Timings for DPathCube.v
Require Import Cubical.DPath.
Require Import Cubical.PathSquare.
Require Import Cubical.DPathSquare.
Require Import Cubical.PathCube.
Declare Scope dcube_scope.
Delimit Scope dcube_scope with dcube.
(* In this file we define a dependent cube *)
(* Dependent cubes *)
Definition DPathCube {A} (B : A -> Type)
{x000 x010 x100 x110 x001 x011 x101 x111 : A}
{p0i0 : x000 = x010} {p1i0 : x100 = x110}
{pi00 : x000 = x100} {pi10 : x010 = x110}
{p0i1 : x001 = x011} {p1i1 : x101 = x111}
{pi01 : x001 = x101} {pi11 : x011 = x111}
{p00i : x000 = x001} {p01i : x010 = x011}
{p10i : x100 = x101} {p11i : x110 = x111}
{s0ii : PathSquare p0i0 p0i1 p00i p01i}
{s1ii : PathSquare p1i0 p1i1 p10i p11i}
{sii0 : PathSquare p0i0 p1i0 pi00 pi10}
{sii1 : PathSquare p0i1 p1i1 pi01 pi11}
{si0i : PathSquare p00i p10i pi00 pi01}
{si1i : PathSquare p01i p11i pi10 pi11}
(cube : PathCube s0ii s1ii sii0 sii1 si0i si1i)
{b000 : B x000} {b010 : B x010} {b100 : B x100} {b110 : B x110}
{b001 : B x001} {b011 : B x011} {b101 : B x101} {b111 : B x111 }
{bp0i0 : DPath B p0i0 b000 b010} {bp1i0 : DPath B p1i0 b100 b110}
{bpi00 : DPath B pi00 b000 b100} {bpi10 : DPath B pi10 b010 b110}
{bp0i1 : DPath B p0i1 b001 b011} {bp1i1 : DPath B p1i1 b101 b111}
{bpi01 : DPath B pi01 b001 b101} {bpi11 : DPath B pi11 b011 b111}
{bp00i : DPath B p00i b000 b001} {bp01i : DPath B p01i b010 b011}
{bp10i : DPath B p10i b100 b101} {bp11i : DPath B p11i b110 b111}
(bs0ii : DPathSquare B s0ii bp0i0 bp0i1 bp00i bp01i)
(bs1ii : DPathSquare B s1ii bp1i0 bp1i1 bp10i bp11i)
(bsii0 : DPathSquare B sii0 bp0i0 bp1i0 bpi00 bpi10)
(bsii1 : DPathSquare B sii1 bp0i1 bp1i1 bpi01 bpi11)
(bsi0i : DPathSquare B si0i bp00i bp10i bpi00 bpi01)
(bsi1i : DPathSquare B si1i bp01i bp11i bpi10 bpi11) : Type.
exact (PathCube bs0ii bs1ii bsii0 bsii1 bsi0i bsi1i).
Definition equiv_dc_const' {A B : Type}
{x000 x010 x100 x110 x001 x011 x101 x111 : A}
{p0i0 : x000 = x010} {p1i0 : x100 = x110}
{pi00 : x000 = x100} {pi10 : x010 = x110}
{p0i1 : x001 = x011} {p1i1 : x101 = x111}
{pi01 : x001 = x101} {pi11 : x011 = x111}
{p00i : x000 = x001} {p01i : x010 = x011}
{p10i : x100 = x101} {p11i : x110 = x111}
{s0ii : PathSquare p0i0 p0i1 p00i p01i}
{s1ii : PathSquare p1i0 p1i1 p10i p11i}
{sii0 : PathSquare p0i0 p1i0 pi00 pi10}
{sii1 : PathSquare p0i1 p1i1 pi01 pi11}
{si0i : PathSquare p00i p10i pi00 pi01}
{si1i : PathSquare p01i p11i pi10 pi11}
{cube : PathCube s0ii s1ii sii0 sii1 si0i si1i}
{b000 b010 b100 b110 b001 b011 b101 b111 : B}
{bp0i0 : DPath (fun _ => B) p0i0 b000 b010}
{bp1i0 : DPath (fun _ => B) p1i0 b100 b110}
{bpi00 : DPath (fun _ => B) pi00 b000 b100}
{bpi10 : DPath (fun _ => B) pi10 b010 b110}
{bp0i1 : DPath (fun _ => B) p0i1 b001 b011}
{bp1i1 : DPath (fun _ => B) p1i1 b101 b111}
{bpi01 : DPath (fun _ => B) pi01 b001 b101}
{bpi11 : DPath (fun _ => B) pi11 b011 b111}
{bp00i : DPath (fun _ => B) p00i b000 b001}
{bp01i : DPath (fun _ => B) p01i b010 b011}
{bp10i : DPath (fun _ => B) p10i b100 b101}
{bp11i : DPath (fun _ => B) p11i b110 b111}
{bs0ii : DPathSquare (fun _ => B) s0ii bp0i0 bp0i1 bp00i bp01i}
{bs1ii : DPathSquare (fun _ => B) s1ii bp1i0 bp1i1 bp10i bp11i}
{bsii0 : DPathSquare (fun _ => B) sii0 bp0i0 bp1i0 bpi00 bpi10}
{bsii1 : DPathSquare (fun _ => B) sii1 bp0i1 bp1i1 bpi01 bpi11}
{bsi0i : DPathSquare (fun _ => B) si0i bp00i bp10i bpi00 bpi01}
{bsi1i : DPathSquare (fun _ => B) si1i bp01i bp11i bpi10 bpi11}
: PathCube
(ds_const'^-1 bs0ii) (ds_const'^-1 bs1ii) (ds_const'^-1 bsii0)
(ds_const'^-1 bsii1) (ds_const'^-1 bsi0i) (ds_const'^-1 bsi1i)
<~> DPathCube (fun _ => B) cube bs0ii bs1ii bsii0 bsii1 bsi0i bsi1i.
