99 lines
3.3 KiB
Text
99 lines
3.3 KiB
Text
/-
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Copyright (c) 2015 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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Authors: Floris van Doorn
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Declaration of the interval
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-/
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import .susp types.eq types.prod cubical.square
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open eq susp unit equiv is_trunc nat prod
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definition interval : Type₀ := susp unit
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namespace interval
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definition zero : interval := north
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definition one : interval := south
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definition seg : zero = one := merid star
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protected definition rec {P : interval → Type} (P0 : P zero) (P1 : P one)
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(Ps : P0 =[seg] P1) (x : interval) : P x :=
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begin
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fapply susp.rec_on x,
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{ exact P0},
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{ exact P1},
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{ intro x, cases x, exact Ps}
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end
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protected definition rec_on [reducible] {P : interval → Type} (x : interval)
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(P0 : P zero) (P1 : P one) (Ps : P0 =[seg] P1) : P x :=
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interval.rec P0 P1 Ps x
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theorem rec_seg {P : interval → Type} (P0 : P zero) (P1 : P one) (Ps : P0 =[seg] P1)
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: apd (interval.rec P0 P1 Ps) seg = Ps :=
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!rec_merid
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protected definition elim {P : Type} (P0 P1 : P) (Ps : P0 = P1) (x : interval) : P :=
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interval.rec P0 P1 (pathover_of_eq Ps) x
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protected definition elim_on [reducible] {P : Type} (x : interval) (P0 P1 : P)
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(Ps : P0 = P1) : P :=
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interval.elim P0 P1 Ps x
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theorem elim_seg {P : Type} (P0 P1 : P) (Ps : P0 = P1)
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: ap (interval.elim P0 P1 Ps) seg = Ps :=
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begin
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apply eq_of_fn_eq_fn_inv !(pathover_constant seg),
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rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑interval.elim,rec_seg],
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end
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protected definition elim_type (P0 P1 : Type) (Ps : P0 ≃ P1) (x : interval) : Type :=
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interval.elim P0 P1 (ua Ps) x
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protected definition elim_type_on [reducible] (x : interval) (P0 P1 : Type) (Ps : P0 ≃ P1)
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: Type :=
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interval.elim_type P0 P1 Ps x
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theorem elim_type_seg (P0 P1 : Type) (Ps : P0 ≃ P1)
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: transport (interval.elim_type P0 P1 Ps) seg = Ps :=
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by rewrite [tr_eq_cast_ap_fn,↑interval.elim_type,elim_seg];apply cast_ua_fn
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definition is_contr_interval [instance] [priority 900] : is_contr interval :=
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is_contr.mk zero (λx, interval.rec_on x idp seg !pathover_eq_r_idp)
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open is_equiv equiv
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definition naive_funext_of_interval : naive_funext :=
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λA P f g p, ap (λ(i : interval) (x : A), interval.elim_on i (f x) (g x) (p x)) seg
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end interval open interval
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definition cube : ℕ → Type₀
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| cube 0 := unit
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| cube (succ n) := cube n × interval
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abbreviation square := cube (succ (succ nat.zero))
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definition cube_one_equiv_interval : cube 1 ≃ interval :=
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!prod_comm_equiv ⬝e !prod_unit_equiv
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definition prod_square {A B : Type} {a a' : A} {b b' : B} (p : a = a') (q : b = b')
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: square (pair_eq p idp) (pair_eq p idp) (pair_eq idp q) (pair_eq idp q) :=
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by cases p; cases q; exact ids
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namespace square
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definition tl : square := (star, zero, zero)
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definition tr : square := (star, one, zero)
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definition bl : square := (star, zero, one )
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definition br : square := (star, one, one )
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-- s stands for "square" in the following definitions
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definition st : tl = tr := pair_eq (pair_eq idp seg) idp
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definition sb : bl = br := pair_eq (pair_eq idp seg) idp
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definition sl : tl = bl := pair_eq idp seg
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definition sr : tr = br := pair_eq idp seg
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definition sfill : square st sb sl sr := !prod_square
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definition fill : st ⬝ sr = sl ⬝ sb := !square_equiv_eq sfill
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end square
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