give alternative definition of free group on a set with decidable equality
This commit is contained in:
Floris van Doorn
parent
743985e3d8
commit
aa191493e9
5 changed files with 613 additions and 21 deletions
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@ -49,7 +49,7 @@ namespace group
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{ reflexivity},
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{ repeat esimp [map], exact cancel2 x},
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{ repeat esimp [map], exact cancel1 x},
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{ repeat esimp [map], apply rflip},
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{ repeat esimp [map], apply fcg_rel.rflip},
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{ rewrite [+map_append], exact resp_append IH₁ IH₂},
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{ exact rtrans IH₁ IH₂}
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end
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@ -60,7 +60,7 @@ namespace group
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{ reflexivity},
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{ repeat esimp [map], exact cancel2 x},
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{ repeat esimp [map], exact cancel1 x},
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{ repeat esimp [map], apply rflip},
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{ repeat esimp [map], apply fcg_rel.rflip},
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{ rewrite [+reverse_append], exact resp_append IH₂ IH₁},
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{ exact rtrans IH₁ IH₂}
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end
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@ -6,9 +6,12 @@ Authors: Floris van Doorn, Egbert Rijke
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Constructions with groups
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-/
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import algebra.group_theory hit.set_quotient types.list types.sum
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import algebra.group_theory hit.set_quotient types.list types.sum ..move_to_lib
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open eq algebra is_trunc set_quotient relation sigma sigma.ops prod sum list trunc function equiv
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prod.ops decidable is_equiv
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universe variable u
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namespace group
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@ -157,6 +160,19 @@ namespace group
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esimp, refine !foldl_append ⬝ _, esimp, apply fgh_helper_mul}
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end
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definition free_group_hom_eq [constructor] {φ ψ : free_group X →g G}
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(H : Πx, φ (free_group_inclusion x) = ψ (free_group_inclusion x)) : φ ~ ψ :=
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begin
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refine set_quotient.rec_prop _, intro l,
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induction l with s l IH,
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{ exact respect_one φ ⬝ (respect_one ψ)⁻¹ },
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{ refine respect_mul φ (class_of [s]) (class_of l) ⬝ _ ⬝
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(respect_mul ψ (class_of [s]) (class_of l))⁻¹,
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refine ap011 mul _ IH, induction s with x x, exact H x,
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refine respect_inv φ (class_of [inl x]) ⬝ ap inv (H x) ⬝
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(respect_inv ψ (class_of [inl x]))⁻¹ }
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end
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definition fn_of_free_group_hom [unfold_full] (φ : free_group X →g G) : X → G :=
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φ ∘ free_group_inclusion
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@ -167,12 +183,323 @@ namespace group
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{ exact fn_of_free_group_hom},
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{ exact free_group_hom},
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{ intro f, apply eq_of_homotopy, intro x, esimp, unfold [foldl], apply one_mul},
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{ intro φ, apply homomorphism_eq, refine set_quotient.rec_prop _, intro l, esimp,
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induction l with s l IH,
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{ esimp [foldl], exact (respect_one φ)⁻¹},
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{ rewrite [foldl_cons, fgh_helper_mul],
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refine _ ⬝ (respect_mul φ (class_of [s]) (class_of l))⁻¹,
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rewrite [IH], induction s: rewrite [▸*, one_mul], rewrite [-respect_inv φ]}}
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{ intro φ, apply homomorphism_eq, apply free_group_hom_eq, intro x, apply one_mul }
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end
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end group
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end group
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/- alternative definition of free group on a set with decidable equality -/
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namespace list
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variables {X : Type.{u}} {v w : X ⊎ X} {l : list (X ⊎ X)}
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inductive is_reduced {X : Type} : list (X ⊎ X) → Type :=
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| nil : is_reduced []
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| singleton : Πv, is_reduced [v]
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| cons : Π⦃v w l⦄, sum.flip v ≠ w → is_reduced (w::l) → is_reduced (v::w::l)
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definition is_reduced_code (H : is_reduced l) : Type.{u} :=
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begin
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cases l with v l, { exact is_reduced.nil = H },
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cases l with w l, { exact is_reduced.singleton v = H },
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exact Σ(pH : sum.flip v ≠ w × is_reduced (w::l)), is_reduced.cons pH.1 pH.2 = H
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end
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definition is_reduced_encode (H : is_reduced l) : is_reduced_code H :=
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begin
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induction H with v v w l p Hl IH,
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{ exact idp },
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{ exact idp },
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{ exact ⟨(p, Hl), idp⟩ }
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end
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definition is_prop_is_reduced (l : list (X ⊎ X)) : is_prop (is_reduced l) :=
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begin
