feat(hott/types): start characterization of pi-types and W-types
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Floris van Doorn
Leonardo de Moura
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153
library/hott/types/W.lean
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library/hott/types/W.lean
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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 W {A : Type} (B : A → Type) :=
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sup : Π(a : A), (B a → W B) → W B
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namespace W
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notation `WW` binders `,` r:(scoped B, W B) := r
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universe variables l k
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variables {A A' : Type.{l}} {B B' : A → Type.{k}} {C : Π(a : A), B a → Type}
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{a a' : A} {f : B a → W B} {f' : B a' → W B} {w w' : WW(a : A), B a}
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definition pr1 (w : WW(a : A), B a) : A :=
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W.rec_on w (λa f IH, a)
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definition pr2 (w : WW(a : A), B a) : B (pr1 w) → WW(a : A), B a :=
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W.rec_on w (λa f IH, f)
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namespace ops
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postfix `.1`:(max+1) := W.pr1
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postfix `.2`:(max+1) := W.pr2
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notation `⟨` a `,` f `⟩`:0 := W.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 : WW 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 B)
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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 B) : (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 B → W B → Type} (w w' : W B)
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(H : ∀ (a a' : A) (f : B a → W B) (f' : B a' → W B),
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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.{k (max 1 l k)}] (n : trunc_index) [HA : is_trunc (n.+1) A]
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: is_trunc (n.+1) (WW 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,
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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 FUN (B a) (λx, W B) n f f'), intro b,
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apply IH
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end
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end W
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59
library/hott/types/pi.lean
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59
library/hott/types/pi.lean
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@ -0,0 +1,59 @@
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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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Ported from Coq HoTT
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Theorems about pi-types (dependent function spaces)
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-/
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import ..trunc ..axioms.funext
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open path equiv is_equiv funext
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namespace pi
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universe variables l k
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variables [H : funext.{l k}] {A : Type.{l}} {B : A → Type.{k}} {C : Πa, B a → Type}
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{D : Πa b, C a b → Type}
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{a a' a'' : A} {b b₁ b₂ : B a} {b' : B a'} {b'' : B a''} {f g h : Πa, B a}
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include H
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/- Paths -/
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/- Paths [p : f ≈ g] in a function type [Πx:X, P x] are equivalent to functions taking values in path types, [H : Πx:X, f x ≈ g x], or concisely, [H : f == g].
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This equivalence, however, is just the combination of [apD10] and function extensionality [funext], and as such, [path_forall], et seq. are given in axioms.funext and path: -/
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/- Now we show how these things compute. -/
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definition equiv_apD10 : (f ≈ g) ≃ (f ∼ g) :=
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equiv.mk _ !funext.ap
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open truncation
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definition trunc_pi [instance] (B : A → Type.{k}) (n : trunc_index)
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[H : ∀a, is_trunc n (B a)] : is_trunc n (Πa, B a) :=
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begin
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reverts (B, H),
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apply (truncation.trunc_index.rec_on n),
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intros (B, H),
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fapply is_contr.mk,
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intro a, apply center, apply H, --remove "apply H" when term synthesis works correctly
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intro f, apply path_forall,
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intro x, apply (contr (f x)),
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intros (n, IH, B, H),
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fapply is_trunc_succ, intros (f, g),
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fapply trunc_equiv',
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apply equiv.symm, apply equiv_apD10,
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apply IH,
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intro a,
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show is_trunc n (f a ≈ g a), from
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succ_is_trunc (f a) (g a)
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end
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definition trunc_path_pi [instance] (n : trunc_index) (f g : Πa, B a)
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[H : ∀a, is_trunc n (f a ≈ g a)] : is_trunc n (f ≈ g) :=
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begin
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apply trunc_equiv', apply equiv.symm,
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apply equiv_apD10,
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apply trunc_pi, exact H,
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end
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end pi
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