65 lines
1.7 KiB
Text
65 lines
1.7 KiB
Text
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open eq is_trunc
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variables {I : Set} {P : I → Type} {i j k : I} {x x₁ x₂ : P i} {y y₁ y₂ : P j} {z : P k}
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{Q : Π⦃i⦄, P i → Type}
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structure heq (x : P i) (y : P j) : Type :=
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(p : i = j)
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(q : x =[p] y)
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namespace eq
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notation x ` ==[`:50 P:0 `] `:0 y:50 := @heq _ P _ _ x y
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infix ` == `:50 := heq -- mostly for printing, since it will be almost always ambiguous what P is
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definition pathover_of_heq {p : i = j} (q : x ==[P] y) : x =[p] y :=
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change_path !is_set.elim (heq.q q)
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definition eq_of_heq (p : x₁ ==[P] x₂) : x₁ = x₂ :=
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eq_of_pathover_idp (pathover_of_heq p)
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definition heq.elim (p : x ==[P] y) (q : Q x) : Q y :=
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begin
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induction p with p r, induction r, exact q
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end
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definition heq.refl [refl] (x : P i) : x ==[P] x :=
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heq.mk idp idpo
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definition heq.rfl : x ==[P] x :=
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heq.refl x
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definition heq.symm [symm] (p : x ==[P] y) : y ==[P] x :=
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begin
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induction p with p q, constructor, exact q⁻¹ᵒ
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end
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definition heq_of_eq (p : x₁ = x₂) : x₁ ==[P] x₂ :=
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heq.mk idp (pathover_idp_of_eq p)
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definition heq.trans [trans] (p : x ==[P] y) (p₂ : y ==[P] z) : x ==[P] z :=
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begin
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induction p with p q, induction p₂ with p₂ q₂, constructor, exact q ⬝o q₂
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end
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infix ` ⬝he `:72 := heq.trans
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postfix `⁻¹ʰᵉ`:(max+10) := heq.symm
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definition heq_of_heq_of_eq (p : x ==[P] y) (p₂ : y = y₂) : x ==[P] y₂ :=
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p ⬝he heq_of_eq p₂
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definition heq_of_eq_of_heq (p : x = x₂) (p₂ : x₂ ==[P] y) : x ==[P] y :=
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heq_of_eq p ⬝he p₂
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infix ` ⬝hep `:73 := concato_eq
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infix ` ⬝phe `:74 := eq_concato
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definition heq_tr (p : i = j) (x : P i) : x ==[P] transport P p x :=
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heq.mk p !pathover_tr
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definition tr_heq (p : i = j) (x : P i) : transport P p x ==[P] x :=
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(heq_tr p x)⁻¹ʰᵉ
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end eq
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