ z) (r : y ≈ x) :
|
||||
q ⬝ p ≈ r → q ≈ r ⬝ p^ :=
|
||||
rec_on p (take q r h, (concat_p1 _)^ ⬝ h ⬝ (concat_p1 _)^) q r
|
||||
path.rec_on p (take q r h, (concat_p1 _)^ ⬝ h ⬝ (concat_p1 _)^) q r
|
||||
|
||||
definition moveL_1M {A : Type} {x y : A} (p q : x ≈ y) :
|
||||
p ⬝ q^ ≈ idp → p ≈ q :=
|
||||
rec_on q (take p h, (concat_p1 _)^ ⬝ h) p
|
||||
path.rec_on q (take p h, (concat_p1 _)^ ⬝ h) p
|
||||
|
||||
definition moveL_M1 {A : Type} {x y : A} (p q : x ≈ y) :
|
||||
q^ ⬝ p ≈ idp → p ≈ q :=
|
||||
rec_on q (take p h, (concat_1p _)^ ⬝ h) p
|
||||
path.rec_on q (take p h, (concat_1p _)^ ⬝ h) p
|
||||
|
||||
definition moveL_1V {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
|
||||
p ⬝ q ≈ idp → p ≈ q^ :=
|
||||
rec_on q (take p h, (concat_p1 _)^ ⬝ h) p
|
||||
path.rec_on q (take p h, (concat_p1 _)^ ⬝ h) p
|
||||
|
||||
definition moveL_V1 {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
|
||||
q ⬝ p ≈ idp → p ≈ q^ :=
|
||||
rec_on q (take p h, (concat_1p _)^ ⬝ h) p
|
||||
path.rec_on q (take p h, (concat_1p _)^ ⬝ h) p
|
||||
|
||||
definition moveR_M1 {A : Type} {x y : A} (p q : x ≈ y) :
|
||||
idp ≈ p^ ⬝ q → p ≈ q :=
|
||||
rec_on p (take q h, h ⬝ (concat_1p _)) q
|
||||
path.rec_on p (take q h, h ⬝ (concat_1p _)) q
|
||||
|
||||
definition moveR_1M {A : Type} {x y : A} (p q : x ≈ y) :
|
||||
idp ≈ q ⬝ p^ → p ≈ q :=
|
||||
rec_on p (take q h, h ⬝ (concat_p1 _)) q
|
||||
path.rec_on p (take q h, h ⬝ (concat_p1 _)) q
|
||||
|
||||
definition moveR_1V {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
|
||||
idp ≈ q ⬝ p → p^ ≈ q :=
|
||||
rec_on p (take q h, h ⬝ (concat_p1 _)) q
|
||||
path.rec_on p (take q h, h ⬝ (concat_p1 _)) q
|
||||
|
||||
definition moveR_V1 {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
|
||||
idp ≈ p ⬝ q → p^ ≈ q :=
|
||||
rec_on p (take q h, h ⬝ (concat_1p _)) q
|
||||
path.rec_on p (take q h, h ⬝ (concat_1p _)) q
|
||||
|
||||
|
||||
-- Transport
|
||||
|
|
@ -205,13 +205,13 @@ definition apD10 {A} {B : A → Type} {f g : Πx, B x} (H : f ≈ g) : f ∼ g :
|
|||
definition ap10 {A B} {f g : A → B} (H : f ≈ g) : f ∼ g := apD10 H
|
||||
|
||||
definition ap11 {A B} {f g : A → B} (H : f ≈ g) {x y : A} (p : x ≈ y) : f x ≈ g y :=
|
||||
rec_on H (rec_on p idp)
|
||||
path.rec_on H (path.rec_on p idp)
|
||||
|
||||
-- TODO: Note that the next line breaks the proof!