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apply is_prop.mk, intro H₁ H₂, induction H₁ with v v w l p Hl IH,
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{ exact is_reduced_encode H₂ },
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{ exact is_reduced_encode H₂ },
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{ cases is_reduced_encode H₂ with pH' q, cases pH' with p' Hl', esimp at q,
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subst q, exact ap011 (λx y, is_reduced.cons x y) !is_prop.elim (IH Hl') }
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end
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definition rlist (X : Type) : Type :=
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Σ(l : list (X ⊎ X)), is_reduced l
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local attribute [instance] is_prop_is_reduced
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definition rlist_eq {l l' : rlist X} (p : l.1 = l'.1) : l = l' :=
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subtype_eq p
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definition is_trunc_rlist {n : ℕ₋₂} {X : Type} (H : is_trunc (n.+2) X) : is_trunc (n.+2) (rlist X) :=
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begin
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apply is_trunc_sigma, { apply is_trunc_list, apply is_trunc_sum },
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intro l, apply is_trunc_succ_of_is_prop
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end
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definition is_reduced_invert (v : X ⊎ X) : is_reduced (v::l) → is_reduced l :=
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begin
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assert H : Π⦃l'⦄, is_reduced l' → l' = v::l → is_reduced l,
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{ intro l' Hl', revert l, induction Hl' with v' v' w' l' p' Hl' IH: intro l p,
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{ cases p },
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{ cases cons_eq_cons p with q r, subst r, apply is_reduced.nil },
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{ cases cons_eq_cons p with q r, subst r, exact Hl' }},
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intro Hl, exact H Hl idp
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end
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definition rnil [constructor] : rlist X :=
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⟨[], !is_reduced.nil⟩
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definition rsingleton [constructor] (x : X ⊎ X) : rlist X :=
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⟨[x], !is_reduced.singleton⟩
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definition is_reduced_doubleton [constructor] {x y : X ⊎ X} (p : flip x ≠ y) :
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is_reduced [x, y] :=
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is_reduced.cons p !is_reduced.singleton
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definition rdoubleton [constructor] {x y : X ⊎ X} (p : flip x ≠ y) : rlist X :=
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⟨[x, y], is_reduced_doubleton p⟩
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definition is_reduced_concat (Hn : sum.flip v ≠ w) (Hl : is_reduced (concat v l)) :
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is_reduced (concat w (concat v l)) :=
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begin
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assert H : Π⦃l'⦄, is_reduced l' → l' = concat v l → is_reduced (concat w l'),
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{ clear Hl, intro l' Hl', revert l, induction Hl' with v' v' w' l' p' Hl' IH: intro l p,
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{ exfalso, exact concat_neq_nil _ _ p⁻¹ },
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{ cases concat_eq_singleton p⁻¹ with q r, subst q,
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exact is_reduced_doubleton Hn },
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{ do 2 esimp [concat], apply is_reduced.cons p', cases l with x l,
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{ cases p },
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{ apply IH l, esimp [concat] at p, revert p, generalize concat v l, intro l'' p,
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cases cons_eq_cons p with q r, exact r }}},
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exact H Hl idp
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end
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definition is_reduced_reverse (H : is_reduced l) : is_reduced (reverse l) :=
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begin
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induction H with v v w l p Hl IH,
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{ apply is_reduced.nil },
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{ apply is_reduced.singleton },
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{ refine is_reduced_concat _ IH, intro q, apply p, subst q, apply flip_flip }
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end
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definition rreverse [constructor] (l : rlist X) : rlist X := ⟨reverse l.1, is_reduced_reverse l.2⟩
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definition is_reduced_flip (H : is_reduced l) : is_reduced (map flip l) :=
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begin
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induction H with v v w l p Hl IH,
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{ apply is_reduced.nil },
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{ apply is_reduced.singleton },
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{ refine is_reduced.cons _ IH, intro q, apply p, exact !flip_flip⁻¹ ⬝ ap flip q ⬝ !flip_flip }
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end
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definition rflip [constructor] (l : rlist X) : rlist X := ⟨map flip l.1, is_reduced_flip l.2⟩
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definition rcons' [decidable_eq X] (v : X ⊎ X) (l : list (X ⊎ X)) : list (X ⊎ X) :=
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begin
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cases l with w l,
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{ exact [v] },
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{ exact if q : sum.flip v = w then l else v::w::l }
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end
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definition is_reduced_rcons [decidable_eq X] (v : X ⊎ X) (Hl : is_reduced l) :
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is_reduced (rcons' v l) :=
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begin
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cases l with w l, apply is_reduced.singleton,
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apply dite (sum.flip v = w): intro q,
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{ have is_set (X ⊎ X), from !is_trunc_sum,
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rewrite [↑rcons', dite_true q _], exact is_reduced_invert w Hl },
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{ rewrite [↑rcons', dite_false q], exact is_reduced.cons q Hl, }
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end
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definition rcons [constructor] [decidable_eq X] (v : X ⊎ X) (l : rlist X) : rlist X :=
