95 lines
3.2 KiB
Text
95 lines
3.2 KiB
Text
/-
|
||
Copyright (c) 2015 Floris van Doorn. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Authors: Floris van Doorn
|
||
|
||
Declaration of the interval
|
||
-/
|
||
|
||
import .suspension types.eq types.prod types.square
|
||
open eq suspension unit equiv equiv.ops is_trunc nat prod
|
||
|
||
definition interval : Type₀ := suspension unit
|
||
|
||
namespace interval
|
||
|
||
definition zero : interval := !north
|
||
definition one : interval := !south
|
||
definition seg : zero = one := merid star
|
||
|
||
protected definition rec {P : interval → Type} (P0 : P zero) (P1 : P one)
|
||
(Ps : P0 =[seg] P1) (x : interval) : P x :=
|
||
begin
|
||
fapply suspension.rec_on x,
|
||
{ exact P0},
|
||
{ exact P1},
|
||
{ intro x, cases x, exact Ps}
|
||
end
|
||
|
||
protected definition rec_on [reducible] {P : interval → Type} (x : interval)
|
||
(P0 : P zero) (P1 : P one) (Ps : P0 =[seg] P1) : P x :=
|
||
interval.rec P0 P1 Ps x
|
||
|
||
theorem rec_seg {P : interval → Type} (P0 : P zero) (P1 : P one) (Ps : P0 =[seg] P1)
|
||
: apdo (interval.rec P0 P1 Ps) seg = Ps :=
|
||
!rec_merid
|
||
|
||
protected definition elim {P : Type} (P0 P1 : P) (Ps : P0 = P1) (x : interval) : P :=
|
||
interval.rec P0 P1 (pathover_of_eq Ps) x
|
||
|
||
protected definition elim_on [reducible] {P : Type} (x : interval) (P0 P1 : P)
|
||
(Ps : P0 = P1) : P :=
|
||
interval.elim P0 P1 Ps x
|
||
|
||
theorem elim_seg {P : Type} (P0 P1 : P) (Ps : P0 = P1)
|
||
: ap (interval.elim P0 P1 Ps) seg = Ps :=
|
||
begin
|
||
apply eq_of_fn_eq_fn_inv !(pathover_constant seg),
|
||
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑interval.elim,rec_seg],
|
||
end
|
||
|
||
protected definition elim_type (P0 P1 : Type) (Ps : P0 ≃ P1) (x : interval) : Type :=
|
||
interval.elim P0 P1 (ua Ps) x
|
||
|
||
protected definition elim_type_on [reducible] (x : interval) (P0 P1 : Type) (Ps : P0 ≃ P1)
|
||
: Type :=
|
||
interval.elim_type P0 P1 Ps x
|
||
|
||
theorem elim_type_seg (P0 P1 : Type) (Ps : P0 ≃ P1)
|
||
: transport (interval.elim_type P0 P1 Ps) seg = Ps :=
|
||
by rewrite [tr_eq_cast_ap_fn,↑interval.elim_type,elim_seg];apply cast_ua_fn
|
||
|
||
definition is_contr_interval [instance] [priority 900] : is_contr interval :=
|
||
is_contr.mk zero (λx, interval.rec_on x idp seg !pathover_eq_r_idp)
|
||
|
||
end interval open interval
|
||
|
||
definition cube : ℕ → Type₀
|
||
| cube 0 := unit
|
||
| cube (succ n) := cube n × interval
|
||
|
||
abbreviation square := cube (succ (succ nat.zero))
|
||
|
||
definition cube_one_equiv_interval : cube 1 ≃ interval :=
|
||
!prod_comm_equiv ⬝e !prod_unit_equiv
|
||
|
||
|
||
definition prod_square {A B : Type} {a a' : A} {b b' : B} (p : a = a') (q : b = b')
|
||
: square (pair_eq p idp) (pair_eq p idp) (pair_eq idp q) (pair_eq idp q) :=
|
||
by cases p; cases q; exact ids
|
||
|
||
namespace square
|
||
|
||
definition tl : square := (star, zero, zero)
|
||
definition tr : square := (star, one, zero)
|
||
definition bl : square := (star, zero, one )
|
||
definition br : square := (star, one, one )
|
||
-- s stands for "square" in the following definitions
|
||
definition st : tl = tr := pair_eq (pair_eq idp seg) idp
|
||
definition sb : bl = br := pair_eq (pair_eq idp seg) idp
|
||
definition sl : tl = bl := pair_eq idp seg
|
||
definition sr : tr = br := pair_eq idp seg
|
||
definition sfill : square st sb sl sr := !prod_square
|
||
definition fill : st ⬝ sr = sl ⬝ sb := !square_equiv_eq sfill
|
||
|
||
end square
|