|
||||
-- opaque_hint (hiding rec_on)
|
||||
-- opaque_hint (hiding path.rec_on)
|
||||
-- set_option pp.implicit true
|
||||
definition apD {A:Type} {B : A → Type} (f : Πa:A, B a) {x y : A} (p : x ≈ y) : p # (f x) ≈ f y :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
|
||||
-- More theorems for moving things around in equations
|
||||
|
|
@ -219,19 +219,19 @@ rec_on p idp
|
|||
|
||||
definition moveR_transport_p {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (u : P x) (v : P y) :
|
||||
u ≈ p^ # v → p # u ≈ v :=
|
||||
rec_on p (take u v, id) u v
|
||||
path.rec_on p (take u v, id) u v
|
||||
|
||||
definition moveR_transport_V {A : Type} (P : A → Type) {x y : A} (p : y ≈ x) (u : P x) (v : P y) :
|
||||
u ≈ p # v → p^ # u ≈ v :=
|
||||
rec_on p (take u v, id) u v
|
||||
path.rec_on p (take u v, id) u v
|
||||
|
||||
definition moveL_transport_V {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (u : P x) (v : P y) :
|
||||
p # u ≈ v → u ≈ p^ # v :=
|
||||
rec_on p (take u v, id) u v
|
||||
path.rec_on p (take u v, id) u v
|
||||
|
||||
definition moveL_transport_p {A : Type} (P : A → Type) {x y : A} (p : y ≈ x) (u : P x) (v : P y) :
|
||||
p^ # u ≈ v → u ≈ p # v :=
|
||||
rec_on p (take u v, id) u v
|
||||
path.rec_on p (take u v, id) u v
|
||||
|
||||
|
||||
-- Functoriality of functions
|
||||
|
|
@ -248,54 +248,54 @@ definition apD_1 {A B} (x : A) (f : forall x : A, B x) : apD f idp ≈ idp :> (f
|
|||
-- Functions commute with concatenation.
|
||||
definition ap_pp {A B : Type} (f : A → B) {x y z : A} (p : x ≈ y) (q : y ≈ z) :
|
||||
ap f (p ⬝ q) ≈ (ap f p) ⬝ (ap f q) :=
|
||||
rec_on q (rec_on p idp)
|
||||
path.rec_on q (path.rec_on p idp)
|
||||
|
||||
definition ap_p_pp {A B : Type} (f : A → B) {w x y z : A} (r : f w ≈ f x) (p : x ≈ y) (q : y ≈ z) :
|
||||
r ⬝ (ap f (p ⬝ q)) ≈ (r ⬝ ap f p) ⬝ (ap f q) :=
|
||||
rec_on p (take r q, rec_on q (concat_p_pp r idp idp)) r q
|
||||
path.rec_on p (take r q, path.rec_on q (concat_p_pp r idp idp)) r q
|
||||
|
||||
definition ap_pp_p {A B : Type} (f : A → B) {w x y z : A} (p : x ≈ y) (q : y ≈ z) (r : f z ≈ f w) :
|
||||
(ap f (p ⬝ q)) ⬝ r ≈ (ap f p) ⬝ (ap f q ⬝ r) :=
|
||||
rec_on p (take q, rec_on q (take r, concat_pp_p _ _ _)) q r
|
||||
path.rec_on p (take q, path.rec_on q (take r, concat_pp_p _ _ _)) q r
|
||||
|
||||
-- Functions commute with path inverses.
|
||||
definition inverse_ap {A B : Type} (f : A → B) {x y : A} (p : x ≈ y) : (ap f p)^ ≈ ap f (p^) :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
definition ap_V {A B : Type} (f : A → B) {x y : A} (p : x ≈ y) : ap f (p^) ≈ (ap f p)^ :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
-- TODO: rename id to idmap?
|
||||
definition ap_idmap {A : Type} {x y : A} (p : x ≈ y) : ap id p ≈ p :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
definition ap_compose {A B C : Type} (f : A → B) (g : B → C) {x y : A} (p : x ≈ y) :
|
||||
ap (g ∘ f) p ≈ ap g (ap f p) :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
-- Sometimes we don't have the actual function [compose].
|
||||
definition ap_compose' {A B C : Type} (f : A → B) (g : B → C) {x y : A} (p : x ≈ y) :
|
||||
ap (λa, g (f a)) p ≈ ap g (ap f p) :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
-- The action of constant maps.
|
||||
definition ap_const {A B : Type} {x y : A} (p : x ≈ y) (z : B) :
|
||||
ap (λu, z) p ≈ idp :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
-- Naturality of [ap].