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⟨rcons' v l.1, is_reduced_rcons v l.2⟩
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definition rcons_eq [decidable_eq X] : is_reduced (v::l) → rcons' v l = v :: l :=
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begin
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assert H : Π⦃l'⦄, is_reduced l' → l' = v::l → rcons' v l = l',
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{ intro l' Hl', revert l, induction Hl' with v' v' w' l' p' Hl' IH: intro l p,
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{ cases p },
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{ cases cons_eq_cons p with q r, subst r, cases p, reflexivity },
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{ cases cons_eq_cons p with q r, subst q, subst r, rewrite [↑rcons', dite_false p'], }},
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intro Hl, exact H Hl idp
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end
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definition rcons_rcons_eq [decidable_eq X] (p : flip v = w) (l : rlist X) :
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rcons v (rcons w l) = l :=
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begin
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have is_set (X ⊎ X), from !is_trunc_sum,
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induction l with l Hl,
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apply rlist_eq, esimp,
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induction l with u l IH,
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{ exact dite_true p _ },
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{ apply dite (sum.flip w = u): intro q,
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{ rewrite [↑rcons' at {1}, dite_true q _], subst p, induction (!flip_flip⁻¹ ⬝ q),
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exact rcons_eq Hl },
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{ rewrite [↑rcons', dite_false q, dite_true p _] }}
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end
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definition reduce_list' [decidable_eq X] (l : list (X ⊎ X)) : list (X ⊎ X) :=
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begin
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induction l with v l IH,
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{ exact [] },
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{ exact rcons' v IH }
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end
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definition is_reduced_reduce_list [decidable_eq X] (l : list (X ⊎ X)) :
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is_reduced (reduce_list' l) :=
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begin
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induction l with v l IH, apply is_reduced.nil,
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apply is_reduced_rcons, exact IH
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end
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definition reduce_list [constructor] [decidable_eq X] (l : list (X ⊎ X)) : rlist X :=
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⟨reduce_list' l, is_reduced_reduce_list l⟩
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definition rappend' [decidable_eq X] (l : list (X ⊎ X)) (l' : rlist X) : rlist X := foldr rcons l' l
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definition rappend_rcons' [decidable_eq X] (x : X ⊎ X) (l₁ : list (X ⊎ X)) (l₂ : rlist X) :
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rappend' (rcons' x l₁) l₂ = rcons x (rappend' l₁ l₂) :=
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begin
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induction l₁ with x' l₁ IH,
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{ reflexivity },
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{ apply dite (sum.flip x = x'): intro p,
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{ have is_set (X ⊎ X), from !is_trunc_sum, rewrite [↑rcons', dite_true p _],
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exact (rcons_rcons_eq p _)⁻¹ },
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{ rewrite [↑rcons', dite_false p] }}
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end
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definition rappend_cons' [decidable_eq X] (x : X ⊎ X) (l₁ : list (X ⊎ X)) (l₂ : rlist X) :
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rappend' (x::l₁) l₂ = rcons x (rappend' l₁ l₂) :=
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idp
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definition rappend [decidable_eq X] (l l' : rlist X) : rlist X := rappend' l.1 l'
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definition rappend_rcons [decidable_eq X] (x : X ⊎ X) (l₁ l₂ : rlist X) :
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rappend (rcons x l₁) l₂ = rcons x (rappend l₁ l₂) :=
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rappend_rcons' x l₁.1 l₂
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definition rappend_assoc [decidable_eq X] (l₁ l₂ l₃ : rlist X) :
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rappend (rappend l₁ l₂) l₃ = rappend l₁ (rappend l₂ l₃) :=
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begin
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induction l₁ with l₁ h, unfold rappend, clear h, induction l₁ with x l₁ IH,
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{ reflexivity },
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{ rewrite [rappend_cons', ▸*, rappend_rcons', IH] }
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end
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definition rappend_rnil [decidable_eq X] (l : rlist X) : rappend l rnil = l :=
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begin
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induction l with l H, apply rlist_eq, esimp, induction H with v v w l p Hl IH,
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{ reflexivity },
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{ reflexivity },
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{ rewrite [↑rappend at *, rappend_cons', ↑rcons, IH, ↑rcons', dite_false p] }
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end
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definition rnil_rappend [decidable_eq X] (l : rlist X) : rappend rnil l = l :=
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by reflexivity
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definition rappend_left_inv [decidable_eq X] (l : rlist X) :
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rappend (rflip (rreverse l)) l = rnil :=
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begin
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induction l with l H, apply rlist_eq, induction l with x l IH,
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{ reflexivity },
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{ have is_set (X ⊎ X), from !is_trunc_sum,
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rewrite [↑rappend, ↑rappend', reverse_cons, map_concat, foldr_concat],
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refine ap (λx, (rappend' _ x).1) (rlist_eq (dite_true !flip_flip _)) ⬝
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IH (is_reduced_invert _ H) }
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end
