This is a formalization of Thierry Coquand's
cubical type theory.
Some of the main results proved so far are:
fill_from_comp_wf which proves that a Kan filling operator
for a cubical type can be constructed from a "composition" operator.
pi_comp_wf2
which shows that from composition operators for A and B we can construct
a composition operator for ΠA B
sigma_comp_wf2
which shows that from composition operators for A and B we can construct
a composition operator for Σ A B
path_comp_wf
which shows that from a composition operator for A we can construct
a composition operator for (Path_A a b)
glue-comp_wf2
which shows that from composition operators for A and T we can construct
a composition operator for Glue [psi ⊢→ (T;equiv-fun(f))] A.
case-type-comp_wf
shows that the "system type" constructed by compatible case has a composition
operator.
compU_wf
shows that the (infinite hierachy of) universes
hava a composition structure.
Using Glue and composition for Glue we can convert and
equivalence between types into a path in the universe.
equiv-path_wf
(Univalence then follows from this.)
⋅
Comment: cubical_type_theory_basics
In this theory (by Thierry Coquand) the basic objects are:
1) Finite sets of names: names(I) (we use natural numbers for the names).
(see names_wf)
2) Morphisms: I ⟶ J == names(J) ⟶ Point(dM(I))
where ⌜dM(I)⌝ is the free De Morgan algebra generated by
⌜<i>⌝ and ⌜<1-i>⌝ for ⌜i ∈ names(I)⌝
(see dM_wf dM_inc_wf dM_opp_wf names-hom_wf)
3) With the appropriate composition,
∀[f:I ⟶ J]. ∀[g:J ⟶ K]. (g ⋅ f ∈ I ⟶ K),
on the morphisms, these constitute
the category of names : CubeCat
(see nh-comp_wf cube-cat_wf)
4) A cubical set is a presheaf over this category:
CubicalSet == Functor(op-cat(names-cat);TypeCat)
(see cubical_set_wf)
5) A cubical set map A ⟶ B between cubical sets A, B is therefore a
presheaf map, a natural transformation
nat-trans(op-cat(names-cat);TypeCat;A;B)
(see cube_set_map_wf)
The basic cubical sets are the following:
a) The formal cube
formal-cube(I) == <λJ.J ⟶ I, λI,J,f,g. g ⋅ f>
that is the representable presheaf defined by I.
(see formal-cube_wf formal-cube-is-rep-pre-sheaf)
If Γ is a cubical set and ⌜ρ ∈ Γ(I)⌝ then ρ induces
a cubical set map <ρ> ∈ formal-cube(I) ⟶ Γ
(see context-map_wf)
b) The interval 𝕀 == <λI.Point(dM(I)), λJ,I,f. dM-lift(I;J;f)>
whose objects are the finitely generated free De Morgan algebras.
(see interval-presheaf_wf)
c) The "face presheaf"
𝔽 == <λI.Point(face_lattice(I)), λJ,I,f. <f>>
whose objects are the finitely generated "face lattices"
which are free distributive lattices generated by
⌜(i=0)⌝,⌜(i=1)⌝ (for ⌜i ∈ names(I)⌝) subject to the relation
⌜(i=0) ∧ (i=1) = 0⌝
(see face-presheaf_wf)
d) A member psi ∈ 𝔽(I) can be seen as a formula about the points
of the formal cube. This defines another cubical set
I,psi (defined as rep-sub-sheaf(names-cat;I;λJ,f. (psi f) = 1))
which is the "cubical subset" of the formal cube I satisfying the
formula psi.
There is a cubical set map iota ∈ I,psi ⟶ formal-cube(I)
that shows that I,psi is a subobject of formal-cube(I)
Also, a map ⌜f ∈ J ⟶ I⌝ induces a cubical set map
subset-trans(I;J;f;psi) ∈ J,(psi)<f> ⟶ I,psi
(see cubical-subset_wf subset-iota_wf subset-trans_wf)
e) A trivial cubical set is the discrete cubical set
discrete-cube(A) == <λJ.A, λJ,K,f,x. x>
that has type A at every dimension and only the identity morphisms.
A special case is () == <λJ.Unit, λJ,K,f,x. ⋅>
which represents the empty context in Martin Lof Type Theory.
(see discrete-cube_wf trivial-cube-set_wf)
⋅
Comment: category of names doc
=========================== cube category =================================
The base category for the cubical set pre-sheaf is
CubeCat == <fset(ℕ), λI,J. I ⟶ J, λI.1, λI,J,K,f,g. g ⋅ f>
The objects are finite sets of names (we can use the natural numbers)
and the morphisms from I to J are maps
I ⟶ J == {n:ℕ| n ∈ J} ⟶ Point(dM(I)).
dM(I) is the free de-Morgan algebra generated by the finite set I.
It is a structure with a type of "points" and meet, join, negation, 0 and 1.
⋅
Definition: names
names(I) == {n:ℕ| n ∈ I}
Lemma: names_wf
∀[I:fset(ℕ)]. (names(I) ∈ Type)
Lemma: names-subtype
∀[I,J:fset(ℕ)]. names(I) ⊆r names(J) supposing I ⊆ J
Definition: names-deq
NamesDeq == IntDeq
Lemma: names-deq_wf
∀[I:fset(ℕ)]. (NamesDeq ∈ EqDecider(names(I)))
Lemma: fset-names-to-list
∀[I:fset(ℕ)]. ∀s:fset(names(I)). ∃L:names(I) List. (L = s ∈ fset(names(I)))
Definition: names-list
names-list(s) == fst((TERMOF{fset-names-to-list:o, 1:l} s))
Lemma: names-list_wf
∀[I:fset(ℕ)]. ∀[s:fset(names(I))]. (names-list(s) ∈ {L:names(I) List| L = s ∈ fset(names(I))} )
Definition: dM
dM(I) == free-DeMorgan-algebra(names(I);NamesDeq)
Lemma: dM_wf
∀[I:fset(ℕ)]. (dM(I) ∈ DeMorganAlgebra)
Lemma: dM-point
∀[I:Top]
(Point(dM(I)) ~ {ac:fset(fset(names(I) + names(I)))|
↑fset-antichain(union-deq(names(I);names(I);NamesDeq;NamesDeq);ac)} )
Lemma: dM-point-subtype
∀[I,J:fset(ℕ)]. Point(dM(I)) ⊆r Point(dM(J)) supposing I ⊆ J
Definition: dM-deq
dM-deq(I) == free-dml-deq(names(I);NamesDeq)
Lemma: dM-deq_wf
∀[I:fset(ℕ)]. (dM-deq(I) ∈ EqDecider(Point(dM(I))))
Definition: dM_inc
<x> == <x>
Lemma: dM_inc_wf
∀[I:fset(ℕ)]. ∀[x:names(I)]. (<x> ∈ Point(dM(I)))
Definition: dM_opp
<1-x> == <1-x>
Lemma: dM_opp_wf
∀[I:fset(ℕ)]. ∀[x:names(I)]. (<1-x> ∈ Point(dM(I)))
Definition: dM0
0 == 0
Lemma: dM0_wf
∀[I:fset(ℕ)]. (0 ∈ Point(dM(I)))
Lemma: dM0-sq-empty
∀[I:Top]. (0 ~ {})
Lemma: dM0-sq-0
∀[I:Top]. (0 ~ 0)
Definition: isdM0
isdM0(x) == null(x)
Lemma: isdM0_wf
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. (isdM0(x) ∈ 𝔹)
Lemma: assert-isdM0
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. uiff(↑isdM0(x);x = 0 ∈ Point(dM(I)))
Definition: dM1
1 == 1
Lemma: dM1_wf
∀[I:fset(ℕ)]. (1 ∈ Point(dM(I)))
Lemma: dM1-sq-singleton-empty
∀[I:Top]. (1 ~ {{}})
Definition: isdM1
isdM1(x) == deq-fset(deq-fset(union-deq(names(I);names(I);NamesDeq;NamesDeq))) x {{}}
Lemma: isdM1_wf
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. (isdM1(x) ∈ 𝔹)
Lemma: dM1-meet
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. (1 ∧ x = x ∈ Point(dM(I)))
Lemma: dM1-meet-dM1
∀[I:Top]. (1 ∧ 1 ~ 1)
Lemma: assert-isdM1
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. uiff(↑isdM1(x);x = 1 ∈ Point(dM(I)))
Lemma: dM_inc_not_1
∀[I:fset(ℕ)]. ∀[x:names(I)]. (¬(<x> = 1 ∈ Point(dM(I))))
Lemma: neg-dM1
∀[J:fset(ℕ)]. (¬(1) = 0 ∈ Point(dM(J)))
Lemma: dma-neg-dM0
∀[I:Top]. (¬(0) ~ 1)
Lemma: dma-neg-dM1
∀[I:Top]. (¬(1) ~ 0)
Lemma: dM-0-not-1
∀[I:fset(ℕ)]. (¬(0 = 1 ∈ Point(dM(I))))
Lemma: dM1-sq-single-empty
∀[I:Top]. (1 ~ {{}})
Lemma: dM1-sq-1
∀[I:Top]. (1 ~ 1)
Definition: dMpair
dMpair(i;j) == {{inl i,inl j}}
Lemma: dMpair-eq-meet
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. (dMpair(i;j) = <i> ∧ <j> ∈ Point(dM(I))) supposing (j ∈ I and i ∈ I)
Lemma: dMpair_wf
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. (dMpair(i;j) ∈ Point(dM(I))) supposing (j ∈ I and i ∈ I)
Lemma: dM-hom-unique
∀[I:fset(ℕ)]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))]. ∀[g,h:Hom(dM(I);L)].
g = h ∈ Hom(dM(I);L) supposing ∀i:names(I). (((g <i>) = (h <i>) ∈ Point(L)) ∧ ((g <1-i>) = (h <1-i>) ∈ Point(L)))
Lemma: neg-dM_inc
∀[I:fset(ℕ)]. ∀[x:names(I)]. (¬(<x>) = <1-x> ∈ Point(dM(I)))
Lemma: dma-neg-dM_inc
∀[I:fset(ℕ)]. ∀[x:names(I)]. (¬(<x>) = <1-x> ∈ Point(dM(I)))
Lemma: neg-dM_opp
∀[I:fset(ℕ)]. ∀[x:names(I)]. (¬(<1-x>) = <x> ∈ Point(dM(I)))
Lemma: dma-neg-dM_opp
∀[I:fset(ℕ)]. ∀[x:names(I)]. (¬(<1-x>) = <x> ∈ Point(dM(I)))
Lemma: dM-neg-properties
∀[I:fset(ℕ)]
((∀[x,y:Point(dM(I))]. (¬(x ∧ y) = ¬(x) ∨ ¬(y) ∈ Point(dM(I))))
∧ (∀[x,y:Point(dM(I))]. (¬(x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(dM(I))))
∧ (∀[x:Point(dM(I))]. (¬(¬(x)) = x ∈ Point(dM(I)))))
Lemma: dma-neg-eq-1-implies-meet-eq-0
∀[K:fset(ℕ)]. ∀[a2,x1:Point(dM(K))]. ((¬(a2) = 1 ∈ Point(dM(K))) ⇒ (0 = a2 ∧ x1 ∈ Point(dM(K))))
Definition: names-hom
I ⟶ J == names(J) ⟶ Point(dM(I))
Lemma: names-hom_wf
∀[I,J:fset(ℕ)]. (I ⟶ J ∈ Type)
Lemma: names-hom-subtype
∀[I1,J1,I2,J2:fset(ℕ)]. (I1 ⟶ J1 ⊆r I2 ⟶ J2) supposing (J2 ⊆ J1 and I1 ⊆ I2)
Definition: dM-lift
dM-lift(I;J;f) == free-dma-lift(names(J);NamesDeq;dM(I);free-dml-deq(names(I);NamesDeq);f)
Lemma: dM-lift_wf
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. (dM-lift(I;J;f) ∈ {g:dma-hom(dM(J);dM(I))| ∀j:names(J). ((g <j>) = (f j) ∈ Point(dM(I)))} \000C)
Lemma: dM-lift_wf2
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. (dM-lift(I;J;f) ∈ Point(dM(J)) ⟶ Point(dM(I)))
Lemma: dM-lift-sq
∀[I',J',I,J,f:Top]. (dM-lift(I;J;f) ~ dM-lift(I';J';f))
Lemma: dM-lift-0-sq
∀[I,J,f:Top]. (dM-lift(I;J;f) 0 ~ 0)
Lemma: dM-lift-1-sq
∀[I,J,f:Top]. (dM-lift(I;J;f) 1 ~ 1)
Lemma: dM-lift-0
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ((dM-lift(I;J;f) 0) = 0 ∈ Point(dM(I)))
Lemma: dM-lift-1
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ((dM-lift(I;J;f) 1) = 1 ∈ Point(dM(I)))
Lemma: dM-lift-inc
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[x:names(J)]. ((dM-lift(I;J;f) <x>) = (f x) ∈ Point(dM(I)))
Lemma: dM-lift-opp
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[x:names(J)]. ((dM-lift(I;J;f) <1-x>) = ¬(f x) ∈ Point(dM(I)))
Lemma: dM-lift-meet
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[x,y:Point(dM(J))].
((dM-lift(I;J;f) x ∧ y) = dM-lift(I;J;f) x ∧ dM-lift(I;J;f) y ∈ Point(dM(I)))
Lemma: dM-lift-join
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[x,y:Point(dM(J))].
((dM-lift(I;J;f) x ∨ y) = dM-lift(I;J;f) x ∨ dM-lift(I;J;f) y ∈ Point(dM(I)))
Lemma: dM-lift-neg
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[x:Point(dM(J))]. ((dM-lift(I;J;f) ¬(x)) = ¬(dM-lift(I;J;f) x) ∈ Point(dM(I)))
Lemma: dM-lift-unique
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[g:dma-hom(dM(J);dM(I))].
dM-lift(I;J;f) = g ∈ dma-hom(dM(J);dM(I)) supposing ∀j:names(J). ((g <j>) = (f j) ∈ Point(dM(I)))
Lemma: dM-lift-unique-fun
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[g:dma-hom(dM(J);dM(I))].
dM-lift(I;J;f) = g ∈ (Point(dM(J)) ⟶ Point(dM(I))) supposing ∀j:names(J). ((g <j>) = (f j) ∈ Point(dM(I)))
Lemma: dM-lift-dMpair
∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[x,y:names(I)]. ((dM-lift(J;I;f) dMpair(x;y)) = f x ∧ f y ∈ Point(dM(J)))
Lemma: dM-basis
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. (x = \/(λs./\(λx.free-dl-inc(x)"(s))"(x)) ∈ Point(dM(I)))
Lemma: dM-hom-basis
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. ∀[l:BoundedLattice].
∀eq:EqDecider(Point(l)). ∀[h:Hom(dM(I);l)]. ((h x) = \/(λs./\(λx.(h free-dl-inc(x))"(s))"(x)) ∈ Point(l))
Lemma: dM-homs-equal
∀[I:fset(ℕ)]. ∀[l:BoundedLattice].
∀eq:EqDecider(Point(l))
∀[h1,h2:Hom(dM(I);l)].
h1 = h2 ∈ (Point(dM(I)) ⟶ Point(l))
supposing ∀i:names(I). (((h1 <i>) = (h2 <i>) ∈ Point(l)) ∧ ((h1 <1-i>) = (h2 <1-i>) ∈ Point(l)))
Lemma: dM-dM-homs-equal
∀[I,J:fset(ℕ)]. ∀[h1,h2:dma-hom(dM(I);dM(J))].
h1 = h2 ∈ (Point(dM(I)) ⟶ Point(dM(J))) supposing ∀i:names(I). ((h1 <i>) = (h2 <i>) ∈ Point(dM(J)))
Lemma: dM-hom-invariant
∀[I:fset(ℕ)]. ∀[J:{J:fset(ℕ)| I ⊆ J} ]. ∀[h:Hom(dM(J);dM(I))].
∀[x:Point(dM(I))]. ((h x) = x ∈ Point(dM(I)))
supposing ∀i:names(I). (((h <i>) = <i> ∈ Point(dM(I))) ∧ ((h <1-i>) = <1-i> ∈ Point(dM(I))))
Lemma: dM-dma-hom-invariant
∀[I:fset(ℕ)]. ∀[J:{J:fset(ℕ)| I ⊆ J} ]. ∀[h:dma-hom(dM(J);dM(I))].
∀[x:Point(dM(I))]. ((h x) = x ∈ Point(dM(I))) supposing ∀i:names(I). ((h <i>) = <i> ∈ Point(dM(I)))
Lemma: dM-lift-is-id
∀[I:fset(ℕ)]. ∀[J:{J:fset(ℕ)| I ⊆ J} ]. ∀[h:I ⟶ J].
∀[x:Point(dM(I))]. ((dM-lift(I;J;h) x) = x ∈ Point(dM(I))) supposing ∀i:names(I). ((h i) = <i> ∈ Point(dM(I)))
Lemma: dM-subobject
∀[I,J:fset(ℕ)]. λv.v ∈ dma-hom(dM(I);dM(J)) supposing I ⊆ J
Lemma: dM-lift-is-id2
∀[I:fset(ℕ)]. ∀[J,K:{K:fset(ℕ)| I ⊆ K} ]. ∀[h:K ⟶ J].
∀[x:Point(dM(I))]. ((dM-lift(K;J;h) x) = x ∈ Point(dM(K))) supposing ∀i:names(I). ((h i) = <i> ∈ Point(dM(K)))
Lemma: dM-meet-inc-opp
∀[i:ℕ]. (<i> ∧ <1-i> ~ {{inr i ,inl i}})
Lemma: dM-join-inc-opp
∀[i:ℕ]. (<i> ∨ <1-i> ~ {{inr i },{inl i}})
Lemma: one-dimensional-dM
The free DeMorgan algebra with one generator i is the free distributive
lattice with two generators, i and 1-i. Hence it has exactly six elements.
The proof is somewhat tedious, but we need this lemma so that we
can prove that all paths in a "discrete cubical type" ⌜discr(T)⌝ are constant.⋅
∀i:ℕ. ∀v:Point(dM({i})).
((v = 0 ∈ Point(dM({i})))
∨ (v = 1 ∈ Point(dM({i})))
∨ (v = <i> ∈ Point(dM({i})))
∨ (v = <1-i> ∈ Point(dM({i})))
∨ (v = <i> ∧ <1-i> ∈ Point(dM({i})))
∨ (v = <i> ∨ <1-i> ∈ Point(dM({i}))))
Definition: nh-id
1 == λx.<x>
Lemma: nh-id_wf
∀[I:fset(ℕ)]. (1 ∈ I ⟶ I)
Definition: nh-comp
g ⋅ f == dma-lift-compose(names(I);names(J);NamesDeq;NamesDeq;f;g)
Lemma: nh-comp_wf
∀[I,J,K:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[g:J ⟶ K]. (g ⋅ f ∈ I ⟶ K)
Lemma: nh-comp-sq
∀[I,J,f,g:Top]. (g ⋅ f ~ dM-lift(I;J;f) o g)
Lemma: nh-id-left
∀I,J:fset(ℕ). ∀f:I ⟶ J. (f ⋅ 1 = f ∈ I ⟶ J)
Lemma: nh-id-right
∀I,J:fset(ℕ). ∀f:I ⟶ J. (1 ⋅ f = f ∈ I ⟶ J)
Lemma: nh-comp-assoc
∀I,J,K,H:fset(ℕ). ∀f:I ⟶ J. ∀g:J ⟶ K. ∀h:K ⟶ H. (h ⋅ g ⋅ f = h ⋅ g ⋅ f ∈ I ⟶ H)
Lemma: nh-comp-cancel
∀I,J,K:fset(ℕ). ∀f:I ⟶ J. ∀g,h:J ⟶ K. ∀x:names(K).
(g ⋅ f x) = (h ⋅ f x) ∈ Point(dM(I)) supposing (g x) = (h x) ∈ Point(dM(J))
Lemma: nh-comp-is-id
∀[I,J:fset(ℕ)].
∀[f:I ⟶ J]. ∀[g:J ⟶ I].
g ⋅ f = 1 ∈ I ⟶ I supposing ∀x:names(I). (((g x) = <x> ∈ Point(dM(J))) ∧ ((f x) = <x> ∈ Point(dM(I))))
supposing I ⊆ J
Definition: cube-cat
CubeCat == <fset(ℕ), λI,J. I ⟶ J, λI.1, λI,J,K,f,g. g ⋅ f>
Lemma: cube-cat_wf
CubeCat ∈ SmallCategory
Lemma: cube-cat-final
Final({})
Comment: names-cat morphisms doc
=========================== names-cat morphisms ===============================
We define a number of special morphisms on the category of names
and prove many equations that they satisfy.
The morphisms we need are
s, r_i, (i0), (i1), m(i;j), e(i;j), g,i=j, and g,i=1-j.
A typical equation
(that is needed in the proof of the composition operation for the Pi type) is
r_i ⋅ f,i=j = f,i=j ⋅ r_j (see nc-e'-r)⋅
Definition: add-name
I+i == fset-add(NamesDeq;i;I)
Lemma: add-name_wf
∀[I:fset(ℕ)]. ∀[i:ℕ]. (I+i ∈ fset(ℕ))
Lemma: fset-member-add-name
∀I:fset(ℕ). ∀i,j:ℕ. uiff(j ∈ I+i;(j = i ∈ ℤ) ∨ j ∈ I)
Lemma: trivial-member-add-name1
∀I:fset(ℕ). ∀i:ℕ. i ∈ I+i
Lemma: trivial-member-add-name2
∀I:fset(ℕ). ∀i:ℕ. ∀[j:ℕ]. i ∈ I+i+j
Lemma: f-subset-add-name1
∀I:fset(ℕ). ∀[J:fset(ℕ)]. ∀i:ℕ. (J ⊆ I ⇒ J ⊆ I+i)
Lemma: f-subset-add-name
∀I:fset(ℕ). ∀i:ℕ. I ⊆ I+i
Lemma: add-name-com
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. (I+i+j = I+j+i ∈ fset(ℕ))
Lemma: not-added-name
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[x:names(I+i)]. x ∈ names(I) supposing x ≠ i
Definition: new-name
new-name(I) == fset-max(λx.x;I) + 1
Lemma: new-name_wf
∀[I:fset(ℕ)]. (new-name(I) ∈ {i:ℕ| ¬i ∈ I} )
Lemma: new-name-property
∀[I:fset(ℕ)]. (¬new-name(I) ∈ I)
Lemma: not-new-name
∀[I:fset(ℕ)]. ∀x:names(I). ((x = new-name(I) ∈ ℤ) ⇒ False)
Definition: nc-s
s == λx.<x>
Lemma: nc-s_wf
∀[I,J:fset(ℕ)]. s ∈ I ⟶ J supposing J ⊆ I
Lemma: dM-lift-s
∀[I,J:fset(ℕ)]. ∀[x:Point(dM(I))]. ((dM-lift(J;I;s) x) = x ∈ Point(dM(J))) supposing I ⊆ J
Definition: nc-r
r_i == λx.if (x =z i) then <1-i> else <x> fi
Lemma: nc-r_wf
∀[I:fset(ℕ)]. ∀[i:ℕ]. r_i ∈ I ⟶ I supposing i ∈ I
Definition: nc-p
(i/z) == λx.if (x =z i) then z else <x> fi
Lemma: nc-p_wf
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[z:Point(dM(I))]. ((i/z) ∈ I ⟶ I+i)
Lemma: s-comp-nc-p
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[z:Point(dM(I))]. s ⋅ (i/z) = 1 ∈ I ⟶ I supposing ¬i ∈ I
Definition: nc-0
(i0) == λx.if (x =z i) then {} else <x> fi
Lemma: nc-0_wf
∀[I:fset(ℕ)]. ∀[i:ℕ]. ((i0) ∈ I ⟶ I+i)
Lemma: nc-0-as-nc-p
∀[I:fset(ℕ)]. ∀[i:ℕ]. ((i0) ~ (i/0))
Definition: nc-1
(i1) == λx.if (x =z i) then {{}} else <x> fi
Lemma: nc-1_wf
∀[I:fset(ℕ)]. ∀[i:ℕ]. ((i1) ∈ I ⟶ I+i)
Lemma: nc-1-lemma
∀J:fset(ℕ). ∀j:ℕ. ∀i:{i:names(J)| ¬i ∈ J} . (((j1) i) = <i> ∈ Point(dM(J)))
Lemma: nc-1-lemma2
∀j:ℕ. ((j1) j ~ {{}})
Lemma: dM-lift-nc-1
∀J:fset(ℕ). ∀j:{i:ℕ| ¬i ∈ J} . ∀v:Point(dM(J)). ((dM-lift(J;J+j;(j1)) v) = v ∈ Point(dM(J)))
Lemma: dM-lift-nc-0
∀J:fset(ℕ). ∀j:{i:ℕ| ¬i ∈ J} . ∀v:Point(dM(J)). ((dM-lift(J;J+j;(j0)) v) = v ∈ Point(dM(J)))
Lemma: nc-1-as-nc-p
∀[I:fset(ℕ)]. ∀[i:ℕ]. ((i1) ~ (i/1))
Lemma: r-comp-nc-1
∀[I:fset(ℕ)]. ∀[i:ℕ]. r_i ⋅ (i1) = (i0) ∈ I ⟶ I+i supposing ¬i ∈ I
Lemma: r-comp-nc-0
∀[I:fset(ℕ)]. ∀[i:ℕ]. r_i ⋅ (i0) = (i1) ∈ I ⟶ I+i supposing ¬i ∈ I
Lemma: r-comp-r
∀[I:fset(ℕ)]. ∀[i:ℕ]. (r_i ⋅ r_i = 1 ∈ I+i ⟶ I+i)
Lemma: s-comp-nc-1
∀[I:fset(ℕ)]. ∀[i:ℕ]. s ⋅ (i1) = 1 ∈ I ⟶ I supposing ¬i ∈ I
Lemma: s-comp-nc-1-new
∀[I:fset(ℕ)]. (s ⋅ (new-name(I)1) = 1 ∈ I ⟶ I)
Lemma: s-comp-nc-0
∀[I:fset(ℕ)]. ∀[i:ℕ]. s ⋅ (i0) = 1 ∈ I ⟶ I supposing ¬i ∈ I
Lemma: s-comp-nc-0-new
∀[I:fset(ℕ)]. (s ⋅ (new-name(I)0) = 1 ∈ I ⟶ I)
Lemma: nc-0-comp-s
∀[I,K:fset(ℕ)]. ∀[i:ℕ]. ∀[f:K ⟶ I+i]. (i0) ⋅ s ⋅ f = f ∈ K ⟶ I+i supposing (f i) = 0 ∈ Point(dM(K))
Lemma: s-comp-nc-0'
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[j:{j:ℕ| ¬j ∈ I} ]. (s ⋅ (j0) = s ∈ I+i ⟶ I)
Lemma: s-comp-nc-1'
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[j:{j:ℕ| ¬j ∈ I} ]. (s ⋅ (j1) = s ∈ I+i ⟶ I)
Lemma: s-comp-nc-0-alt
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ I+i} ]. (s ⋅ (i0) = s ∈ I+j ⟶ I)
Lemma: s-comp-s
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. (s ⋅ s = s ∈ I+i+j ⟶ I)
Definition: nc-m
m(i;j) == λx.if (x =z i) then dMpair(i;j) else <x> fi
Lemma: nc-m_wf
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. m(i;j) ∈ I+j ⟶ I supposing i ∈ I
Lemma: nh-comp-nc-m
∀[I,K:fset(ℕ)]. ∀[i,j:ℕ]. ∀[f:K ⟶ I+j]. (m(i;j) ⋅ f i) = f i ∧ f j ∈ Point(dM(K)) supposing i ∈ I
Lemma: nh-comp-nc-m-eq
∀[I,K:fset(ℕ)]. ∀[i,j:ℕ]. ∀[f:K ⟶ I+i+j]. (i0) ⋅ s ⋅ s ⋅ f = m(i;j) ⋅ f ∈ K ⟶ I+i supposing (f i) = 0 ∈ Point(dM(K))
Lemma: nh-comp-nc-m-eq2
∀[I,K:fset(ℕ)]. ∀[i,j:ℕ]. ∀[f:K ⟶ I+i+j]. (i0) ⋅ s ⋅ f = m(i;j) ⋅ f ∈ K ⟶ I+i supposing (f i) = 0 ∈ Point(dM(K))
Lemma: nc-m-nc-1
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ I+i} ]. (m(i;j) ⋅ (j1) = 1 ∈ I+i ⟶ I+i)
Lemma: nc-m-nc-0
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ I+i} ]. (m(i;j) ⋅ (j0) = (i0) ⋅ s ∈ I+i ⟶ I+i)
Definition: nc-e
e(i;j) == λx.if (x =z i) then <j> else <x> fi
Lemma: nc-e_wf
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. (e(i;j) ∈ I+j ⟶ I+i)
Lemma: nc-e-same
∀I:fset(ℕ). ∀z:ℕ. (e(z;z) = 1 ∈ I+z ⟶ I+z)
Lemma: nc-e-comp
∀I:fset(ℕ). ∀i,j,k:ℕ. ((¬j ∈ I) ⇒ (e(k;j) ⋅ e(j;i) = e(k;i) ∈ I+i ⟶ I+k))
Definition: nc-e'
g,i=j == λx.if (x =z i) then <j> else g x fi
Lemma: nc-e'_wf
∀[I,J:fset(ℕ)]. ∀[i,j:ℕ]. ∀[g:J ⟶ I]. (g,i=j ∈ J+j ⟶ I+i)
Lemma: nc-e'-lemma1
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[J:fset(ℕ)]. ∀[g:J ⟶ I]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ((i1) ⋅ g = g,i=j ⋅ (j1) ∈ J ⟶ I+i)
Lemma: nc-e'-lemma2
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[J:fset(ℕ)]. ∀[g:J ⟶ I]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ((i0) ⋅ g = g,i=j ⋅ (j0) ∈ J ⟶ I+i)
Lemma: nc-e'-lemma3
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[J:fset(ℕ)]. ∀[g:J ⟶ I]. ∀[j:{j:ℕ| ¬j ∈ J} ]. (s ⋅ g,i=j = g ⋅ s ∈ J+j ⟶ I)
Lemma: nc-e'-1
∀[I,i,j:Top]. (1,i=j ~ e(i;j))
Lemma: nc-e-comp-e'
∀[I,K:fset(ℕ)]. ∀[f:K ⟶ I]. ∀[z,z1,v:ℕ]. f,z=v = e(z;z1) ⋅ f,z1=v ∈ K+v ⟶ I+z supposing ¬z1 ∈ I
Lemma: nc-e'-comp-e
∀[I,J:fset(ℕ)]. ∀[g:J ⟶ I]. ∀[j,k,v:ℕ]. g,j=v ⋅ e(v;k) = g,j=k ∈ J+k ⟶ I+j supposing ¬v ∈ J
Lemma: nc-e-comp-nc-p
∀[I:fset(ℕ)]. ∀[i,j:{j:ℕ| ¬j ∈ I} ]. ∀[z:Point(dM(I))]. (e(i;j) ⋅ (j/z) = (i/z) ∈ I ⟶ I+i)
Lemma: nc-e-comp-nc-0
∀[I:fset(ℕ)]. ∀[i,j:{j:ℕ| ¬j ∈ I} ]. ((i0) = e(i;j) ⋅ (j0) ∈ I ⟶ I+i)
Lemma: nc-e-comp-nc-1
∀[I:fset(ℕ)]. ∀[i,j:{j:ℕ| ¬j ∈ I} ]. ((i1) = e(i;j) ⋅ (j1) ∈ I ⟶ I+i)
Lemma: nh-comp-nc-m-eq3
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ I+i} ]. (s,i=j ⋅ (j0) = m(i;j) ⋅ (j0) ∈ I+i ⟶ I+i)
Lemma: nh-comp-nc-m-s
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:ℕ]. (s ⋅ m(i;j) = s ⋅ s ∈ I+i+j ⟶ I)
Lemma: nc-s-comp-e
∀I:fset(ℕ). ∀i,j:ℕ. ((¬i ∈ I) ⇒ (s ⋅ e(i;j) = s ∈ I+j ⟶ I))
Lemma: nc-s-comp-e'
∀I:fset(ℕ). ∀i,j:ℕ. ((¬i ∈ I) ⇒ (s ⋅ 1,i=j = s ∈ I+j ⟶ I))
Lemma: nc-e'-comp-m
∀[I,J:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[g:J ⟶ I]. ∀[k:{k:ℕ| ¬k ∈ I+i} ]. ∀[l:{l:ℕ| ¬l ∈ J+j} ].