|
||||
definition concat_Ap {A B : Type} {f g : A → B} (p : forall x, f x ≈ g x) {x y : A} (q : x ≈ y) :
|
||||
(ap f q) ⬝ (p y) ≈ (p x) ⬝ (ap g q) :=
|
||||
rec_on q (concat_1p _ ⬝ (concat_p1 _)^)
|
||||
path.rec_on q (concat_1p _ ⬝ (concat_p1 _)^)
|
||||
|
||||
-- Naturality of [ap] at identity.
|
||||
definition concat_A1p {A : Type} {f : A → A} (p : forall x, f x ≈ x) {x y : A} (q : x ≈ y) :
|
||||
(ap f q) ⬝ (p y) ≈ (p x) ⬝ q :=
|
||||
rec_on q (concat_1p _ ⬝ (concat_p1 _)^)
|
||||
path.rec_on q (concat_1p _ ⬝ (concat_p1 _)^)
|
||||
|
||||
definition concat_pA1 {A : Type} {f : A → A} (p : forall x, x ≈ f x) {x y : A} (q : x ≈ y) :
|
||||
(p x) ⬝ (ap f q) ≈ q ⬝ (p y) :=
|
||||
rec_on q (concat_p1 _ ⬝ (concat_1p _)^)
|
||||
path.rec_on q (concat_p1 _ ⬝ (concat_1p _)^)
|
||||
|
||||
--TODO: note that the Coq proof for the preceding is
|
||||
--
|
||||
|
|
@ -310,7 +310,7 @@ definition concat_pA_pp {A B : Type} {f g : A → B} (p : forall x, f x ≈ g x)
|
|||
{x y : A} (q : x ≈ y)
|
||||
{w z : B} (r : w ≈ f x) (s : g y ≈ z) :
|
||||
(r ⬝ ap f q) ⬝ (p y ⬝ s) ≈ (r ⬝ p x) ⬝ (ap g q ⬝ s) :=
|
||||
rec_on q (take s, rec_on s (take r, idp)) s r
|
||||
path.rec_on q (take s, path.rec_on s (take r, idp)) s r
|
||||
|
||||
-- Action of [apD10] and [ap10] on paths
|
||||
-- -------------------------------------
|
||||
|
|
@ -321,11 +321,11 @@ definition apD10_1 {A} {B : A → Type} (f : Πx, B x) (x : A) : apD10 (idpath f
|
|||
|
||||
definition apD10_pp {A} {B : A → Type} {f f' f'' : Πx, B x} (h : f ≈ f') (h' : f' ≈ f'') (x : A) :
|
||||
apD10 (h ⬝ h') x ≈ apD10 h x ⬝ apD10 h' x :=
|
||||
rec_on h (take h', rec_on h' idp) h'
|
||||
path.rec_on h (take h', path.rec_on h' idp) h'
|
||||
|
||||
definition apD10_V {A : Type} {B : A → Type} {f g : Πx : A, B x} (h : f ≈ g) (x : A) :
|
||||
apD10 (h^) x ≈ (apD10 h x)^ :=
|
||||
rec_on h idp
|
||||
path.rec_on h idp
|
||||
|
||||
definition ap10_1 {A B} {f : A → B} (x : A) : ap10 (idpath f) x ≈ idp := idp
|
||||
|
||||
|
|
@ -337,7 +337,7 @@ definition ap10_V {A B} {f g : A→B} (h : f ≈ g) (x:A) : ap10 (h^) x ≈ (ap1
|
|||
-- [ap10] also behaves nicely on paths produced by [ap]
|
||||
definition ap_ap10 {A B C} (f g : A → B) (h : B → C) (p : f ≈ g) (a : A) :
|
||||
ap h (ap10 p a) ≈ ap10 (ap (λ f', h ∘ f') p) a:=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
|
||||
-- Transport and the groupoid structure of paths
|
||||
|
|
@ -349,7 +349,7 @@ rec_on p idp
|
|||
|
||||
definition transport_pp {A : Type} (P : A → Type) {x y z : A} (p : x ≈ y) (q : y ≈ z) (u : P x) :
|
||||
p ⬝ q # u ≈ q # p # u :=
|
||||
rec_on q (rec_on p idp)
|
||||
path.rec_on q (path.rec_on p idp)
|
||||
|
||||
definition transport_pV {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (z : P y) :
|
||||
p # p^ # z ≈ z :=
|
||||
|
|
@ -369,27 +369,27 @@ theorem double_induction
|
|||
{C : Π(x y z : A), Π(p : x ≈ y), Π(q : y ≈ z), Type}
|
||||
(H : C x x x (idpath x) (idpath x)) :