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end list open list
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namespace group
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open sigma.ops
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variables (X : Type) [decidable_eq X] {G : Group}
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definition group_dfree_group [constructor] : group (rlist X) :=
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group.mk (is_trunc_rlist _) rappend rappend_assoc rnil rnil_rappend rappend_rnil
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(rflip ∘ rreverse) rappend_left_inv
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definition dfree_group [constructor] : Group :=
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Group.mk _ (group_dfree_group X)
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variable {X}
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definition fgh_helper_rcons (f : X → G) (g : G) (x : X ⊎ X) {l : list (X ⊎ X)} :
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foldl (fgh_helper f) g (rcons' x l) = foldl (fgh_helper f) g (x :: l) :=
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begin
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cases l with x' l, reflexivity,
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apply dite (sum.flip x = x'): intro q,
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{ have is_set (X ⊎ X), from !is_trunc_sum,
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rewrite [↑rcons', dite_true q _, foldl_cons, foldl_cons, -q],
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induction x with x, rewrite [↑fgh_helper,mul_inv_cancel_right],
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rewrite [↑fgh_helper,inv_mul_cancel_right] },
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{ rewrite [↑rcons', dite_false q] }
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end
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definition fgh_helper_rappend (f : X → G) (g : G) (l l' : rlist X) :
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foldl (fgh_helper f) g (rappend l l').1 = foldl (fgh_helper f) g (l.1 ++ l'.1) :=
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begin
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revert g, induction l with l lH, unfold rappend, clear lH,
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induction l with v l IH: intro g, reflexivity,
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rewrite [rappend_cons', ↑rcons, fgh_helper_rcons, foldl_cons, IH]
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end
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local attribute [instance] is_prop_is_reduced
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-- definition dfree_group_hom' [constructor] {G : InfGroup} (f : X → G) :
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-- Σ(f : dfree_group X → G), is_mul_hom f :=
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-- ⟨λx, foldl (fgh_helper f) 1 x.1,
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-- begin intro l₁ l₂, exact !fgh_helper_rappend ⬝ !foldl_append ⬝ !fgh_helper_mul end⟩
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-- definition dfree_group_hom_eq' [constructor] {φ ψ : dfree_group X →g G}
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-- (H : Πx, φ (rsingleton (inl x)) = ψ (rsingleton (inl x))) : φ ~ ψ :=
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-- begin
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-- assert H2 : Πv, φ (rsingleton v) = ψ (rsingleton v),
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-- { intro v, induction v with x x, exact H x,
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-- exact respect_inv φ _ ⬝ ap inv (H x) ⬝ (respect_inv ψ _)⁻¹ },
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-- intro l, induction l with l Hl,
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-- induction Hl with v v w l p Hl IH,
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-- { exact respect_one φ ⬝ (respect_one ψ)⁻¹ },
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-- { exact H2 v },
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-- { refine ap φ (rlist_eq (rcons_eq (is_reduced.cons p Hl))⁻¹) ⬝
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-- respect_mul φ (rsingleton v) ⟨_, Hl⟩ ⬝ ap011 mul (H2 v) IH ⬝ _,
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-- symmetry, exact ap ψ (rlist_eq (rcons_eq (is_reduced.cons p Hl))⁻¹) ⬝
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-- respect_mul ψ (rsingleton v) ⟨_, Hl⟩ }
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-- end
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definition dfree_group_hom [constructor] {G : Group} (f : X → G) : dfree_group X →g G :=
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homomorphism.mk (λx, foldl (fgh_helper f) 1 x.1)
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begin intro l₁ l₂, exact !fgh_helper_rappend ⬝ !foldl_append ⬝ !fgh_helper_mul end
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local attribute [instance] is_prop_is_reduced
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definition dfree_group_hom_eq [constructor] {φ ψ : dfree_group X →g G}
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(H : Πx, φ (rsingleton (inl x)) = ψ (rsingleton (inl x))) : φ ~ ψ :=
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begin
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assert H2 : Πv, φ (rsingleton v) = ψ (rsingleton v),
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{ intro v, induction v with x x, exact H x,
|
||||
exact respect_inv φ _ ⬝ ap inv (H x) ⬝ (respect_inv ψ _)⁻¹ },
|
||||
intro l, induction l with l Hl,
|
||||
induction Hl with v v w l p Hl IH,
|
||||
{ exact respect_one φ ⬝ (respect_one ψ)⁻¹ },
|
||||
{ exact H2 v },
|
||||
{ refine ap φ (rlist_eq (rcons_eq (is_reduced.cons p Hl))⁻¹) ⬝
|
||||
respect_mul φ (rsingleton v) ⟨_, Hl⟩ ⬝ ap011 mul (H2 v) IH ⬝ _,
|
||||
symmetry, exact ap ψ (rlist_eq (rcons_eq (is_reduced.cons p Hl))⁻¹) ⬝
|
||||
respect_mul ψ (rsingleton v) ⟨_, Hl⟩ }
|
||||
end
|
||||
|
||||
variable (X)
|
||||
|
||||
definition free_group_of_dfree_group [constructor] : dfree_group X →g free_group X :=
|
||||
dfree_group_hom free_group_inclusion
|
||||
|
||||
definition dfree_group_of_free_group [constructor] : free_group X →g dfree_group X :=
|
||||
free_group_hom (λx, rsingleton (inl x))
|
||||
|
||||
definition dfree_group_isomorphism : dfree_group X ≃g free_group X :=
|
||||
begin
|
||||
apply isomorphism.MK (free_group_of_dfree_group X) (dfree_group_of_free_group X),
|
||||
{ apply free_group_hom_eq, intro x, reflexivity },
|
||||
{ apply dfree_group_hom_eq, intro x, reflexivity }
|
||||
end
|
||||
|
||||
end group
|
||||
|
|
|
|||
|
|
@ -1,6 +1,6 @@
|
|||
-- definitions, theorems and attributes which should be moved to files in the HoTT library
|
||||
|
||||
import homotopy.sphere2 homotopy.cofiber homotopy.wedge hit.prop_trunc hit.set_quotient eq2 types.pointed2
|
||||
import homotopy.sphere2 homotopy.cofiber homotopy.wedge hit.prop_trunc hit.set_quotient eq2 types.pointed2 algebra.graph
|
||||
|
||||
open eq nat int susp pointed sigma is_equiv equiv fiber algebra trunc pi group
|
||||
is_trunc function unit prod bool
|
||||
|
|
@ -11,7 +11,7 @@ attribute ap010 [unfold 7]
|
|||
attribute tro_invo_tro [unfold 9] -- TODO: move
|
||||
-- TODO: homotopy_of_eq and apd10 should be the same
|
||||
-- TODO: there is also apd10_eq_of_homotopy in both pi and eq(?)
