986 lines
59 KiB
Text
986 lines
59 KiB
Text
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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Leonardo de Moura
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import logic
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namespace N1
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definition transport {A : Type} {a b : A} {P : A → Type} (p : a = b) (H : P a) : P b
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:= eq_rec H p
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theorem transport_refl {A : Type} {a : A} {P : A → Type} (H : P a) : transport (refl a) H = H
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:= refl H
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opaque_hint (hiding transport)
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theorem transport_proof_irrel {A : Type} {a b : A} {P : A → Type} (p1 p2 : a = b) (H : P a) : transport p1 H = transport p2 H
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:= refl (transport p1 H)
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theorem transport_eq {A : Type} {a : A} {P : A → Type} (p : a = a) (H : P a) : transport p H = H
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:= calc transport p H = transport (refl a) H : transport_proof_irrel p (refl a) H
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... = H : transport_refl H
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theorem dcongr {A : Type} {B : A → Type} {a b : A} (f : Π x, B x) (p : a = b) : transport p (f a) = f b
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:= have H1 : ∀ p1 : a = a, transport p1 (f a) = f a, from
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assume p1 : a = a, transport_eq p1 (f a),
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eq_rec H1 p p
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theorem transport_trans {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p, calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans1 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans2 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans3 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans4 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans5 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans6 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans7 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans8 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans9 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans10 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans11 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans12 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans13 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans14 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans15 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans16 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans17 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans18 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans19 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans20 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans21 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans22 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans23 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans24 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans25 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans26 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans27 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans28 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans29 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p, calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans30 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans31 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans32 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans33 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans34 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans35 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans36 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans37 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans38 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans39 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans40 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans41 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans42 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans43 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans44 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans45 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans46 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans47 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans48 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans49 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans50 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans51 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans52 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans53 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans54 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans55 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans56 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans57 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans58 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
end
|
||
|
|
|
||
|
|
namespace N2
|
||
|
|
definition transport {A : Type} {a b : A} {P : A → Type} (p : a = b) (H : P a) : P b
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:= eq_rec H p
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theorem transport_refl {A : Type} {a : A} {P : A → Type} (H : P a) : transport (refl a) H = H
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:= refl H
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opaque_hint (hiding transport)
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theorem transport_proof_irrel {A : Type} {a b : A} {P : A → Type} (p1 p2 : a = b) (H : P a) : transport p1 H = transport p2 H
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:= refl (transport p1 H)
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theorem transport_eq {A : Type} {a : A} {P : A → Type} (p : a = a) (H : P a) : transport p H = H
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:= calc transport p H = transport (refl a) H : transport_proof_irrel p (refl a) H
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... = H : transport_refl H
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theorem dcongr {A : Type} {B : A → Type} {a b : A} (f : Π x, B x) (p : a = b) : transport p (f a) = f b
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:= have H1 : ∀ p1 : a = a, transport p1 (f a) = f a, from
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assume p1 : a = a, transport_eq p1 (f a),
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eq_rec H1 p p
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theorem transport_trans {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p, calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans1 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans2 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans3 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans4 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans5 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans6 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
|
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans7 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
|
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans8 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
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transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
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... = transport (trans p1 p) H : refl (transport p1 H),
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eq_rec H1 p2 p2
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theorem transport_trans9 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
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transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
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take p : b = b,
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||
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
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... = transport (trans p1 p) H : refl (transport p1 H),
|
||
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eq_rec H1 p2 p2
|
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theorem transport_trans10 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
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|
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transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
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take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
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theorem transport_trans11 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
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transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
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take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
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... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
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eq_rec H1 p2 p2
|
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theorem transport_trans12 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
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|
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transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
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take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
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... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
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eq_rec H1 p2 p2
|
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theorem transport_trans13 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
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transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
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take p : b = b,
|
||
|
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calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
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... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
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eq_rec H1 p2 p2
|
||
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theorem transport_trans14 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
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transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
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:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
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... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
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eq_rec H1 p2 p2
|
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theorem transport_trans15 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
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transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
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eq_rec H1 p2 p2
|
||
|
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|
||
|
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theorem transport_trans16 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
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transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
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theorem transport_trans17 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
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|
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theorem transport_trans18 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans19 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans20 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans21 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans22 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans23 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans24 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans25 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans26 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans27 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans28 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans29 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p, calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans30 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans31 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans32 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans33 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans34 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans35 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans36 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans37 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans38 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans39 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans40 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans41 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans42 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans43 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans44 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans45 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans46 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans47 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans48 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans49 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans50 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans51 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans52 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans53 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans54 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans55 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans56 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans57 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
|
||
|
|
theorem transport_trans58 {A : Type} {a b c : A} {P : A → Type} (p1 : a = b) (p2 : b = c) (H : P a) :
|
||
|
|
transport p1 (transport p2 H) = transport (trans p1 p2) H
|
||
|
|
:= have H1 : ∀ p : b = b, transport p1 (transport p H) = transport (trans p1 p) H, from
|
||
|
|
take p : b = b,
|
||
|
|
calc transport p1 (transport p H) = transport p1 H : {transport_eq p H}
|
||
|
|
... = transport (trans p1 p) H : refl (transport p1 H),
|
||
|
|
eq_rec H1 p2 p2
|
||
|
|
end
|