(g,i=j ⋅ m(j;l) = m(i;k) ⋅ g,i=j,k=l ∈ J+j+l ⟶ I+i)
Lemma: nc-e'-comp-m-2
∀[I,J:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[g:J ⟶ I]. ∀[l:{i:ℕ| ¬i ∈ J+j} ].
(s ⋅ g,i=j ⋅ m(j;l) = g ⋅ s ⋅ s ∈ J+j+l ⟶ I)
Lemma: nc-e'-lemma4
∀[I,J:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[g:J ⟶ I]. ∀[k:{i1:ℕ| ¬i1 ∈ I+i} ]. ∀[l:{i:ℕ| ¬i ∈ J+j} ].
((i0) ⋅ s ⋅ g,i=j,k=l = g,i=j ⋅ (j0) ⋅ s ∈ J+j+l ⟶ I+i)
Lemma: nc-e'-lemma5
∀[I,J:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[g:J ⟶ I]. ∀[k:{i1:ℕ| ¬i1 ∈ I+i} ]. ∀[l:{i:ℕ| ¬i ∈ J+j} ].
(s ⋅ g,i=j,k=l = g ⋅ s ∈ J+j+l ⟶ I)
Lemma: nc-e'-lemma6
∀[I,J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. ∀[z,v,j:ℕ]. f,z=j ⋅ g,j=v = f ⋅ g,z=v ∈ K+v ⟶ I+z supposing ¬j ∈ J
Lemma: nc-e'-lemma7
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀j:ℕ. (1,i=j = 1 ⋅ s,i=j ∈ I+j ⟶ I+i)
Definition: nc-r'
g,i=1-j == λx.if (x =z i) then <1-j> else g x fi
Lemma: nc-r'_wf
∀[I,J:fset(ℕ)]. ∀[i,j:ℕ]. ∀[g:J ⟶ I]. (g,i=1-j ∈ J+j ⟶ I+i)
Lemma: nc-r'-nc-0
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{i:ℕ| ¬i ∈ J} ]. ((i1) ⋅ f = f,i=1-j ⋅ (j0) ∈ J ⟶ I+\000Ci)
Lemma: nc-r'-nc-1
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{i:ℕ| ¬i ∈ J} ]. ((i0) ⋅ f = f,i=1-j ⋅ (j1) ∈ J ⟶ I+\000Ci)
Lemma: nc-r'-r
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{i:ℕ| ¬i ∈ J} ]. (f,i=1-j ⋅ r_j = f,i=j ∈ J+j ⟶ I+i)
Lemma: nc-r'-to-e'
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{i:ℕ| ¬i ∈ J} ]. (f,i=1-j = f,i=j ⋅ r_j ∈ J+j ⟶ I+i)
Lemma: nc-e'-r
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{i:ℕ| ¬i ∈ J} ].
(r_i ⋅ f,i=j = f,i=j ⋅ r_j ∈ J+j ⟶ I+i)
Lemma: nc-r-e'-r
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{i:ℕ| ¬i ∈ J} ].
(r_i ⋅ f,i=j ⋅ r_j = f,i=j ∈ J+j ⟶ I+i)
Lemma: nc-r'-to-e'2
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{i:ℕ| ¬i ∈ J} ]. (f,i=1-j = r_i ⋅ f,i=j ∈ J+j ⟶ I+i)
Lemma: nc-p-s-commute
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. ∀[z:Point(dM(I))]. ((i/z) ⋅ s = s ⋅ (i/z) ∈ I+j ⟶ I+i)
Lemma: nc-1-s-commute
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. ((i1) ⋅ s = s ⋅ (i1) ∈ I+j ⟶ I+i)
Lemma: nc-0-s-commute
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. ((i0) ⋅ s = s ⋅ (i0) ∈ I+j ⟶ I+i)
Lemma: nc-s-i1-j0
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[j:{j:ℕ| ¬j ∈ I} ]. ((i1) = s ⋅ (i1) ⋅ (j0) ∈ I ⟶ I+i)
Lemma: nc-s-i1-j1
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[j:{j:ℕ| ¬j ∈ I} ]. ((i1) = s ⋅ (i1) ⋅ (j1) ∈ I ⟶ I+i)
Lemma: nc-e'-p
∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[z:Point(dM(I))]. ∀[i:ℕ]. ∀[j:{j:ℕ| ¬j ∈ J} ].
((i/z) ⋅ f = f,i=j ⋅ (j/dM-lift(J;I;f) z) ∈ J ⟶ I+i)
Lemma: nc-se'-p
∀[J:fset(ℕ)]. ∀[k,z:ℕ]. ∀[n:{n:ℕ| ¬n ∈ J} ]. (s,z=n ⋅ (n/<k>) = e(z;k) ∈ J+k ⟶ J+z)
Lemma: nc-0-s-0
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. ((i0) ⋅ s ⋅ (i0) = s ⋅ (i0) ∈ I+j ⟶ I+i)
Lemma: s-comp-1-e'
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{j:ℕ| ¬j ∈ I} ]. ∀[k:{j:ℕ| ¬j ∈ J} ].
(s ⋅ (i1) ⋅ f,j=k = (i1) ⋅ f ⋅ s ∈ J+k ⟶ I+i)
Lemma: s-comp-0-e'
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{j:ℕ| ¬j ∈ I} ]. ∀[k:{j:ℕ| ¬j ∈ J} ].
(s ⋅ (i0) ⋅ f,j=k = (i0) ⋅ f ⋅ s ∈ J+k ⟶ I+i)
Lemma: nc-e'-s-lemma1
∀[I,J:fset(ℕ)]. ∀[i,z:ℕ]. ∀[g:J ⟶ I]. ∀[j,k:ℕ]. g,i=z ⋅ s = s ⋅ g,j=k,i=z ∈ J+z+k ⟶ I+i supposing ¬j ∈ I
Lemma: s-comp-if-lemma1
∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀x:Point(dM(J)). (s ⋅ λj.if (j =z new-name(I)) then x else 1 ⋅ f j fi = f ∈ J ⟶ I)
Comment: basic cubical sets doc
=========================== basic cubical sets ===============================
This section defines CubicalSet, I-cubes A(I), and their restrictions
cubical set maps A ⟶ B, the interval pre-sheaf 𝕀, formal-cube(I)
and the maps <rho> and facts about them.
Most of the definitions here are just specializations of definitions
from Error :presheaf models of type theory, where take the category C
in the general definition to be CubeCat. ⋅
Definition: cubical_set
CubicalSet == __⊢
Lemma: cubical_set_wf
j⊢ ∈ 𝕌{[i | j'']}
Definition: small_cubical_set
SmallCubicalSet == small_ps_context{i:l}(CubeCat)
Lemma: small_cubical_set_wf
SmallCubicalSet ∈ 𝕌'
Lemma: small_cubical_set_subtype
small_cubical_set{j:l} ⊆r CubicalSet{j}
Lemma: cubical_set_cumulativity
CubicalSet{j} ⊆r cubical_set{j':l}
Lemma: cubical_set_cumulativity-i-j
CubicalSet ⊆r cubical_set{[j | i]:l}
Lemma: cubical_sets_equal
∀[X,Y:j⊢].
X = Y ∈ CubicalSet{j} supposing X = Y ∈ (F:fset(ℕ) ⟶ 𝕌{j'} × (x:fset(ℕ) ⟶ y:fset(ℕ) ⟶ y ⟶ x ⟶ (F x) ⟶ (F y)))
Definition: ext-eq-cs
X ≡ Y == X ≡ Y
Lemma: ext-eq-cs_wf
∀[X,Y:j⊢]. (X ≡ Y ∈ ℙ{[i | j']})
Definition: I_cube
A(I) == A I
Lemma: I_cube_wf
∀[A:j⊢]. ∀[I:fset(ℕ)]. (A(I) ∈ 𝕌{j'})
Lemma: I_cube_pair_redex_lemma
∀I,G,F:Top. (<F, G>(I) ~ F I)
Definition: cube-set-restriction
f(s) == (snd(X)) I J f s
Lemma: cube-set-restriction_wf
∀[X:j⊢]. ∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[s:X(I)]. (f(s) ∈ X(J))
Lemma: cube_set_restriction_pair_lemma
∀s,f,J,I,G,F:Top. (f(s) ~ G I J f s)
Lemma: cubical_set-ext
{FM:F:fset(ℕ) ⟶ 𝕌{j'} × (I:fset(ℕ) ⟶ J:fset(ℕ) ⟶ J ⟶ I ⟶ (F I) ⟶ (F J))|
let F,M = FM
in (∀I:fset(ℕ). ∀s:FM(I). (1(s) = s ∈ FM(I)))
∧ (∀I,J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀s:FM(I). (f ⋅ g(s) = g(f(s)) ∈ FM(K)))} ≡ CubicalSet{j}
Definition: cs-predicate
cs-predicate(X;I,rho.P[I; rho]) == psc-predicate(CubeCat; X; I,rho.P[I; rho])
Lemma: cs-predicate_wf
∀[X:j⊢]. ∀[P:I:fset(ℕ) ⟶ X(I) ⟶ ℙ{[i | j']}]. (cs-predicate(X;I,rho.P[I;rho]) ∈ ℙ{[i | j']})
Definition: sub-cubical-set
X | I,rho.P[I; rho] == X|I,rho.P[I; rho]
Lemma: sub-cubical-set_wf
∀[X:j⊢]. ∀[P:I:fset(ℕ) ⟶ X(I) ⟶ ℙ{j'}]. X | I,rho.P[I;rho] j⊢ supposing cs-predicate(X;I,rho.P[I;rho])
Lemma: sub-cubical-set_functionality
∀[X:j⊢]. ∀[P,Q:I:fset(ℕ) ⟶ X(I) ⟶ ℙ].
X | I,rho.P[I;rho] ≡ X | I,rho.Q[I;rho]
supposing (∀I:fset(ℕ). ∀rho:X(I). (P[I;rho] ⇐⇒ Q[I;rho])) ∧ cs-predicate(X;I,rho.P[I;rho])
Lemma: sub-cubical-set-true
∀[X:j⊢]. X | I,rho.True ≡ X
Lemma: sub-cubical-set-and
∀[X:j⊢]. ∀[P,Q:I:fset(ℕ) ⟶ X(I) ⟶ ℙ].
X | I,rho.P[I;rho] | I,rho.Q[I;rho] ≡ X | I,rho.P[I;rho] ∧ Q[I;rho]
supposing cs-predicate(X;I,rho.P[I;rho]) ∧ cs-predicate(X;I,rho.Q[I;rho])
Definition: cubes
cubes(X) == sets(CubeCat; X)
Lemma: cubes_wf
∀[X:j⊢]. (cubes(X) ∈ small-category{[i | j']:l})
Lemma: cubes-ob
∀[X:Top]. (cat-ob(cubes(X)) ~ I:fset(ℕ) × X(I))
Lemma: cubes-id
∀[X:Top]. (cat-id(cubes(X)) ~ λ_.1)
Lemma: cubes-arrow
∀[X:Top]. (cat-arrow(cubes(X)) ~ λAa,Bb. let A,a = Aa in let B,b = Bb in {f:A ⟶ B| f(b) = a ∈ (X A)} )
Lemma: cubes-comp
∀[X:Top]. (cat-comp(cubes(X)) ~ λAa,Bb,_,f,g. g ⋅ f)
Definition: interval-presheaf
This is the cubical set that assigns to a finite set of names I the
points of the free DeMorgan algebra generated by I.
Since, a morphism f : I->J maps J -> dM(I), it lifts to a
homomorphism dM(J)->dM(I).⋅
𝕀 == <λI.Point(dM(I)), λJ,I,f. dM-lift(I;J;f)>
Lemma: interval-presheaf_wf
𝕀 ∈ SmallCubicalSet
Lemma: interval-presheaf-restriction
∀[I,K,f,r:Top]. (f(r) ~ dM-lift(K;I;f) r)
Lemma: nc-e'-p2
∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[r:Point(dM(I))]. ∀[i:ℕ]. ∀[j:{j:ℕ| ¬j ∈ J} ]. (f,i=j ⋅ (j/f(r)) = (i/r) ⋅ f ∈ J ⟶ I+i)
Definition: formal-cube
formal-cube(I) == <λJ.J ⟶ I, λI,J,f,g. g ⋅ f>
Lemma: formal-cube-is-rep-pre-sheaf
∀[I:Top]. (formal-cube(I) ~ rep-pre-sheaf(CubeCat;I))
Lemma: formal-cube_wf
∀[I:fset(ℕ)]. formal-cube(I) ⊢
Lemma: formal-cube_wf1
∀[I:fset(ℕ)]. formal-cube(I) j⊢
Lemma: formal-cube-restriction
∀[I,f,s,A,B:Top]. (f(s) ~ s ⋅ f)
Lemma: formal-cube-singleton1
∀x:ℕ. (λA,f. (f x) ∈ nat-trans(op-cat(CubeCat);TypeCat';formal-cube({x});𝕀))
Lemma: formal-cube-singleton2
∀x:ℕ. (λA,u,x. u ∈ nat-trans(op-cat(CubeCat);TypeCat';𝕀;formal-cube({x})))
Lemma: formal-cube-singleton-iso-interval-presheaf
∀x:ℕ. cat-isomorphic(FUN(op-cat(CubeCat);TypeCat');formal-cube({x});𝕀)
Definition: representable-cube-set
representable-cube-set() == {G:cubical_set{1:l}| ∃I:fset(ℕ). (G = formal-cube(I) ∈ cubical_set{1:l})}
Lemma: representable-cube-set_wf
representable-cube-set() ∈ 𝕌{3}
Definition: trivial-cube-set
() == <λJ.Unit, λJ,K,f,x. ⋅>
Lemma: trivial-cube-set_wf
() j⊢
Definition: discrete-cube
discrete-cube(A) == <λJ.A, λJ,K,f,x. x>
Lemma: discrete-cube_wf
∀[A:𝕌{j}]. discrete-cube(A) j⊢
Definition: cubical-false
False == discrete-cube(Void)
Lemma: cubical-false_wf
False j⊢
Definition: cube_set_map
A ⟶ B == A ⟶ B
Lemma: cube_set_map_wf
∀[A,B:j⊢]. (A j⟶ B ∈ 𝕌{j''})
Lemma: cube_set_map_cumulativity
∀[G,H:j⊢]. (H j⟶ G ⊆r cube_set_map{j':l}(H; G))
Lemma: cube_set_map_cumulativity-i-j
∀[G,H:⊢]. (H ⟶ G ⊆r H ij⟶ G)
Definition: context-map
<rho> == λA,f. (Gamma I A f rho)
Lemma: context-map_wf
∀I:fset(ℕ). ∀Gamma:j⊢. ∀rho:Gamma(I). (<rho> ∈ formal-cube(I) j⟶ Gamma)
Lemma: csm-equal
∀[A,B:j⊢]. ∀[f:A j⟶ B]. ∀[g:I:fset(ℕ) ⟶ A(I) ⟶ B(I)]. f = g ∈ A j⟶ B supposing f = g ∈ (I:fset(ℕ) ⟶ A(I) ⟶ B(I))
Lemma: csm-equal2
∀[A,B:j⊢]. ∀[f,g:A j⟶ B]. f = g ∈ A j⟶ B supposing ∀K:fset(ℕ). ∀x:A(K). ((f K x) = (g K x) ∈ B(K))
Lemma: cube-set-map-subtype
∀[A,B:j⊢]. (A j⟶ B ⊆r (I:fset(ℕ) ⟶ A(I) ⟶ B(I)))
Definition: csm-ap
(s)x == s I x
Lemma: csm-ap_wf
∀[X,Y:j⊢]. ∀[s:X j⟶ Y]. ∀[I:fset(ℕ)]. ∀[x:X(I)]. ((s)x ∈ Y(I))
Lemma: csm-ap-context-map
∀[I,J,a,rho,G:Top]. ((<rho>)a ~ a(rho))
Definition: csm-comp
G o F == λx.((G x) o (F x))
Lemma: csm-comp-sq
∀[A,B,C,F,G:Top]. (G o F ~ F o G)
Lemma: csm-comp_wf
∀[A,B,C:j⊢]. ∀[F:A j⟶ B]. ∀[G:B j⟶ C]. (G o F ∈ A j⟶ C)
Lemma: csm-comp-assoc
∀[A,B,C,D:j⊢]. ∀[F:A j⟶ B]. ∀[G:B j⟶ C]. ∀[H:C j⟶ D]. (H o G o F = H o G o F ∈ A j⟶ D)
Definition: csm-id
1(X) == λA,x. x
Lemma: csm-id-sq
∀[X:j⊢]. (1(X) ~ identity-trans(op-cat(CubeCat);type-cat{j':l};X))
Lemma: csm-id_wf
∀[X:j⊢]. (1(X) ∈ X j⟶ X)
Lemma: csm-id-comp
∀[A,B:j⊢]. ∀[sigma:A j⟶ B]. ((sigma o 1(A) = sigma ∈ A j⟶ B) ∧ (1(B) o sigma = sigma ∈ A j⟶ B))
Lemma: csm-id-comp-sq
∀[A,B,C,D,x,y,X,Y,Z:Top]. (1(D) o x o y ~ x o y)
Lemma: csm-ap-csm-comp
∀Gamma,Delta,Z,s1,s2,I,alpha:Top. ((s2 o s1)alpha ~ (s2)(s1)alpha)
Lemma: context-map-1
∀[I:fset(ℕ)]. (<1> = 1(formal-cube(I)) ∈ formal-cube(I) ⟶ formal-cube(I))
Lemma: context-map-comp
∀[I,J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. (<f ⋅ g> = <f> o <g> ∈ formal-cube(K) ⟶ formal-cube(I))
Lemma: cube-set-restriction-id
∀[X:j⊢]. ∀[I:fset(ℕ)]. ∀[s:X(I)]. (1(s) = s ∈ X(I))
Lemma: cube-set-restriction-when-id
∀[X:j⊢]. ∀[I:fset(ℕ)]. ∀[s:X(I)]. ∀[f:I ⟶ I]. f(s) = s ∈ X(I) supposing f = 1 ∈ I ⟶ I
Lemma: cube-set-restriction-comp
∀X:j⊢. ∀I,J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀a:X(I). (g(f(a)) = f ⋅ g(a) ∈ X(K))
Lemma: cube-set-restriction-comp2
∀X:j⊢. ∀I,J1,J2,K:fset(ℕ). ∀f:J1 ⟶ I. ∀g:K ⟶ J1. ∀a:X(I). g(f(a)) = f ⋅ g(a) ∈ X(K) supposing J1 = J2 ∈ fset(ℕ)
Lemma: cube-set-map-is
∀[A,B:j⊢].
(A j⟶ B ~ {trans:I:fset(ℕ) ⟶ A(I) ⟶ B(I)|
∀I,J:fset(ℕ). ∀g:J ⟶ I. ((λs.g(trans I s)) = (λs.(trans J g(s))) ∈ (A(I) ⟶ B(J)))} )
Lemma: csm-ap-restriction
∀X,Y:j⊢. ∀s:X j⟶ Y. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀a:X(I). (f((s)a) = (s)f(a) ∈ Y(J))
Definition: sub_cubical_set
Y ⊆ X == 1(Y) ∈ Y ⟶ X
Lemma: sub_cubical_set_wf
∀[Y,X:j⊢]. (sub_cubical_set{j:l}(Y; X) ∈ 𝕌{[i | j'']})
Lemma: sub_cubical_set-cumulativity1
∀[Y,X:j⊢]. sub_cubical_set{[j | i]:l}(Y; X) supposing sub_cubical_set{j:l}(Y; X)
Lemma: sub_cubical_set_transitivity
∀[X,Y,Z:j⊢]. sub_cubical_set{j:l}(Z; X) supposing sub_cubical_set{j:l}(Z; Y) ∧ sub_cubical_set{j:l}(Y; X)
Lemma: sub_cubical_set_weakening
∀[X,Z:j⊢]. sub_cubical_set{j:l}(Z; X) supposing Z = X ∈ CubicalSet{j}
Lemma: sub_cubical_set_self
∀[X:j⊢]. sub_cubical_set{j:l}(X; X)
Lemma: implies-sub_cubical_set
∀[Y,X:j⊢].
(sub_cubical_set{j:l}(Y; X)) supposing
((∀A,B:fset(ℕ). ∀g:B ⟶ A. ∀rho:Y(A). (g(rho) = g(rho) ∈ X(B))) and
(∀I:fset(ℕ). (Y(I) ⊆r X(I))))
Lemma: subset-I_cube
∀[X,Y:j⊢]. ∀I:fset(ℕ). (Y(I) ⊆r X(I)) supposing sub_cubical_set{j:l}(Y; X)
Lemma: subset-restriction
∀[X,Y:j⊢]. ∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[a:Y(I)]. (f(a) = f(a) ∈ X(J)) supposing sub_cubical_set{j:l}(Y; X)
Lemma: cube_set_map_subtype
∀[X,Y,Z:j⊢]. Z j⟶ X ⊆r Z j⟶ Y supposing sub_cubical_set{j:l}(X; Y)
Lemma: cube_set_map_subtype2
∀[X,Y,Z:j⊢]. X j⟶ Z ⊆r Y j⟶ Z supposing sub_cubical_set{j:l}(Y; X)
Lemma: cube_set_map_subtype3
∀[X,Y,Z,U:j⊢]. (X j⟶ Z ⊆r Y j⟶ U) supposing (sub_cubical_set{j:l}(Y; X) and sub_cubical_set{j:l}(Z; U))
Lemma: cube_set_map_cumulativity2
∀[X,Y,Z,U:j⊢]. (X j⟶ Z ⊆r Y j⟶ U) supposing (sub_cubical_set{j:l}(Y; X) and sub_cubical_set{j:l}(Z; U))
Comment: face lattice doc
================================= face lattice ================================
The face lattice for names I is
⌜face-lattice(names(I);NamesDeq)⌝ then free distributive lattice
on generators ⌜(i=0)⌝ and ⌜(i=1)⌝ with ⌜(i=0) ∧ (i=1) = 0 ∈ Point(face-lattice(names(I);NamesDeq))⌝⋅
Definition: face_lattice
face_lattice(I) == face-lattice(names(I);NamesDeq)
Lemma: face_lattice_wf
∀[I:fset(ℕ)]. (face_lattice(I) ∈ BoundedDistributiveLattice)
Lemma: face_lattice-point_wf
∀[I:fset(ℕ)]. (Point(face_lattice(I)) ∈ Type)
Lemma: respects-equality-face-lattice-point
∀[I,J:fset(ℕ)]. respects-equality(Point(face_lattice(I));Point(face_lattice(J)))
Lemma: respects-equality-face-lattice-point-2
∀[I,J:fset(ℕ)]. respects-equality(fset(fset(names(I) + names(I)));Point(face_lattice(J)))
Definition: face_lattice-deq
face_lattice-deq() == deq-fset(deq-fset(sumdeq(NamesDeq;NamesDeq)))
Lemma: face_lattice-deq_wf
∀[I:fset(ℕ)]. (face_lattice-deq() ∈ EqDecider(Point(face_lattice(I))))
Lemma: decidable__equal_face_lattice
∀[I:fset(ℕ)]. ∀x,y:Point(face_lattice(I)). Dec(x = y ∈ Point(face_lattice(I)))
Lemma: face-lattice-0-not-1
∀[J:fset(ℕ)]. (¬(0 = 1 ∈ Point(face_lattice(J))))
Lemma: face_lattice-1-join-irreducible
∀I:fset(ℕ). ∀x,y:Point(face_lattice(I)).
(x ∨ y = 1 ∈ Point(face_lattice(I)) ⇐⇒ (x = 1 ∈ Point(face_lattice(I))) ∨ (y = 1 ∈ Point(face_lattice(I))))
Lemma: face_lattice-fset-join-eq-1
∀I:fset(ℕ). ∀s:fset(Point(face_lattice(I))).
(\/(s) = 1 ∈ Point(face_lattice(I)) ⇐⇒ ∃x:Point(face_lattice(I)). (x ∈ s ∧ (x = 1 ∈ Point(face_lattice(I)))))
Lemma: face_lattice-point-subtype
∀[I,J:fset(ℕ)]. Point(face_lattice(I)) ⊆r Point(face_lattice(J)) supposing I ⊆ J
Lemma: face_lattice-join-invariant
∀[v,y,I,J:Top]. (v ∨ y ~ v ∨ y)
Lemma: face_lattice-meet-invariant
∀[v,y,I,J:Top]. (v ∧ y ~ v ∧ y)
Definition: fl0
(x=0) == (x=0)
Lemma: fl0_wf
∀[I:fset(ℕ)]. ∀[x:names(I)]. ((x=0) ∈ Point(face_lattice(I)))
Definition: fl1
(x=1) == (x=1)
Lemma: fl1_wf
∀[I:fset(ℕ)]. ∀[x:names(I)]. ((x=1) ∈ Point(face_lattice(I)))
Definition: fl-eq
(x==y) == free-dml-deq(names(I);NamesDeq) x y
Lemma: fl-eq_wf
∀[I:fset(ℕ)]. ∀[x,y:Point(face_lattice(I))]. ((x==y) ∈ 𝔹)
Lemma: fl-eq-0-1-false
∀I:Top. ((0==1) ~ ff)
Lemma: assert-fl-eq
∀[I:fset(ℕ)]. ∀[x,y:Point(face_lattice(I))]. uiff(↑(x==y);x = y ∈ Point(face_lattice(I)))
Definition: fl-deq
Deq(F(I)) == λx,y. (x==y)
Lemma: face_lattice-induction
∀I:fset(ℕ)
∀[P:Point(face_lattice(I)) ⟶ ℙ]
((∀x:Point(face_lattice(I)). SqStable(P[x]))
⇒ P[0]
⇒ P[1]
⇒ (∀x,y:Point(face_lattice(I)). (P[x] ⇒ P[y] ⇒ P[x ∨ y]))
⇒ (∀x:Point(face_lattice(I)). (P[x] ⇒ (∀i:names(I). (P[(i=0) ∧ x] ∧ P[(i=1) ∧ x]))))
⇒ {∀x:Point(face_lattice(I)). P[x]})
Lemma: face_lattice-hom-fixes-sublattice
∀[I,J:fset(ℕ)].
∀[f:Hom(face_lattice(J);face_lattice(I))]. ∀[x:Point(face_lattice(J))].
(f x) = x ∈ Point(face_lattice(I))
supposing ∀i:names(J)
(((f (i=0)) = (i=0) ∈ Point(face_lattice(I))) ∧ ((f (i=1)) = (i=1) ∈ Point(face_lattice(I))))
supposing J ⊆ I
Lemma: face_lattice-hom-fixes-sublattice2
∀[I,J:fset(ℕ)].
∀[f:Hom(face_lattice(I);face_lattice(J))]. ∀[x:Point(face_lattice(J))].
(f x) = x ∈ Point(face_lattice(J))
supposing ∀i:names(J)
(((f (i=0)) = (i=0) ∈ Point(face_lattice(J))) ∧ ((f (i=1)) = (i=1) ∈ Point(face_lattice(J))))
supposing J ⊆ I
Lemma: fl-deq_wf
∀[I:fset(ℕ)]. (Deq(F(I)) ∈ EqDecider(Point(face_lattice(I))))
Lemma: fl0-not-1
∀[I:fset(ℕ)]. ∀[x:names(I)]. (¬((x=0) = 1 ∈ Point(face_lattice(I))))
Lemma: fl1-not-1
∀[I:fset(ℕ)]. ∀[x:names(I)]. (¬((x=1) = 1 ∈ Point(face_lattice(I))))
Lemma: FL-meet-0-1
∀[I:fset(ℕ)]. ∀[x:names(I)].
(((x=0) ∧ (x=1) = 0 ∈ Point(face_lattice(I))) ∧ ((x=1) ∧ (x=0) = 0 ∈ Point(face_lattice(I))))
Lemma: face_lattice-hom-equal
∀[I,J:fset(ℕ)]. ∀[g,h:Hom(face_lattice(I);face_lattice(J))].
g = h ∈ Hom(face_lattice(I);face_lattice(J))
supposing (∀x:names(I). ((g (x=0)) = (h (x=0)) ∈ Point(face_lattice(J))))
∧ (∀x:names(I). ((g (x=1)) = (h (x=1)) ∈ Point(face_lattice(J))))
Definition: irr_face
irr_face(I;as;bs) == /\(λa.(a=0)"(as) ⋃ λb.(b=1)"(bs))
Lemma: irr_face_wf
∀[I:fset(ℕ)]. ∀[as,bs:fset(names(I))]. (irr_face(I;as;bs) ∈ Point(face_lattice(I)))
Definition: face_lattice_components
face_lattice_components(I;x) == λs.<fset-mapfilter(λx.outl(x);λx.isl(x);s), fset-mapfilter(λx.outr(x);λx.isr(x);s)>"(x)
Lemma: face_lattice_components_wf
∀I:fset(ℕ). ∀x:Point(face_lattice(I)).
(face_lattice_components(I;x) ∈ {fs:fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )\000C|
x = \/(λpr.irr_face(I;fst(pr);snd(pr))"(fs)) ∈ Point(face_lattice(I))} )
Lemma: face_lattice-basis
∀I:fset(ℕ). ∀x:Point(face_lattice(I)).