|
||||
|
||||
universe variable u
|
||||
|
||||
|
||||
namespace algebra
|
||||
|
|
@ -24,6 +24,16 @@ end algebra
|
|||
|
||||
namespace eq
|
||||
|
||||
definition pathover_tr_pathover_idp_of_eq {A : Type} {B : A → Type} {a a' : A} {b : B a} {b' : B a'} {p : a = a'}
|
||||
(q : b =[p] b') :
|
||||
pathover_tr p b ⬝o pathover_idp_of_eq (tr_eq_of_pathover q) = q :=
|
||||
begin induction q; reflexivity end
|
||||
|
||||
-- rename pathover_of_tr_eq_idp
|
||||
definition pathover_of_tr_eq_idp' {A : Type} {B : A → Type} {a a₂ : A} (p : a = a₂) (b : B a) :
|
||||
pathover_of_tr_eq idp = pathover_tr p b :=
|
||||
by induction p; constructor
|
||||
|
||||
definition homotopy.symm_symm {A : Type} {P : A → Type} {f g : Πx, P x} (H : f ~ g) :
|
||||
H⁻¹ʰᵗʸ⁻¹ʰᵗʸ = H :=
|
||||
begin apply eq_of_homotopy, intro x, apply inv_inv end
|
||||
|
|
@ -130,6 +140,11 @@ begin apply eq_of_homotopy, intro x, apply inv_inv end
|
|||
homotopy_group_isomorphism_of_pequiv n f ⬝g
|
||||
ghomotopy_group_ptrunc_of_le H B
|
||||
|
||||
definition fundamental_group_isomorphism {X : Type*} {G : Group}
|
||||
(e : Ω X ≃ G) (hom_e : Πp q, e (p ⬝ q) = e p * e q) : π₁ X ≃g G :=
|
||||
isomorphism_of_equiv (trunc_equiv_trunc 0 e ⬝e (trunc_equiv 0 G))
|
||||
begin intro p q, induction p with p, induction q with q, exact hom_e p q end
|
||||
|
||||
definition equiv_pathover2 {A : Type} {a a' : A} (p : a = a')
|
||||
{B : A → Type} {C : A → Type} (f : B a ≃ C a) (g : B a' ≃ C a')
|
||||
(r : to_fun f =[p] to_fun g) : f =[p] g :=
|
||||
|
|
@ -545,10 +560,17 @@ namespace lift
|
|||
[H : is_trunc n A] : is_trunc n (plift A) :=
|
||||
is_trunc_lift A n
|
||||
|
||||
definition lift_functor2 {A B C : Type} (f : A → B → C) (x : lift A) (y : lift B) : lift C :=
|
||||
up (f (down x) (down y))
|
||||
|
||||
end lift
|
||||
|
||||
namespace trunc
|
||||
open trunc_index
|
||||
|
||||
definition Prop_eq {P Q : Prop} (H : P ↔ Q) : P = Q :=
|
||||
tua (equiv_of_is_prop (iff.mp H) (iff.mpr H))
|
||||
|
||||
definition trunc_index_equiv_nat [constructor] : ℕ₋₂ ≃ ℕ :=
|
||||
equiv.MK add_two sub_two add_two_sub_two sub_two_add_two
|
||||
|
||||
|
|
@ -676,6 +698,19 @@ end is_trunc
|
|||
namespace sigma
|
||||
open sigma.ops
|
||||
|
||||
definition eq.rec_sigma {A : Type} {B : A → Type} {a : A} {b : B a}
|
||||
(P : Π⦃a'⦄ {b' : B a'}, ⟨a, b⟩ = ⟨a', b'⟩ → Type)
|
||||
(IH : P idp) ⦃a' : A⦄ {b' : B a'} (p : ⟨a, b⟩ = ⟨a', b'⟩) : P p :=
|
||||
begin
|
||||
apply transport (λp, P p) (to_left_inv !sigma_eq_equiv p),
|
||||
generalize !sigma_eq_equiv p, esimp, intro q,
|
||||
induction q with q₁ q₂, induction q₂, exact IH
|
||||
end
|
||||
|
||||
definition ap_dpair_eq_dpair_pr {A A' : Type} {B : A → Type} {a a' : A} {b : B a} {b' : B a'} (f : Πa, B a → A') (p : a = a') (q : b =[p] b')
|
||||
: ap (λx, f x.1 x.2) (dpair_eq_dpair p q) = apd011 f p q :=
|
||||
by induction q; reflexivity
|
||||
|
||||
definition sigma_eq_equiv_of_is_prop_right [constructor] {A : Type} {B : A → Type} (u v : Σa, B a)
|
||||
[H : Π a, is_prop (B a)] : u = v ≃ u.1 = v.1 :=
|
||||
!sigma_eq_equiv ⬝e !sigma_equiv_of_is_contr_right
|
||||
|
|
@ -762,6 +797,49 @@ by induction q'; reflexivity
|
|||
end sigma open sigma
|
||||
|
||||
namespace group
|
||||
|
||||
definition isomorphism.MK [constructor] {G H : Group} (φ : G →g H) (ψ : H →g G)
|
||||
(p : φ ∘g ψ ~ gid H) (q : ψ ∘g φ ~ gid G) : G ≃g H :=
|
||||
isomorphism.mk φ (adjointify φ ψ p q)
|
||||
|
||||
protected definition homomorphism.sigma_char [constructor]
|
||||
(A B : Group) : (A →g B) ≃ Σ(f : A → B), is_mul_hom f :=
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
{intro F, exact ⟨F, _⟩ },
|
||||
{intro p, cases p with f H, exact (homomorphism.mk f H) },
|
||||
{intro p, cases p, reflexivity },
|
||||
{intro F, cases F, reflexivity },
|
||||
end
|
||||
|
||||
definition homomorphism_pathover {A : Type} {a a' : A} (p : a = a')
|
||||
{B : A → Group} {C : A → Group} (f : B a →g C a) (g : B a' →g C a')
|
||||
