76 lines
3.8 KiB
Text
76 lines
3.8 KiB
Text
@include logic.pg
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section Category
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variable
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(Obj : ★)
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(~> : Obj → Obj → ★);
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infixr 10 ~>;
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variable
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(id : forall (A : Obj), A ~> A)
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(comp : forall (A B C : Obj), A ~> B → B ~> C → A ~> C);
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hypothesis
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(id_l : forall (A B : Obj) (f : A ~> B), eq (A ~> B) (comp A A B (id A) f) f)
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(id_r : forall (A B : Obj) (f : B ~> A), eq (A ~> B) (comp B A A f (id A)) f)
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(assoc : forall (A B C D : Obj) (f : A ~> B) (g : B ~> C) (h : C ~> D),
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eq (A ~> D)
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(comp A B D f (comp B C D g h))
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(comp A C D (comp A B C f g) h));
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def initial (A : Obj) := forall (B : Obj), exists_uniq_t (A ~> B);
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def terminal (A : Obj) := forall (B : Obj), exists_uniq_t (B ~> A);
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def × (A B C : Obj) (piA : C ~> A) (piB : C ~> B) :=
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forall (D : Obj) (f : D ~> A) (g : D ~> B),
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exists_uniq (D ~> C) (fun (fxg : D ~> C) =>
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eq (D ~> A) (comp D C A fxg piA) f
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∧ eq (D ~> B) (comp D C B fxg piB) g);
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section Inverses
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variable
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(A B : Obj)
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(f : A ~> B)
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(g : B ~> A);
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def inv_l := eq (A ~> A) (comp A B A f g) (id A);
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def inv_r := eq (B ~> B) (comp B A B g f) (id B);
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def inv := inv_l ∧ inv_r;
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end Inverses
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def ≅ (A B : Obj) := exists (A ~> B) (fun (f : A ~> B) => exists (B ~> A) (inv A B f));
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infixl 20 ≅;
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def initial_uniq (A B : Obj) (hA : initial A) (hB : initial B) : A ≅ B :=
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exists_uniq_t_elim (A ~> B) (A ≅ B) (hA B) (fun (f : A ~> B) (f_uniq : forall (y : A ~> B), eq (A ~> B) f y) =>
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exists_uniq_t_elim (B ~> A) (A ≅ B) (hB A) (fun (g : B ~> A) (g_uniq : forall (y : B ~> A), eq (B ~> A) g y) =>
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exists_uniq_t_elim (A ~> A) (A ≅ B) (hA A) (fun (a : A ~> A) (a_uniq : forall (y : A ~> A), eq (A ~> A) a y) =>
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exists_uniq_t_elim (B ~> B) (A ≅ B) (hB B) (fun (b : B ~> B) (b_uniq : forall (y : B ~> B), eq (B ~> B) b y) =>
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exists_intro (A ~> B) (fun (f : A ~> B) => exists (B ~> A) (inv A B f)) f
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(exists_intro (B ~> A) (inv A B f) g
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(and_intro (inv_l A B f g) (inv_r A B f g)
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(eq_trans (A ~> A) (comp A B A f g) a (id A)
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(eq_sym (A ~> A) a (comp A B A f g) (a_uniq (comp A B A f g)))
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(a_uniq (id A)))
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(eq_trans (B ~> B) (comp B A B g f) b (id B)
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(eq_sym (B ~> B) b (comp B A B g f) (b_uniq (comp B A B g f)))
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(b_uniq (id B)))))))));
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def terminal_uniq (A B : Obj) (hA : terminal A) (hB : terminal B) : A ≅ B :=
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exists_uniq_t_elim (A ~> B) (A ≅ B) (hB A) (fun (f : A ~> B) (f_uniq : forall (y : A ~> B), eq (A ~> B) f y) =>
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exists_uniq_t_elim (B ~> A) (A ≅ B) (hA B) (fun (g : B ~> A) (g_uniq : forall (y : B ~> A), eq (B ~> A) g y) =>
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exists_uniq_t_elim (A ~> A) (A ≅ B) (hA A) (fun (a : A ~> A) (a_uniq : forall (y : A ~> A), eq (A ~> A) a y) =>
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exists_uniq_t_elim (B ~> B) (A ≅ B) (hB B) (fun (b : B ~> B) (b_uniq : forall (y : B ~> B), eq (B ~> B) b y) =>
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exists_intro (A ~> B) (fun (f : A ~> B) => exists (B ~> A) (inv A B f)) f
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(exists_intro (B ~> A) (inv A B f) g
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(and_intro (inv_l A B f g) (inv_r A B f g)
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(eq_trans (A ~> A) (comp A B A f g) a (id A)
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(eq_sym (A ~> A) a (comp A B A f g) (a_uniq (comp A B A f g)))
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(a_uniq (id A)))
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(eq_trans (B ~> B) (comp B A B g f) b (id B)
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(eq_sym (B ~> B) b (comp B A B g f) (b_uniq (comp B A B g f)))
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(b_uniq (id B)))))))));
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end Category
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