|
||||
C x y z p q :=
|
||||
rec_on p (take z q, rec_on q H) z q
|
||||
path.rec_on p (take z q, path.rec_on q H) z q
|
||||
|
||||
theorem double_induction2
|
||||
{A : Type} {x y z : A} (p : x ≈ y) (q : z ≈ y)
|
||||
{C : Π(x y z : A), Π(p : x ≈ y), Π(q : z ≈ y), Type}
|
||||
(H : C z z z (idpath z) (idpath z)) :
|
||||
C x y z p q :=
|
||||
rec_on p (take y q, rec_on q H) y q
|
||||
path.rec_on p (take y q, path.rec_on q H) y q
|
||||
|
||||
theorem double_induction2'
|
||||
{A : Type} {x y z : A} (p : x ≈ y) (q : z ≈ y)
|
||||
{C : Π(x y z : A), Π(p : x ≈ y), Π(q : z ≈ y), Type}
|
||||
(H : C z z z (idpath z) (idpath z)) : C x y z p q :=
|
||||
rec_on p (take y q, rec_on q H) y q
|
||||
path.rec_on p (take y q, path.rec_on q H) y q
|
||||
|
||||
theorem triple_induction
|
||||
{A : Type} {x y z w : A} (p : x ≈ y) (q : y ≈ z) (r : z ≈ w)
|
||||
{C : Π(x y z w : A), Π(p : x ≈ y), Π(q : y ≈ z), Π(r: z ≈ w), Type}
|
||||
(H : C x x x x (idpath x) (idpath x) (idpath x)) :
|
||||
C x y z w p q r :=
|
||||
rec_on p (take z q, rec_on q (take w r, rec_on r H)) z q w r
|
||||
path.rec_on p (take z q, path.rec_on q (take w r, path.rec_on r H)) z q w r
|
||||
|
||||
-- try this again
|
||||
definition concat_pV_p_new {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) : (p ⬝ q^) ⬝ q ≈ p :=
|
||||
|
|
@ -406,13 +406,13 @@ triple_induction p q r (take u, idp) u
|
|||
-- Here is another coherence lemma for transport.
|
||||
definition transport_pVp {A} (P : A → Type) {x y : A} (p : x ≈ y) (z : P x) :
|
||||
transport_pV P p (transport P p z) ≈ ap (transport P p) (transport_Vp P p z)
|
||||
:= rec_on p idp
|
||||
:= path.rec_on p idp
|
||||
|
||||
-- Dependent transport in a doubly dependent type.
|
||||
definition transportD {A : Type} (B : A → Type) (C : Π a : A, B a → Type)
|
||||
{x1 x2 : A} (p : x1 ≈ x2) (y : B x1) (z : C x1 y) :
|
||||
C x2 (p # y) :=
|
||||
rec_on p z
|
||||
path.rec_on p z
|
||||
|
||||
-- Transporting along higher-dimensional paths
|
||||
definition transport2 {A : Type} (P : A → Type) {x y : A} {p q : x ≈ y} (r : p ≈ q) (z : P x) :
|
||||
|
|
@ -422,28 +422,28 @@ definition transport2 {A : Type} (P : A → Type) {x y : A} {p q : x ≈ y} (r :
|
|||
definition transport2_is_ap10 {A : Type} (Q : A → Type) {x y : A} {p q : x ≈ y} (r : p ≈ q)
|
||||
(z : Q x) :
|
||||
transport2 Q r z ≈ ap10 (ap (transport Q) r) z :=
|
||||
rec_on r idp
|
||||
path.rec_on r idp
|
||||
|
||||
definition transport2_p2p {A : Type} (P : A → Type) {x y : A} {p1 p2 p3 : x ≈ y}
|
||||
(r1 : p1 ≈ p2) (r2 : p2 ≈ p3) (z : P x) :
|
||||
transport2 P (r1 ⬝ r2) z ≈ transport2 P r1 z ⬝ transport2 P r2 z :=
|
||||
rec_on r1 (rec_on r2 idp)
|
||||
path.rec_on r1 (path.rec_on r2 idp)
|
||||
|
||||
-- TODO: another interesting case
|
||||
definition transport2_V {A : Type} (Q : A → Type) {x y : A} {p q : x ≈ y} (r : p ≈ q) (z : Q x) :
|
||||
transport2 Q (r^) z ≈ ((transport2 Q r z)^) :=
|
||||