∃fs:fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )
(x = \/(λpr.irr_face(I;fst(pr);snd(pr))"(fs)) ∈ Point(face_lattice(I)))
Definition: dM-to-FL
dM-to-FL(I;z) ==
lattice-extend(face_lattice(I);union-deq(names(I);names(I);NamesDeq;NamesDeq);face_lattice-deq();λx.case x
of inl(a) =>
(a=1)
| inr(a) =>
(a=0);z)
Lemma: dM-to-FL_wf
∀[I:fset(ℕ)]. ∀[z:Point(dM(I))]. (dM-to-FL(I;z) ∈ Point(face_lattice(I)))
Lemma: dM-to-FL-sq
∀[I,J,v:Top]. (dM-to-FL(I;v) ~ dM-to-FL(J;v))
Lemma: dm-neg-sq
∀[I,J,v:Top]. (¬(v) ~ ¬(v))
Lemma: dM-to-FL-inc
∀[I:fset(ℕ)]. ∀[x:names(I)]. (dM-to-FL(I;<x>) = (x=1) ∈ Point(face_lattice(I)))
Lemma: dM-to-FL-opp
∀[I:fset(ℕ)]. ∀[x:names(I)]. (dM-to-FL(I;<1-x>) = (x=0) ∈ Point(face_lattice(I)))
Lemma: dM-to-FL-is-hom
∀[I:fset(ℕ)]. (λz.dM-to-FL(I;z) ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I)))
Lemma: dM-to-FL-unique
∀[I:fset(ℕ)]. ∀[g:Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I))].
∀[x:Point(dM(I))]. (dM-to-FL(I;x) = (g x) ∈ Point(face_lattice(I)))
supposing ∀i:names(I). (((g <i>) = (i=1) ∈ Point(face_lattice(I))) ∧ ((g <1-i>) = (i=0) ∈ Point(face_lattice(I))))
Lemma: dM-to-FL-properties
∀[I:fset(ℕ)]
((∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
(dM-to-FL(I;x ∨ y) = dM-to-FL(I;x) ∨ dM-to-FL(I;y) ∈ Point(face_lattice(I))))
∧ (∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
(dM-to-FL(I;x ∧ y) = dM-to-FL(I;x) ∧ dM-to-FL(I;y) ∈ Point(face_lattice(I))))
∧ (dM-to-FL(I;0) = 0 ∈ Point(face_lattice(I)))
∧ (dM-to-FL(I;1) = 1 ∈ Point(face_lattice(I))))
Lemma: dM-to-FL-eq-1
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. uiff(dM-to-FL(I;x) = 1 ∈ Point(face_lattice(I));x = 1 ∈ Point(dM(I)))
Lemma: dM-to-FL-1
∀[I:fset(ℕ)]. (dM-to-FL(I;1) = 1 ∈ Point(face_lattice(I)))
Lemma: dM-to-FL-dM1
∀[I:fset(ℕ)]. (dM-to-FL(I;1) = 1 ∈ Point(face_lattice(I)))
Lemma: dM-to-FL-dM0
∀[I:fset(ℕ)]. (dM-to-FL(I;0) = 0 ∈ Point(face_lattice(I)))
Lemma: dM-to-FL-neg
∀[I:fset(ℕ)]
∀x:Point(free-DeMorgan-lattice(names(I);NamesDeq)). (dM-to-FL(I;x) ∧ dM-to-FL(I;¬(x)) = 0 ∈ Point(face_lattice(I)))
Lemma: dM-to-FL-neg2
∀[I:fset(ℕ)]
∀x:Point(free-DeMorgan-lattice(names(I);NamesDeq)). (dM-to-FL(I;¬(x)) ∧ dM-to-FL(I;x) = 0 ∈ Point(face_lattice(I)))
Lemma: dM-to-FL-neg-is-1
∀[I:fset(ℕ)]
∀x:Point(free-DeMorgan-lattice(names(I);NamesDeq))
dM-to-FL(I;x) = 0 ∈ Point(face_lattice(I)) supposing dM-to-FL(I;¬(x)) = 1 ∈ Point(face_lattice(I))
Lemma: face-lattice-hom-is-id
∀I:fset(ℕ)
∀[h:Hom(face_lattice(I);face_lattice(I))]
h = (λx.x) ∈ Hom(face_lattice(I);face_lattice(I))
supposing ∀x:names(I). (((h (x=0)) = (x=0) ∈ Point(face_lattice(I))) ∧ ((h (x=1)) = (x=1) ∈ Point(face_lattice(I))))
Definition: fl-morph
<f> == fl-lift(names(J);NamesDeq;face_lattice(I);face_lattice-deq();λj.dM-to-FL(I;¬(f j));λj.dM-to-FL(I;f j))
Lemma: fl-morph_wf
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. (<f> ∈ Hom(face_lattice(J);face_lattice(I)))
Lemma: fl-morph-id
∀I:fset(ℕ). (<1> = (λx.x) ∈ (Point(face_lattice(I)) ⟶ Point(face_lattice(I))))
Lemma: apply-fl-morph-id
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))]. ((phi)<1> = phi ∈ Point(face_lattice(I)))
Lemma: fl-morph-face-lattice0
∀[J,I:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[x:names(I)]. (((x=0))<f> = dM-to-FL(J;¬(f x)) ∈ Point(face_lattice(J)))
Lemma: fl-morph-face-lattice1
∀[J,I:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[x:names(I)]. (((x=1))<f> = dM-to-FL(J;f x) ∈ Point(face_lattice(J)))
Lemma: fl-morph-fl0
∀[J,I:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[x:names(I)]. (((x=0))<f> = dM-to-FL(J;¬(f x)) ∈ Point(face_lattice(J)))
Lemma: fl-morph-fl0-is-1
∀[J,I:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[x:names(I)]. uiff(((x=0))<f> = 1 ∈ Point(face_lattice(J));(f x) = 0 ∈ Point(dM(J)))
Lemma: fl-s-fl0
∀[I:fset(ℕ)]. ∀[x:names(I)]. (((x=0))<s> = (x=0) ∈ Point(face_lattice(I+x)))
Lemma: fl-morph-fl1
∀[J,I:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[x:names(I)]. (((x=1))<f> = dM-to-FL(J;f x) ∈ Point(face_lattice(J)))
Lemma: fl-morph-fl1-is-1
∀[J,I:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[x:names(I)]. uiff(((x=1))<f> = 1 ∈ Point(face_lattice(J));(f x) = 1 ∈ Point(dM(J)))
Lemma: fl-s-fl1
∀[I:fset(ℕ)]. ∀[x:names(I)]. (((x=1))<s> = (x=1) ∈ Point(face_lattice(I+x)))
Lemma: fl-morph-comp-1
∀[J,K:fset(ℕ)]. ∀[f:K ⟶ J]. ∀[z:Point(dM(J))]. ∀[h:dma-hom(dM(J);dM(K))].
(dM-to-FL(J;z))<f> = dM-to-FL(K;h z) ∈ Point(face_lattice(K)) supposing ∀i:names(J). ((h <i>) = (f i) ∈ Point(dM(K)))
Lemma: fl-morph-comp-dM-lift
∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[z:Point(dM(I))].
((dM-to-FL(I;z))<f> = dM-to-FL(J;dM-lift(J;I;f) z) ∈ Point(face_lattice(J)))
Lemma: fl-morph-comp
∀[I,J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J].
(<f ⋅ g> = (<g> o <f>) ∈ (Point(face_lattice(I)) ⟶ Point(face_lattice(K))))
Lemma: fl-morph-comp2
∀[I,J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. ∀[x:Point(face_lattice(I))].
(((x)<f>)<g> = (x)<f ⋅ g> ∈ Point(face_lattice(K)))
Lemma: fl-morph-1
∀[A,B:fset(ℕ)]. ∀[g:A ⟶ B]. ((1)<g> = 1 ∈ Point(face_lattice(A)))
Lemma: fl-morph-0
∀[A,B:fset(ℕ)]. ∀[g:A ⟶ B]. ((0)<g> = 0 ∈ Point(face_lattice(A)))
Lemma: fl-morph-join
∀[A,B:fset(ℕ)]. ∀[g:A ⟶ B]. ∀[x,y:Point(face_lattice(B))]. ((x ∨ y)<g> = (x)<g> ∨ (y)<g> ∈ Point(face_lattice(A)))
Lemma: fl-morph-meet
∀[A,B:fset(ℕ)]. ∀[g:A ⟶ B]. ∀[x,y:Point(face_lattice(B))]. ((x ∧ y)<g> = (x)<g> ∧ (y)<g> ∈ Point(face_lattice(A)))
Lemma: fl-morph-fset-meet
∀[A,B:fset(ℕ)]. ∀[g:A ⟶ B]. ∀[x:fset(Point(face_lattice(B)))]. ((/\(x))<g> = /\(<g>"(x)) ∈ Point(face_lattice(A)))
Lemma: fl-morph-fset-join
∀[A,B:fset(ℕ)]. ∀[g:A ⟶ B]. ∀[x:fset(Point(face_lattice(B)))]. ((\/(x))<g> = \/(<g>"(x)) ∈ Point(face_lattice(A)))
Definition: irr-face-morph
irr-face-morph(I;as;bs) == λi.if i ∈b as then 0 if i ∈b bs then 1 else <i> fi
Lemma: irr-face-morph_wf
∀[I:fset(ℕ)]. ∀[as,bs:fset(names(I))]. (irr-face-morph(I;as;bs) ∈ I ⟶ I)
Definition: face-presheaf
𝔽 == <λI.Point(face_lattice(I)), λJ,I,f. <f>>
Lemma: face-presheaf_wf1
𝔽 ∈ SmallCubicalSet
Lemma: face-presheaf_wf
𝔽 ∈ small_cubical_set{j:l}
Lemma: face-presheaf_wf2
𝔽 j⊢
Lemma: face-presheaf-point-subtype
∀[I:fset(ℕ)]. (𝔽(I) ⊆r Point(face_lattice(I)))
Lemma: subtype-face-presheaf-point
∀[I:fset(ℕ)]. (Point(face_lattice(I)) ⊆r 𝔽(I))
Lemma: face-fl-morph-id
∀[I:fset(ℕ)]. ∀[phi:𝔽(I)]. ((phi)<1> = phi ∈ 𝔽(I))
Lemma: face-presheaf-restriction-1
∀[A,B:fset(ℕ)]. ∀[g:B ⟶ A]. (g(1) = 1 ∈ Point(face_lattice(B)))
Lemma: fl-morph-restriction
∀[I,J,K:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[g:J ⟶ K]. ∀[phi:𝔽(K)]. ((g(phi))<f> = g ⋅ f(phi) ∈ 𝔽(I))
Definition: fl-join
fl-join(I;x;y) == x ∨ y
Lemma: fl-join_wf
∀[I:fset(ℕ)]. ∀[x,y:𝔽(I)]. (fl-join(I;x;y) ∈ 𝔽(I))
Definition: dM_FL
dM_FL() == λI,z. dM-to-FL(I;z)
Lemma: dM_FL_wf
dM_FL() ∈ 𝕀 j⟶ 𝔽
Lemma: face_lattice_hom_subtype
∀[I,J:fset(ℕ)]. Hom(face_lattice(J);face_lattice(I)) ⊆r Hom(face_lattice(I);face_lattice(I)) supposing I ⊆ J
Definition: fl-all-hom
fl-all-hom(I;i) ==
fl-lift(names(I+i);NamesDeq;face_lattice(I);face_lattice-deq();λx.if (x =z i) then 0 else (x=0) fi ;λx.if (x =z i)
then 0
else (x=1)
fi )
Lemma: fl-all-hom_wf1
∀[I:fset(ℕ)]. ∀[i:ℕ].
(fl-all-hom(I;i) ∈ {g:Hom(face_lattice(I+i);face_lattice(I))|
(∀j:names(I)
((¬(j = i ∈ ℤ))
⇒ (((g (j=0)) = (j=0) ∈ Point(face_lattice(I)))
∧ ((g (j=1)) = (j=1) ∈ Point(face_lattice(I))))))
∧ ((g (i=0)) = 0 ∈ Point(face_lattice(I)))
∧ ((g (i=1)) = 0 ∈ Point(face_lattice(I)))} )
Lemma: fl-all-hom_wf
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ].
(fl-all-hom(I;i) ∈ {g:Hom(face_lattice(I+i);face_lattice(I))|
(∀x:Point(face_lattice(I)). ((g x) = x ∈ Point(face_lattice(I))))
∧ ((g (i=0)) = 0 ∈ Point(face_lattice(I)))
∧ ((g (i=1)) = 0 ∈ Point(face_lattice(I)))} )
Definition: fl_all
(∀i.phi) == fl-all-hom(I;i) phi
Lemma: fl_all_wf
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[phi:Point(face_lattice(I+i))]. ((∀i.phi) ∈ Point(face_lattice(I)))
Lemma: fl_all-1
∀I,J,i:Top. ((∀i.1) ~ 1)
Lemma: fl_all-0
∀I,J,i:Top. ((∀i.0) ~ 0)
Lemma: fl_all-fl1
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[x:names(I+i)]. ((∀i.(x=1)) = if (x =z i) then 0 else (x=1) fi ∈ Point(face_lattice(I)))
Lemma: fl_all-fl0
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[x:names(I+i)]. ((∀i.(x=0)) = if (x =z i) then 0 else (x=0) fi ∈ Point(face_lattice(I)))
Lemma: fl_all-id
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[phi:Point(face_lattice(I))]. ((∀i.phi) = phi ∈ Point(face_lattice(I)))
Lemma: fl_all-meet
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[a,b:Point(face_lattice(I+i))]. ((∀i.a ∧ b) = (∀i.a) ∧ (∀i.b) ∈ Point(face_lattice(I)))
Lemma: fl_all-join
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[a,b:Point(face_lattice(I+i))]. ((∀i.a ∨ b) = (∀i.a) ∨ (∀i.b) ∈ Point(face_lattice(I)))
Lemma: fl_all_com
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. ∀[phi:Point(face_lattice(I+i+j))]. ((∀i.(∀j.phi)) = (∀j.(∀i.phi)) ∈ Point(face_lattice(I)))
Lemma: fl_all_decomp
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[phi:Point(face_lattice(I+i))].
(phi = (∀i.phi) ∨ phi ∧ (i=0) ∨ phi ∧ (i=1) ∈ Point(face_lattice(I+i)))
Lemma: fl_all-implies-instance
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. ∀[i:ℕ]. ∀[v:Point(face_lattice(I+i))].
(((∀i.v) = 1 ∈ Point(face_lattice(I))) ⇒ ((v)<(i/x)> = 1 ∈ Point(face_lattice(I))))
Definition: dM-fl-all
dM-fl-all(I;phi;z) ==
lattice-extend(face_lattice(I);union-deq(names(I);names(I);NamesDeq;NamesDeq);face_lattice-deq();λx.case x
of inl(a) =>
(∀a.phi)
| inr(a) =>
(∀a.phi);z)
Lemma: dM-fl-all_wf
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))]. ∀[z:Point(dM(I))]. (dM-fl-all(I;phi;z) ∈ Point(face_lattice(I)))
Lemma: fl-morph-fl_all
∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{i:ℕ| ¬i ∈ J} ]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[psi:Point(face_lattice(I+i))].
(((∀i.psi))<f> = (∀j.(psi)<f,i=j>) ∈ Point(face_lattice(J)))
Definition: face-lattice-tube
face-lattice-tube(I;phi;j) == s(phi) ∨ (j=0) ∨ (j=1)
Lemma: face-lattice-tube_wf
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))]. ∀[j:ℕ]. (face-lattice-tube(I;phi;j) ∈ Point(face_lattice(I+j)))
Lemma: fl-morph-face-lattice-tube1
∀[J:fset(ℕ)]. ∀[k:names(J)]. ∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))]. ∀[j:ℕ]. ∀[g:J ⟶ I+j].
(face-lattice-tube(I;phi;j))<g> = (s(phi))<g> ∨ (k=0) ∨ (k=1) ∈ Point(face_lattice(J))
supposing (g j) = <k> ∈ Point(dM(J))
Comment: cubical subset doc
================================= cubical subset ===============================
We next define the subset of the formal cube I,psi where psi ∈ 𝔽(I).
Later, we will define a more general subset operation G, phi on any context G
where phi ∈ {G ⊢ _:𝔽} is a cubical term of type ⌜𝔽⌝⋅
Definition: name-morph-satisfies
(psi f) = 1 == (psi)<f> = 1 ∈ Point(face_lattice(J))
Lemma: name-morph-satisfies_wf
∀[I,J:fset(ℕ)]. ∀[psi:Point(face_lattice(I))]. ∀[f:J ⟶ I]. ((psi f) = 1 ∈ ℙ)
Lemma: name-morph-satisfies-comp
∀[I,J,K:fset(ℕ)]. ∀[psi:Point(face_lattice(I))]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. uiff((f(psi) g) = 1;(psi f ⋅ g) = 1)
Lemma: implies-nh-comp-satisfies
∀[I,J,K:fset(ℕ)]. ∀[psi:Point(face_lattice(I))]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. (psi f ⋅ g) = 1 supposing (psi f) = 1
Lemma: name-morph-satisfies-join
∀I,J:fset(ℕ). ∀a,b:Point(face_lattice(I)). ∀f:J ⟶ I. ((fl-join(I;a;b) f) = 1 ⇐⇒ (a f) = 1 ∨ (b f) = 1)
Lemma: name-morph-satisfies-fset-join
∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀s:fset(Point(face_lattice(I))).
((\/(s) f) = 1 ⇐⇒ ↓∃a:Point(face_lattice(I)). (a ∈ s ∧ (a f) = 1))
Lemma: name-morph-satisfies-fl0
∀[I,J:fset(ℕ)]. ∀[i:names(I)]. ∀[f:J ⟶ I]. uiff(((i=0) f) = 1;(f i) = 0 ∈ Point(dM(J)))
Lemma: name-morph-satisfies-fl1
∀[I,J:fset(ℕ)]. ∀[i:names(I)]. ∀[f:J ⟶ I]. uiff(((i=1) f) = 1;(f i) = 1 ∈ Point(dM(J)))
Lemma: name-morph-satisfies-0
∀[I:fset(ℕ)]. ∀[i:ℕ]. ((i=0) (i0)) = 1
Lemma: name-morph-satisfies-0'
∀[I:fset(ℕ)]. ∀[i:ℕ]. ((i=0) (i0)) = 1
Lemma: name-morph-1-satisfies
∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. (1 f) = 1
Lemma: satisfies-irr-face
∀[I,J:fset(ℕ)]. ∀[as,bs:fset(names(I))]. ∀[g:J ⟶ I].
uiff((irr_face(I;as;bs) g) = 1;(∀a:names(I). (a ∈ as ⇒ ((g a) = 0 ∈ Point(dM(J)))))
∧ (∀b:names(I). (b ∈ bs ⇒ ((g b) = 1 ∈ Point(dM(J))))))
Lemma: irr-face-morph-satisfies
∀[I:fset(ℕ)]. ∀[as,bs:fset(names(I))].
(irr_face(I;as;bs) irr-face-morph(I;as;bs)) = 1 supposing ↑fset-disjoint(NamesDeq;as;bs)
Lemma: irr-face-morph-property
∀[I:fset(ℕ)]. ∀[as,bs:fset(names(I))]. ∀[J:fset(ℕ)]. ∀[g:J ⟶ I].
((irr_face(I;as;bs) g) = 1 ⇒ (g = irr-face-morph(I;as;bs) ⋅ g ∈ J ⟶ I))
Lemma: face_lattice-le
∀[I:fset(ℕ)]. ∀[x,y:Point(face_lattice(I))].
uiff(x ≤ y;∀f:I ⟶ I. (((x)<f> = 1 ∈ Point(face_lattice(I))) ⇒ ((y)<f> = 1 ∈ Point(face_lattice(I)))))
Lemma: satisfies-face-lattice-tube
∀I:fset(ℕ). ∀i:{i:ℕ| ¬i ∈ I} . ∀phi:𝔽(I). ∀j:{j:ℕ| ¬j ∈ I+i} . ∀K:fset(ℕ). ∀g:K ⟶ I+j.
((face-lattice-tube(I;phi;j) g) = 1 ⇐⇒ (phi s ⋅ g) = 1 ∨ ((g j) = 0 ∈ Point(dM(K))) ∨ ((g j) = 1 ∈ Point(dM(K))))
Lemma: satisfies-s-face-lattice-tube
∀I:fset(ℕ). ∀i:{i:ℕ| ¬i ∈ I} . ∀phi:𝔽(I). ∀j:{j:ℕ| ¬j ∈ I+i} . ∀K:fset(ℕ). ∀f:K ⟶ I+j+i.
((s(face-lattice-tube(I;phi;j)) f) = 1
⇐⇒ (s(phi) s ⋅ f) = 1 ∨ ((s ⋅ f j) = 0 ∈ Point(dM(K))) ∨ ((s ⋅ f j) = 1 ∈ Point(dM(K))))
Definition: cubical-subset
I,psi == rep-sub-sheaf(CubeCat;I;λJ,f. (psi f) = 1)
Lemma: cubical-subset_wf
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. I,psi j⊢
Lemma: cubical-subset-I_cube
∀[I,psi,J:Top]. (I,psi(J) ~ {f:J ⟶ I| (psi f) = 1} )
Lemma: member-cubical-subset-I_cube
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. f ∈ I,psi(J) supposing (psi f) = 1
Lemma: cubical-subset-I_cube-member
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. ∀[J:fset(ℕ)]. ∀[f:I,psi(J)]. ((f ∈ J ⟶ I) ∧ (psi f) = 1)
Lemma: cubical-subset-restriction
∀[I,J,K,f,g,phi:Top]. (g(f) ~ f ⋅ g)
Definition: subset-iota
iota == λK,g. g
Lemma: subset-iota_wf
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. (iota ∈ I,psi j⟶ formal-cube(I))
Definition: subset_iota
subset_iota(psi) == λK,g. g
Lemma: subset_iota_wf
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. (subset_iota(psi) ∈ I,psi ⟶ formal-cube(I))
Definition: subset-trans
subset-trans(I;J;f;x) == λK,g. f ⋅ g
Lemma: subset-trans_wf
∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[psi:𝔽(I)]. (subset-trans(I;J;f;psi) ∈ J,(psi)<f> j⟶ I,psi)
Lemma: subset-trans-iota-lemma
∀Gamma:j⊢. ∀I:fset(ℕ). ∀rho:Gamma(I). ∀phi:𝔽(I). ∀J:fset(ℕ). ∀f:J ⟶ I.
(<rho> o iota o subset-trans(I;J;f;phi) = <f(rho)> o iota ∈ J,(phi)<f> j⟶ Gamma)
Comment: Rules of MLTT without type formers
The rules from Figure 1 in Bezem, Coquand, Huber
"A model of type theory in cubical sets".
But now we interpret everything in the new cubical type theory.
(First those without type formers: 20 lemmas)
Γ ⊢
------------- this is csm-id_wf
1: Γ ⟶ Γ
σ: Δ ⟶ Γ δ: Θ -> Δ
------------------------ this is csm-comp_wf
σ δ : Θ ⟶ Γ
Γ ⊢ A σ : Δ ⟶ Γ
---------------------- this is csm-ap-type_wf
Δ ⊢ A σ
Γ ⊢ t: A σ: Δ ⟶ Γ
----------------------- this is csm-ap-term_wf
Δ ⊢ t σ: A σ
------------- this is trivial-cube-set_wf
() ⊢
Γ ⊢ Γ ⊢ A
----------------- this is cube-context-adjoin_wf
Γ.A ⊢
Γ ⊢ A
------------------ this is cc-fst_wf
p: Γ.A ⟶ Γ
Γ ⊢ A
----------------- this is cc-snd_wf
Γ.A ⊢ q: Ap
σ:Δ ⟶ Γ Γ ⊢ A Δ ⊢ u:A s
------------------------------ this is csm-adjoin_wf
(σ,u) ⊢ Δ ⟶ Γ.A
1 σ = σ 1 = σ this is csm-id-comp
(σ δ) ν = σ (δ ν) this is csm-comp-assoc
[u] = (1,u) this is csm-id-adjoin
A 1 = A this is csm-ap-id-type
(A σ) δ = A (σ δ) this is csm-ap-comp-type
u 1 = u this is csm-ap-id-term
(u σ) δ = u (σ δ) this is csm-ap-comp-term
(σ, u) δ = (σ δ, u δ) this is csm-adjoin-ap
p (σ, u) = σ this is cc-fst-csm-adjoin
q (σ, u) = u this is cc-snd-csm-adjoin
(p,q) = 1 this is csm-adjoin-fst-snd
⋅
Comment: Rules of MLTT type formers
The rules about type formers Π and ⌜Σ⌝ from Figure 1 in Bezem, Coquand, Huber
"A model of type theory in cubical sets".
(Interpreted in the new cubical type theory: 19 lemmas)
Γ.A ⊢ B
------------- this is cubical-pi_wf
Γ ⊢ Π A B
Γ.A ⊢ B Γ.A ⊢ b:B
------------------------ this is cubical-lambda_wf
Γ ⊢ λb : Π A B
Γ.A ⊢ B
------------- this is cubical-sigma_wf
Γ ⊢ Σ A B
Γ.A ⊢ B Γ ⊢ u:A Γ ⊢ v:B[u]
------------------------------ this is cubical-pair_wf
Γ ⊢ (u,v): Σ A B
Γ ⊢ w: Σ A B
---------------- this is cubical-fst_wf
Γ ⊢ w.1: A
Γ ⊢ w: Σ A B
----------------- this is cubical-snd_wf
Γ ⊢ w.2: B[w.1]
Γ ⊢ w:Π A B Γ ⊢ u:A
------------------------ this is cubical-app_wf
Γ ⊢ app(w,u): B[u]
(Π A B)σ = Π (Aσ) (B(σp,q)) this is csm-cubical-pi
(λb)σ = λ(b(σp,q)) this is csm-ap-cubical-lambda
app(w,u)δ = app(wδ, uδ) this is csm-ap-cubical-app
app(λb, u) = b[u] this is cubical-beta
w = λ(app(wp,q)) this is cubical-eta
(Σ A B)σ = (Aσ) (B(σp,q)) this is csm-cubical-sigma
(w.1)σ = (wσ).1 this is csm-ap-cubical-fst
(w.2)σ = (wσ).2 this is csm-ap-cubical-snd
(u,v)σ = (uσ, vσ) this is csm-ap-cubical-pair
(u,v).1 = u this is cubical-fst-pair
(u,v).2 = v this is cubical-snd-pair
(w.1,w.2) = w this is cubical-pair-eta
⋅
Definition: cubical_type
cubical_type{i:l}(X) == presheaf{i':l}(cubes(X))
Lemma: cubical_type_wf
∀[X:j⊢]. (cubical_type{i:l}(X) ∈ 𝕌{[i'' | j']})
Definition: cubical-type
{X ⊢ _} ==
{AF:A:I:fset(ℕ) ⟶ X(I) ⟶ Type × (I:fset(ℕ) ⟶ J:fset(ℕ) ⟶ f:J ⟶ I ⟶ a:X(I) ⟶ (A I a) ⟶ (A J f(a)))|
let A,F = AF
in (∀I:fset(ℕ). ∀a:X(I). ∀u:A I a. ((F I I 1 a u) = u ∈ (A I a)))
∧ (∀I,J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀a:X(I). ∀u:A I a.
((F I K f ⋅ g a u) = (F J K g f(a) (F I J f a u)) ∈ (A K f ⋅ g(a))))}
Lemma: cubical-type_wf
∀[X:j⊢]. (X ⊢ ∈ 𝕌{[i' | j']})
Lemma: cubical-type-sq-presheaf-type
∀[X:Top]. ({X ⊢ _} ~ {X ⊢ _})
Lemma: cubical-type-equal
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:A:I:fset(ℕ) ⟶ X(I) ⟶ Type × (I:fset(ℕ)
⟶ J:fset(ℕ)
⟶ f:J ⟶ I
⟶ a:X(I)
⟶ (A I a)
⟶ (A J f(a)))].
A = B ∈ {X ⊢ _}
supposing A
= B
∈ (A:I:fset(ℕ) ⟶ X(I) ⟶ Type × (I:fset(ℕ) ⟶ J:fset(ℕ) ⟶ f:J ⟶ I ⟶ a:X(I) ⟶ (A I a) ⟶ (A J f(a))))
Lemma: cubical-type-equal2
∀[X:j⊢]. ∀[A,B:{X ⊢ _}].
A = B ∈ {X ⊢ _}
supposing A
= B
∈ (A:I:fset(ℕ) ⟶ X(I) ⟶ Type × (I:fset(ℕ) ⟶ J:fset(ℕ) ⟶ f:J ⟶ I ⟶ a:X(I) ⟶ (A I a) ⟶ (A J f(a))))
Definition: cubical-type-iso
cubical-type-iso(X) == λF.<λI,rho. (F <I, rho>), λI,J,f,a. (F <I, a> <J, f(a)> f)>
Lemma: cubical-type-iso_wf
∀[X:j⊢]. (cubical-type-iso(X) ∈ cubical_type{i:l}(X) ⟶ {X ⊢' _})
Definition: cubical-type-rev-iso
cubical-type-rev-iso(X) ==
λF.Presheaf(Set(p) =(fst(F)) (fst(p)) (snd(p));
Morphism(p,q,f,a) = (snd(F)) (fst(p)) (fst(q)) f (snd(p)) a)
Lemma: cubical-type-rev-iso_wf
∀[X:j⊢]. (cubical-type-rev-iso(X) ∈ {X ⊢ _} ⟶ cubical_type{i:l}(X))
Lemma: cubical-type-iso-inverse
∀[X:j⊢]. ((cubical-type-rev-iso(X) o cubical-type-iso(X)) = (λx.x) ∈ (cubical_type{i:l}(X) ⟶ cubical_type{i:l}(X)))
Lemma: cubical-type-iso-inverse2
∀[X:j⊢]. ((cubical-type-iso(X) o cubical-type-rev-iso(X)) = (λx.x) ∈ ({X ⊢ _} ⟶ {X ⊢ _}))
Lemma: cubical-type-cumulativity2
∀[X:j⊢]. ({X ⊢ _} ⊆r cubical-type{[j | i]:l}(X))
Lemma: cubical-type-cumulativity
∀[X:j⊢]. ({X ⊢ _} ⊆r {X ⊢' _})
Lemma: cubical-type-cumulativity-i-j
∀[X:j⊢]. ({X ⊢j _} ⊆r cubical-type{[j | i]:l}(X))
Definition: cubical-type-at
A(a) == (fst(A)) I a
Lemma: cubical-type-at_wf1
∀[X:j⊢]. ∀[A:{X ⊢j _}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. (A(a) ∈ 𝕌{j})
Lemma: cubical-type-at_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. (A(a) ∈ Type)
Lemma: istype-cubical-type-at
∀[X:j⊢]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[A:{X ⊢ _}]. istype(A(a))
Lemma: cubical_type_at_pair_lemma
∀a,I,B,A:Top. (<A, B>(a) ~ A I a)
Definition: cubical-type-ap-morph
(u a f) == (snd(A)) I J f a u
Lemma: cubical-type-ap-morph_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[a:X(I)]. ∀[u:A(a)]. ((u a f) ∈ A(f(a)))
Lemma: cubical_type_ap_morph_pair_lemma
∀u,a,f,J,I,G,F:Top. ((u a f) ~ G I J f a u)
Lemma: cubical-type-ap-morph-comp-general
∀[X:j⊢]. ∀[A:{X ⊢j _}]. ∀[I,J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. ∀[a:X(I)]. ∀[u:A(a)].
