154 lines
5.1 KiB
Text
154 lines
5.1 KiB
Text
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/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Floris van Doorn
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Theorems about W-types (well-founded trees)
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-/
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import .sigma .pi
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open path sigma sigma.ops equiv is_equiv
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inductive Wtype {A : Type} (B : A → Type) :=
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sup : Π(a : A), (B a → Wtype B) → Wtype B
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namespace Wtype
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notation `W` binders `,` r:(scoped B, Wtype B) := r
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universe variables u v
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variables {A A' : Type.{u}} {B B' : A → Type.{v}} {C : Π(a : A), B a → Type}
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{a a' : A} {f : B a → W a, B a} {f' : B a' → W a, B a} {w w' : W(a : A), B a}
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definition pr1 (w : W(a : A), B a) : A :=
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Wtype.rec_on w (λa f IH, a)
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definition pr2 (w : W(a : A), B a) : B (pr1 w) → W(a : A), B a :=
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Wtype.rec_on w (λa f IH, f)
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namespace ops
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postfix `.1`:(max+1) := Wtype.pr1
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postfix `.2`:(max+1) := Wtype.pr2
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notation `⟨` a `,` f `⟩`:0 := Wtype.sup a f --input ⟨ ⟩ as \< \>
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end ops
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open ops
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protected definition eta (w : W a, B a) : ⟨w.1 , w.2⟩ ≈ w :=
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cases_on w (λa f, idp)
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definition path_W_sup (p : a ≈ a') (q : p ▹ f ≈ f') : ⟨a, f⟩ ≈ ⟨a', f'⟩ :=
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path.rec_on p (λf' q, path.rec_on q idp) f' q
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definition path_W (p : w.1 ≈ w'.1) (q : p ▹ w.2 ≈ w'.2) : w ≈ w' :=
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cases_on w
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(λw1 w2, cases_on w' (λ w1' w2', path_W_sup))
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p q
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definition pr1_path (p : w ≈ w') : w.1 ≈ w'.1 :=
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path.rec_on p idp
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definition pr2_path (p : w ≈ w') : pr1_path p ▹ w.2 ≈ w'.2 :=
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path.rec_on p idp
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namespace ops
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postfix `..1`:(max+1) := pr1_path
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postfix `..2`:(max+1) := pr2_path
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end ops
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open ops
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definition sup_path_W (p : w.1 ≈ w'.1) (q : p ▹ w.2 ≈ w'.2)
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: dpair (path_W p q)..1 (path_W p q)..2 ≈ dpair p q :=
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begin
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reverts (p, q),
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apply (cases_on w), intros (w1, w2),
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apply (cases_on w'), intros (w1', w2', p), generalize w2', --change to revert
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apply (path.rec_on p), intros (w2', q),
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apply (path.rec_on q), apply idp
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end
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definition pr1_path_W (p : w.1 ≈ w'.1) (q : p ▹ w.2 ≈ w'.2) : (path_W p q)..1 ≈ p :=
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(!sup_path_W)..1
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definition pr2_path_W (p : w.1 ≈ w'.1) (q : p ▹ w.2 ≈ w'.2)
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: pr1_path_W p q ▹ (path_W p q)..2 ≈ q :=
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(!sup_path_W)..2
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definition eta_path_W (p : w ≈ w') : path_W (p..1) (p..2) ≈ p :=
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begin
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apply (path.rec_on p),
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apply (cases_on w), intros (w1, w2),
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apply idp
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end
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definition transport_pr1_path_W {B' : A → Type} (p : w.1 ≈ w'.1) (q : p ▹ w.2 ≈ w'.2)
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: transport (λx, B' x.1) (path_W p q) ≈ transport B' p :=
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begin
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reverts (p, q),
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apply (cases_on w), intros (w1, w2),
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apply (cases_on w'), intros (w1', w2', p), generalize w2',
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apply (path.rec_on p), intros (w2', q),
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apply (path.rec_on q), apply idp
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end
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definition path_W_uncurried (pq : Σ(p : w.1 ≈ w'.1), p ▹ w.2 ≈ w'.2) : w ≈ w' :=
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destruct pq path_W
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definition sup_path_W_uncurried (pq : Σ(p : w.1 ≈ w'.1), p ▹ w.2 ≈ w'.2)
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: dpair (path_W_uncurried pq)..1 (path_W_uncurried pq)..2 ≈ pq :=
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destruct pq sup_path_W
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definition pr1_path_W_uncurried (pq : Σ(p : w.1 ≈ w'.1), p ▹ w.2 ≈ w'.2)
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: (path_W_uncurried pq)..1 ≈ pq.1 :=
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(!sup_path_W_uncurried)..1
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definition pr2_path_W_uncurried (pq : Σ(p : w.1 ≈ w'.1), p ▹ w.2 ≈ w'.2)
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: (pr1_path_W_uncurried pq) ▹ (path_W_uncurried pq)..2 ≈ pq.2 :=
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(!sup_path_W_uncurried)..2
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definition eta_path_W_uncurried (p : w ≈ w') : path_W_uncurried (dpair p..1 p..2) ≈ p :=
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!eta_path_W
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definition transport_pr1_path_W_uncurried {B' : A → Type} (pq : Σ(p : w.1 ≈ w'.1), p ▹ w.2 ≈ w'.2)
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: transport (λx, B' x.1) (@path_W_uncurried A B w w' pq) ≈ transport B' pq.1 :=
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destruct pq transport_pr1_path_W
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definition isequiv_path_W /-[instance]-/ (w w' : W a, B a)
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: is_equiv (@path_W_uncurried A B w w') :=
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adjointify path_W_uncurried
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(λp, dpair (p..1) (p..2))
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eta_path_W_uncurried
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sup_path_W_uncurried
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definition equiv_path_W (w w' : W a, B a) : (Σ(p : w.1 ≈ w'.1), p ▹ w.2 ≈ w'.2) ≃ (w ≈ w') :=
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equiv.mk path_W_uncurried !isequiv_path_W
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definition double_induction_on {P : (W a, B a) → (W a, B a) → Type} (w w' : W a, B a)
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(H : ∀ (a a' : A) (f : B a → W a, B a) (f' : B a' → W a, B a),
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(∀ (b : B a) (b' : B a'), P (f b) (f' b')) → P (sup a f) (sup a' f')) : P w w' :=
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begin
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revert w',
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apply (rec_on w), intros (a, f, IH, w'),
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apply (cases_on w'), intros (a', f'),
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apply H, intros (b, b'),
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apply IH
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end
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/- truncatedness -/
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open truncation
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definition trunc_W [FUN : funext.{v (max 1 u v)}] (n : trunc_index) [HA : is_trunc (n.+1) A]
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: is_trunc (n.+1) (W a, B a) :=
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begin
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fapply is_trunc_succ, intros (w, w'),
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apply (double_induction_on w w'), intros (a, a', f, f', IH),
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fapply trunc_equiv',
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apply equiv_path_W,
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apply trunc_sigma,
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fapply (succ_is_trunc n),
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intro p, revert IH, generalize f', --change to revert after simpl
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apply (path.rec_on p), intros (f', IH),
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apply pi.trunc_path_pi, intro b,
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apply IH
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end
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end Wtype
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