-- rec_on r idp -- doesn't work
|
||||
rec_on r (idpath (inverse (transport2 Q (idpath p) z)))
|
||||
-- path.rec_on r idp -- doesn't work
|
||||
path.rec_on r (idpath (inverse (transport2 Q (idpath p) z)))
|
||||
|
||||
definition concat_AT {A : Type} (P : A → Type) {x y : A} {p q : x ≈ y} {z w : P x} (r : p ≈ q)
|
||||
(s : z ≈ w) :
|
||||
ap (transport P p) s ⬝ transport2 P r w ≈ transport2 P r z ⬝ ap (transport P q) s :=
|
||||
rec_on r (concat_p1 _ ⬝ (concat_1p _)^)
|
||||
path.rec_on r (concat_p1 _ ⬝ (concat_1p _)^)
|
||||
|
||||
-- TODO (from Coq library): What should this be called?
|
||||
definition ap_transport {A} {P Q : A → Type} {x y : A} (p : x ≈ y) (f : Πx, P x → Q x) (z : P x) :
|
||||
f y (p # z) ≈ (p # (f x z)) :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
|
||||
-- Transporting in particular fibrations
|
||||
|
|
@ -461,34 +461,34 @@ subdirectory. Here we consider only the most basic cases.
|
|||
-- Transporting in a constant fibration.
|
||||
definition transport_const {A B : Type} {x1 x2 : A} (p : x1 ≈ x2) (y : B) :
|
||||
transport (λx, B) p y ≈ y :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
definition transport2_const {A B : Type} {x1 x2 : A} {p q : x1 ≈ x2} (r : p ≈ q) (y : B) :
|
||||
transport_const p y ≈ transport2 (λu, B) r y ⬝ transport_const q y :=
|
||||
rec_on r (concat_1p _)^
|
||||
path.rec_on r (concat_1p _)^
|
||||
|
||||
-- Transporting in a pulled back fibration.
|
||||
definition transport_compose {A B} {x y : A} (P : B → Type) (f : A → B) (p : x ≈ y) (z : P (f x)) :
|
||||
transport (λx, P (f x)) p z ≈ transport P (ap f p) z :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
definition transport_precompose {A B C} (f : A → B) (g g' : B → C) (p : g ≈ g') :
|
||||
transport (λh : B → C, g ∘ f ≈ h ∘ f) p idp ≈ ap (λh, h ∘ f) p :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
definition apD10_ap_precompose {A B C} (f : A → B) (g g' : B → C) (p : g ≈ g') (a : A) :
|
||||
apD10 (ap (λh : B → C, h ∘ f) p) a ≈ apD10 p (f a) :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
definition apD10_ap_postcompose {A B C} (f : B → C) (g g' : A → B) (p : g ≈ g') (a : A) :
|
||||
apD10 (ap (λh : A → B, f ∘ h) p) a ≈ ap f (apD10 p a) :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
-- TODO: another example where a term has to be given explicitly
|
||||
-- A special case of [transport_compose] which seems to come up a lot.
|
||||
definition transport_idmap_ap A (P : A → Type) x y (p : x ≈ y) (u : P x) :
|
||||
transport P p u ≈ transport (λz, z) (ap P p) u :=
|
||||
rec_on p (idpath (transport (λ (z : Type), z) (ap P (idpath x)) u))
|
||||
path.rec_on p (idpath (transport (λ (z : Type), z) (ap P (idpath x)) u))
|
||||
|
||||
|
||||
-- The behavior of [ap] and [apD]
|
||||
|
|
@ -497,7 +497,7 @@ rec_on p (idpath (transport (λ (z : Type), z) (ap P (idpath x)) u))
|
|||
-- In a constant fibration, [apD] reduces to [ap], modulo [transport_const].