(((u a f) f(a) g) = (u a f ⋅ g) ∈ A(f ⋅ g(a)))
Lemma: cubical-type-ap-morph-comp
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[I,J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. ∀[a:X(I)]. ∀[u:A(a)].
(((u a f) f(a) g) = (u a f ⋅ g) ∈ A(f ⋅ g(a)))
Lemma: cubical-type-ap-morph-comp-eq-general
∀[X:j⊢]. ∀[A:{X ⊢j _}]. ∀[I,J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. ∀[a:X(I)]. ∀[b:X(J)]. ∀[u:A(a)].
((u a f) b g) = (u a f ⋅ g) ∈ A(f ⋅ g(a)) supposing b = f(a) ∈ X(J)
Lemma: cubical-type-ap-morph-comp-eq
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[I,J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. ∀[a:X(I)]. ∀[b:X(J)]. ∀[u:A(a)].
((u a f) b g) = (u a f ⋅ g) ∈ A(f ⋅ g(a)) supposing b = f(a) ∈ X(J)
Lemma: cubical-type-ap-morph-id
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[I:fset(ℕ)]. ∀[f:I ⟶ I]. ∀[a:X(I)]. ∀[u:A(a)]. (u a f) = u ∈ A(a) supposing f = 1 ∈ I ⟶ I
Lemma: cubical-type-equal3
∀[X:j⊢]. ∀[A,B:{X ⊢ _}].
(A = B ∈ {X ⊢ _}) supposing
((∀I:fset(ℕ). ∀[rho:X(I)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[u:A(rho)]. ((u rho f) = (u rho f) ∈ A(f(rho)))) and
(∀I:fset(ℕ). ∀[rho:X(I)]. (A(rho) = B(rho) ∈ Type)))
Definition: csm-ap-type
(AF)s == let A,F = AF in <λI,a. (A I (s)a), λI,J,f,a,u. (F I J f (s)a u)>
Lemma: csm-ap-type_wf1
∀[Gamma,Delta:j⊢]. ∀[A:{Gamma ⊢j _}]. ∀[s:Delta j⟶ Gamma]. Delta ⊢j (A)s
Lemma: csm-ap-type_wf
∀[Gamma,Delta:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[s:Delta j⟶ Gamma]. Delta ⊢ (A)s
Lemma: csm-ap-type-subset-iota
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[s:Top]. (((A)s)iota ~ (A)s)
Definition: csm-type-ap
csm-type-ap(A;s) == (A)s
Lemma: csm-type-ap_wf
∀[Gamma,Delta:j⊢]. ∀[s:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. Delta ⊢ csm-type-ap(A;s)
Lemma: context-map-ap-type
∀[I:fset(ℕ)]. ∀[Gamma:j⊢]. ∀[rho:Gamma(I)]. ∀[A:{Gamma ⊢ _}]. formal-cube(I) ⊢ (A)<rho>
Lemma: csm-ap-type-at
∀[B,s,x,K:Top]. ((B)s(x) ~ B((s)x))
Lemma: csm-ap-id-type
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ((A)1(Gamma) = A ∈ {Gamma ⊢ _})
Lemma: csm-ap-type-is-id
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[s:Gamma j⟶ Gamma]. (A)s = A ∈ {Gamma ⊢ _} supposing s = 1(Gamma) ∈ Gamma j⟶ Gamma
Lemma: csm-ap-type-iota
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ((A)iota = A ∈ {X ⊢ _})
Lemma: csm-comp-type
∀[Gamma,Delta,Z,s1,s2,A:Top]. ((A)s2 o s1 ~ ((A)s2)s1)
Lemma: csm-type-comp
∀[Gamma,A,Delta,Z,s1,s2:Top]. (((A)s2)s1 ~ (A)s2 o s1)
Lemma: csm-ap-comp-type
∀[Gamma,Delta,Z:j⊢]. ∀[s1:Z j⟶ Delta]. ∀[s2:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ((A)s2 o s1 = ((A)s2)s1 ∈ {Z ⊢ _})
Lemma: csm-ap-comp-type-sq
∀[Gamma,Delta,Z,s1,s2,A:Top]. ((A)s2 o s1 ~ ((A)s2)s1)
Lemma: csm-ap-comp-type-sq2
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[s1,s2:Top]. (((A)s2)s1 ~ (A)s2 o s1)
Lemma: csm-cubical-type-ap-morph
∀[A,I,J,f,a,u,s:Top]. ((u a f) ~ (u (s)a f))
Definition: closed-cubical-type
{ * ⊢ _} ==
{AF:A:I:fset(ℕ) ⟶ Type × (I:fset(ℕ) ⟶ J:fset(ℕ) ⟶ f:J ⟶ I ⟶ (A I) ⟶ (A J))|
let A,F = AF
in (∀I:fset(ℕ). ∀u:A I. ((F I I 1 u) = u ∈ (A I)))
∧ (∀I,J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀u:A I. ((F I K f ⋅ g u) = (F J K g (F I J f u)) ∈ (A K)))}
Lemma: closed-cubical-type_wf
{ * ⊢ _} ∈ 𝕌'
Lemma: closed-cubical-type-cumulativity
{ * ⊢ _} ⊆r { * ⊢' _}
Definition: closed-type-to-type
closed-type-to-type(T) == let A,F = T in <λI,a. (A I), λI,J,f,a,u. (F I J f u)>
Lemma: closed-type-to-type_wf
∀[T:{ * ⊢ _}]. ∀[X:j⊢]. X ⊢ closed-type-to-type(T)
Lemma: csm-closed-type
∀[T:{ * ⊢ _}]. ∀[s:Top]. ((closed-type-to-type(T))s ~ closed-type-to-type(T))
Definition: cubical-term
{X ⊢ _:A} == {u:I:fset(ℕ) ⟶ a:X(I) ⟶ A(a)| ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀a:X(I). ((u I a a f) = (u J f(a)) ∈ A(f(a)))}
Lemma: cubical-term_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ({X ⊢ _:A} ∈ 𝕌{[i | j']})
Lemma: cubical-term-eqcd
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. {X ⊢ _:A} = {X ⊢ _:B} ∈ 𝕌{[i | j']} supposing A = B ∈ {X ⊢ _}
Lemma: istype-cubical-term
∀[X:j⊢]. ∀[A:{X ⊢ _}]. istype({X ⊢ _:A})
Lemma: istype-cubical-term-closed-type
∀[X:j⊢]. ∀[T:{ * ⊢ _}]. istype({X ⊢ _:closed-type-to-type(T)})
Lemma: cubical-term-sq-presheaf-term
∀[X,A:Top]. ({X ⊢ _:A} ~ {X ⊢ _:A})
Definition: cubical-term-at
u(a) == u I a
Lemma: cubical-term-at_wf1
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. (u(a) ∈ A(a))
Lemma: cubical-term-at_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. (u(a) ∈ A(a))
Lemma: cubical-term-at-morph1
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I].
((u(a) a f) = u(f(a)) ∈ A(f(a)))
Lemma: cubical-term-at-morph
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I].
((u(a) a f) = u(f(a)) ∈ A(f(a)))
Lemma: cubical-term-equal
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[z:I:fset(ℕ) ⟶ a:X(I) ⟶ A(a)].
u = z ∈ {X ⊢ _:A} supposing u = z ∈ (I:fset(ℕ) ⟶ a:X(I) ⟶ A(a))
Lemma: cubical-term-equal2
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u,z:{X ⊢ _:A}]. u = z ∈ {X ⊢ _:A} supposing ∀I:fset(ℕ). ∀a:X(I). ((u I a) = (z I a) ∈ A(a))
Definition: closed-cubical-term
closed-cubical-term(T) ==
{u:I:fset(ℕ) ⟶ ((fst(T)) I)| ∀I,J:fset(ℕ). ∀f:J ⟶ I. (((snd(T)) I J f (u I)) = (u J) ∈ ((fst(T)) J))}
Lemma: closed-cubical-term_wf
∀[T:{ * ⊢ _}]. (closed-cubical-term(T) ∈ Type)
Definition: closed-term-to-term
closed-term-to-term(t) == λI,a. (t I)
Lemma: closed-term-to-term_wf
∀[T:{ * ⊢ _}]. ∀[t:closed-cubical-term(T)]. ∀[X:j⊢]. (closed-term-to-term(t) ∈ {X ⊢ _:closed-type-to-type(T)})
Definition: canonical-section
canonical-section(Gamma;A;I;rho;a) == λK,f. (a rho f)
Lemma: canonical-section_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[a:A(rho)].
(canonical-section(Gamma;A;I;rho;a) ∈ {formal-cube(I) ⊢ _:(A)<rho>})
Lemma: canonical-section-at
∀[Gamma,A,I,rho,a,J,x:Top]. (canonical-section(Gamma;A;I;rho;a)(x) ~ (a rho x))
Definition: trivial-section
The trivial section, () must satisfy I,0 |- ():A subset-iota().
But since the "cubical subset" I,0 is empty, everything satisfies this
property; so we could define it to be anything we like.
(We chose ⌜⋅⌝.)⋅
() == ⋅
Lemma: trivial-section_wf
∀[I:fset(ℕ)]. ∀[X:Top]. ∀[phi:Point(face_lattice(I))]. () ∈ {I,phi ⊢ _:X} supposing phi = 0 ∈ Point(face_lattice(I))
Definition: csm-ap-term
(t)s == λI,a. (t I (s)a)
Lemma: csm-ap-term_wf1
∀[Delta,Gamma:j⊢]. ∀[A:{Gamma ⊢j _}]. ∀[s:Delta j⟶ Gamma]. ∀[t:{Gamma ⊢ _:A}]. ((t)s ∈ {Delta ⊢ _:(A)s})
Lemma: csm-ap-term_wf
∀[Delta,Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[s:Delta j⟶ Gamma]. ∀[t:{Gamma ⊢ _:A}]. ((t)s ∈ {Delta ⊢ _:(A)s})
Lemma: csm-ap-term-at
∀[J,s,rho,t:Top]. ((t)s(rho) ~ t((s)rho))
Lemma: csm-ap-id-term
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[t:{Gamma ⊢ _:A}]. ((t)1(Gamma) = t ∈ {Gamma ⊢ _:A})
Lemma: csm-comp-term
∀[Gamma,Delta,Z,s1,s2,t:Top]. ((t)s2 o s1 ~ ((t)s2)s1)
Lemma: csm-ap-comp-term-sq
∀[Gamma,Delta,Z,s1,s2,t:Top]. ((t)s2 o s1 ~ ((t)s2)s1)
Lemma: csm-ap-comp-term-sq2
∀[s1,s2,t:Top]. (((t)s2)s1 ~ (t)s2 o s1)
Lemma: csm-ap-comp-term
∀[Gamma,Delta,Z:j⊢]. ∀[s1:Z j⟶ Delta]. ∀[s2:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[t:{Gamma ⊢ _:A}].
((t)s2 o s1 = ((t)s2)s1 ∈ {Z ⊢ _:(A)s2 o s1})
Lemma: context-map-comp2
∀[G:j⊢]. ∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[a:G(I)]. (<a> o <f> = <f(a)> ∈ formal-cube(J) ij⟶ G)
Lemma: csm-comp-context-map
∀[Gamma,Delta:j⊢]. ∀[sigma:Delta j⟶ Gamma]. ∀[I:fset(ℕ)]. ∀[rho:Delta(I)].
(sigma o <rho> = <(sigma)rho> ∈ formal-cube(I) j⟶ Gamma)
Definition: cube-context-adjoin
X.A == <λI.(alpha:X(I) × A(alpha)), λI,J,f,p. <f(fst(p)), (snd(p) fst(p) f)>>
Lemma: cube-context-adjoin-wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢' _}]. Gamma.A ij⊢
Lemma: cube-context-adjoin_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. Gamma.A ij⊢
Lemma: cube-context-adjoin-I_cube
∀[Gamma,A,I:Top]. (Gamma.A(I) ~ alpha:Gamma(I) × A(alpha))
Definition: cc-adjoin-cube
(v;u) == <v, u>
Lemma: cc-adjoin-cube_wf
∀X:j⊢. ∀A:{X ⊢ _}. ∀J:fset(ℕ). ∀v:X(J). ∀u:A(v). ((v;u) ∈ X.A(J))
Lemma: cc-adjoin-cube-restriction
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[J,K,g,v,u:Top]. (g((v;u)) ~ (g(v);(u v g)))
Definition: cc-fst
p == λI,p. (fst(p))
Lemma: cc-fst_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (p ∈ Gamma.A ij⟶ Gamma)
Definition: cc-snd
q == λI,p. (snd(p))
Lemma: cc-snd_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (q ∈ {Gamma.A ⊢ _:(A)p})
Lemma: cc_fst_adjoin_cube_lemma
∀y,x,J:Top. ((p)(x;y) ~ x)
Lemma: cc_snd_adjoin_cube_lemma
∀y,x,J:Top. ((q)(x;y) ~ y)
Lemma: csm_comp_fst_adjoin_cube_lemma
∀y,x,K,s,X,B,A:Top. ((s o p)(x;y) ~ (s)x)
Lemma: csm_comp_snd_adjoin-cube_lemma
∀y,x,K,s,X,B,A:Top. ((s o q)(x;y) ~ (s)y)
Definition: cc-fstfst
pp == λI,rho. (fst(fst(rho)))
Lemma: cc-fstfst_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. (pp ∈ Gamma.A.B ij⟶ Gamma)
Definition: typed-cc-fst
tp{i:l} == p
Lemma: typed-cc-fst_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. (tp{i:l} ∈ G.A ij⟶ G)
Definition: typed-cc-snd
tq == q
Lemma: typed-cc-snd_wf
∀G:j⊢. ∀A:{G ⊢ _}. (tq ∈ {G.A ⊢ _:(A)tp{i:l}})
Definition: cc-m2
q2 == (q)p
Lemma: cc-m2_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. (q2 ∈ {X.A.B ⊢ _:((A)p)p})
Definition: cc-m3
q3 == (q2)p
Lemma: cc-m3_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[C:{X.A.B ⊢ _}]. (q3 ∈ {X.A.B.C ⊢ _:(((A)p)p)p})
Definition: cc-m4
q4 == (q3)p
Lemma: cc-m4_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[C:{X.A.B ⊢ _}]. ∀[D:{X.A.B.C ⊢ _}]. (q4 ∈ {X.A.B.C.D ⊢ _:((((A)p)p)p)p})
Lemma: csm-ap-term-p
∀[Gamma:j⊢]. ∀[A,T:{Gamma ⊢ _}]. ∀[t:{Gamma ⊢ _:A}]. ((t)p ∈ {Gamma.T ⊢ _:(A)p})
Lemma: csm-type-at
∀s,A,I,alpha:Top. ((A)s(alpha) ~ A((s)alpha))
Definition: csm-adjoin
(s;u) == λI,a. <(s)a, u I a>
Lemma: csm-adjoin-wf
∀[Gamma,Delta:j⊢]. ∀[A:{Gamma ⊢' _}]. ∀[sigma:Delta j⟶ Gamma]. ∀[u:{Delta ⊢ _:(A)sigma}].
((sigma;u) ∈ Delta ij⟶ Gamma.A)
Lemma: csm-adjoin_wf
∀[Gamma,Delta:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[sigma:Delta j⟶ Gamma]. ∀[u:{Delta ⊢ _:(A)sigma}].
((sigma;u) ∈ Delta ij⟶ Gamma.A)
Lemma: csm-adjoin-comp
∀[Gamma,Delta,X,A,sigma,u,g:Top]. ((sigma;u) o g ~ (sigma o g;(u)g))
Lemma: csm-adjoin-ap
∀[sigma,u,I,del:Top]. (((sigma;u))del ~ ((sigma)del;(u)del))
Definition: csm-id-adjoin
[u] == (1(X);u)
Lemma: csm-id-adjoin-wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢' _}]. ∀[u:{Gamma ⊢ _:A}]. ([u] ∈ Gamma ij⟶ Gamma.A)
Lemma: csm-id-adjoin_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[u:{Gamma ⊢ _:A}]. ([u] ∈ Gamma ij⟶ Gamma.A)
Lemma: csm-id-adjoin-ap
∀X,u,I,a:Top. (([u])a ~ (a;u I a))
Lemma: cc-fst-csm-adjoin-sq
∀[Gamma,Delta,X,sigma,u:Top]. (p o (sigma;u) ~ sigma o 1(Delta))
Lemma: cc-fst-csm-adjoin
∀[Gamma,Delta:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[sigma:Delta j⟶ Gamma]. ∀[u:{Delta ⊢ _:(A)sigma}].
(p o (sigma;u) = sigma ∈ Delta j⟶ Gamma)
Lemma: cc_snd_csm_id_adjoin_lemma
∀u,G:Top. ((q)[u] ~ (u)1(G))
Lemma: typed_cc_snd_id_adjoin_lemma
∀A,G,u,X:Top. ((tq)[u] ~ (u)1(X))
Lemma: csm_id_adjoin_fst_type_lemma
∀A,t,X:Top. (((A)p)[t] ~ (A)1(X))
Lemma: csm_id_adjoin_fst_term_lemma
∀a,t,X:Top. (((a)p)[t] ~ (a)1(X))
Lemma: csm_id_ap_term_lemma
∀t,X,s:Top. (((t)1(X))s ~ (t)s)
Lemma: cc-snd-csm-adjoin-sq
∀[G,sigma,u:Top]. ((q)(sigma;u) ~ (u)1(G))
Lemma: cc-snd-csm-adjoin
∀[Gamma,Delta:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[sigma:Delta j⟶ Gamma]. ∀[u:{Delta ⊢ _:(A)sigma}].
((q)(sigma;u) = u ∈ {Delta ⊢ _:(A)sigma})
Lemma: csm-adjoin-fst-snd
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ((p;q) = 1(Gamma.A) ∈ Gamma.A ij⟶ Gamma.A)
Lemma: csm-ap-type-fst-adjoin
∀[X:j⊢]. ∀[B:{X ⊢ _}]. ∀[s,u:Top]. (((B)p)(s;u) ~ (B)s)
Lemma: csm-ap-type-fst-id-adjoin
∀[X:j⊢]. ∀[B:{X ⊢ _}]. ∀[u:Top]. (((B)p)[u] = B ∈ {X ⊢ _})
Lemma: csm-id-adjoin-ap-type
∀Gamma,Delta:j⊢. ∀A:{Gamma ⊢ _}. ∀B:{Gamma.A ⊢ _}. ∀sigma:Delta j⟶ Gamma. ∀u:{Delta ⊢ _:(A)sigma}.
(((B)(sigma o p;q))[u] = (B)(sigma;u) ∈ {Delta ⊢ _})
Lemma: csm_ap_term_fst_adjoin_lemma
∀zz,yy,xx:Top. (((zz)p)(xx;yy) ~ (zz)xx)
Lemma: csm-ap-term-snd-adjoin
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[xx:Top]. ((q)(xx;u) = u ∈ {X ⊢ _:A})
Lemma: csm-adjoin-id-adjoin
∀[B,xx,s,X,Y,Z:Top]. (((B)(s o p;q))[xx] ~ (B)(s;xx))
Definition: csm+
tau+ == (tau o p;q)
Lemma: csm+_wf
∀[H,K:j⊢]. ∀[A:{H ⊢ _}]. ∀[tau:K j⟶ H]. (tau+ ∈ K.(A)tau ij⟶ H.A)
Lemma: csm+-comp-csm+-sq
∀[H,K,X,A,tau,s:Top]. (tau+ o s+ ~ tau o s+)
Lemma: csm+-id
∀[G:j⊢]. ∀[A:{G ⊢ _}]. (1(G)+ = 1(G.A) ∈ G.A ij⟶ G.A)
Lemma: csm+-comp-csm+
∀[H,K,X:j⊢]. ∀[A:{H ⊢ _}]. ∀[tau:K j⟶ H]. ∀[s:X j⟶ K]. (tau+ o s+ = tau o s+ ∈ X.((A)tau)s ij⟶ H.A)
Lemma: csm+_wf+
∀[H,K:j⊢]. ∀[A:{H ⊢ _}]. ∀[tau:K ij⟶ H]. ∀[B:{H.A ⊢ _}]. (tau++ ∈ K.(A)tau.(B)tau+ ij⟶ H.A.B)
Lemma: p-csm+-type
∀[H,K,A,B,tau:Top]. (((A)p)tau+ ~ ((A)tau)p)
Lemma: csm+-p-type
∀[C,H,K,A,B,tau:Top]. (((A)p)(tau+ o p;q) ~ ((A)tau+)p)
Lemma: p-csm+-term
∀[H,K,A,t,tau:Top]. (((t)p)tau+ ~ ((t)tau)p)
Lemma: csm+-p-term
∀[C,H,K,t,B,tau:Top]. (((t)p)(tau+ o p;q) ~ ((t)tau+)p)
Lemma: q-csm+
∀[H,K,A,tau:Top]. ((q)tau+ ~ q)
Lemma: csm-adjoin-p-q
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ((B)(p;q) = B ∈ {X.A ⊢ _})
Definition: csm-swap
csm-swap(G;A;B) == (p+;(q)p)
Lemma: csm-swap_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. (csm-swap(G;A;B) ∈ G.A.(B)p ij⟶ G.B.(A)p)
Definition: cubical-pi-family
cubical-pi-family(X;A;B;I;a) ==
{w:J:fset(ℕ) ⟶ f:J ⟶ I ⟶ u:A(f(a)) ⟶ B((f(a);u))|
∀J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀u:A(f(a)). ((w J f u (f(a);u) g) = (w K f ⋅ g (u f(a) g)) ∈ B(g((f(a);u))))}
Lemma: cubical-pi-family_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. (cubical-pi-family(X;A;B;I;a) ∈ Type)
Lemma: cubical-pi-family-comp
∀X,Delta:j⊢. ∀s:Delta j⟶ X. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀a:Delta(I). ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}.
∀w:cubical-pi-family(X;A;B;I;(s)a).
(λK,g. (w K f ⋅ g) ∈ cubical-pi-family(X;A;B;J;(s)f(a)))
Lemma: csm-cubical-pi-family
∀X,Delta:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta j⟶ X. ∀I:fset(ℕ). ∀a:Delta(I).
(cubical-pi-family(X;A;B;I;(s)a) = cubical-pi-family(Delta;(A)s;(B)(s o p;q);I;a) ∈ Type)
Definition: cubical-fun-family
cubical-fun-family(X; A; B; I; a) ==
{w:J:fset(ℕ) ⟶ f:J ⟶ I ⟶ u:A(f(a)) ⟶ B(f(a))|
∀J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀u:A(f(a)). ((w J f u f(a) g) = (w K f ⋅ g (u f(a) g)) ∈ B(g(f(a))))}
Lemma: cubical-fun-family_wf
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. (cubical-fun-family(X; A; B; I; a) ∈ Type)
Lemma: cubical-fun-family-comp
∀X,Delta:j⊢. ∀s:Delta j⟶ X. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀a:Delta(I). ∀A,B:{X ⊢ _}.
∀w:cubical-fun-family(X; A; B; I; (s)a).
(λK,g. (w K f ⋅ g) ∈ cubical-fun-family(X; A; B; J; (s)f(a)))
Lemma: csm-cubical-fun-family
∀X,Delta:j⊢. ∀A,B:{X ⊢ _}. ∀s:Delta j⟶ X. ∀I:fset(ℕ). ∀a:Delta(I).
(cubical-fun-family(X; A; B; I; (s)a) = cubical-fun-family(Delta; (A)s; (B)s; I; a) ∈ Type)
Definition: cubical-pi
ΠA B == <λI,a. cubical-pi-family(X;A;B;I;a), λI,J,f,a,w,K,g. (w K f ⋅ g)>
Lemma: cubical-pi_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. X ⊢ ΠA B
Lemma: csm-cubical-pi
∀X,Delta:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta j⟶ X. ((ΠA B)s = Delta ⊢ Π(A)s (B)(s o p;q) ∈ {Delta ⊢ _})
Lemma: cubical-pi-p
∀X:j⊢. ∀T,A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ((ΠA B)p = X.T ⊢ Π(A)p (B)(p o p;q) ∈ {X.T ⊢ _})
Definition: csm-dependent
(s)dep == (s o tp{i:l};tq)
Lemma: csm-dependent_wf
∀X,Delta:j⊢. ∀A:{X ⊢ _}. ∀s:Delta j⟶ X. ((s)dep ∈ Delta.(A)s j⟶ X.A)
Lemma: csm_dep_p_rewrite_lemma
∀T,s,A,Delta,X:Top. (((T)p)(s)dep ~ ((T)s)p)
Lemma: csm_dep_typed_p_lemma
∀T,s,A,Delta,X:Top. (((T)tp{i:l})(s)dep ~ ((T)s)tp{i:l})
Lemma: csm-dep_term_p_lemma
∀t,s,A,Delta,X:Top. (((t)tp{i:l})(s)dep ~ ((t)s)tp{i:l})
Lemma: csm_dep_term_q_lemma
∀s,A,Delta,X:Top. ((tq)(s)dep ~ tq)
Lemma: csm-cubical-pi-typed
∀X,Delta:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta ij⟶ X. ((ΠA B)s = Delta ⊢ Π(A)s (B)(s)dep ∈ {Delta ⊢ _})
Definition: cubical-fun
(A ⟶ B) == <λI,a. cubical-fun-family(X; A; B; I; a), λI,J,f,a,w,K,g. (w K f ⋅ g)>
Lemma: cubical-fun_wf
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. X ⊢ (A ⟶ B)
Lemma: csm-cubical-fun
∀X,Delta:j⊢. ∀A,B:{X ⊢ _}. ∀s:Delta j⟶ X. (((A ⟶ B))s = (Delta ⊢ (A)s ⟶ (B)s) ∈ {Delta ⊢ _})
Lemma: cubical-fun-equal
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[f,g:{X ⊢ _:(A ⟶ B)}].
f = g ∈ {X ⊢ _:(A ⟶ B)}
supposing ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[J:fset(ℕ)]. ∀[h:J ⟶ I]. ∀[u:A(h(a))]. ((f(a) J h u) = (g(a) J h u) ∈ B(h(a)))
Lemma: cubical-fun-equal2
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[f:{X ⊢ _:(A ⟶ B)}].
∀[g:I:fset(ℕ) ⟶ a:X(I) ⟶ J:fset(ℕ) ⟶ h:J ⟶ I ⟶ u:A(h(a)) ⟶ B(h(a))].
f = g ∈ {X ⊢ _:(A ⟶ B)}
supposing ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[J:fset(ℕ)]. ∀[h:J ⟶ I]. ∀[u:A(h(a))]. ((f(a) J h u) = (g(a) J h u) ∈ B(h(a)))
Lemma: cubical-fun-p
∀X:j⊢. ∀A,B,T:{X ⊢ _}. (((A ⟶ B))p = (X.T ⊢ (A)p ⟶ (B)p) ∈ {X.T ⊢ _})
Lemma: member-cubical-fun-p
∀[X:j⊢]. ∀[A,B,T:{X ⊢ _}]. ∀[x:{X ⊢ _:(A ⟶ B)}]. ((x)p ∈ {X.T ⊢ _:((A)p ⟶ (B)p)})
Lemma: cc-snd_wf-cubical-fun
∀X:j⊢. ∀A,B:{X ⊢ _}. (q ∈ {X.(A ⟶ B) ⊢ _:((A)p ⟶ (B)p)})
Lemma: cubical-fun-as-cubical-pi
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ((X ⊢ A ⟶ B) = X ⊢ ΠA (B)p ∈ {X ⊢ _})
Definition: cubical-lambda
(λb) == λI,a,J,f,u. (b J (f(a);u))
Lemma: cubical-lambda_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[b:{X.A ⊢ _:B}]. ((λb) ∈ {X ⊢ _:ΠA B})
Lemma: csm-cubical-lambda
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[b:{X.A ⊢ _:B}]. ∀[H:j⊢]. ∀[s:H j⟶ X]. (((λb))s = (λ(b)s+) ∈ {H ⊢ _:(ΠA B)s})
Definition: cubical-lam
cubical-lam(X;b) == (λb)
Lemma: cubical-lam_wf
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[b:{X.A ⊢ _:(B)p}]. (cubical-lam(X;b) ∈ {X ⊢ _:(A ⟶ B)})
Lemma: csm-cubical-lam
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[b:{X.A ⊢ _:(B)p}]. ∀[H:j⊢]. ∀[s:H j⟶ X].
((cubical-lam(X;b))s = cubical-lam(H;(b)s+) ∈ {H ⊢ _:((A ⟶ B))s})
Definition: cubical-id-fun
cubical-id-fun(X) == cubical-lam(X;q)
Lemma: cubical-id-fun_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. (cubical-id-fun(X) ∈ {X ⊢ _:(A ⟶ A)})
Lemma: csm-cubical-id-fun
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[H:j⊢]. ∀[s:H j⟶ X]. ((cubical-id-fun(X))s = cubical-id-fun(H) ∈ {H ⊢ _:((A)s ⟶ (A)s)})
Definition: cubical-app
app(w; u) == λI,a. (w I a I 1 (u I a))
Lemma: cubical-app_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[u:{X ⊢ _:A}]. (app(w; u) ∈ {X ⊢ _:(B)[u]})
Lemma: cubical-app_wf-csm
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[H:j⊢]. ∀[tau:H j⟶ X]. ∀[u:{H ⊢ _:(A)tau}].