Notation dc_const' := equiv_dc_const'.
Definition equiv_dc_const {A B : Type}
{x000 x010 x100 x110 x001 x011 x101 x111 : A}
{p0i0 : x000 = x010} {p1i0 : x100 = x110}
{pi00 : x000 = x100} {pi10 : x010 = x110}
{p0i1 : x001 = x011} {p1i1 : x101 = x111}
{pi01 : x001 = x101} {pi11 : x011 = x111}
{p00i : x000 = x001} {p01i : x010 = x011}
{p10i : x100 = x101} {p11i : x110 = x111}
{s0ii : PathSquare p0i0 p0i1 p00i p01i}
{s1ii : PathSquare p1i0 p1i1 p10i p11i}
{sii0 : PathSquare p0i0 p1i0 pi00 pi10}
{sii1 : PathSquare p0i1 p1i1 pi01 pi11}
{si0i : PathSquare p00i p10i pi00 pi01}
{si1i : PathSquare p01i p11i pi10 pi11}
{cube : PathCube s0ii s1ii sii0 sii1 si0i si1i}
{b000 b010 b100 b110 b001 b011 b101 b111 : B}
{bp0i0 : b000 = b010} {bp1i0 : b100 = b110}
{bpi00 : b000 = b100} {bpi10 : b010 = b110}
{bp0i1 : b001 = b011} {bp1i1 : b101 = b111}
{bpi01 : b001 = b101} {bpi11 : b011 = b111}
{bp00i : b000 = b001} {bp01i : b010 = b011}
{bp10i : b100 = b101} {bp11i : b110 = b111}
{bs0ii : PathSquare bp0i0 bp0i1 bp00i bp01i}
{bs1ii : PathSquare bp1i0 bp1i1 bp10i bp11i}
{bsii0 : PathSquare bp0i0 bp1i0 bpi00 bpi10}
{bsii1 : PathSquare bp0i1 bp1i1 bpi01 bpi11}
{bsi0i : PathSquare bp00i bp10i bpi00 bpi01}
{bsi1i : PathSquare bp01i bp11i bpi10 bpi11}
: PathCube bs0ii bs1ii bsii0 bsii1 bsi0i bsi1i
<~> DPathCube (fun _ => B) cube (ds_const bs0ii) (ds_const bs1ii) (ds_const bsii0)
(ds_const bsii1) (ds_const bsi0i) (ds_const bsi1i).
Notation dc_const := equiv_dc_const.