|
||||
definition apD_const {A B} {x y : A} (f : A → B) (p: x ≈ y) :
|
||||
apD f p ≈ transport_const p (f x) ⬝ ap f p :=
|
||||
rec_on p idp
|
||||
path.rec_on p idp
|
||||
|
||||
|
||||
-- The 2-dimensional groupoid structure
|
||||
|
|
@ -506,13 +506,13 @@ rec_on p idp
|
|||
-- Horizontal composition of 2-dimensional paths.
|
||||
definition concat2 {A} {x y z : A} {p p' : x ≈ y} {q q' : y ≈ z} (h : p ≈ p') (h' : q ≈ q') :
|
||||
p ⬝ q ≈ p' ⬝ q' :=
|
||||
rec_on h (rec_on h' idp)
|
||||
path.rec_on h (path.rec_on h' idp)
|
||||
|
||||
infixl `⬝⬝`:75 := concat2
|
||||
|
||||
-- 2-dimensional path inversion
|
||||
definition inverse2 {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) : p^ ≈ q^ :=
|
||||
rec_on h idp
|
||||
path.rec_on h idp
|
||||
|
||||
-- Whiskering
|
||||
-- ----------
|
||||
|
|
@ -528,47 +528,47 @@ h ⬝⬝ idp
|
|||
-- -------------------------------
|
||||
|
||||
definition cancelL {A} {x y z : A} (p : x ≈ y) (q r : y ≈ z) : (p ⬝ q ≈ p ⬝ r) → (q ≈ r) :=
|
||||
rec_on p (take r, rec_on r (take q a, (concat_1p q)^ ⬝ a)) r q
|
||||
path.rec_on p (take r, path.rec_on r (take q a, (concat_1p q)^ ⬝ a)) r q
|
||||
|
||||
definition cancelR {A} {x y z : A} (p q : x ≈ y) (r : y ≈ z) : (p ⬝ r ≈ q ⬝ r) → (p ≈ q) :=
|
||||
rec_on r (take p, rec_on p (take q a, a ⬝ concat_p1 q)) p q
|
||||
path.rec_on r (take p, path.rec_on p (take q a, a ⬝ concat_p1 q)) p q
|
||||
|
||||
-- Whiskering and identity paths.
|
||||
|
||||
definition whiskerR_p1 {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
|
||||
(concat_p1 p)^ ⬝ whiskerR h idp ⬝ concat_p1 q ≈ h :=
|
||||
rec_on h (rec_on p idp)
|
||||
path.rec_on h (path.rec_on p idp)
|
||||
|
||||
definition whiskerR_1p {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) :
|
||||
whiskerR idp q ≈ idp :> (p ⬝ q ≈ p ⬝ q) :=
|
||||
rec_on q idp
|
||||
path.rec_on q idp
|
||||
|
||||
definition whiskerL_p1 {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) :
|
||||
whiskerL p idp ≈ idp :> (p ⬝ q ≈ p ⬝ q) :=
|
||||
rec_on q idp
|
||||
path.rec_on q idp
|
||||
|
||||
definition whiskerL_1p {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
|
||||
(concat_1p p) ^ ⬝ whiskerL idp h ⬝ concat_1p q ≈ h :=
|
||||
rec_on h (rec_on p idp)
|
||||
path.rec_on h (path.rec_on p idp)
|
||||
|
||||
definition concat2_p1 {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
|
||||
h ⬝⬝ idp ≈ whiskerR h idp :> (p ⬝ idp ≈ q ⬝ idp) :=
|
||||
rec_on h idp
|
||||
path.rec_on h idp
|
||||
|
||||
definition concat2_1p {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
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idp ⬝⬝ h ≈ whiskerL idp h :> (idp ⬝ p ≈ idp ⬝ q) :=
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rec_on h idp
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path.rec_on h idp
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-- TODO: note, 4 inductions
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-- The interchange law for concatenation.