(r : homomorphism.φ f =[p] homomorphism.φ g) : f =[p] g :=
|
||||
begin
|
||||
fapply pathover_of_fn_pathover_fn,
|
||||
{ intro a, apply homomorphism.sigma_char },
|
||||
{ fapply sigma_pathover, exact r, apply is_prop.elimo }
|
||||
end
|
||||
|
||||
protected definition isomorphism.sigma_char [constructor]
|
||||
(A B : Group) : (A ≃g B) ≃ Σ(f : A →g B), is_equiv f :=
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
{intro F, exact ⟨F, _⟩ },
|
||||
{intro p, cases p with f H, exact (isomorphism.mk f H) },
|
||||
{intro p, cases p, reflexivity },
|
||||
{intro F, cases F, reflexivity },
|
||||
end
|
||||
|
||||
definition isomorphism_pathover {A : Type} {a a' : A} (p : a = a')
|
||||
{B : A → Group} {C : A → Group} (f : B a ≃g C a) (g : B a' ≃g C a')
|
||||
(r : pathover (λa, B a → C a) f p g) : f =[p] g :=
|
||||
begin
|
||||
fapply pathover_of_fn_pathover_fn,
|
||||
{ intro a, apply isomorphism.sigma_char },
|
||||
{ fapply sigma_pathover, apply homomorphism_pathover, exact r, apply is_prop.elimo }
|
||||
end
|
||||
|
||||
-- definition is_equiv_isomorphism
|
||||
|
||||
|
||||
|
|
@ -816,9 +894,10 @@ namespace group
|
|||
end group open group
|
||||
|
||||
namespace fiber
|
||||
open pointed sigma
|
||||
|
||||
definition is_contr_pfiber_pid (A : Type*) : is_contr (pfiber (pid A)) :=
|
||||
is_contr.mk pt begin intro x, induction x with a p, esimp at p, cases p, reflexivity end
|
||||
definition is_contr_pfiber_pid (A : Type*) : is_contr (pfiber (pid A)) :=
|
||||
is_contr.mk pt begin intro x, induction x with a p, esimp at p, cases p, reflexivity end
|
||||
|
||||
definition fiber_functor [constructor] {A A' B B' : Type} {f : A → B} {f' : A' → B'} {b : B} {b' : B'}
|
||||
(g : A → A') (h : B → B') (H : hsquare g h f f') (p : h b = b') (x : fiber f b) : fiber f' b' :=
|
||||
|
|
@ -1142,6 +1221,10 @@ namespace sum
|
|||
{f₁₀ : A₀₀ → A₂₀} {f₁₂ : A₀₂ → A₂₂} {f₀₁ : A₀₀ → A₀₂} {f₂₁ : A₂₀ → A₂₂}
|
||||
{g₁₀ : B₀₀ → B₂₀} {g₁₂ : B₀₂ → B₂₂} {g₀₁ : B₀₀ → B₀₂} {g₂₁ : B₂₀ → B₂₂}
|
||||
{h₀₁ : B₀₀ → A₀₂} {h₂₁ : B₂₀ → A₂₂}
|
||||
|
||||
definition flip_flip (x : A ⊎ B) : flip (flip x) = x :=
|
||||
begin induction x: reflexivity end
|
||||
|
||||
definition sum_rec_hsquare [unfold 16] (h : hsquare f₁₀ f₁₂ f₀₁ f₂₁)
|
||||
(k : hsquare g₁₀ f₁₂ h₀₁ h₂₁) : hsquare (f₁₀ +→ g₁₀) f₁₂ (sum.rec f₀₁ h₀₁) (sum.rec f₂₁ h₂₁) :=
|
||||
begin intro x, induction x with a b, exact h a, exact k b end
|
||||
|
|
@ -1195,3 +1278,181 @@ namespace equiv
|
|||
end
|
||||
|
||||
end equiv
|
||||
|
||||
|
||||
namespace paths
|
||||
|
||||
variables {A : Type} {R : A → A → Type} {a₁ a₂ a₃ a₄ : A}
|
||||
inductive all (T : Π⦃a₁ a₂ : A⦄, R a₁ a₂ → Type) : Π⦃a₁ a₂ : A⦄, paths R a₁ a₂ → Type :=
|
||||
| nil {} : Π{a : A}, all T (@nil A R a)
|
||||
| cons : Π{a₁ a₂ a₃ : A} {r : R a₂ a₃} {p : paths R a₁ a₂}, T r → all T p → all T (cons r p)
|
||||
|
||||
inductive Exists (T : Π⦃a₁ a₂ : A⦄, R a₁ a₂ → Type) : Π⦃a₁ a₂ : A⦄, paths R a₁ a₂ → Type :=
|
||||
| base : Π{a₁ a₂ a₃ : A} {r : R a₂ a₃} (p : paths R a₁ a₂), T r → Exists T (cons r p)
|
||||
| cons : Π{a₁ a₂ a₃ : A} (r : R a₂ a₃) {p : paths R a₁ a₂}, Exists T p → Exists T (cons r p)
|
||||
|
||||
inductive mem (l : R a₃ a₄) : Π⦃a₁ a₂ : A⦄, paths R a₁ a₂ → Type :=
|
||||
| base : Π{a₂ : A} (p : paths R a₂ a₃), mem l (cons l p)
|
||||
| cons : Π{a₁ a₂ a₃ : A} (r : R a₂ a₃) {p : paths R a₁ a₂}, mem l p → mem l (cons r p)
|
||||
|
||||
definition len (p : paths R a₁ a₂) : ℕ :=
|
||||
begin
|
||||
induction p with a a₁ a₂ a₃ r p IH,
|
||||
{ exact 0 },
|
||||
{ exact nat.succ IH }
|
||||
end
|
||||
|
||||
definition mem_equiv_Exists (l : R a₁ a₂) (p : paths R a₃ a₄) :
|
||||
mem l p ≃ Exists (λa a' r, ⟨a₁, a₂, l⟩ = ⟨a, a', r⟩) p :=
|
||||
sorry
|
||||
|
||||
end paths
|
||||
|
||||
namespace list
|
||||
open is_trunc trunc sigma.ops prod.ops lift
|
||||
variables {A B X : Type}
|
||||
|
||||
definition foldl_homotopy {f g : A → B → A} (h : f ~2 g) (a : A) : foldl f a ~ foldl g a :=