(app((w)tau; u) ∈ {H ⊢ _:((B)tau+)[u]})
Definition: cubical-apply
cubical-apply(w;u) == app(w; u)
Lemma: cubical-apply_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[u:{X ⊢ _:A}]. (cubical-apply(w;u) ∈ {X ⊢ _:(B)[u]})
Lemma: cubical-app_wf_fun
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[w:{X ⊢ _:(A ⟶ B)}]. ∀[u:{X ⊢ _:A}]. (app(w; u) ∈ {X ⊢ _:B})
Definition: cubical-fun-comp
(f o g) == cubical-lam(G;app((f)p; app((g)p; q)))
Lemma: cubical-fun-comp_wf
∀[X:j⊢]. ∀[A,B,C:{X ⊢ _}]. ∀[g:{X ⊢ _:(A ⟶ B)}]. ∀[f:{X ⊢ _:(B ⟶ C)}]. ((f o g) ∈ {X ⊢ _:(A ⟶ C)})
Definition: cubical-sigma
Σ A B == <λI,a. (u:A(a) × B((a;u))), λI,J,f,a,p. <(fst(p) a f), (snd(p) (a;fst(p)) f)>>
Lemma: cubical-sigma_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. X ⊢ Σ A B
Lemma: cubical-sigma-at
∀[X,A,B,I,a:Top]. (Σ A B(a) ~ u:A(a) × B((a;u)))
Lemma: csm-cubical-sigma
∀X,Delta:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta j⟶ X. ((Σ A B)s = Σ (A)s (B)(s o p;q) ∈ {Delta ⊢ _})
Lemma: cubical-sigma-p
∀X:j⊢. ∀T,A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ((Σ A B)p = Σ (A)p (B)(p o p;q) ∈ {X.T ⊢ _})
Lemma: cubical-sigma-p-p
∀X:j⊢. ∀T,A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀C:{X.T ⊢ _}. (((Σ A B)p)p = Σ ((A)p)p (B)(p o p o p;q) ∈ {X.T.C ⊢ _})
Lemma: csm-cubical-sigma-typed
∀X,Delta:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta j⟶ X. ((Σ A B)s = Σ (A)s (B)(s)dep ∈ {Delta ⊢ _})
Definition: cubical-fst
p.1 == λI,a. (fst((p I a)))
Lemma: cubical-fst_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[p:{X ⊢ _:Σ A B}]. (p.1 ∈ {X ⊢ _:A})
Lemma: csm-cubical-fst
∀[p,s:Top]. ((p.1)s ~ (p)s.1)
Lemma: csm-ap-cubical-fst
∀[X,Delta:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[p:{X ⊢ _:Σ A B}]. ∀[s:Delta j⟶ X]. ((p.1)s = (p)s.1 ∈ {Delta ⊢ _:(A)s})
Definition: cubical-snd
p.2 == λI,a. (snd((p I a)))
Lemma: cubical-snd_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[p:{X ⊢ _:Σ A B}]. (p.2 ∈ {X ⊢ _:(B)[p.1]})
Lemma: cubical-snd-at
∀[a,I,p:Top]. (p.2(a) ~ snd(p(a)))
Lemma: cubical-fst-at
∀[a,I,p:Top]. (p.1(a) ~ fst(p(a)))
Lemma: csm-cubical-snd
∀[p,s:Top]. ((p.2)s ~ (p)s.2)
Lemma: csm-ap-cubical-snd
∀[X,Delta:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[p:{X ⊢ _:Σ A B}]. ∀[s:Delta j⟶ X].
((p.2)s = (p)s.2 ∈ {Delta ⊢ _:((B)[p.1])s})
Definition: cubical-pair
cubical-pair(u;v) == λI,a. <u I a, v I a>
Lemma: cubical-pair_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[v:{X ⊢ _:(B)[u]}]. (cubical-pair(u;v) ∈ {X ⊢ _:Σ A B})
Lemma: csm-cubical-pair
∀[u,v,s:Top]. ((cubical-pair(u;v))s ~ cubical-pair((u)s;(v)s))
Definition: sigma-elim-csm
SigmaElim == λI,x. let a,u,v = x in ((a;u);v)
Lemma: sigma-elim-csm_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. (SigmaElim ∈ X.Σ A B ij⟶ X.A.B)
Definition: sigma-unelim-csm
SigmaUnElim == λI,x. let au,v = x in let a,u = au in (a;(u;v))
Lemma: sigma-unelim-csm_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. (SigmaUnElim ∈ X.A.B ij⟶ X.Σ A B)
Lemma: sigma-elim-unelim
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. (SigmaUnElim o SigmaElim = 1(X.Σ A B) ∈ X.Σ A B ij⟶ X.Σ A B)
Lemma: sigma-unelim-elim
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. (SigmaElim o SigmaUnElim = 1(X.A.B) ∈ X.A.B ij⟶ X.A.B)
Lemma: sigma-unelim-elim-type
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[T:{X.Σ A B ⊢ _}]. (((T)SigmaUnElim)SigmaElim = T ∈ {X.Σ A B ⊢ _})
Lemma: sigma-unelim-elim-term
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[T:{X.Σ A B ⊢ _}]. ∀[t:{X.Σ A B ⊢ _:T}].
(((t)SigmaUnElim)SigmaElim = t ∈ {X.Σ A B ⊢ _:T})
Lemma: sigma-unelim-p
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. (p o SigmaUnElim = p o p ∈ X.A.B ij⟶ X)
Lemma: sigma-unelim-p-type
∀[T:Top]. (((T)p)SigmaUnElim ~ ((T)p)p)
Lemma: sigma-unelim-p-term
∀[t:Top]. (((t)p)SigmaUnElim ~ ((t)p)p)
Lemma: sigma-elim-rule
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[T:{X.Σ A B ⊢ _}]. ∀[t:{X.A.B ⊢ _:(T)SigmaUnElim}].
((t)SigmaElim ∈ {X.Σ A B ⊢ _:T})
Lemma: sigma-elim-equality-rule
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[T:{X.Σ A B ⊢ _}]. ∀[t1,t2:{X.A.B ⊢ _:(T)SigmaUnElim}].
(t1)SigmaElim = (t2)SigmaElim ∈ {X.Σ A B ⊢ _:T} supposing t1 = t2 ∈ {X.A.B ⊢ _:(T)SigmaUnElim}
Lemma: sigma-elim-equality-rule2
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[T:{X.Σ A B ⊢ _}]. ∀[t1:{X.A.B ⊢ _:(T)SigmaUnElim}]. ∀[t2:{X.Σ A B ⊢ _:T}].
(t1)SigmaElim = t2 ∈ {X.Σ A B ⊢ _:T} supposing t1 = (t2)SigmaUnElim ∈ {X.A.B ⊢ _:(T)SigmaUnElim}
Lemma: csm-ap-cubical-pair
∀[X,Delta:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[v:{X ⊢ _:(B)[u]}]. ∀[s:Delta j⟶ X].
((cubical-pair(u;v))s = cubical-pair((u)s;(v)s) ∈ {Delta ⊢ _:(Σ A B)s})
Lemma: csm-cubical-app
∀[w,u,s:Top]. ((app(w; u))s ~ app((w)s; (u)s))
Lemma: csm-ap-cubical-app
∀[X,Delta:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[u:{X ⊢ _:A}]. ∀[s:Delta j⟶ X].
((app(w; u))s = app((w)s; (u)s) ∈ {Delta ⊢ _:((B)[u])s})
Lemma: csm-ap-cubical-app-fun
∀[X,Delta:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[w:{X ⊢ _:(A ⟶ B)}]. ∀[u:{X ⊢ _:A}]. ∀[s:Delta j⟶ X].
((app(w; u))s = app((w)s; (u)s) ∈ {Delta ⊢ _:(B)s})
Lemma: csm-ap-cubical-lambda
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[b:{X.A ⊢ _:B}]. ∀[H:j⊢]. ∀[s:H j⟶ X]. (((λb))s = (λ(b)s+) ∈ {H ⊢ _:(ΠA B)s})
Lemma: cubical-snd-pair
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[v:{X ⊢ _:(B)[u]}]. (cubical-pair(u;v).2 = v ∈ {X ⊢ _:(B)[u]})
Lemma: cubical-fst-pair
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[v:Top]. (cubical-pair(u;v).1 = u ∈ {X ⊢ _:A})
Lemma: cubical-pair-eta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:Σ A B}]. (cubical-pair(w.1;w.2) = w ∈ {X ⊢ _:Σ A B})
Lemma: cubical-sigma-equal
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w,y:{X ⊢ _:Σ A B}].
(w = y ∈ {X ⊢ _:Σ A B}) supposing ((w.2 = y.2 ∈ {X ⊢ _:(B)[w.1]}) and (w.1 = y.1 ∈ {X ⊢ _:A}))
Lemma: cubical-eta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ((λapp((w)p; q)) = w ∈ {X ⊢ _:ΠA B})
Lemma: cubical-fun-eta
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[w:{X ⊢ _:(A ⟶ B)}]. (cubical-lam(X;app((w)p; q)) = w ∈ {X ⊢ _:(A ⟶ B)})
Lemma: cubical-beta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[b:{X.A ⊢ _:B}]. ∀[u:{X ⊢ _:A}]. (app((λb); u) = (b)[u] ∈ {X ⊢ _:(B)[u]})
Lemma: cubical-app-id-fun
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. (app(cubical-id-fun(X); u) = u ∈ {X ⊢ _:A})
Definition: cubical-sigma-fun
cubical-sigma-fun(G;A;B;f) == cubical-lam(G;cubical-pair(q.1;app(app(((λf))p; q.1); q.2)))
Lemma: cubical-sigma-fun_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[B,B':{G.A ⊢ _}]. ∀[f:{G.A ⊢ _:(B ⟶ B')}].
(cubical-sigma-fun(G;A;B;f) ∈ {G ⊢ _:(Σ A B ⟶ Σ A B')})
Definition: discrete-cubical-type
discr(T) == <λI,alpha. T, λI,J,f,alpha,x. x>
Lemma: discrete-cubical-type_wf
∀[T:Type]. ∀[X:j⊢]. X ⊢ discr(T)
Lemma: csm-discrete-cubical-type
∀[T,s:Top]. ((discr(T))s ~ discr(T))
Definition: discrete-cubical-term
discr(t) == λI,alpha. t
Lemma: discrete-cubical-term_wf
∀[T:Type]. ∀[t:T]. ∀[X:j⊢]. (discr(t) ∈ {X ⊢ _:discr(T)})
Lemma: discrete-cubical-term-at-morph
∀[T:Type]. ∀[X:j⊢]. ∀[t:{X ⊢ _:discr(T)}]. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀a:X(I). (t(a) = t(f(a)) ∈ T)
Lemma: discrete-unary_wf
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[X:j⊢]. ∀[t:{X ⊢ _:discr(A)}]. (discrete-unary(t;x.f[x]) ∈ {X ⊢ _:discr(B)})
Lemma: csm-discrete-cubical-term
∀[t,s:Top]. ((discr(t))s ~ discr(t))
Lemma: discrete-cubical-term-is-map
∀[T:Type]. ∀[X:j⊢]. {X ⊢ _:discr(T)} ≡ X ij⟶ discrete-cube(T)
Lemma: discrete-map-is-constant
∀[T:𝕌{j}]. ∀[I:fset(ℕ)]. ∀[s:formal-cube(I) ij⟶ discrete-cube(T)].
(s = (λJ,g. (s I 1)) ∈ formal-cube(I) ij⟶ discrete-cube(T))
Lemma: discrete-cubical-term-is-constant
Not every term of a discrete cubical type is constant, but when the
context is the formal-cube(I) -- Yoneda(I) -- then it is.⋅
∀[T:Type]. ∀[I:fset(ℕ)]. ∀[t:{formal-cube(I) ⊢ _:discr(T)}]. (t = discr(t(1)) ∈ {formal-cube(I) ⊢ _:discr(T)})
Lemma: discrete-map-is-constant2
∀[T:Type]. ∀[I:fset(ℕ)]. ∀[phi:𝔽(I)]. ∀[s:I,phi ⟶ discrete-cube(T)]. ∀[f:{f:I ⟶ I| (phi f) = 1} ].
s = (λJ,g. (s I f)) ∈ I,phi ⟶ discrete-cube(T)
supposing ∀[J:fset(ℕ)]. ∀[g:J ⟶ I]. ((phi g) = 1 ⇒ (g = f ⋅ g ∈ J ⟶ I))
Lemma: discrete-cubical-term-is-constant-on-irr-face
Not every term of a discrete cubical type is constant, but when the
context is the formal-cube(I) -- Yoneda(I) -- then it is.⋅
∀[T:Type]. ∀[I:fset(ℕ)]. ∀[as,bs:fset(names(I))].
∀[t:{I,irr_face(I;as;bs) ⊢ _:discr(T)}]. (t = discr(t(irr-face-morph(I;as;bs))) ∈ {I,irr_face(I;as;bs) ⊢ _:discr(T)})
supposing ↑fset-disjoint(NamesDeq;as;bs)
Definition: cubical-unit
1 == discr(Unit)
Lemma: cubical-unit_wf
∀[X:j⊢]. X ⊢ 1
Lemma: csm-cubical-unit
∀[s:Top]. ((1)s ~ 1)
Definition: cubical-it
* == discr(⋅)
Lemma: cubical-it_wf
∀[X:j⊢]. (* ∈ {X ⊢ _:1})
Lemma: cubical-it-unique
∀[X:j⊢]. ∀[t:{X ⊢ _:1}]. (t = * ∈ {X ⊢ _:1})
Definition: constant-cubical-type
(X) == <λI,alpha. X(I), λI,J,f,alpha,w. f(w)>
Lemma: constant-cubical-type_wf
∀[X:SmallCubicalSet]. ∀[Gamma:j⊢]. Gamma ⊢ (X)
Lemma: csm-constant-cubical-type
∀[X,s:Top]. (((X))s ~ (X))
Lemma: constant-cubical-type-at
∀[X,I,a:Top]. ((X)(a) ~ X(I))
Lemma: unit-cube-cubical-type
∀[I:fset(ℕ)]
({formal-cube(I) ⊢ _} ~ {AF:A:L:fset(ℕ) ⟶ L ⟶ I ⟶ Type × (L:fset(ℕ)
⟶ J:fset(ℕ)
⟶ f:J ⟶ L
⟶ a:L ⟶ I
⟶ (A L a)
⟶ (A J a ⋅ f))|
let A,F = AF
in (∀K:fset(ℕ). ∀a:K ⟶ I. ∀u:A K a. ((F K K 1 a u) = u ∈ (A K a)))
∧ (∀L,J,K:fset(ℕ). ∀f:J ⟶ L. ∀g:K ⟶ J. ∀a:L ⟶ I. ∀u:A L a.
((F L K f ⋅ g a u) = (F J K g a ⋅ f (F L J f a u)) ∈ (A K a ⋅ f ⋅ g)))} )
Lemma: empty-cubical-subset-I_cube
∀[I,J:fset(ℕ)]. (¬I,0(J))
Lemma: empty-cubical-subset
∀[I:fset(ℕ)]. ∀[A,B:Top]. I,0 ⊢ <A, B>
Lemma: empty-cubical-subset-term
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))].
∀[X,A,B:Top]. (A = B ∈ {I,phi ⊢ _:X}) supposing phi = 0 ∈ Point(face_lattice(I))
Lemma: member-empty-cubical-subset
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))].
∀[X,A:Top]. (A ∈ {I,phi ⊢ _:X}) supposing phi = 0 ∈ Point(face_lattice(I))
Definition: section-iota
section-iota(Gamma;A;I;rho;a) == (canonical-section(Gamma;A;I;rho;a))iota
Lemma: section-iota_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[a:A(rho)]. ∀[psi:𝔽(I)].
(section-iota(Gamma;A;I;rho;a) ∈ {I,psi ⊢ _:((A)<rho>)iota})
Lemma: cubical-term-at-comp
∀Gamma:j⊢. ∀T:{Gamma ⊢ _}. ∀t:{Gamma ⊢ _:T}. ∀I:fset(ℕ). ∀rho:Gamma(I). ∀J:fset(ℕ). ∀f:J ⟶ I. ∀K:fset(ℕ). ∀g:K ⟶ J.
(t(f ⋅ g(rho)) = t(g(f(rho))) ∈ T(g(f(rho))))
Lemma: subset-cubical-type-general
∀[X,Y:j⊢]. {X ⊢j _} ⊆r {Y ⊢j _} supposing sub_cubical_set{j:l}(Y; X)
Lemma: subset-cubical-type
∀[X,Y:j⊢]. {X ⊢ _} ⊆r {Y ⊢ _} supposing sub_cubical_set{j:l}(Y; X)
Lemma: subset-cubical-term
∀[X,Y:j⊢]. ∀[A:{X ⊢ _}]. ({X ⊢ _:A} ⊆r {Y ⊢ _:A}) supposing sub_cubical_set{j:l}(Y; X)
Lemma: subset-cubical-term2
∀[X,Y:j⊢]. ∀[A,B:{X ⊢ _}]. {X ⊢ _:A} ⊆r {Y ⊢ _:B} supposing A = B ∈ {X ⊢ _} supposing sub_cubical_set{j:l}(Y; X)
Lemma: csm-subset-codomain
∀[X,Y,Z:j⊢]. X j⟶ Y ⊆r X j⟶ Z supposing sub_cubical_set{j:l}(Y; Z)
Lemma: csm-subset-domain
∀[X,Y,Z:j⊢]. Y j⟶ X ⊆r Z j⟶ X supposing sub_cubical_set{j:l}(Z; Y)
Lemma: csm-subset-subtype
∀[A,B,Y,Z:j⊢]. (Y j⟶ A ⊆r Z j⟶ B) supposing (sub_cubical_set{j:l}(A; B) and sub_cubical_set{j:l}(Z; Y))
Lemma: sub_cubical_set_functionality
∀[Y,X:j⊢]. ∀[A:{X ⊢ _}]. sub_cubical_set{[i | j]:l}(Y.A; X.A) supposing sub_cubical_set{j:l}(Y; X)
Lemma: cubical-pi-intro
∀G:j⊢. ∀A:{G ⊢ _}. ∀B:{G.A ⊢ _}. ({G.A ⊢ _:B} ⇒ {G ⊢ _:ΠA B})
Lemma: cubical-sigma-intro
∀G:j⊢. ∀A:{G ⊢ _}. ∀B:{G.A ⊢ _}. ((∃a:{G ⊢ _:A}. {G ⊢ _:(B)[a]}) ⇒ {G ⊢ _:Σ A B})
Lemma: cubical-pi-implies-sigma
∀G:j⊢. ∀A:{G ⊢ _}. ∀B:{G.A ⊢ _}. ({G ⊢ _:A} ⇒ {G ⊢ _:ΠA B} ⇒ {G ⊢ _:Σ A B})
Lemma: csm_typedp_id_adjoin_lemma
∀E,s,F,G,u,H:Top. ((((E)s)tp{i:l})[u] ~ (E)s)
Lemma: cubical-fun-family-discrete
∀[A,B:Type]. ∀[X:j⊢]. ∀[I:fset(ℕ)]. ∀[a:X(I)].
(cubical-fun-family(X; discr(A); discr(B); I; a)
= {w:J:fset(ℕ) ⟶ f:J ⟶ I ⟶ u:A ⟶ B| ∀J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀u:A. ((w J f u) = (w K f ⋅ g u) ∈ B)}
∈ Type)
Definition: dcff-inj
dcff-inj(I;w) == w I 1
Lemma: dcff-inj_wf
∀[A,B:Type]. ∀[X:j⊢]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[w:cubical-fun-family(X; discr(A); discr(B); I; a)].
(dcff-inj(I;w) ∈ A ⟶ B)
Lemma: dcff-inj-injection
∀[A,B:Type]. ∀[X:j⊢]. ∀[I:fset(ℕ)]. ∀[a:X(I)].
Inj(cubical-fun-family(X; discr(A); discr(B); I; a);A ⟶ B;λw.dcff-inj(I;w))
Lemma: dcff-inj-bijection
∀[A,B:Type]. ∀[X:j⊢]. ∀[I:fset(ℕ)]. ∀[a:X(I)].
Bij(cubical-fun-family(X; discr(A); discr(B); I; a);A ⟶ B;λw.dcff-inj(I;w))
Comment: interval type doc
================================= interval type ===============================
From the interval presheaf 𝕀 we get a cubical type 𝕀 and operations on it.⋅
Definition: interval-type
𝕀 == (𝕀)
Lemma: interval-type_wf
∀[Gamma:j⊢]. Gamma ⊢ 𝕀
Lemma: interval-type-ap-morph
∀[I,J,f,rho,u:Top]. ((u rho f) ~ dM-lift(J;I;f) u)
Lemma: interval-type-ap-inc
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[x:names(J)]. ∀[rho:Top]. ((<x> rho f) = (f x) ∈ Point(dM(I)))
Lemma: interval-type-at
∀[J,rho:Top]. (𝕀(rho) ~ 𝕀(J))
Lemma: interval-type-at-is-point
∀[J,rho:Top]. (𝕀(rho) ~ Point(dM(J)))
Lemma: csm-interval-type
∀[s:Top]. ((𝕀)s ~ 𝕀)
Lemma: csm+-comp-csm+-sq-interval
∀[H,K,X,tau,s:Top]. (tau+ o s+ ~ tau o s+)
Definition: interval-0
0(𝕀) == λI,rho. 0
Lemma: interval-0_wf
∀[Gamma:j⊢]. (0(𝕀) ∈ {Gamma ⊢ _:𝕀})
Lemma: csm-id-adjoin_wf-interval-0
∀[Gamma:j⊢]. ([0(𝕀)] ∈ Gamma j⟶ Gamma.𝕀)
Lemma: csm-interval-0
∀[s:Top]. ((0(𝕀))s ~ 0(𝕀))
Lemma: cc-fst_wf_interval
∀[Gamma:j⊢]. (p ∈ Gamma.𝕀 j⟶ Gamma)
Lemma: cc-snd-0
∀[X:Top]. ((q)[0(𝕀)] ~ 0(𝕀))
Lemma: interval-0-at
∀[I,a:Top]. (0(𝕀)(a) ~ 0)
Definition: interval-1
1(𝕀) == λI,rho. 1
Lemma: interval-1_wf
∀[Gamma:j⊢]. (1(𝕀) ∈ {Gamma ⊢ _:𝕀})
Lemma: csm-id-adjoin_wf-interval-1
∀[Gamma:j⊢]. ([1(𝕀)] ∈ Gamma j⟶ Gamma.𝕀)
Lemma: csm-interval-1
∀[s:Top]. ((1(𝕀))s ~ 1(𝕀))
Lemma: cc-snd-1
∀[X:Top]. ((q)[1(𝕀)] ~ 1(𝕀))
Lemma: csm-ap-interval-1-adjoin-lemma
∀H:j⊢. ∀I:fset(ℕ). ∀v:H(I). ∀j:{i:ℕ| ¬i ∈ I} . ((j1)((s(v);<j>)) = ([1(𝕀)])v ∈ H.𝕀(I))
Lemma: interval-1-at
∀[I,a:Top]. (1(𝕀)(a) ~ 1)
Definition: interval-rev
1-(r) == λI,rho. ¬(r(rho))
Lemma: interval-rev_wf
∀[Gamma:j⊢]. ∀[r:{Gamma ⊢ _:𝕀}]. (1-(r) ∈ {Gamma ⊢ _:𝕀})
Lemma: interval-rev-0
∀[Gamma:j⊢]. (1-(0(𝕀)) = 1(𝕀) ∈ {Gamma ⊢ _:𝕀})
Lemma: interval-rev-1
∀[Gamma:j⊢]. (1-(1(𝕀)) = 0(𝕀) ∈ {Gamma ⊢ _:𝕀})
Definition: interval-meet
r ∧ s == λI,rho. r(rho) ∧ s(rho)
Lemma: interval-meet_wf
∀[Gamma:j⊢]. ∀[r,s:{Gamma ⊢ _:𝕀}]. (r ∧ s ∈ {Gamma ⊢ _:𝕀})
Lemma: csm-interval-meet
∀[h,r,s:Top]. ((r ∧ s)h ~ (r)h ∧ (s)h)
Lemma: interval-meet-1
∀[X:j⊢]. ∀[z:{X ⊢ _:𝕀}]. (z ∧ 1(𝕀) = z ∈ {X ⊢ _:𝕀})
Lemma: interval-meet-0
∀[X:j⊢]. ∀[z:{X ⊢ _:𝕀}]. (z ∧ 0(𝕀) = 0(𝕀) ∈ {X ⊢ _:𝕀})
Lemma: interval-meet-comm
∀[X:j⊢]. ∀[z,y:{X ⊢ _:𝕀}]. (z ∧ y = y ∧ z ∈ {X ⊢ _:𝕀})
Definition: interval-join
r ∨ s == λI,rho. r(rho) ∨ s(rho)
Lemma: interval-join_wf
∀[Gamma:j⊢]. ∀[r,s:{Gamma ⊢ _:𝕀}]. (r ∨ s ∈ {Gamma ⊢ _:𝕀})
Lemma: csm-interval-join
∀[h,r,s:Top]. ((r ∨ s)h ~ (r)h ∨ (s)h)
Lemma: csm+_wf_interval
∀[H,K:j⊢]. ∀[tau:K j⟶ H]. (tau+ ∈ K.𝕀 j⟶ H.𝕀)
Lemma: csm+-comp-csm+-interval
∀[H,K,X:j⊢]. ∀[tau:K j⟶ H]. ∀[s:X j⟶ K]. (tau+ o s+ = tau o s+ ∈ X.𝕀 j⟶ H.𝕀)
Definition: cube+
cube+(I;i) == λA,fx. let f,x = fx in λj.if (j =z i) then x else f j fi
Lemma: cube+_wf
∀[I:fset(ℕ)]. ∀[i:ℕ]. (cube+(I;i) ∈ formal-cube(I).𝕀 j⟶ formal-cube(I+i))
Definition: cube-
cube-(I;i) == λA,f. <f, f i>
Lemma: cube-_wf
∀[I:fset(ℕ)]. ∀[i:ℕ]. (cube-(I;i) ∈ formal-cube(I+i) j⟶ formal-cube(I).𝕀)
Lemma: cube+-
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ].
(cube-(I;i) o cube+(I;i) = 1(formal-cube(I).𝕀) ∈ formal-cube(I).𝕀 j⟶ formal-cube(I).𝕀)
Lemma: cube-+
∀[I:fset(ℕ)]. ∀[i:ℕ]. (cube+(I;i) o cube-(I;i) = 1(formal-cube(I+i)) ∈ formal-cube(I+i) j⟶ formal-cube(I+i))
Lemma: cube+_interval-0
∀[I:fset(ℕ)]. ∀[i:ℕ]. (cube+(I;i) o [0(𝕀)] = <(i0)> ∈ formal-cube(I) j⟶ formal-cube(I+i))
Lemma: cube+_interval-1
∀[I:fset(ℕ)]. ∀[i:ℕ]. (cube+(I;i) o [1(𝕀)] = <(i1)> ∈ formal-cube(I) j⟶ formal-cube(I+i))
Lemma: context-map-cube+-csm+
∀[Gamma:j⊢]. ∀[I,J:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[g:J ⟶ I]. ∀[rho:Gamma(I+i)].
(<g,i=j(rho)> o cube+(J;j) = <rho> o cube+(I;i) o <g>+ ∈ formal-cube(J).𝕀 j⟶ Gamma)
Definition: swap-interval
swap-interval(G;A) == csm-swap(G;𝕀;A)
Lemma: swap-interval_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (swap-interval(Gamma;A) ∈ Gamma.𝕀.(A)p j⟶ Gamma.A.𝕀)
Lemma: p+-swap-interval
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[B:{Gamma ⊢ _}]. (((A)p+)swap-interval(Gamma;B) = (A)p ∈ {Gamma.𝕀.(B)p ⊢ _})
Definition: swap-intervals
swap-intervals(G) == csm-swap(G;𝕀;𝕀)
Lemma: swap-intervals_wf
∀[Gamma:j⊢]. (swap-intervals(Gamma) ∈ Gamma.𝕀.𝕀 j⟶ Gamma.𝕀.𝕀)
Definition: example-bool-term
example-bool-term(X) == discrete-bool-term(I,alpha.isdM0(snd(λi.0(alpha))))
Comment: face type doc
=================================== face type =================================
From the face presheaf 𝔽 we get a cubcial type 𝔽 and operations on it.
⋅
Definition: face-type
𝔽 == (𝔽)
Lemma: face-type_wf
∀[Gamma:j⊢]. Gamma ⊢ 𝔽
Lemma: face-type-sq
𝔽 ~ <λI,alpha. Point(face_lattice(I)), λI,J,f,alpha,w. (w)<f>>
Lemma: face-type-ap-morph
∀[I,J,f,rho,u:Top]. ((u rho f) ~ (u)<f>)
Lemma: face-type-ap-morph-ps
∀[I,J,f,rho,u:Top]. ((u rho f) ~ (u)<f>)
Lemma: face-type-at
∀[J,rho:Top]. (𝔽(rho) ~ Point(face_lattice(J)))
Lemma: cubical-type-at_wf_face-type
∀[J:fset(ℕ)]. ∀[rho:Top]. (𝔽(rho) ∈ Type)
Lemma: csm-face-type
∀[s:Top]. ((𝔽)s ~ 𝔽)
Definition: face-0
0(𝔽) == λI,rho. 0
Lemma: face-0_wf
∀[Gamma:j⊢]. (0(𝔽) ∈ {Gamma ⊢ _:𝔽})
Lemma: csm-face-0
∀[s:Top]. ((0(𝔽))s ~ 0(𝔽))
Definition: face-1
1(𝔽) == λI,rho. 1
Lemma: face-1_wf
∀[Gamma:j⊢]. (1(𝔽) ∈ {Gamma ⊢ _:𝔽})
Lemma: csm-face-1
∀[s:Top]. ((1(𝔽))s ~ 1(𝔽))
Definition: face-and
(a ∧ b) == λI,rho. a(rho) ∧ b(rho)
Lemma: face-and_wf
∀[Gamma:j⊢]. ∀[r,s:{Gamma ⊢ _:𝔽}]. ((r ∧ s) ∈ {Gamma ⊢ _:𝔽})
Lemma: face-and-com
∀[Gamma:j⊢]. ∀[r,s:{Gamma ⊢ _:𝔽}]. ((r ∧ s) = (s ∧ r) ∈ {Gamma ⊢ _:𝔽})
Lemma: face-and-at
∀[r,s,I,rho:Top]. ((r ∧ s)(rho) ~ r(rho) ∧ s(rho))
Lemma: face-and-eq-1
∀[Gamma:j⊢]
∀r,s:{Gamma ⊢ _:𝔽}. ∀I:fset(ℕ). ∀rho:Gamma(I).
((r ∧ s)(rho) = 1 ∈ Point(face_lattice(I))
⇐⇒ (r(rho) = 1 ∈ Point(face_lattice(I))) ∧ (s(rho) = 1 ∈ Point(face_lattice(I))))
Definition: face-or
(a ∨ b) == λI,rho. a(rho) ∨ b(rho)
Lemma: face-or_wf
∀[Gamma:j⊢]. ∀[r,s:{Gamma ⊢ _:𝔽}]. ((r ∨ s) ∈ {Gamma ⊢ _:𝔽})
Lemma: face-or-at
∀[r,s,I,rho:Top]. ((r ∨ s)(rho) ~ r(rho) ∨ s(rho))
Lemma: face-or-1
∀[X:j⊢]. ∀[psi:{X ⊢ _:𝔽}]. ((psi ∨ 1(𝔽)) = 1(𝔽) ∈ {X ⊢ _:𝔽})
Lemma: face-or-0
∀[X:j⊢]. ∀[psi:{X ⊢ _:𝔽}]. ((psi ∨ 0(𝔽)) = psi ∈ {X ⊢ _:𝔽})
Lemma: face-1-or
∀[X:j⊢]. ∀[psi:{X ⊢ _:𝔽}]. ((1(𝔽) ∨ psi) = 1(𝔽) ∈ {X ⊢ _:𝔽})
Lemma: face-0-or
∀[X:j⊢]. ∀[psi:{X ⊢ _:𝔽}]. ((0(𝔽) ∨ psi) = psi ∈ {X ⊢ _:𝔽})
Lemma: csm-face-or
∀[r,s,sigma:Top]. (((r ∨ s))sigma ~ ((r)sigma ∨ (s)sigma))
Lemma: csm-face-and
∀[r,s,sigma:Top]. (((r ∧ s))sigma ~ ((r)sigma ∧ (s)sigma))
Lemma: face-or-eq-1
∀[Gamma:j⊢]
∀r,s:{Gamma ⊢ _:𝔽}. ∀I:fset(ℕ). ∀rho:Gamma(I).