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definition concat_concat2 {A : Type} {x y z : A} {p p' p'' : x ≈ y} {q q' q'' : y ≈ z}
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(a : p ≈ p') (b : p' ≈ p'') (c : q ≈ q') (d : q' ≈ q'') :
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(a ⬝⬝ c) ⬝ (b ⬝⬝ d) ≈ (a ⬝ b) ⬝⬝ (c ⬝ d) :=
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rec_on d (rec_on c (rec_on b (rec_on a idp)))
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path.rec_on d (path.rec_on c (path.rec_on b (path.rec_on a idp)))
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definition concat_whisker {A} {x y z : A} (p p' : x ≈ y) (q q' : y ≈ z) (a : p ≈ p') (b : q ≈ q') :
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(whiskerR a q) ⬝ (whiskerL p' b) ≈ (whiskerL p b) ⬝ (whiskerR a q') :=
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rec_on b (rec_on a (concat_1p _)^)
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path.rec_on b (path.rec_on a (concat_1p _)^)
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-- Structure corresponding to the coherence equations of a bicategory.
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|
@ -578,12 +578,12 @@ definition pentagon {A : Type} {v w x y z : A} (p : v ≈ w) (q : w ≈ x) (r :
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⬝ concat_p_pp p (q ⬝ r) s
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⬝ whiskerR (concat_p_pp p q r) s
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≈ concat_p_pp p q (r ⬝ s) ⬝ concat_p_pp (p ⬝ q) r s :=
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rec_on p (take q, rec_on q (take r, rec_on r (take s, rec_on s idp))) q r s
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path.rec_on p (take q, path.rec_on q (take r, path.rec_on r (take s, path.rec_on s idp))) q r s
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-- The 3-cell witnessing the left unit triangle.
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||||
definition triangulator {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) :
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concat_p_pp p idp q ⬝ whiskerR (concat_p1 p) q ≈ whiskerL p (concat_1p q) :=
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rec_on p (take q, rec_on q idp) q
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path.rec_on p (take q, path.rec_on q idp) q
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||||
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||||
definition eckmann_hilton {A : Type} {x:A} (p q : idp ≈ idp :> (x ≈ x)) : p ⬝ q ≈ q ⬝ p :=
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||||
(whiskerR_p1 p ⬝⬝ whiskerL_1p q)^
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||||
|
|
@ -597,20 +597,20 @@ definition eckmann_hilton {A : Type} {x:A} (p q : idp ≈ idp :> (x ≈ x)) : p
|
|||
|
||||
-- The action of functions on 2-dimensional paths
|
||||
definition ap02 {A B : Type} (f:A → B) {x y : A} {p q : x ≈ y} (r : p ≈ q) : ap f p ≈ ap f q :=
|
||||
rec_on r idp
|
||||
path.rec_on r idp
|
||||
|
||||
definition ap02_pp {A B} (f : A → B) {x y : A} {p p' p'' : x ≈ y} (r : p ≈ p') (r' : p' ≈ p'') :
|
||||
ap02 f (r ⬝ r') ≈ ap02 f r ⬝ ap02 f r' :=
|
||||
rec_on r (rec_on r' idp)
|
||||
path.rec_on r (path.rec_on r' idp)
|
||||
|
||||
definition ap02_p2p {A B} (f : A→B) {x y z : A} {p p' : x ≈ y} {q q' :y ≈ z} (r : p ≈ p')
|
||||
(s : q ≈ q') :
|
||||
ap02 f (r ⬝⬝ s) ≈ ap_pp f p q
|
||||
⬝ (ap02 f r ⬝⬝ ap02 f s)
|
||||
⬝ (ap_pp f p' q')^ :=
|
||||
rec_on r (rec_on s (rec_on q (rec_on p idp)))
|
||||
path.rec_on r (path.rec_on s (path.rec_on q (path.rec_on p idp)))
|
||||
|
||||
definition apD02 {A : Type} {B : A → Type} {x y : A} {p q : x ≈ y} (f : Π x, B x) (r : p ≈ q) :
|
||||
apD f p ≈ transport2 B r (f x) ⬝ apD f q :=
|
||||
rec_on r (concat_1p _)^
|
||||
path.rec_on r (concat_1p _)^
|
||||
end path
|
||||
|
|
|
|||
Loading…
Reference in a new issue