|
||||
begin
|
||||
intro bs, revert a, induction bs with b bs p: intro a, reflexivity, esimp [foldl],
|
||||
exact p (f a b) ⬝ ap010 (foldl g) (h a b) bs
|
||||
end
|
||||
|
||||
definition cons_eq_cons {x x' : X} {l l' : list X} (p : x::l = x'::l') : x = x' × l = l' :=
|
||||
begin
|
||||
refine lift.down (list.no_confusion p _), intro q r, split, exact q, exact r
|
||||
end
|
||||
|
||||
definition concat_neq_nil (x : X) (l : list X) : concat x l ≠ nil :=
|
||||
begin
|
||||
intro p, cases l: cases p,
|
||||
end
|
||||
|
||||
definition concat_eq_singleton {x x' : X} {l : list X} (p : concat x l = [x']) :
|
||||
x = x' × l = [] :=
|
||||
begin
|
||||
cases l with x₂ l,
|
||||
{ cases cons_eq_cons p with q r, subst q, split: reflexivity },
|
||||
{ exfalso, esimp [concat] at p, apply concat_neq_nil x l, revert p, generalize (concat x l),
|
||||
intro l' p, cases cons_eq_cons p with q r, exact r }
|
||||
end
|
||||
|
||||
definition foldr_concat (f : A → B → B) (b : B) (a : A) (l : list A) :
|
||||
foldr f b (concat a l) = foldr f (f a b) l :=
|
||||
begin
|
||||
induction l with a' l p, reflexivity, rewrite [concat_cons, foldr_cons, p]
|
||||
end
|
||||
|
||||
definition iterated_prod (X : Type.{u}) (n : ℕ) : Type.{u} :=
|
||||
iterate (prod X) n (lift unit)
|
||||
|
||||
definition is_trunc_iterated_prod {k : ℕ₋₂} {X : Type} {n : ℕ} (H : is_trunc k X) :
|
||||
is_trunc k (iterated_prod X n) :=
|
||||
begin
|
||||
induction n with n IH,
|
||||
{ apply is_trunc_of_is_contr, apply is_trunc_lift },
|
||||
{ exact @is_trunc_prod _ _ _ H IH }
|
||||
end
|
||||
|
||||
definition list_of_iterated_prod {n : ℕ} (x : iterated_prod X n) : list X :=
|
||||
begin
|
||||
induction n with n IH,
|
||||
{ exact [] },
|
||||
{ exact x.1::IH x.2 }
|
||||
end
|
||||
|
||||
definition list_of_iterated_prod_succ {n : ℕ} (x : X) (xs : iterated_prod X n) :
|
||||
@list_of_iterated_prod X (succ n) (x, xs) = x::list_of_iterated_prod xs :=
|
||||
by reflexivity
|
||||
|
||||
definition iterated_prod_of_list (l : list X) : Σn, iterated_prod X n :=
|
||||
begin
|
||||
induction l with x l IH,
|
||||
{ exact ⟨0, up ⋆⟩ },
|
||||
{ exact ⟨succ IH.1, (x, IH.2)⟩ }
|
||||
end
|
||||
|
||||
definition iterated_prod_of_list_cons (x : X) (l : list X) :
|
||||
iterated_prod_of_list (x::l) =
|
||||
⟨succ (iterated_prod_of_list l).1, (x, (iterated_prod_of_list l).2)⟩ :=
|
||||
by reflexivity
|
||||
|
||||
protected definition sigma_char [constructor] (X : Type) : list X ≃ Σ(n : ℕ), iterated_prod X n :=
|
||||
begin
|
||||
apply equiv.MK iterated_prod_of_list (λv, list_of_iterated_prod v.2),
|
||||
{ intro x, induction x with n x, esimp, induction n with n IH,
|
||||
{ induction x with x, induction x, reflexivity },
|
||||
{ revert x, change Π(x : X × iterated_prod X n), _, intro xs, cases xs with x xs,
|
||||
rewrite [list_of_iterated_prod_succ, iterated_prod_of_list_cons],
|
||||
apply sigma_eq (ap succ (IH xs)..1),
|
||||
apply pathover_ap, refine prod_pathover _ _ _ _ (IH xs)..2,
|
||||
apply pathover_of_eq, reflexivity }},
|
||||
{ intro l, induction l with x l IH,
|
||||
{ reflexivity },
|
||||
{ exact ap011 cons idp IH }}
|
||||
end
|
||||
|
||||
local attribute [instance] is_trunc_iterated_prod
|
||||
definition is_trunc_list [instance] {n : ℕ₋₂} {X : Type} (H : is_trunc (n.+2) X) :
|
||||
is_trunc (n.+2) (list X) :=
|
||||
begin
|
||||
assert H : is_trunc (n.+2) (Σ(k : ℕ), iterated_prod X k),
|
||||
{ apply is_trunc_sigma, apply is_trunc_succ_succ_of_is_set,
|
||||
intro, exact is_trunc_iterated_prod H },
|
||||
apply is_trunc_equiv_closed_rev _ (list.sigma_char X),
|
||||
end
|
||||
|
||||
end list
|
||||
|
||||
/- namespace logic? -/
|
||||
namespace decidable
|
||||
definition double_neg_elim {A : Type} (H : decidable A) (p : ¬ ¬ A) : A :=
|
||||
begin induction H, assumption, contradiction end
|
||||
|
||||
|
||||
definition dite_true {C : Type} [H : decidable C] {A : Type}
|
||||
{t : C → A} {e : ¬ C → A} (c : C) (H' : is_prop C) : dite C t e = t c :=