((r ∨ s)(rho) = 1 ∈ Point(face_lattice(I))
⇐⇒ (r(rho) = 1 ∈ Point(face_lattice(I))) ∨ (s(rho) = 1 ∈ Point(face_lattice(I))))
Definition: face-or-list
\/(L) == reduce(λa,b. (a ∨ b);0(𝔽);L)
Lemma: face-or-list_wf
∀[Gamma:j⊢]. ∀[L:{Gamma ⊢ _:𝔽} List]. (\/(L) ∈ {Gamma ⊢ _:𝔽})
Lemma: face-or-list-eq-1
∀[Gamma:j⊢]
∀L:{Gamma ⊢ _:𝔽} List. ∀I:fset(ℕ). ∀rho:Gamma(I).
(\/(L)(rho) = 1 ∈ Point(face_lattice(I)) ⇐⇒ (∃x∈L. x(rho) = 1 ∈ Point(face_lattice(I))))
Definition: face-one
(i=1) == λI,rho. dM-to-FL(I;i(rho))
Lemma: face-one_wf
∀[Gamma:j⊢]. ∀[i:{Gamma ⊢ _:𝕀}]. ((i=1) ∈ {Gamma ⊢ _:𝔽})
Definition: face-zero
(i=0) == λI,rho. dM-to-FL(I;¬(i(rho)))
Lemma: face-zero_wf
∀[Gamma:j⊢]. ∀[i:{Gamma ⊢ _:𝕀}]. ((i=0) ∈ {Gamma ⊢ _:𝔽})
Lemma: face-zero-as-face-one
∀[Gamma:j⊢]. ∀[i:{Gamma ⊢ _:𝕀}]. ((i=0) = (1-(i)=1) ∈ {Gamma ⊢ _:𝔽})
Lemma: csm-face-zero
∀[r,sigma:Top]. (((r=0))sigma ~ ((r)sigma=0))
Lemma: csm-face-one
∀[r,sigma:Top]. (((r=1))sigma ~ ((r)sigma=1))
Lemma: face-one-and-zero
∀[X:j⊢]. ∀[z:{X ⊢ _:𝕀}]. (((z=1) ∧ (z=0)) = 0(𝔽) ∈ {X ⊢ _:𝔽})
Lemma: face-zero-and-one
∀[X:j⊢]. ∀[z:{X ⊢ _:𝕀}]. (((z=0) ∧ (z=1)) = 0(𝔽) ∈ {X ⊢ _:𝔽})
Lemma: face-zero-eq-1
∀[H:j⊢]. ∀[z:{H ⊢ _:𝕀}]. ∀[I:fset(ℕ)]. ∀[a:H(I)]. z(a) = 0 ∈ 𝕀(a) supposing (z=0)(a) = 1 ∈ Point(face_lattice(I))
Lemma: face-zero-interval-0
∀[H:j⊢]. ((0(𝕀)=0) = 1(𝔽) ∈ {H ⊢ _:𝔽})
Lemma: face-zero-interval-1
∀[H:j⊢]. ((1(𝕀)=0) = 0(𝔽) ∈ {H ⊢ _:𝔽})
Lemma: face-one-interval-1
∀[H:j⊢]. ((1(𝕀)=1) = 1(𝔽) ∈ {H ⊢ _:𝔽})
Lemma: face-one-interval-0
∀[H:j⊢]. ((0(𝕀)=1) = 0(𝔽) ∈ {H ⊢ _:𝔽})
Lemma: face-one-eq-1
∀[H:j⊢]. ∀[z:{H ⊢ _:𝕀}]. ∀[I:fset(ℕ)]. ∀[a:H(I)]. z(a) = 1 ∈ 𝕀(a) supposing (z=1)(a) = 1 ∈ Point(face_lattice(I))
Lemma: face-term-at-restriction
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[a:Gamma(I)]. (phi(f(a)) = f(phi(a)) ∈ 𝔽(J))
Lemma: face-term-at-restriction-eq-1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A,B:fset(ℕ)]. ∀[g:B ⟶ A]. ∀[v:Gamma(A)].
phi(g(v)) = 1 ∈ Point(face_lattice(B)) supposing phi(v) = 1 ∈ Point(face_lattice(A))
Definition: context-subset
Gamma, phi == <λI.{rho:Gamma(I)| phi(rho) = 1 ∈ Point(face_lattice(I))} , λI,J,f,rho. f(rho)>
Lemma: context-subset_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. Gamma, phi j⊢
Lemma: context-subset-restriction
∀[Gamma,phi,I,J,a,f:Top]. (f(a) ~ f(a))
Lemma: context-subset-map
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[Z:j⊢]. ∀[s:Z j⟶ Gamma]. (s ∈ Z, (phi)s j⟶ Gamma, phi)
Lemma: context-map-subset
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[I:fset(ℕ)]. ∀[a:G, phi(I)]. (<a> = <a> ∈ formal-cube(I) j⟶ G, phi)
Lemma: cubical-fun-subset
∀[G,phi,T,A:Top]. ((G, phi ⊢ T ⟶ A) ~ (G ⊢ T ⟶ A))
Lemma: cubical-fun-subset-adjoin
∀[G,B,phi,T,A:Top]. ((G, phi.B ⊢ T ⟶ A) ~ (G.B ⊢ T ⟶ A))
Lemma: subset-iota_wf2
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. (iota ∈ Gamma, phi j⟶ Gamma)
Lemma: cubical-subset-is-context-subset
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. (I,psi = formal-cube(I), λJ,f. f(psi) ∈ CubicalSet{j})
Lemma: cubical-subset-is-context-subset-canonical
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. (I,psi = formal-cube(I), canonical-section(();𝔽;I;⋅;psi) ∈ CubicalSet{j})
Lemma: context-subset-is-cubical-subset
∀[I:fset(ℕ)]. ∀[phi:{formal-cube(I) ⊢ _:𝔽}]. (formal-cube(I), phi = I,phi(1) ∈ CubicalSet{j})
Definition: face-term-implies
Gamma ⊢ (phi ⇒ psi) ==
∀I:fset(ℕ). ∀rho:Gamma(I). ((phi(rho) = 1 ∈ Point(face_lattice(I))) ⇒ (psi(rho) = 1 ∈ Point(face_lattice(I))))
Lemma: face-term-implies_wf
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. (Gamma ⊢ (phi ⇒ psi) ∈ ℙ{[i | j']})
Lemma: csm-face-term-implies
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}].
∀[H:j⊢]. ∀[s:H j⟶ Gamma]. H ⊢ ((phi)s ⇒ (psi)s) supposing Gamma ⊢ (phi ⇒ psi)
Lemma: face-term-implies-1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. Gamma ⊢ (phi ⇒ 1(𝔽))
Lemma: face-term-implies-same
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. Gamma ⊢ (phi ⇒ phi)
Lemma: face-term-and-implies1
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. Gamma ⊢ ((phi ∧ psi) ⇒ phi)
Lemma: face-term-and-implies2
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. Gamma ⊢ ((phi ∧ psi) ⇒ psi)
Lemma: face-term-and-implies
∀[Gamma:j⊢]. ∀[phi,psi,phi',psi':{Gamma ⊢ _:𝔽}].
(Gamma ⊢ ((phi ∧ psi) ⇒ (phi' ∧ psi'))) supposing (Gamma ⊢ (phi ⇒ phi') and Gamma ⊢ (psi ⇒ psi'))
Lemma: face-term-implies-or1
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. Gamma ⊢ (phi ⇒ (phi ∨ psi))
Lemma: face-term-implies-or2
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. Gamma ⊢ (phi ⇒ (psi ∨ phi))
Lemma: face-term-or-implies
∀[Gamma:j⊢]. ∀[a,b,c:{Gamma ⊢ _:𝔽}]. (Gamma ⊢ ((a ∨ b) ⇒ c)) supposing (Gamma ⊢ (a ⇒ c) and Gamma ⊢ (b ⇒ c))
Lemma: face-term-implies-and
∀[Gamma:j⊢]. ∀[a,b,c:{Gamma ⊢ _:𝔽}]. (Gamma ⊢ (c ⇒ (a ∧ b))) supposing (Gamma ⊢ (c ⇒ a) and Gamma ⊢ (c ⇒ b))
Lemma: face-term-implies-or
∀[Gamma:j⊢]. ∀[a,b,c:{Gamma ⊢ _:𝔽}]. ((Gamma ⊢ (c ⇒ a) ∨ Gamma ⊢ (c ⇒ b)) ⇒ Gamma ⊢ (c ⇒ (a ∨ b)))
Definition: face-term-iff
Gamma ⊢ (phi ⇐⇒ psi) == Gamma ⊢ (phi ⇒ psi) ∧ Gamma ⊢ (psi ⇒ phi)
Lemma: face-term-iff_wf
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. (Gamma ⊢ (phi ⇐⇒ psi) ∈ ℙ{[i | j']})
Lemma: face-term-implies_functionality
∀[Gamma:j⊢]. ∀[a,b,c,d:{Gamma ⊢ _:𝔽}].
(Gamma ⊢ (a ⇐⇒ c) ⇒ Gamma ⊢ (b ⇐⇒ d) ⇒ (Gamma ⊢ (a ⇒ b) ⇐⇒ Gamma ⊢ (c ⇒ d)))
Lemma: face-term-distrib1
∀[Gamma:j⊢]. ∀[a,b,c:{Gamma ⊢ _:𝔽}]. Gamma ⊢ ((a ∧ (b ∨ c)) ⇐⇒ ((a ∧ b) ∨ (a ∧ c)))
Lemma: face-term-distrib2
∀[Gamma:j⊢]. ∀[a,b,c:{Gamma ⊢ _:𝔽}]. Gamma ⊢ ((a ∨ (b ∧ c)) ⇐⇒ ((a ∨ b) ∧ (a ∨ c)))
Lemma: face-term-distrib3
∀[Gamma:j⊢]. ∀[a,b,c:{Gamma ⊢ _:𝔽}]. Gamma ⊢ (((b ∨ c) ∧ a) ⇐⇒ ((b ∧ a) ∨ (c ∧ a)))
Lemma: face-term-distrib4
∀[Gamma:j⊢]. ∀[a,b,c:{Gamma ⊢ _:𝔽}]. Gamma ⊢ (((b ∧ c) ∨ a) ⇐⇒ ((b ∨ a) ∧ (c ∨ a)))
Definition: face-forall
(∀ phi) == λJ,rho. (∀new-name(J).phi((s(rho);<new-name(J)>)))
Lemma: face-forall_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma.𝕀 ⊢ _:𝔽}]. ((∀ phi) ∈ {Gamma ⊢ _:𝔽})
Lemma: face-forall-1
∀[Gamma:j⊢]. ∀[phi:{Gamma.𝕀 ⊢ _:𝔽}]. (Gamma ⊢ ∀ phi) = 1(𝔽) ∈ {Gamma ⊢ _:𝔽} supposing phi = 1(𝔽) ∈ {Gamma.𝕀 ⊢ _:𝔽}
Lemma: csm-face-forall
∀[Gamma,Delta:j⊢]. ∀[sigma:Delta j⟶ Gamma]. ∀[phi:{Gamma.𝕀 ⊢ _:𝔽}].
(((∀ phi))sigma = (Delta ⊢ ∀ (phi)sigma+) ∈ {Delta ⊢ _:𝔽})
Lemma: face-forall-or
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma.𝕀 ⊢ _:𝔽}]. ((Gamma ⊢ ∀ (phi ∨ psi)) = ((∀ phi) ∨ (∀ psi)) ∈ {Gamma ⊢ _:𝔽})
Lemma: face-forall-q=0
∀[Gamma:j⊢]. ((Gamma ⊢ ∀ (q=0)) = 0(𝔽) ∈ {Gamma ⊢ _:𝔽})
Lemma: face-forall-q=1
∀[Gamma:j⊢]. ((Gamma ⊢ ∀ (q=1)) = 0(𝔽) ∈ {Gamma ⊢ _:𝔽})
Lemma: face-forall-q=0-or-q=1
∀[Gamma:j⊢]. ((Gamma ⊢ ∀ ((q=0) ∨ (q=1))) = 0(𝔽) ∈ {Gamma ⊢ _:𝔽})
Lemma: face-forall-and
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma.𝕀 ⊢ _:𝔽}]. ((Gamma ⊢ ∀ (phi ∧ psi)) = ((∀ phi) ∧ (∀ psi)) ∈ {Gamma ⊢ _:𝔽})
Lemma: csm-canonical-section-face
∀[X,I,rho,phi,s:Top]. ((canonical-section(X;𝔽;I;rho;phi))s ~ λJ,a. (phi)<(s)a>)
Lemma: csm-canonical-section-face-type
∀[I,K:fset(ℕ)]. ∀[f:K ⟶ I]. ∀[phi:𝔽(I)].
(canonical-section(();𝔽;K;⋅;f(phi)) = (canonical-section(();𝔽;I;⋅;phi))<f> ∈ {formal-cube(K) ⊢ _:𝔽})
Lemma: csm-canonical-section-face-type-0
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[phi:𝔽(I)].
((canonical-section(();𝔽;I+i;⋅;s(phi)))<(i0)> = canonical-section(();𝔽;I;⋅;phi) ∈ {formal-cube(I) ⊢ _:𝔽})
Lemma: csm-canonical-section-face-type-1
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[phi:𝔽(I)].
((canonical-section(();𝔽;I+i;⋅;s(phi)))<(i1)> = canonical-section(();𝔽;I;⋅;phi) ∈ {formal-cube(I) ⊢ _:𝔽})
Lemma: context-subset-subtype
∀[Gamma:j⊢]. ∀[phi1,phi2:{Gamma ⊢ _:𝔽}].
{Gamma, phi1 ⊢ _} ⊆r {Gamma, phi2 ⊢ _}
supposing ∀I:fset(ℕ). ∀rho:Gamma(I).
((phi2(rho) = 1 ∈ Point(face_lattice(I))) ⇒ (phi1(rho) = 1 ∈ Point(face_lattice(I))))
Lemma: context-subset_functionality
∀[Gamma:j⊢]. ∀[a,b:{Gamma ⊢ _:𝔽}]. (Gamma ⊢ (a ⇐⇒ b) ⇒ {Gamma, a ⊢ _} ≡ {Gamma, b ⊢ _})
Lemma: context-subset-is-subset
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. sub_cubical_set{j:l}(Gamma, phi; Gamma)
Lemma: cubical-term-restriction-is-1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I].
((phi(rho) = 1 ∈ Point(face_lattice(I))) ⇒ (phi(f(rho)) = 1 ∈ Point(face_lattice(J))))
Lemma: face-term-implies-subset
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. sub_cubical_set{j:l}(Gamma, phi; Gamma, psi) supposing Gamma ⊢ (phi ⇒ psi)
Lemma: cubical-subset_functionality_wrt_le
∀J:fset(ℕ). ∀a,b:Point(face_lattice(J)). (a ≤ b ⇒ sub_cubical_set{j:l}(J,a; J,b))
Lemma: context-subset-subtype-and
∀[Gamma:j⊢]. ∀[phi1,phi2:{Gamma ⊢ _:𝔽}]. ({Gamma, phi1 ⊢ _} ⊆r {Gamma, (phi1 ∧ phi2) ⊢ _})
Lemma: context-subset-subtype-and2
∀[Gamma:j⊢]. ∀[phi1,phi2:{Gamma ⊢ _:𝔽}]. ({Gamma, phi2 ⊢ _} ⊆r {Gamma, (phi1 ∧ phi2) ⊢ _})
Lemma: context-subset-subtype-or
∀[Gamma:j⊢]. ∀[phi1,phi2:{Gamma ⊢ _:𝔽}]. ({Gamma, (phi1 ∨ phi2) ⊢ _} ⊆r {Gamma, phi1 ⊢ _})
Lemma: context-subset-subtype-or2
∀[Gamma:j⊢]. ∀[phi1,phi2:{Gamma ⊢ _:𝔽}]. ({Gamma, (phi1 ∨ phi2) ⊢ _} ⊆r {Gamma, phi2 ⊢ _})
Lemma: cubical-term-at-comp-constant-type
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[J:fset(ℕ)]. ∀[f:{f:J ⟶ I|
phi(f(rho))
= 1
∈ Point(face_lattice(J))} ].
∀[K:fset(ℕ)]. ∀[g:K ⟶ J].
(phi(g(f(rho))) = phi(f ⋅ g(rho)) ∈ Point(face_lattice(K)))
Lemma: cubical-term-at-comp-is-1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[K:fset(ℕ)]. ∀[g:K ⟶ J].
((phi(f(rho)) = 1 ∈ Point(face_lattice(J))) ⇒ (phi(f ⋅ g(rho)) = 1 ∈ Point(face_lattice(K))))
Lemma: subset-context-1
∀[Gamma:j⊢]. sub_cubical_set{j:l}(Gamma, 1(𝔽); Gamma)
Lemma: context-1-subset
∀[Gamma:j⊢]. sub_cubical_set{j:l}(Gamma; Gamma, 1(𝔽))
Lemma: face-1-implies-subset
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. sub_cubical_set{j:l}(Gamma; Gamma, phi) supposing Gamma ⊢ (1(𝔽) ⇒ phi)
Lemma: context-subset-1
∀[Gamma:j⊢]. {Gamma, 1(𝔽) ⊢ _} ≡ {Gamma ⊢ _}
Lemma: context-subset-1-subtype
∀[Gamma:j⊢]. ({Gamma, 1(𝔽) ⊢ _} ⊆r {Gamma ⊢ _})
Lemma: context-subset-subtype-simple
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ({Gamma ⊢ _} ⊆r {Gamma, phi ⊢ _})
Lemma: thin-context-subset
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[t:{Gamma ⊢ _}]. Gamma, phi ⊢ t
Lemma: thin-context-subset-adjoin
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma ⊢ _}]. ∀[t:{Gamma.T ⊢ _}]. Gamma, phi.T ⊢ t
Lemma: term-context-subset-subtype
∀[Gamma:j⊢]. ∀[phi1,phi2:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma ⊢ _}].
{Gamma, phi1 ⊢ _:A} ⊆r {Gamma, phi2 ⊢ _:A} supposing Gamma ⊢ (phi2 ⇒ phi1)
Lemma: context-subset-term-subtype
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ({Gamma ⊢ _:A} ⊆r {Gamma, phi ⊢ _:A})
Lemma: respects-equality-context-subset-term
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[phi:{X ⊢ _:𝔽}]. respects-equality({X ⊢ _:A};{X, phi ⊢ _:A})
Lemma: face-term-implies-subtype
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi,psi:{Gamma ⊢ _:𝔽}].
{Gamma, phi ⊢ _:A} ⊆r {Gamma, psi ⊢ _:A} supposing Gamma ⊢ (psi ⇒ phi)
Lemma: context-iterated-subset
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}].
(sub_cubical_set{j:l}(Gamma, (phi ∧ psi); Gamma, psi, phi)
∧ sub_cubical_set{j:l}(Gamma, psi, phi; Gamma, (phi ∧ psi)))
Lemma: context-iterated-subset0
∀[X:j⊢]. ∀[xx,yy:{X ⊢ _:𝔽}]. sub_cubical_set{j:l}(X, xx, yy; X)
Lemma: context-iterated-subset1
∀[X:j⊢]. ∀[xx,yy:{X ⊢ _:𝔽}]. sub_cubical_set{j:l}(X, xx, yy; X, yy)
Lemma: context-iterated-subset2
∀[X:j⊢]. ∀[xx,yy:{X ⊢ _:𝔽}]. sub_cubical_set{j:l}(X, yy, xx; X, yy)
Lemma: context-subset-swap
∀[X:j⊢]. ∀[a,b:{X ⊢ _:𝔽}]. sub_cubical_set{j:l}(X, b, a; X, a, b)
Lemma: context-subset-term-1
∀[Gamma:j⊢]. ∀[T:{Gamma ⊢ _}]. ({Gamma ⊢ _:T} ⊆r {Gamma, 1(𝔽) ⊢ _:T})
Lemma: context-subset-term-0
∀[Gamma:j⊢]. ∀[T:{Gamma ⊢ _}]. (Top ⊆r {Gamma, 0(𝔽) ⊢ _:T})
Lemma: sub_cubical_set_functionality2
∀[Y,X:j⊢]. ∀[phi:{X ⊢ _:𝔽}]. sub_cubical_set{j:l}(Y, phi; X, phi) supposing sub_cubical_set{j:l}(Y; X)
Lemma: csm-ap-term-subset
∀[Gamma,K:j⊢]. ∀[s:K j⟶ Gamma]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[Z:{Gamma, phi ⊢ _:A}].
((Z)s ∈ {K, (phi)s ⊢ _:(A)s})
Lemma: csm-ap-term-subset-subset
∀[Gamma,K:j⊢]. ∀[s:K j⟶ Gamma]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[psi:{Gamma, phi ⊢ _:𝔽}]. ∀[A:{Gamma, phi, psi ⊢ _}].
∀[Z:{Gamma, phi, psi ⊢ _:A}].
((Z)s ∈ {K, (phi)s, (psi)s ⊢ _:(A)s})
Lemma: context-subset-type-subtype
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ({G ⊢j _} ⊆r {G, phi ⊢j _})
Definition: constrained-cubical-term
{Gamma ⊢ _:A[phi |⟶ t]} == {a:{Gamma ⊢ _:A}| t = a ∈ {Gamma, phi ⊢ _:A}}
Lemma: constrained-cubical-term_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[t:{Gamma, phi ⊢ _:A}]. ({Gamma ⊢ _:A[phi |⟶ t]} ∈ 𝕌{[i | j']})
Lemma: constrained-cubical-term-eqcd
∀[Gamma:j⊢]. ∀[A,B:{Gamma ⊢ _}]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[t:{Gamma, phi ⊢ _:A}]. ∀[t':{Gamma, psi ⊢ _:B}].
({Gamma ⊢ _:A[phi |⟶ t]} = {Gamma ⊢ _:B[psi |⟶ t']} ∈ 𝕌{[i | j']}) supposing
((t' = t ∈ {Gamma, phi ⊢ _:A}) and
(phi = psi ∈ {Gamma ⊢ _:𝔽}) and
(A = B ∈ {Gamma ⊢ _}))
Lemma: constrained-cubical-term-0
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[t:Top]. ∀[x:{Gamma ⊢ _:A}]. (x ∈ {Gamma ⊢ _:A[0(𝔽) |⟶ t]})
Lemma: csm-context-subset-subtype
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[X:j⊢]. (X j⟶ Gamma, phi.𝕀 ⊆r X j⟶ Gamma.𝕀)
Lemma: subset-constrained-cubical-term
∀[X,Y:j⊢].
∀[A:{X ⊢ _}]. ∀[phi:{X ⊢ _:𝔽}]. ∀[t:{X, phi ⊢ _:A}]. ({X ⊢ _:A[phi |⟶ t]} ⊆r {Y ⊢ _:A[phi |⟶ t]})
supposing sub_cubical_set{j:l}(Y; X)
Lemma: csm-context-subset-subtype2
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[X:j⊢]. (Gamma j⟶ X ⊆r Gamma, phi j⟶ X)
Lemma: csm-context-subset-subtype3
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma ⊢ _}]. ∀[X:j⊢]. (Gamma.A ij⟶ X ⊆r Gamma, phi.A ij⟶ X)
Lemma: csm-constrained-cubical-term
∀[Gamma,K:j⊢]. ∀[s:K j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[t:{Gamma, phi ⊢ _:A}].
∀[v:{Gamma ⊢ _:A[phi |⟶ t]}].
((v)s ∈ {K ⊢ _:(A)s[(phi)s |⟶ (t)s]})
Definition: same-cubical-type
Gamma ⊢ A = B == A = B ∈ {Gamma ⊢ _}
Lemma: same-cubical-type_wf
∀[Gamma:j⊢]. ∀[A,B:{Gamma ⊢ _}]. (Gamma ⊢ A = B ∈ 𝕌{[i' | j']})
Lemma: csm-same-cubical-type
∀[Gamma:j⊢]. ∀[A,B:{Gamma ⊢ _}]. ∀[H:j⊢]. ∀[s:H j⟶ Gamma]. H ⊢ (A)s = (B)s supposing Gamma ⊢ A = B
Definition: same-cubical-term
X ⊢ u=v:A == u = v ∈ {X ⊢ _:A}
Lemma: same-cubical-term_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u,v:{X ⊢ _:A}]. (X ⊢ u=v:A ∈ 𝕌{[i | j']})
Lemma: csm-same-cubical-term
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u,v:{X ⊢ _:A}]. ∀[Y:j⊢]. ∀[s:Y j⟶ X]. Y ⊢ (u)s=(v)s:(A)s supposing X ⊢ u=v:A
Lemma: face-one-context-implies
∀[X:j⊢]. ∀[i:{X ⊢ _:𝕀}]. X, (i=1) ⊢ i=1(𝕀):𝕀
Lemma: face-zero-context-implies
∀[X:j⊢]. ∀[i:{X ⊢ _:𝕀}]. X, (i=0) ⊢ i=0(𝕀):𝕀
Lemma: same-cubical-type-0
∀[Gamma:j⊢]. ∀[A,B:{Gamma, 0(𝔽) ⊢ _}]. Gamma, 0(𝔽) ⊢ A = B
Lemma: same-cubical-type-zero-and-one
∀[G:j⊢]. ∀[A,B:{G, 0(𝔽) ⊢ _}]. ∀[i:{G ⊢ _:𝕀}]. G, ((i=0) ∧ (i=1)) ⊢ A = B
Lemma: subtype-context-subset-0
∀[X,Y:j⊢]. ({X ⊢ _} ⊆r {Y, 0(𝔽) ⊢ _})
Lemma: same-cubical-type-by-cases
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A,B:{Gamma, (phi ∨ psi) ⊢ _}].
(Gamma, (phi ∨ psi) ⊢ A = B) supposing (Gamma, phi ⊢ A = B and Gamma, psi ⊢ A = B)
Lemma: same-cubical-term-by-cases
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, (phi ∨ psi) ⊢ _}]. ∀[a,b:{Gamma, (phi ∨ psi) ⊢ _:A}].
(Gamma, (phi ∨ psi) ⊢ a=b:A) supposing (Gamma, phi ⊢ a=b:A and Gamma, psi ⊢ a=b:A)
Lemma: same-cubical-term-by-cases2
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma ⊢ _}]. ∀[a,b:{Gamma, (phi ∨ psi) ⊢ _:A}].
(Gamma, (phi ∨ psi) ⊢ a=b:A) supposing (Gamma, phi ⊢ a=b:A and Gamma, psi ⊢ a=b:A)
Lemma: same-cubical-type-by-list-cases
∀[Gamma:j⊢]. ∀L:{Gamma ⊢ _:𝔽} List. ∀A,B:{Gamma, \/(L) ⊢ _}. Gamma, \/(L) ⊢ A = B supposing (∀phi∈L.Gamma, phi ⊢ A = B)
Lemma: context-subset-adjoin-subtype
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ({Gamma.A ⊢ _} ⊆r {Gamma, phi.A ⊢ _})
Lemma: face-forall-implies-0
∀[H:j⊢]. ∀[phi:{H.𝕀 ⊢ _:𝔽}]. ∀[X:Top]. H ⊢ ((∀ phi) ⇒ (phi)[0(𝕀)])
Lemma: face-forall-implies-1
∀[H:j⊢]. ∀[phi:{H.𝕀 ⊢ _:𝔽}]. ∀[X:Top]. H ⊢ ((∀ phi) ⇒ (phi)[1(𝕀)])
Lemma: face-forall-implies
∀[H:j⊢]. ∀[phi:{H.𝕀 ⊢ _:𝔽}]. H.𝕀 ⊢ (((∀ phi))p ⇒ phi)
Lemma: face-forall-implies-csm+
∀[H:j⊢]. ∀[phi:{H.𝕀 ⊢ _:𝔽}]. ∀[K:j⊢]. ∀[tau:K j⟶ H]. K.𝕀 ⊢ ((((∀ phi))tau)p ⇒ (phi)tau+)
Lemma: face-forall-decomp
∀[H:j⊢]. ∀[phi:{H.𝕀 ⊢ _:𝔽}]. H.𝕀 ⊢ (phi ⇐⇒ (((∀ phi))p ∨ ((phi ∧ (q=0)) ∨ (phi ∧ (q=1)))))
Lemma: face-forall-type-subtype
∀[H:j⊢]. ∀[phi:{H.𝕀 ⊢ _:𝔽}]. ({H.𝕀, phi ⊢ _} ⊆r {H.𝕀, ((∀ phi))p ⊢ _})
Lemma: csm-ap-term-wf-subset
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[A:{H ⊢ _}]. ∀[t:{H, phi ⊢ _:A}]. ∀[K:j⊢]. ∀[psi:{K ⊢ _:𝔽}]. ∀[B:{K ⊢ _}]. ∀[tau:K j⟶ H].
((t)tau ∈ {K, psi ⊢ _:B}) supposing (K, psi ⊢ B = (A)tau and K ⊢ (psi ⇒ (phi)tau))
Definition: partial-term-0
u[0] == (u)[0(𝕀)]
Lemma: partial-term-0_wf
∀[H:j⊢]. ∀[A:{H.𝕀 ⊢ _}]. ∀[phi:{H ⊢ _:𝔽}]. ∀[u:{H.𝕀, (phi)p ⊢ _:A}]. (u[0] ∈ {H, phi ⊢ _:(A)[0(𝕀)]})
Definition: partial-term-1
u[1] == (u)[1(𝕀)]
Lemma: partial-term-1_wf
∀[H:j⊢]. ∀[A:{H.𝕀 ⊢ _}]. ∀[phi:{H ⊢ _:𝔽}]. ∀[u:{H.𝕀, (phi)p ⊢ _:A}]. (u[1] ∈ {H, phi ⊢ _:(A)[1(𝕀)]})
Lemma: partial-term-1-subset
∀[H,xx,u:Top]. (partial-term-1(H, xx;u) ~ partial-term-1(H;u))
Lemma: partial-term-0-subset
∀[H,xx,u:Top]. (partial-term-0(H, xx;u) ~ partial-term-0(H;u))
Lemma: context-adjoin-subset0
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀T:{H ⊢ _}. sub_cubical_set{k:l}(H.T, (phi)p; H, phi.T)
Lemma: context-adjoin-subset3
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀T:{H ⊢ _}. sub_cubical_set{k:l}(H, phi.T; H.T, (phi)p)
Lemma: context-adjoin-subset4
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}].