(** Dependent Kan fillers *)
Context {A} {x000 x010 x100 x110 x001 x011 x101 x111 : A}
{p0i0 : x000 = x010} {p1i0 : x100 = x110} {pi00 : x000 = x100}
{pi10 : x010 = x110} {p0i1 : x001 = x011} {p1i1 : x101 = x111}
{pi01 : x001 = x101} {pi11 : x011 = x111} {p00i : x000 = x001}
{p01i : x010 = x011} {p10i : x100 = x101} {p11i : x110 = x111}
{s1ii : PathSquare p1i0 p1i1 p10i p11i} {s0ii : PathSquare p0i0 p0i1 p00i p01i}
{sii0 : PathSquare p0i0 p1i0 pi00 pi10} {sii1 : PathSquare p0i1 p1i1 pi01 pi11}
{si0i : PathSquare p00i p10i pi00 pi01} {si1i : PathSquare p01i p11i pi10 pi11}
(c : PathCube s0ii s1ii sii0 sii1 si0i si1i)
{P : A -> Type} {y000 y010 y100 y110 y001 y011 y101 y111}
{q0i0 : DPath P p0i0 y000 y010} {q1i0 : DPath P p1i0 y100 y110} {qi00 : DPath P pi00 y000 y100}
{qi10 : DPath P pi10 y010 y110} {q0i1 : DPath P p0i1 y001 y011} {q1i1 : DPath P p1i1 y101 y111}
{qi01 : DPath P pi01 y001 y101} {qi11 : DPath P pi11 y011 y111} {q00i : DPath P p00i y000 y001}
{q01i : DPath P p01i y010 y011} {q10i : DPath P p10i y100 y101} {q11i : DPath P p11i y110 y111}.
Definition dc_fill_left
(t1ii : DPathSquare P s1ii q1i0 q1i1 q10i q11i)
(tii0 : DPathSquare P sii0 q0i0 q1i0 qi00 qi10) (tii1 : DPathSquare P sii1 q0i1 q1i1 qi01 qi11)
(ti0i : DPathSquare P si0i q00i q10i qi00 qi01) (ti1i : DPathSquare P si1i q01i q11i qi10 qi11)
: {t0ii : DPathSquare P s0ii q0i0 q0i1 q00i q01i & DPathCube P c t0ii t1ii tii0 tii1 ti0i ti1i}.
Definition dc_fill_right
(t0ii : DPathSquare P s0ii q0i0 q0i1 q00i q01i)
(tii0 : DPathSquare P sii0 q0i0 q1i0 qi00 qi10) (tii1 : DPathSquare P sii1 q0i1 q1i1 qi01 qi11)
(ti0i : DPathSquare P si0i q00i q10i qi00 qi01) (ti1i : DPathSquare P si1i q01i q11i qi10 qi11)
: {t1ii : DPathSquare P s1ii q1i0 q1i1 q10i q11i & DPathCube P c t0ii t1ii tii0 tii1 ti0i ti1i}.
Definition dc_fill_top
(t0ii : DPathSquare P s0ii q0i0 q0i1 q00i q01i) (t1ii : DPathSquare P s1ii q1i0 q1i1 q10i q11i)
(tii1 : DPathSquare P sii1 q0i1 q1i1 qi01 qi11)
(ti0i : DPathSquare P si0i q00i q10i qi00 qi01) (ti1i : DPathSquare P si1i q01i q11i qi10 qi11)
: {tii0 : DPathSquare P sii0 q0i0 q1i0 qi00 qi10 & DPathCube P c t0ii t1ii tii0 tii1 ti0i ti1i}.
Definition dc_fill_bottom
(t0ii : DPathSquare P s0ii q0i0 q0i1 q00i q01i) (t1ii : DPathSquare P s1ii q1i0 q1i1 q10i q11i)
(tii0 : DPathSquare P sii0 q0i0 q1i0 qi00 qi10)
(ti0i : DPathSquare P si0i q00i q10i qi00 qi01) (ti1i : DPathSquare P si1i q01i q11i qi10 qi11)
: {tii1 : DPathSquare P sii1 q0i1 q1i1 qi01 qi11 & DPathCube P c t0ii t1ii tii0 tii1 ti0i ti1i}.
Definition dc_fill_front
(t0ii : DPathSquare P s0ii q0i0 q0i1 q00i q01i) (t1ii : DPathSquare P s1ii q1i0 q1i1 q10i q11i)
(tii0 : DPathSquare P sii0 q0i0 q1i0 qi00 qi10) (tii1 : DPathSquare P sii1 q0i1 q1i1 qi01 qi11)
(ti1i : DPathSquare P si1i q01i q11i qi10 qi11)
: {ti0i : DPathSquare P si0i q00i q10i qi00 qi01 & DPathCube P c t0ii t1ii tii0 tii1 ti0i ti1i}.
Definition dc_fill_back
(t0ii : DPathSquare P s0ii q0i0 q0i1 q00i q01i) (t1ii : DPathSquare P s1ii q1i0 q1i1 q10i q11i)
(tii0 : DPathSquare P sii0 q0i0 q1i0 qi00 qi10) (tii1 : DPathSquare P sii1 q0i1 q1i1 qi01 qi11)
(ti0i : DPathSquare P si0i q00i q10i qi00 qi01)
: {ti1i : DPathSquare P si1i q01i q11i qi10 qi11 & DPathCube P c t0ii t1ii tii0 tii1 ti0i ti1i}.