|
||||
begin
|
||||
induction H with H H,
|
||||
exact ap t !is_prop.elim,
|
||||
contradiction
|
||||
end
|
||||
|
||||
definition dite_false {C : Type} [H : decidable C] {A : Type}
|
||||
{t : C → A} {e : ¬ C → A} (c : ¬ C) : dite C t e = e c :=
|
||||
begin
|
||||
induction H with H H,
|
||||
contradiction,
|
||||
exact ap e !is_prop.elim,
|
||||
end
|
||||
|
||||
definition decidable_eq_of_is_prop (A : Type) [is_prop A] : decidable_eq A :=
|
||||
λa a', decidable.inl !is_prop.elim
|
||||
|
||||
definition decidable_eq_sigma [instance] {A : Type} (B : A → Type) [HA : decidable_eq A]
|
||||
[HB : Πa, decidable_eq (B a)] : decidable_eq (Σa, B a) :=
|
||||
begin
|
||||
intro v v', induction v with a b, induction v' with a' b',
|
||||
cases HA a a' with p np,
|
||||
{ induction p, cases HB a b b' with q nq,
|
||||
induction q, exact decidable.inl idp,
|
||||
apply decidable.inr, intro p, apply nq, apply @eq_of_pathover_idp A B,
|
||||
exact change_path !is_prop.elim p..2 },
|
||||
{ apply decidable.inr, intro p, apply np, exact p..1 }
|
||||
end
|
||||
|
||||
open sum
|
||||
definition decidable_eq_sum [instance] (A B : Type) [HA : decidable_eq A] [HB : decidable_eq B] :
|
||||
decidable_eq (A ⊎ B) :=
|
||||
begin
|
||||
intro v v', induction v with a b: induction v' with a' b',
|
||||
{ cases HA a a' with p np,
|
||||
{ exact decidable.inl (ap sum.inl p) },
|
||||
{ apply decidable.inr, intro p, cases p, apply np, reflexivity }},
|
||||
{ apply decidable.inr, intro p, cases p },
|
||||
{ apply decidable.inr, intro p, cases p },
|
||||
{ cases HB b b' with p np,
|
||||
{ exact decidable.inl (ap sum.inr p) },
|
||||
{ apply decidable.inr, intro p, cases p, apply np, reflexivity }},
|
||||
end
|
||||
end decidable
|
||||
|
|
|
|||
|
|
@ -599,6 +599,13 @@ namespace pointed
|
|||
pfiber f ≃* psigma_gen (λa, f a = pt) (respect_pt f) :=
|
||||
pequiv_of_equiv (fiber.sigma_char f pt) idp
|
||||
|
||||
definition fiberpt [constructor] {A B : Type*} {f : A →* B} : fiber f pt :=
|
||||
fiber.mk pt (respect_pt f)
|
||||
|
||||
definition psigma_fiber_pequiv [constructor] {A B : Type*} (f : A →* B) :
|
||||
psigma_gen (fiber f) fiberpt ≃* A :=
|
||||
pequiv_of_equiv (sigma_fiber_equiv f) idp
|
||||
|
||||
definition ppmap.sigma_char [constructor] (A B : Type*) :
|
||||
ppmap A B ≃* @psigma_gen (A →ᵘ* B) (λf, f pt = pt) idp :=
|
||||
pequiv_of_equiv pmap.sigma_char idp
|
||||
|
|
|
|||
|
|
@ -3,7 +3,7 @@ Copyright (c) 2017 Jeremy Avigad. All rights reserved.
|
|||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Jeremy Avigad
|
||||
-/
|
||||
import types.trunc .logic
|
||||
import types.trunc .logic .move_to_lib
|
||||
open funext eq trunc is_trunc logic
|
||||
|
||||
definition property (X : Type) := X → Prop
|
||||
|
|
@ -18,9 +18,8 @@ definition mem (x : X) (a : property X) := a x
|
|||
infix ∈ := mem
|
||||
notation a ∉ b := ¬ mem a b
|
||||
|
||||
/-theorem ext {a b : property X} (H : ∀x, x ∈ a ↔ x ∈ b) : a = b :=
|
||||
eq_of_homotopy (take x, propext (H x))
|
||||
-/
|
||||
theorem ext {X : Type} {a b : property X} (H : ∀x, x ∈ a ↔ x ∈ b) : a = b :=
|
||||
eq_of_homotopy (take x, Prop_eq (H x))
|
||||
|
||||
definition subproperty (a b : property X) : Prop := Prop.mk (∀⦃x⦄, x ∈ a → x ∈ b) _
|
||||
infix ⊆ := subproperty
|
||||
|
|
@ -33,10 +32,8 @@ theorem subproperty.refl (a : property X) : a ⊆ a := take x, assume H, H
|
|||
theorem subproperty.trans {a b c : property X} (subab : a ⊆ b) (subbc : b ⊆ c) : a ⊆ c :=
|
||||
take x, assume ax, subbc (subab ax)
|
||||
|
||||
/-
|
||||
theorem subproperty.antisymm {a b : property X} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
|
||||
theorem subproperty.antisymm {X : Type} {a b : property X} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
|
||||
ext (λ x, iff.intro (λ ina, h₁ ina) (λ inb, h₂ inb))
|
||||
-/
|
||||
|
||||
-- an alterantive name
|
||||
/-
|
||||
|
|
|
|||
Loading…
Reference in a new issue