∀T:{H ⊢j _}. ∀[psi:{H.T ⊢ _:𝔽}]. ((psi = (phi)p ∈ {H.T ⊢ _:𝔽}) ⇒ sub_cubical_set{j:l}(H, phi.T; H.T, psi))
Lemma: context-adjoin-subset1
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. sub_cubical_set{j:l}(H.𝕀, (phi)p; H, phi.𝕀)
Lemma: context-adjoin-subset2
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. sub_cubical_set{j:l}(H, phi.𝕀; H.𝕀, (phi)p)
Lemma: context-adjoin-subset
∀[X,Y:j⊢]. ∀[A:{Y ⊢ _}]. sub_cubical_set{[i | j]:l}(X.A; Y.A) supposing sub_cubical_set{j:l}(X; Y)
Lemma: csm-subtype-iso-instance1
∀[X,H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. (X j⟶ H.𝕀, (phi)p ⊆r X j⟶ H, phi.𝕀)
Lemma: csm-subtype-iso-instance2
∀[X,H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. (H.𝕀, (phi)p j⟶ X ⊆r H, phi.𝕀 j⟶ X)
Lemma: context-subset-map-equal
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[X:j⊢]. ∀[f,g:X j⟶ H.𝕀]. ((f = g ∈ X j⟶ H.𝕀) ⇒ (f = g ∈ X, ((phi)p)f j⟶ H, phi.𝕀))
Lemma: face-forall-subset
∀[Gamma,phi,xx:Top]. ((Gamma, xx ⊢ ∀ phi) ~ (Gamma ⊢ ∀ phi))
Lemma: face-forall-map
∀[G,H:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[s:H.𝕀 j⟶ G]. (s ∈ H, (∀ (phi)s).𝕀 j⟶ G, phi)
Lemma: csm+-ap-term-wf
∀[H,K:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[A:{H, phi.𝕀 ⊢ _}]. ∀[tau:K j⟶ H]. ∀[t:{H, phi.𝕀 ⊢ _:A}].
((t)tau+ ∈ {K, (phi)tau.𝕀 ⊢ _:(A)tau+})
Lemma: csm-id-adjoin-subset
∀[Gamma,phi,u:Top]. ([u] ~ [u])
Lemma: cubical-sigma-normalize
∀[X,A,B:Top]. (Σ A B ~ Σ A B)
Lemma: cubical-sigma-subset
∀[X,A,B,phi:Top]. (Σ A B ~ Σ A B)
Lemma: cubical-sigma-subset-adjoin
∀[X,A,B,T,phi:Top]. (Σ A B ~ Σ A B)
Lemma: cubical-sigma-subset-adjoin2
∀[X,A,B,T1,T2,phi:Top]. (Σ A B ~ Σ A B)
Lemma: term-p+0
∀[X,Y,Z,W,t,A:Top]. (((t)p+)[0(𝕀)] ~ ((t)[0(𝕀)])p)
Lemma: term-p+1
∀[X,Y,Z,W,t,A:Top]. (((t)p+)[1(𝕀)] ~ ((t)[1(𝕀)])p)
Lemma: face-1-in-context-subset
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. (phi = 1(𝔽) ∈ {G, phi ⊢ _:𝔽})
Lemma: cubical-pi-context-subset
∀[X,phi,A,B:Top]. (X, phi ⊢ ΠA B ~ X ⊢ ΠA B)
Lemma: cubical-pi-subset
∀[X,phi,A,B,C:Top]. (X, phi.C ⊢ ΠA B ~ X.C ⊢ ΠA B)
Lemma: cubical-pi-subset-adjoin2
∀[X,phi,A,B,C,D:Top]. (X, phi.C.D ⊢ ΠA B ~ X.C.D ⊢ ΠA B)
Lemma: cubical-pi-subset-adjoin3
∀[X,phi,A,B,C,D,E:Top]. (X, phi.C.D.E ⊢ ΠA B ~ X.C.D.E ⊢ ΠA B)
Lemma: cubical-lambda-subset
∀[X,phi,b:Top]. ((λb) ~ (λb))
Lemma: cubical-lambda-subset2
∀[X,A,phi,b:Top]. ((λb) ~ (λb))
Lemma: implies-face-forall-holds
∀H:j⊢. ∀phi:{H.𝕀 ⊢ _:𝔽}. (H.𝕀 ⊢ (1(𝔽) ⇒ phi) ⇒ H ⊢ (1(𝔽) ⇒ (∀ phi)))
Lemma: map-to-context-subset-disjoint
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[H:j⊢].
∀[sigma:H j⟶ Gamma, (phi ∧ psi)]. ∀[I:fset(ℕ)]. (¬H(I)) supposing Gamma ⊢ ((phi ∧ psi) ⇒ 0(𝔽))
Comment: system type doc
================================= system type =================================
System types allow definitions by cases.⋅
Definition: case-cube
case-cube(phi;A;B;I;rho) == if (phi(rho)==1) then A(rho) else B(rho) fi
Lemma: case-cube_wf
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[I:fset(ℕ)].
∀[rho:Gamma, (phi ∨ psi)(I)].
(case-cube(phi;A;B;I;rho) ∈ Type)
Definition: case-type
(if phi then A else B) == <λI,rho. case-cube(phi;A;B;I;rho), λI,J,f,rho,c. if (phi(rho)==1) then (c rho f) else (c rho \000Cf) fi >
Lemma: case-type_wf
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}].
Gamma, (phi ∨ psi) ⊢ (if phi then A else B) supposing Gamma, (phi ∧ psi) ⊢ A = B
Lemma: case-type-partition
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}].
(∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. Gamma ⊢ (if phi then A else B)) supposing
(Gamma ⊢ (1(𝔽) ⇒ (phi ∨ psi)) and
Gamma ⊢ ((phi ∧ psi) ⇒ 0(𝔽)))
Lemma: csm-case-type
∀[phi,A,B,s:Top]. (((if phi then A else B))s ~ (if (phi)s then (A)s else (B)s))
Lemma: case-type-same1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[psi:{Gamma ⊢ _:𝔽}]. ∀[B:{Gamma, psi ⊢ _}].
Gamma, phi ⊢ (if phi then A else B) = A
Lemma: case-type-same2
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[psi:{Gamma ⊢ _:𝔽}]. ∀[B:{Gamma, psi ⊢ _}].
Gamma, psi ⊢ (if phi then A else B) = B supposing Gamma, (phi ∧ psi) ⊢ A = B
Definition: compatible-system
compatible-system{i:l}(Gamma;sys) ==
(∀phiA1,phiA2∈sys. let phi1,A1 = phiA1
in let phi2,A2 = phiA2
in A1 = A2 ∈ {Gamma, (phi1 ∧ phi2) ⊢ _})
Lemma: compatible-system_wf
∀[Gamma:j⊢]. ∀[sys:(phi:{Gamma ⊢ _:𝔽} × {Gamma, phi ⊢ _}) List]. (compatible-system{i:l}(Gamma;sys) ∈ ℙ{[i' | j']})
Definition: system-type
system-type(sys) == reduce(λphi_A,B. let phi,A = phi_A in (if phi then A else B);discr(Top);sys)
Lemma: system-type-properties
∀Gamma:j⊢. ∀sys:(phi:{Gamma ⊢ _:𝔽} × {Gamma, phi ⊢ _}) List.
(compatible-system{i:l}(Gamma;sys)
⇒ (Gamma, \/(map(λp.(fst(p));sys)) ⊢ system-type(sys)
∧ (∀i:ℕ||sys||. (system-type(sys) = (snd(sys[i])) ∈ {Gamma, fst(sys[i]) ⊢ _}))))
Lemma: system-type_wf
∀[Gamma:j⊢]. ∀[sys:(phi:{Gamma ⊢ _:𝔽} × {Gamma, phi ⊢ _}) List].
Gamma, \/(map(λp.(fst(p));sys)) ⊢ system-type(sys) supposing compatible-system{i:l}(Gamma;sys)
Lemma: system-type-same
∀[Gamma:j⊢]. ∀[sys:(phi:{Gamma ⊢ _:𝔽} × {Gamma, phi ⊢ _}) List].
∀i:ℕ||sys||. Gamma, fst(sys[i]) ⊢ system-type(sys) = snd(sys[i]) supposing compatible-system{i:l}(Gamma;sys)
Definition: case-term
(u ∨ v) == λI,rho. if (phi(rho)==1) then u(rho) else v(rho) fi
Lemma: case-term_wf
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, (phi ∨ psi) ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}]. ∀[v:{Gamma, psi ⊢ _:A}].
(u ∨ v) ∈ {Gamma, (phi ∨ psi) ⊢ _:A} supposing Gamma, (phi ∧ psi) ⊢ u=v:A
Lemma: case-term_wf2
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, (phi ∨ psi) ⊢ _}]. ∀[v:{Gamma, psi ⊢ _:A}].
∀[u:{Gamma, phi ⊢ _:A[psi |⟶ v]}].
((u ∨ v) ∈ {Gamma, (phi ∨ psi) ⊢ _:A})
Lemma: case-term-same
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}]. ((u ∨ u) = u ∈ {Gamma, phi ⊢ _:A})
Lemma: case-term-same2
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}]. ∀[v:{Gamma, psi ⊢ _:A}].
∀[w:{Gamma ⊢ _:A}].
(Gamma, (phi ∨ psi) ⊢ (u ∨ v)=w:A) supposing (Gamma, phi ⊢ u=w:A and Gamma, psi ⊢ v=w:A)
Lemma: case-term-equal-left
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}]. ∀[v:Top]. Gamma, phi ⊢ (u ∨ v)=u:A
Lemma: case-term-equal-left'
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}]. ∀[v:Top]. ∀[x:{Gamma, phi ⊢ _:A}].
Gamma, phi ⊢ (u ∨ v)=x:A supposing x = u ∈ {Gamma, phi ⊢ _:A}
Lemma: case-term-1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma ⊢ _}]. ∀[u:{Gamma ⊢ _:A}]. ∀[v:Top].
Gamma ⊢ (u ∨ v)=u:A supposing phi = 1(𝔽) ∈ {Gamma ⊢ _:𝔽}
Lemma: case-term-1'
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma ⊢ _}]. ∀[u,x:{Gamma ⊢ _:A}]. ∀[v:Top].
(Gamma ⊢ (u ∨ v)=x:A) supposing ((x = u ∈ {Gamma ⊢ _:A}) and (phi = 1(𝔽) ∈ {Gamma ⊢ _:𝔽}))
Lemma: case-term-equal-right
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, (phi ∨ psi) ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}]. ∀[v:{Gamma, psi ⊢ _:A}].
Gamma, psi ⊢ (u ∨ v)=v:A supposing Gamma, (phi ∧ psi) ⊢ u=v:A
Lemma: case-term-equal-right'
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, (phi ∨ psi) ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}].
∀[v,x:{Gamma, psi ⊢ _:A}].
(Gamma, psi ⊢ (u ∨ v)=x:A) supposing ((x = v ∈ {Gamma, psi ⊢ _:A}) and Gamma, (phi ∧ psi) ⊢ u=v:A)
Lemma: case-term-equality-right
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, (phi ∨ psi) ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}]. ∀[v:{Gamma, psi ⊢ _:A}].
∀[x:{Gamma, (phi ∨ psi) ⊢ _:A}]. Gamma, psi ⊢ v=x:A supposing Gamma, (phi ∨ psi) ⊢ (u ∨ v)=x:A
supposing Gamma, (phi ∧ psi) ⊢ u=v:A
Lemma: case-term-wf3
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, (phi ∨ psi) ⊢ _}]. ∀[v:{Gamma, psi ⊢ _:A}].
∀[u:{Gamma, phi ⊢ _:A[psi |⟶ v]}].
((u ∨ v) ∈ {t:{Gamma, (phi ∨ psi) ⊢ _:A}| Gamma, psi ⊢ t=v:A} )
Lemma: case-term-wf4
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}].
∀[v:{Gamma, psi ⊢ _:B}].
((u ∨ v) ∈ {Gamma, (phi ∨ psi) ⊢ _:(if phi then A else B)}) supposing
(Gamma, (phi ∧ psi) ⊢ u=v:A and
Gamma, (phi ∧ psi) ⊢ A = B)
Lemma: csm-case-term
∀[phi,u,v,s:Top]. (((u ∨ v))s ~ ((u)s ∨ (v)s))
Comment: metric space type doc
We define a cubical type for any metric space X that is ⌜WCE(X;d)⌝⋅
Definition: metric-type
metric-type(X;d) == <λI,rho. X, λI,J,f,rho,c. f>
Comment: cubical path type doc
================================= path type ================================
The path type (Path_A a b) can now be defined as a subspace of the free path
space Path(A) == (𝕀 ⟶ A). To do that we use a general defintion of a subset
of a cubical type: {t:T | ∀I,alpha. psi[I;
alpha;
t]}.⋅
Comment: The Type (Path_A a b)
We used to define (Path_A a b) using an equivalence class (quotient type)
of paths whose I-cubes were (I+z)-cubes of A where z was a fresh coordinate
name (and such that z:=0 maps to a and z:=1 maps to b).
But we later realized that we can start with the
"free path space" Path(A) == (𝕀 ⟶ A)
and define the "subspace" of paths p such that p(0) = a and p(1) = b.
⋅
Definition: cubical-type-restriction
cubical-type-restriction(X;T;I,a,x.psi[I;
a;
x]) ==
∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀a:X(I). ∀x:T(a).
(psi[I;
a;
x]
⇒ psi[J;
f(a);
(x a f)])
Lemma: cubical-type-restriction_wf
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[psi:I:fset(ℕ) ⟶ alpha:X(I) ⟶ T(alpha) ⟶ ℙ{[i' | j']}].
(cubical-type-restriction(X;T;I,a,t.psi[I;a;t]) ∈ ℙ{[i' | j']})
Lemma: cubical-type-restriction-and
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[psi,phi:I:fset(ℕ) ⟶ alpha:X(I) ⟶ T(alpha) ⟶ ℙ].
(cubical-type-restriction(X;T;I,a,t.psi[I;a;t])
⇒ cubical-type-restriction(X;T;I,a,t.phi[I;a;t])
⇒ cubical-type-restriction(X;T;I,a,t.psi[I;a;t] ∧ phi[I;a;t]))
Lemma: cubical-type-restriction-eq
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[g:I:fset(ℕ) ⟶ alpha:X(I) ⟶ T(alpha) ⟶ A(alpha)].
cubical-type-restriction(X;T;I,alpha,t.(g I alpha t) = a(alpha) ∈ A(alpha))
supposing ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀alpha:X(I). ∀t:T(alpha).
((g J f(alpha) (t alpha f)) = (g I alpha t alpha f) ∈ A(f(alpha)))
Definition: cubical-subset
{t:T | ∀I,alpha. psi[I;
alpha;
t]} ==
<λI,alpha. {t:T(alpha)|
psi[I;
alpha;
t]}
, λI,J,f,alpha,p. (p alpha f)
>
Lemma: cubical-subset_wf
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[psi:I:fset(ℕ) ⟶ alpha:X(I) ⟶ T(alpha) ⟶ ℙ].
X ⊢ {t:T | ∀I,alpha. psi[I;alpha;t]} supposing cubical-type-restriction(X;T;I,a,t.psi[I;a;t])
Lemma: cubical-subset-term
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[psi:I:fset(ℕ) ⟶ alpha:X(I) ⟶ T(alpha) ⟶ ℙ].
∀[t:{X ⊢ _:T}]
t ∈ {X ⊢ _:{t:T | ∀I,alpha. psi[I;alpha;t]}} supposing ∀I:fset(ℕ). ∀alpha:X(I). psi[I;alpha;t(alpha)]
supposing cubical-type-restriction(X;T;I,a,t.psi[I;a;t])
Lemma: csm-cubical-subset
∀[T,psi,s:Top]. (({t:T | ∀I,alpha. psi[I;alpha;t]})s ~ {t:(T)s | ∀I,alpha. psi[I;(s)alpha;t]})
Definition: pathtype
Path(A) == (𝕀 ⟶ A)
Lemma: pathtype_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. X ⊢ Path(A)
Definition: cubicalpath-app
pth @ r == app(pth; r)
Lemma: cubicalpath-app_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[t:{X ⊢ _:Path(A)}]. ∀[r:{X ⊢ _:𝕀}]. (t @ r ∈ {X ⊢ _:A})
Lemma: path-restriction
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}].
cubical-type-restriction(X;Path(A);I,a1,p.((p I 1 0) = a(a1) ∈ A(a1)) ∧ ((p I 1 1) = b(a1) ∈ A(a1)))
Definition: path-type
(Path_A a b) == {p:Path(A) | ∀I,alpha. ((p I 1 0) = a(alpha) ∈ A(alpha)) ∧ ((p I 1 1) = b(alpha) ∈ A(alpha))}
Lemma: path-type_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. X ⊢ (Path_A a b)
Lemma: path-type-at-subtype
∀X:j⊢. ∀A:{X ⊢ _}. ∀a,b:{X ⊢ _:A}. ∀I:fset(ℕ). ∀rho:X(I). ((Path_A a b)(rho) ⊆r Path(A)(rho))
Lemma: path-type-subtype
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ({X ⊢ _:(Path_A a b)} ⊆r {X ⊢ _:Path(A)})
Lemma: pathtype-subset
∀[X,A,phi:Top]. (Path(A) ~ Path(A))
Lemma: path-type-subset
∀[X,A,a,b,phi:Top]. ((X, phi ⊢ Path_A a b) ~ (X ⊢ Path_A a b))
Lemma: path-type-subset-adjoin
∀[X,T,A,w,a,phi:Top]. ((X, phi.T ⊢ Path_A a w) ~ (X.T ⊢ Path_A a w))
Lemma: path-type-subset-adjoin2
∀[X,T1,T2,A,w,a,phi:Top]. ((X, phi.T1.T2 ⊢ Path_A a w) ~ (X.T1.T2 ⊢ Path_A a w))
Lemma: path-type-subset-adjoin3
∀[X,T1,T2,T3,A,w,a,phi:Top]. ((X, phi.T1.T2.T3 ⊢ Path_A a w) ~ (X.T1.T2.T3 ⊢ Path_A a w))
Lemma: path-type-subset-adjoin4
∀[X,T1,T2,T3,T4,A,w,a,phi:Top]. ((X, phi.T1.T2.T3.T4 ⊢ Path_A a w) ~ (X.T1.T2.T3.T4 ⊢ Path_A a w))
Lemma: path-type-ap-morph
∀[X,A,a,b,I,J,f,alpha,u:Top]. ((u alpha f) ~ λK,g. (u K f ⋅ g))
Lemma: path-type-at
∀[X,A,a,b,I,alpha:Top].
((Path_A a b)(alpha) ~ {p:{w:J:fset(ℕ) ⟶ f:J ⟶ I ⟶ u:Point(dM(J)) ⟶ A(f(alpha))|
∀J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀u:Point(dM(J)).
((w J f u f(alpha) g) = (w K f ⋅ g (u f(alpha) g)) ∈ A(g(f(alpha))))} |
((p I 1 0) = a(alpha) ∈ A(alpha)) ∧ ((p I 1 1) = b(alpha) ∈ A(alpha))} )
Lemma: csm-pathtype
∀X,Delta:j⊢. ∀s:Delta j⟶ X. ∀A:{X ⊢ _}. ((Path(A))s = Path((A)s) ∈ {Delta ⊢ _})
Lemma: csm-path-type
∀X,Delta:j⊢. ∀s:Delta j⟶ X. ∀A:{X ⊢ _}. ∀a,b:{X ⊢ _:A}.
(((Path_A a b))s = (Delta ⊢ Path_(A)s (a)s (b)s) ∈ {Delta ⊢ _})
Lemma: csm-path-type-id-adjoin
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[a,b:{X.B ⊢ _:(A)p}]. ∀[u:{X ⊢ _:B}].
(((Path_(A)p a b))[u] = (X ⊢ Path_A (a)[u] (b)[u]) ∈ {X ⊢ _})
Lemma: path-type-q-id-adjoin
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,u:{X ⊢ _:A}]. (((Path_(A)p (a)p q))[u] = (X ⊢ Path_A a u) ∈ {X ⊢ _})
Lemma: path-type-q-csm-adjoin
∀[X,H:j⊢]. ∀[s:H j⟶ X]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[u:{H ⊢ _:(A)s}].
(((Path_(A)p (a)p q))(s;u) = (H ⊢ Path_(A)s (a)s u) ∈ {H ⊢ _})
Lemma: path-type-p
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ((X.B ⊢ Path_(A)p (a)p (b)p) = ((Path_A a b))p ∈ {X.B ⊢ _})
Definition: path-eta
path-eta(pth) == (pth)p @ q
Lemma: path-eta_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[pth:{G ⊢ _:Path(A)}]. (path-eta(pth) ∈ {G.𝕀 ⊢ _:(A)p})
Definition: path-elim
path-elim(pth) == (pth)p @ q
Lemma: path-elim_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[pth:{G ⊢ _:Path(A)}]. (path-elim(pth) ∈ {G.𝕀 ⊢ _:(A)p})
Definition: cubical-path-app
pth @ r == pth @ r
Lemma: cubical-path-app_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[t:{X ⊢ _:(Path_A a b)}]. ∀[r:{X ⊢ _:𝕀}]. (t @ r ∈ {X ⊢ _:A})
Lemma: cubical-path-app-sq
∀[t,r:Top]. (t @ r ~ t @ r)
Lemma: csm-cubicalpath-app
∀[t,r,s:Top]. ((t @ r)s ~ (t)s @ (r)s)
Lemma: path-eta-id-adjoin
∀[G,u,pth:Top]. ((path-eta(pth))[u] ~ pth @ u)
Lemma: path-eta-0
∀[G,pth:Top]. ((path-eta(pth))[0(𝕀)] ~ pth @ 0(𝕀))
Lemma: path-eta-1
∀[G,pth:Top]. ((path-eta(pth))[1(𝕀)] ~ pth @ 1(𝕀))
Lemma: csm-cubical-path-app
∀[t,r,s:Top]. ((t @ r)s ~ (t)s @ (r)s)
Lemma: csm-path-eta
∀[pth,s:Top]. ((path-eta(pth))s ~ ((pth)p)s @ (q)s)
Lemma: cubical-path-app-0
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[t:{X ⊢ _:(Path_A a b)}]. (t @ 0(𝕀) = a ∈ {X ⊢ _:A})
Lemma: cubical-path-app-1
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[t:{X ⊢ _:(Path_A a b)}]. (t @ 1(𝕀) = b ∈ {X ⊢ _:A})
Lemma: cubical-path-ap-id-adjoin
∀[pth,x,Z:Top]. (((pth)p @ q)[x] ~ pth @ x)
Lemma: cubical-path-ap-id-adjoin2
∀[pth,x,Z:Top]. (((pth)p @ q)[x] ~ pth @ x)
Lemma: path-type-ext-eq
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}].
{t:{X ⊢ _:Path(A)}| (t @ 0(𝕀) = a ∈ {X ⊢ _:A}) ∧ (t @ 1(𝕀) = b ∈ {X ⊢ _:A})} ≡ {X ⊢ _:(Path_A a b)}
Lemma: path-type-sub-pathtype
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ({X ⊢ _:(Path_A a b)} ⊆r {X ⊢ _:Path(A)})
Lemma: csm-path-type-sub-pathtype
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[H:j⊢]. ∀[tau:H j⟶ X]. ({H ⊢ _:((Path_A a b))tau} ⊆r {H ⊢ _:(Path(A))tau})
Lemma: paths-equal
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[p:{X ⊢ _:(Path_A a b)}]. ∀[q:{X ⊢ _:Path(A)}].
p = q ∈ {X ⊢ _:(Path_A a b)} supposing p = q ∈ {X ⊢ _:Path(A)}
Lemma: csm-paths-equal
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[p:{X ⊢ _:(Path_A a b)}]. ∀[H:j⊢]. ∀[tau:H j⟶ X]. ∀[q:{H ⊢ _:(Path(A))tau}].
(p)tau = q ∈ {H ⊢ _:((Path_A a b))tau} supposing (p)tau = q ∈ {H ⊢ _:(Path(A))tau}
Definition: term-to-path
<>(a) == (λa)
Lemma: term-to-path_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀a:{X.𝕀 ⊢ _:(A)p}. (<>(a) ∈ {X ⊢ _:(Path_A (a)[0(𝕀)] (a)[1(𝕀)])})
Lemma: term-to-path-subset
∀[X,phi,a:Top]. (X, phi ⊢ <>(a) ~ X ⊢ <>(a))
Lemma: csm-term-to-path
∀[G:j⊢]. ∀[A:{G ⊢ _}].
∀a:{G.𝕀 ⊢ _:(A)p}
∀[H:j⊢]. ∀[sigma:H j⟶ G].
((<>(a))sigma = H ⊢ <>((a)sigma+) ∈ {H ⊢ _:(Path_(A)sigma ((a)sigma+)[0(𝕀)] ((a)sigma+)[1(𝕀)])})
Lemma: term-to-path-beta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[b:{X.𝕀 ⊢ _:(A)p}]. ∀[r:{X ⊢ _:𝕀}]. (<>(b) @ r = (b)[r] ∈ {X ⊢ _:A})
Lemma: term-to-path-eta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}]. (X ⊢ <>((pth)p @ q) = pth ∈ {X ⊢ _:(Path_A a b)})
Lemma: equal-paths-eta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[p,q:{X ⊢ _:Path(A)}].
p = q ∈ {X ⊢ _:Path(A)} supposing path-eta(p) = path-eta(q) ∈ {X.𝕀 ⊢ _:(A)p}
Lemma: paths-equal-eta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[p:{X ⊢ _:(Path_A a b)}]. ∀[q:{X ⊢ _:Path(A)}].
p = q ∈ {X ⊢ _:(Path_A a b)} supposing path-eta(p) = path-eta(q) ∈ {X.𝕀 ⊢ _:(A)p}
Lemma: term-to-path-app-snd
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ((<>((a)p))p @ q = (a)p ∈ {X.𝕀 ⊢ _:(A)p})
Lemma: term-to-path-wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u,v:{X ⊢ _:A}].
∀a:{X.𝕀 ⊢ _:(A)p}. (<>(a) ∈ {X ⊢ _:(Path_A u v)}) supposing (X ⊢ (a)[0(𝕀)]=u:A and X ⊢ (a)[1(𝕀)]=v:A)
Lemma: term-to-path-equal
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u,v:{X ⊢ _:A}].
∀a:{X.𝕀 ⊢ _:(A)p}
(∀[b:{X.𝕀 ⊢ _:(A)p}]. X ⊢ <>(a) = X ⊢ <>(b) ∈ {X ⊢ _:(Path_A u v)} supposing a = b ∈ {X.𝕀 ⊢ _:(A)p}) supposing
(X ⊢ (a)[0(𝕀)]=u:A and
X ⊢ (a)[1(𝕀)]=v:A)
Definition: term-to-pathtype
<>a == <>(a)
Lemma: term-to-pathtype_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀a:{X.𝕀 ⊢ _:(A)p}. (<>a ∈ {X ⊢ _:Path(A)})
Lemma: term-to-pathtype-eta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀pth:{X ⊢ _:Path(A)}. (<>(pth)p @ q = pth ∈ {X ⊢ _:Path(A)})
Lemma: csm-term-to-pathtype
∀[G:j⊢]. ∀[A:{G ⊢ _}].
∀a:{G.𝕀 ⊢ _:(A)p}. ∀[H:j⊢]. ∀[sigma:H j⟶ G]. ((<>a)sigma = <>(a)sigma+ ∈ {H ⊢ _:Path((A)sigma)})
Lemma: term-to-pathtype-beta
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀r:{G ⊢ _:𝕀}. ∀a:{G.𝕀 ⊢ _:(A)p}. (<>a @ r = (a)[r] ∈ {G ⊢ _:A})
Definition: map-path
map-path(G;f;pth) == <>(app((f)p; path-eta(pth)))
Lemma: map-path_wf
∀[G:j⊢]. ∀[S,T:{G ⊢ _}]. ∀[f:{G ⊢ _:(S ⟶ T)}]. ∀[a,b:{G ⊢ _:S}]. ∀[pth:{G ⊢ _:(Path_S a b)}].
(map-path(G;f;pth) ∈ {G ⊢ _:(Path_T app(f; a) app(f; b))})
Definition: cubical-refl
refl(a) == <>((a)p)
Lemma: cubical-refl_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. (refl(a) ∈ {X ⊢ _:(Path_A a a)})
Lemma: term-to-path-is-refl
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀a:{X ⊢ _:A}. (X ⊢ <>((a)p) = refl(a) ∈ {X ⊢ _:(Path_A a a)})
Lemma: csm-cubical-refl
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[H:j⊢]. ∀[s:H j⟶ X]. ((refl(a))s = refl((a)s) ∈ {H ⊢ _:(Path_(A)s (a)s (a)s)})
Lemma: cubical-refl-p
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. (refl((a)p) = (refl(a))p ∈ {X.B ⊢ _:((Path_A a a))p})
Lemma: cubical-refl-p-p
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[C:{X.B ⊢ _}].
(refl(((a)p)p) = ((refl(a))p)p ∈ {X.B.C ⊢ _:(((Path_A a a))p)p})
Lemma: refl-path-app
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[r:{X ⊢ _:𝕀}]. (refl(a) @ r = a ∈ {X ⊢ _:A})
Lemma: cubical-refl-app-snd
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ((refl(a))p @ q = (a)p ∈ {X.𝕀 ⊢ _:(A)p})
Definition: rev-path
rev-path(G;pth) == <>((pth)p @ 1-(q))
Lemma: rev-path_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[b,a:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}]. (rev-path(X;pth) ∈ {X ⊢ _:(Path_A b a)})
Definition: contractible-type
Contractible(A) == Σ A Π(A)p (Path_((A)p)p (q)p q)
Lemma: contractible-type_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. X ⊢ Contractible(A)
Lemma: contractible-type-subset
∀[X,A,phi:Top]. (X, phi ⊢ Contractible(A) ~ X ⊢ Contractible(A))
Lemma: contractible-type-subset2
∀[X,A,B,C,phi:Top]. (X, phi.B.C ⊢ Contractible(A) ~ X.B.C ⊢ Contractible(A))
Lemma: csm-contractible-type
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[Z:j⊢]. ∀[s:Z j⟶ X]. ((Contractible(A))s = Z ⊢ Contractible((A)s) ∈ {Z ⊢ _})
Lemma: contractible-type-at
∀[X,A,I,rho:Top]. (Contractible(A)(rho) ~ u:A(rho) × cubical-pi-family(X.A;(A)p;(Path_((A)p)p (q)p q);I;(rho;u)))
Definition: contr-center
contr-center(c) == c.1
Lemma: contr-center_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[c:{X ⊢ _:Contractible(A)}]. (contr-center(c) ∈ {X ⊢ _:A})
Lemma: csm-contr-center
∀[c,s:Top]. ((contr-center(c))s ~ contr-center((c)s))
Definition: contr-path
contr-path(c;x) == app(c.2; x)
Lemma: contr-path_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[c:{X ⊢ _:Contractible(A)}]. ∀[x:{X ⊢ _:A}].
(contr-path(c;x) ∈ {X ⊢ _:(Path_A contr-center(c) x)})
Lemma: csm-contr-path
∀[s,c,x:Top]. ((contr-path(c;x))s ~ contr-path((c)s;(x)s))
Definition: contr-witness
contr-witness(X;c;p) == cubical-pair(c;(λp))
Lemma: contr-witness_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[c:{X ⊢ _:A}]. ∀[p:{X.A ⊢ _:(Path_(A)p (c)p q)}].
(contr-witness(X;c;p) ∈ {X ⊢ _:Contractible(A)})
Definition: cubical-fiber
Fiber(w;a) == Σ T (Path_(A)p (a)p app((w)p; q))
Lemma: cubical-fiber_wf
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. X ⊢ Fiber(w;a)
Lemma: cubical-fiber-subset
∀[X,T,A,w,a,phi:Top]. (X, phi ⊢ Fiber(w;a) ~ X ⊢ Fiber(w;a))
Lemma: cubical-fiber-subset-adjoin
∀[X,T,A,B,w,a,phi:Top]. (X, phi.B ⊢ Fiber(w;a) ~ X.B ⊢ Fiber(w;a))
Lemma: cubical-fiber-subset-adjoin2
∀[X,T,A,B1,B2,w,a,phi:Top]. (X, phi.B1.B2 ⊢ Fiber(w;a) ~ X.B1.B2 ⊢ Fiber(w;a))
Lemma: csm-cubical-fiber
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[Z:j⊢]. ∀[s:Z j⟶ X].
((Fiber(w;a))s = Z ⊢ Fiber((w)s;(a)s) ∈ {Z ⊢ _})
Lemma: cubical-fiber-at
∀[X,T,A,w,a,I,rho:Top]. (Fiber(w;a)(rho) ~ u:T(rho) × (Path_(A)p (a)p app((w)p; q))((rho;u)))
Definition: fiber-point
fiber-point(t;c) == cubical-pair(t;c)
Lemma: fiber-point_wf
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[f:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[t:{X ⊢ _:T}]. ∀[c:{X ⊢ _:(Path_A a app(f; t))}].
(fiber-point(t;c) ∈ {X ⊢ _:Fiber(f;a)})
Lemma: csm-fiber-point
∀[s,t,c:Top]. ((fiber-point(t;c))s ~ fiber-point((t)s;(c)s))
Lemma: fiber-subset
∀[X,T,A,f,a,phi:Top]. (X, phi ⊢ Fiber(f;a) ~ X ⊢ Fiber(f;a))
Definition: fiber-member
fiber-member(p) == p.1
Lemma: fiber-member_wf
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[p:{X ⊢ _:Fiber(w;a)}]. (fiber-member(p) ∈ {X ⊢ _:T})
Lemma: csm-fiber-member
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[p:{X ⊢ _:Fiber(w;a)}]. ∀[H:j⊢]. ∀[s:H j⟶ X].
((fiber-member(p))s = fiber-member((p)s) ∈ {H ⊢ _:(T)s})
Lemma: fiber-member-fiber-point
∀[X,a,b:Top]. (fiber-member(fiber-point(a;b)) ~ (a)1(X))
Definition: fiber-path
fiber-path(p) == p.2
Lemma: fiber-path_wf
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[p:{X ⊢ _:Fiber(w;a)}].
(fiber-path(p) ∈ {X ⊢ _:(Path_A a app(w; fiber-member(p)))})
Lemma: csm-fiber-path
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[p:{X ⊢ _:Fiber(w;a)}]. ∀[H:j⊢]. ∀[s:H j⟶ X].
((fiber-path(p))s = fiber-path((p)s) ∈ {H ⊢ _:(Path_(A)s (a)s app((w)s; fiber-member((p)s)))})
Lemma: fiber-path-fiber-point
∀[X,a,b:Top]. (fiber-path(fiber-point(a;b)) ~ (b)1(X))
Lemma: sigma-path-fst
∀X:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀x,y:{X ⊢ _:Σ A B}. ({X ⊢ _:(Path_Σ A B x y)} ⇒ {X ⊢ _:(Path_A x.1 y.1)})
Definition: is-cubical-equiv
IsEquiv(T;A;w) == ΠA Contractible(Fiber((w)p;q))
Lemma: is-cubical-equiv_wf
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. X ⊢ IsEquiv(T;A;w)
Lemma: csm-is-cubical-equiv
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[Z:j⊢]. ∀[s:Z j⟶ X].
((IsEquiv(T;A;w))s = IsEquiv((T)s;(A)s;(w)s) ∈ {Z ⊢ _})
Lemma: is-cubical-equiv-at
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[A,w,I,rho:Top].
(IsEquiv(T;A;w)(rho) ~ {eqv:J:fset(ℕ)
⟶ f:J ⟶ I
⟶ u:A(f(rho))
⟶ (u1:u1:T((p)(f(rho);u)) × (Path_((A)p)p (q)p app(((w)p)p; q))(((f(rho);u);u1))
× cubical-pi-family(X.A.Fiber((w)p;q);(Fiber((w)p;q))p;(Path_((Fiber((w)p;q))p)p (q)p
q);J;((f(rho);u);u1)))|
∀J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀u:A(f(rho)).
((eqv J f u (f(rho);u) g)
= (eqv K f ⋅ g (u f(rho) g))
∈ (u1:u1:T((p)g((f(rho);u))) × (Path_((A)p)p (q)p app(((w)p)p; q))((g((f(rho);u));u1))
× cubical-pi-family(X.A.Fiber((w)p;q);(Fiber((w)p;q))p;(Path_((Fiber((w)p;q))p)p (q)p
q);K;(g((f(rho);u));u1))))} )
Definition: is-equiv-witness
is-equiv-witness(G;A;c;p) == (λcontr-witness(G.A;c;p))
Lemma: is-equiv-witness_wf
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[c:{X.A ⊢ _:Fiber((w)p;q)}].
∀[p:{X.A.Fiber((w)p;q) ⊢ _:(Path_(Fiber((w)p;q))p (c)p q)}].
(is-equiv-witness(X;A;c;p) ∈ {X ⊢ _:IsEquiv(T;A;w)})
Definition: cubical-equiv
Equiv(T;A) == Σ (T ⟶ A) IsEquiv((T)p;(A)p;q)
Lemma: cubical-equiv_wf
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. X ⊢ Equiv(T;A)
Lemma: csm-cubical-equiv
∀[H,K:j⊢]. ∀[tau:K j⟶ H]. ∀[A,E:{H ⊢ _}]. ((Equiv(A;E))tau = Equiv((A)tau;(E)tau) ∈ {K ⊢ _})
Lemma: cubical-equiv-p
∀[H:j⊢]. ∀[T,A,E:{H ⊢ _}]. ((Equiv(A;E))p = Equiv((A)p;(E)p) ∈ {H.T ⊢ _})
Lemma: equal-cubical-equiv-at
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[x:Equiv(T;A)(a)]. ∀[y:u:(T ⟶ A)(a) × IsEquiv((T)p;(A)p;q)((a;u))].
x = y ∈ Equiv(T;A)(a) supposing x = y ∈ (u:(T ⟶ A)(a) × IsEquiv((T)p;(A)p;q)((a;u)))
Lemma: cubical-equiv-subset
∀[X,T,A,phi:Top]. (Equiv(T;A) ~ Equiv(T;A))
Definition: equiv-fun
equiv-fun(f) == f.1
Lemma: equiv-fun_wf
∀[G:j⊢]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. (equiv-fun(f) ∈ {G ⊢ _:(T ⟶ A)})
Lemma: csm-equiv-fun
∀[s,f:Top]. ((equiv-fun(f))s ~ equiv-fun((f)s))
Definition: equiv-contr
equiv-contr(f;a) == app(f.2; a)
Lemma: equiv-contr_wf
∀[G:j⊢]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[a:{G ⊢ _:A}].
(equiv-contr(f;a) ∈ {G ⊢ _:Contractible(Fiber(equiv-fun(f);a))})
Lemma: csm-equiv-contr
∀[s,f,a:Top]. ((equiv-contr(f;a))s ~ equiv-contr((f)s;(a)s))
Definition: equiv-witness
equiv-witness(f;cntr) == cubical-pair(f;cntr)
Lemma: equiv-witness_wf
∀[G:j⊢]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:(T ⟶ A)}]. ∀[iseq:{G ⊢ _:IsEquiv(T;A;f)}].
(equiv-witness(f;iseq) ∈ {G ⊢ _:Equiv(T;A)})
Lemma: cubical-fiber-id-fun
∀X:j⊢. ∀T:{X ⊢ _}. ∀[u:{X ⊢ _:T}]. (X ⊢ Fiber(cubical-id-fun(X);u) = Σ T (Path_(T)p (u)p q) ∈ {X ⊢ _})
Definition: path-point
path-point(pth) == (pth)p @ q
Lemma: path-point_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}]. (path-point(pth) ∈ {X.𝕀 ⊢ _:(A)p})
Lemma: path-point-0
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}]. ((path-point(pth))[0(𝕀)] = a ∈ {X ⊢ _:A})
Lemma: path-point-1
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}]. ((path-point(pth))[1(𝕀)] = b ∈ {X ⊢ _:A})
Definition: path-contraction
path-contraction(X;pth) == <>(((pth)p)p @ (q)p ∧ q)
Lemma: path-contraction_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}].
(path-contraction(X;pth) ∈ {X.𝕀 ⊢ _:(Path_(A)p (a)p path-point(pth))})
Lemma: path-contraction-0
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}].
((path-contraction(X;pth))[0(𝕀)] = refl(a) ∈ {X ⊢ _:(Path_A a a)})
Lemma: path-contraction-1
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}].
((path-contraction(X;pth))[1(𝕀)] = pth ∈ {X ⊢ _:(Path_A a b)})
Definition: singleton-contraction
singleton-contraction(X;pth) == <>(cubical-pair((pth)p @ q;path-contraction(X;pth)))
Lemma: singleton-contraction_wf
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[a,b:{X ⊢ _:T}]. ∀[pth:{X ⊢ _:(Path_T a b)}].
(singleton-contraction(X;pth) ∈ {X ⊢ _:(Path_Σ T (Path_(T)p (a)p q) cubical-pair(a;refl(a)) cubical-pair(b;pth))})
Definition: singleton-type
Singleton(a) == Σ A (Path_(A)p (a)p q)
Lemma: singleton-type_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. X ⊢ Singleton(a)
Definition: singleton-center
singleton-center(X;a) == cubical-pair(a;refl(a))
Lemma: singleton-center_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. (singleton-center(X;a) ∈ {X ⊢ _:Singleton(a)})
Lemma: csm-singleton-center
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[H:j⊢]. ∀[s:H j⟶ X].
((singleton-center(X;a))s = singleton-center(H;(a)s) ∈ {H ⊢ _:Singleton((a)s)})
Lemma: singleton-contraction_wf2
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[a:{X ⊢ _:T}]. ∀[s:{X ⊢ _:Singleton(a)}].
(singleton-contraction(X;s.2) ∈ {X ⊢ _:(Path_Singleton(a) singleton-center(X;a) s)})
Lemma: csm-singleton-type
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[H:j⊢]. ∀[s:H j⟶ X]. ((Singleton(a))s = Singleton((a)s) ∈ {H ⊢ _})
Definition: singleton-contr
singleton-contr(X;A;a) == contr-witness(X;singleton-center(X;a);singleton-contraction(X.Singleton(a);q.2))
Lemma: singleton-contr_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. (singleton-contr(X;A;a) ∈ {X ⊢ _:Contractible(Singleton(a))})
Lemma: contractibilty-of-singleton
∀X:j⊢. ∀A:{X ⊢ _}. ∀a:{X ⊢ _:A}. {X ⊢ _:Contractible(Singleton(a))}
Definition: refl-contraction
refl-contraction(X;A;a) == singleton-contraction(X;refl(a))
Definition: id-fiber-center
id-fiber-center(X;T) == cubical-pair(q;refl(q))
Lemma: id-fiber-center_wf
∀[X:j⊢]. ∀[T:{X ⊢ _}]. (id-fiber-center(X;T) ∈ {X.T ⊢ _:Σ (T)p (Path_((T)p)p (q)p q)})
Lemma: csm-id-fiber-center
∀[G,K:j⊢]. ∀[tau:K j⟶ G]. ∀[A:{G ⊢ _}].
(id-fiber-center(K;(A)tau) = (id-fiber-center(G;A))tau+ ∈ {K.(A)tau ⊢ _:Fiber((cubical-id-fun(K))p;q)})
Definition: id-fiber-contraction
id-fiber-contraction(X;T) == (singleton-contraction(X.T.(T)p.(Path_((T)p)p (q)p q);q))SigmaElim
Lemma: id-fiber-contraction_wf
∀[X:j⊢]. ∀[T:{X ⊢ _}].
(id-fiber-contraction(X;T) ∈ {X.T.Σ (T)p (Path_((T)p)p (q)p q) ⊢ _
:(Path_(Σ (T)p (Path_((T)p)p (q)p q))p (id-fiber-center(X;T))p q)})
Lemma: csm-id-fiber-contraction
∀[G,K:j⊢]. ∀[tau:K j⟶ G]. ∀[A:{G ⊢ _}].
(id-fiber-contraction(K;(A)tau)
= (id-fiber-contraction(G;A))tau++
∈ {K.(A)tau.Σ ((A)tau)p (Path_(((A)tau)p)p (q)p q) ⊢ _
:(Path_(Σ ((A)tau)p (Path_(((A)tau)p)p (q)p q))p (id-fiber-center(K;(A)tau))p q)})
Definition: cubical-id-is-equiv
cubical-id-is-equiv(X;T) == (λcontr-witness(X.T;id-fiber-center(X;T);id-fiber-contraction(X;T)))
Lemma: cubical-id-is-equiv_wf
∀X:j⊢. ∀T:{X ⊢ _}. (cubical-id-is-equiv(X;T) ∈ {X ⊢ _:IsEquiv(T;T;cubical-id-fun(X))})
Lemma: csm-cubical-id-is-equiv
∀[G,K:j⊢]. ∀[tau:K j⟶ G]. ∀[A:{G ⊢ _}].
(cubical-id-is-equiv(K;(A)tau) = (cubical-id-is-equiv(G;A))tau ∈ {K ⊢ _:IsEquiv((A)tau;(A)tau;cubical-id-fun(K))})
Definition: cubical-id-equiv
IdEquiv(X;T) == equiv-witness(cubical-id-fun(X);cubical-id-is-equiv(X;T))
Lemma: cubical-id-equiv_wf
∀X:j⊢. ∀T:{X ⊢ _}. (IdEquiv(X;T) ∈ {X ⊢ _:Equiv(T;T)})
Lemma: csm-cubical-id-equiv
∀[G,K:j⊢]. ∀[tau:K j⟶ G]. ∀[A:{G ⊢ _}]. ((IdEquiv(G;A))tau = IdEquiv(K;(A)tau) ∈ {K ⊢ _:Equiv((A)tau;(A)tau)})
Lemma: cubical-id-equiv-subset
∀[G,phi,A:Top]. (IdEquiv(G, phi;A) ~ IdEquiv(G;A))
Lemma: equiv-fun-cubical-id-equiv
∀[G,A:Top]. (equiv-fun(IdEquiv(G;A)) ~ cubical-id-fun(G))
Lemma: unit-pathtype
∀[G:j⊢]. ∀a,b:{G ⊢ _:Path(1)}. (a = b ∈ {G ⊢ _:Path(1)})
Lemma: unit-path-type
∀[G:j⊢]. ∀[x,y:{G ⊢ _:1}]. ∀a,b:{G ⊢ _:(Path_1 x y)}. (a = b ∈ {G ⊢ _:(Path_1 x y)})
Definition: h-level
h-level(X;n;A) == primrec(n;λA.Contractible(A);λi,r,A. (r ΠA Π(A)p (Path_((A)p)p (q)p q))) A
Lemma: h-level_wf
∀[n:ℕ]. ∀[X:j⊢]. ∀[A:{X ⊢ _}]. X ⊢ h-level(X;n;A)
Definition: is-prop
isProp(A) == ΠA Π(A)p (Path_((A)p)p (q)p q)
Lemma: is-prop_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. X ⊢ isProp(A)
Lemma: is-prop-contractible
∀X:j⊢. ∀A:{X ⊢ _}. ({X ⊢ _:isProp(A)} ⇒ {X ⊢ _:A} ⇒ {X ⊢ _:Contractible(A)})
Definition: cubical-fun-ext
cubical-fun-ext(X;e) == <>((λapp(((e)p)p; q) @ (q)p))
Lemma: cubical-fun-ext_wf
∀X:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀f,g:{X ⊢ _:ΠA B}. ∀e:{X ⊢ _:ΠA (Path_B app((f)p; q) app((g)p; q))}.
(cubical-fun-ext(X;e) ∈ {X ⊢ _:(Path_ΠA B f g)})
Lemma: discrete-path-endpoints
∀[X:j⊢]. ∀[T:Type].
∀a:{X ⊢ _:discr(T)}. ∀[b:{X ⊢ _:discr(T)}]. ∀[p:{X ⊢ _:(Path_discr(T) a b)}]. (a = b ∈ {X ⊢ _:discr(T)})
Lemma: discrete-pathtype
∀[T:Type]. ∀[X:j⊢]. ∀[pth:{X ⊢ _:Path(discr(T))}]. (pth = refl(pth @ 0(𝕀)) ∈ {X ⊢ _:Path(discr(T))})
Lemma: discrete-pathtype-uniform
∀[T:Type]. ∀[X,Z:j⊢]. ∀[s:Z j⟶ X]. ∀[pth:{Z ⊢ _:(Path(discr(T)))s}].
(pth = refl(pth @ 0(𝕀)) ∈ {Z ⊢ _:(Path(discr(T)))s})
Lemma: discrete-path
∀[T:Type]. ∀[X:j⊢]. ∀[a,b:{X ⊢ _:discr(T)}]. ∀[p:{X ⊢ _:(Path_discr(T) a b)}].
(p = refl(a) ∈ {X ⊢ _:(Path_discr(T) a b)})
Lemma: discrete-path-equal
∀[T:Type]. ∀[X:j⊢]. ∀[a,b:{X ⊢ _:discr(T)}]. ∀[p1,p2:{X ⊢ _:(Path_discr(T) a b)}].
(p1 = p2 ∈ {X ⊢ _:(Path_discr(T) a b)})
Lemma: equal-fiber-when-discrete
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[f:{X ⊢ _:(A ⟶ T)}]. ∀[z:{X ⊢ _:T}]. ∀[a,b:{X ⊢ _:Fiber(f;z)}].
a = b ∈ {X ⊢ _:Fiber(f;z)} ⇐⇒ a.1 = b.1 ∈ {X ⊢ _:A}
supposing ∀x:{X ⊢ _:Path(T)}. (x = refl(x @ 0(𝕀)) ∈ {X ⊢ _:Path(T)})
Lemma: equal-fiber-discrete
∀[B:Type]. ∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[f:{X ⊢ _:(A ⟶ discr(B))}]. ∀[z:{X ⊢ _:discr(B)}]. ∀[a,b:{X ⊢ _:Fiber(f;z)}].
(a = b ∈ {X ⊢ _:Fiber(f;z)} ⇐⇒ a.1 = b.1 ∈ {X ⊢ _:A})
Lemma: fiber-discrete-equal
∀[B:Type]. ∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[f:{X ⊢ _:(A ⟶ discr(B))}]. ∀[z:{X ⊢ _:discr(B)}]. ∀[fbr:{X ⊢ _:Fiber(f;z)}].
(app(f; fiber-member(fbr)) = z ∈ {X ⊢ _:discr(B)})
Definition: discrete-fun
discrete-fun(f) == λI,alpha. (f I alpha I 1)
Lemma: discrete-fun_wf
∀[A,B:Type]. ∀[X:j⊢]. ∀[f:{X ⊢ _:(discr(A) ⟶ discr(B))}]. (discrete-fun(f) ∈ {X ⊢ _:discr(A ⟶ B)})
Lemma: discrete-fun-at-app
∀[f,a,alpha,I:Top]. (app(f; discr(a))(alpha) ~ discrete-fun(f)(alpha) a)
Lemma: discrete-fun-at
∀[A,B:Type]. ∀[f:{() ⊢ _:(discr(A) ⟶ discr(B))}]. ∀[I:fset(ℕ)]. ∀[a:()(I)].
((f I a) = (λJ,h,u. (f {} ⋅ {} 1 u)) ∈ (discr(A) ⟶ discr(B))(a))
Lemma: discrete-fun-invariant
∀[A,B:Type]. ∀[f:{() ⊢ _:(discr(A) ⟶ discr(B))}]. ∀[I:fset(ℕ)]. ∀[a:()(I)].
((f I a) = (f {} ⋅) ∈ (discr(A) ⟶ discr(B))(a))
Lemma: discrete-fun-app-invariant
∀[A,B:Type]. ∀[f:{() ⊢ _:(discr(A) ⟶ discr(B))}]. ∀[I:fset(ℕ)]. ∀[a:()(I)]. ∀[t:A].
(app(f; discr(t))(a) = app(f; discr(t))(⋅) ∈ B)
Lemma: discrete-fun-bijection
∀[A,B:Type]. Bij({() ⊢ _:(discr(A) ⟶ discr(B))};{() ⊢ _:discr(A ⟶ B)};λf.discrete-fun(f))
Definition: discrete-family
discrete-family(A;a.B[a]) == <λI,alpha. B[snd(alpha)], λI,J,f,alpha,x. x>
Lemma: discrete-family_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. X.discr(A) ⊢ discrete-family(A;a.B[a])
Lemma: csm-discrete-family
∀[A,B,s:Top]. ((discrete-family(A;a.B[a]))(s;q) ~ discrete-family(A;a.B[a]))
Lemma: csm-discrete-pi
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X,Y:j⊢]. ∀[s:Y j⟶ X].
((Πdiscr(A) discrete-family(A;a.B[a]))s = Y ⊢ Πdiscr(A) discrete-family(A;a.B[a]) ∈ {Y ⊢ _})
Lemma: csm-discrete-sigma
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X,Y:j⊢]. ∀[s:Y j⟶ X].
((Σ discr(A) discrete-family(A;a.B[a]))s = Σ discr(A) discrete-family(A;a.B[a]) ∈ {Y ⊢ _})
Definition: discrete-function
discrete-function(f) == λI,alpha. (f I alpha I 1)
Lemma: discrete-function_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. ∀[f:{X ⊢ _:Πdiscr(A) discrete-family(A;a.B[a])}].
(discrete-function(f) ∈ {X ⊢ _:discr(a:A ⟶ B[a])})
Definition: discrete-function-inv
discrete-function-inv(X; b) == (λλI,alpha. (b(fst(alpha)) q(alpha)))
Lemma: discrete-function-inv_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. ∀[b:{X ⊢ _:discr(a:A ⟶ B[a])}].
(discrete-function-inv(X; b) ∈ {X ⊢ _:Πdiscr(A) discrete-family(A;a.B[a])})
Lemma: discrete-function-inv-property
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. ∀[b:{X ⊢ _:discr(a:A ⟶ B[a])}].
(discrete-function(discrete-function-inv(X; b)) = b ∈ {X ⊢ _:discr(a:A ⟶ B[a])})
Lemma: discrete-function-injection
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢].
∀f,g:{X ⊢ _:Πdiscr(A) discrete-family(A;a.B[a])}.
f = g ∈ {X ⊢ _:Πdiscr(A) discrete-family(A;a.B[a])}
supposing discrete-function(f) = discrete-function(g) ∈ {X ⊢ _:discr(a:A ⟶ B[a])}
Lemma: discrete-pi-bijection
∀[A:Type]. ∀[B:A ⟶ Type]. ∀X:j⊢. {X ⊢ _:Πdiscr(A) discrete-family(A;a.B[a])} ~ {X ⊢ _:discr(a:A ⟶ B[a])}
Lemma: discrete-pi-equiv
∀A:Type. ∀B:A ⟶ Type. ∀X:j⊢. {X ⊢ _:Equiv(Πdiscr(A) discrete-family(A;a.B[a]);discr(a:A ⟶ B[a]))}
Definition: discrete-pair
discrete-pair(p) == λI,alpha. <p.1(alpha), p.2(alpha)>
Lemma: discrete-pair_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. ∀[p:{X ⊢ _:Σ discr(A) discrete-family(A;a.B[a])}].
(discrete-pair(p) ∈ {X ⊢ _:discr(a:A × B[a])})
Definition: discrete-pair-inv
discrete-pair-inv(X;b) == cubical-pair(λI,alpha. (fst(b(alpha)));λI,alpha. (snd(b(alpha))))
Lemma: discrete-pair-inv_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. ∀[b:{X ⊢ _:discr(a:A × B[a])}].
(discrete-pair-inv(X;b) ∈ {X ⊢ _:Σ discr(A) discrete-family(A;a.B[a])})
Lemma: discrete-pair-inv-property
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. ∀[b:{X ⊢ _:discr(a:A × B[a])}].
(discrete-pair(discrete-pair-inv(X;b)) = b ∈ {X ⊢ _:discr(a:A × B[a])})
Lemma: discrete-pair-injection
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢].
∀f,g:{X ⊢ _:Σ discr(A) discrete-family(A;a.B[a])}.
f = g ∈ {X ⊢ _:Σ discr(A) discrete-family(A;a.B[a])}
supposing discrete-pair(f) = discrete-pair(g) ∈ {X ⊢ _:discr(a:A × B[a])}
Lemma: discrete-sigma-bijection
∀[A:Type]. ∀[B:A ⟶ Type]. ∀X:j⊢. {X ⊢ _:Σ discr(A) discrete-family(A;a.B[a])} ~ {X ⊢ _:discr(a:A × B[a])}
Lemma: discrete-sigma-equiv
∀A:Type. ∀B:A ⟶ Type. ∀X:j⊢. {X ⊢ _:Equiv(Σ discr(A) discrete-family(A;a.B[a]);discr(a:A × B[a]))}
Definition: bijection-equiv
bijection-equiv(X;A;B;f;g) ==
equiv-witness(cubical-lam(X;λI,alpha. (f (snd(alpha))));
(λcontr-witness(X.discr(B);fiber-point(λI,alpha. (g (snd(alpha)));refl(q));refl(q))))
Lemma: bijection-equiv_wf
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[g:B ⟶ A].
(∀[X:j⊢]. (bijection-equiv(X;A;B;f;g) ∈ {X ⊢ _:Equiv(discr(A);discr(B))})) supposing
((∀a:A. ((g (f a)) = a ∈ A)) and
(∀b:B. ((f (g b)) = b ∈ B)))
Lemma: equiv-discrete-type
∀A,B:Type. ∀f:A ⟶ B. (Bij(A;B;f) ⇒ (∀X:j⊢. {X ⊢ _:Equiv(discr(A);discr(B))}))
Definition: equiv-bijection
equiv-bijection(e) == discrete-fun(equiv-fun(e))(⋅)
Lemma: equiv-bijection_wf
∀[A,B:Type]. ∀[e:{() ⊢ _:Equiv(discr(A);discr(B))}]. (equiv-bijection(e) ∈ A ⟶ B)
Definition: equiv-bijection-is
equiv-bijection-is(e) == <λa1,a2,eq. Ax, λb.<equiv-contr(e;discr(b)).1.1(⋅), Ax>>
Lemma: equiv-bijection-is_wf
∀[A,B:Type]. ∀[e:{() ⊢ _:Equiv(discr(A);discr(B))}]. (equiv-bijection-is(e) ∈ Bij(A;B;equiv-bijection(e)))
Lemma: equiv-bijection-property
∀A,B:Type. ∀e:{() ⊢ _:Equiv(discr(A);discr(B))}. Bij(A;B;equiv-bijection(e))
Lemma: discrete-equiv-iff
∀A,B:Type. ({() ⊢ _:Equiv(discr(A);discr(B))} ⇐⇒ A ~ B)
Lemma: equiv-bijection-equiv
∀[A,B:Type]. ∀[e:A ~ B].
((((λe.<equiv-bijection(e), equiv-bijection-is(e)>) o (λe.bijection-equiv(();A;B;fst(e);bij_inv(snd(e))))) e)
= e
∈ A ~ B)
Lemma: paths-are-refl-iff
∀[X:j⊢]. ∀[A:{X ⊢ _}].
uiff(∀Z:j⊢. ∀s:Z j⟶ X. ∀p:{Z ⊢ _:Path((A)s)}. (p = refl(p @ 0(𝕀)) ∈ {Z ⊢ _:Path((A)s)});∀Z:j⊢. ∀s:Z j⟶ X.
∀p:{Z ⊢ _:Path((A)s)}.
∀[x,y:{Z ⊢ _:𝕀}].
(p @ x
= p @ y
∈ {Z ⊢ _:(A)s}))
Lemma: paths-are-refl-iff2
∀[X:j⊢]. ∀[A:{X ⊢ _}].
uiff(∀Z:j⊢. ∀s:Z j⟶ X. ∀p:{Z ⊢ _:Path((A)s)}. (p = refl(p @ 0(𝕀)) ∈ {Z ⊢ _:Path((A)s)});∀Z:j⊢. ∀s:Z j⟶ X.
∀p:{Z.𝕀 ⊢ _:((A)s)p}.
∀[x,y:{Z ⊢ _:𝕀}].
((p)[x]
= (p)[y]
∈ {Z ⊢ _:(A)s}))
Definition: refl-map
refl-map(X;A) == cubical-lam(X;discr(refl(q)))
Lemma: refl-map_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. (refl-map(X;A) ∈ {X ⊢ _:(A ⟶ discr({X.A ⊢ _:Path((A)p)}))})
Definition: refl-inv
refl-inv(b) == λI,alpha. (b(alpha) @ 0(𝕀))p(fst(alpha))
Comment: composition structure doc
=========================== composition structure =============================
In this section we define composition and filling operators for
cubical types.
composition-op
filling-op.
We prove that fill_from_comp constructs a filling operator from
a composition operator.
(Proved in the lemmas fill_from_comp_wf1, fill_from_comp_property1,
fill_from_comp_wf2, and fill_from_comp_wf.)
One main result is that pi-comp constructs a composition operator
for ΠA B from composition operators for A and B.
This is proved in the series of lemmas
pi-comp_wf1, pi-comp_wf2, pi-comp_wf3,
pi-comp-property, and pi-comp-uniformity, and finally pi-comp_wf.
⋅
Comment: definitions of composition
Coquand's original formulation of a composition structure
for a type ⌜Gamma ⊢ A⌝ is defined in composition-op.
The type, Gamma ⊢ CompOp(A), is the type of composition operations.
Thought of as a proposition, it says
⌜∀I:fset(ℕ). ∀i:{i:ℕ| ¬i ∈ I} . ∀rho:Gamma(I+i). ∀phi:𝔽(I). ∀u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}. ∀a0:A((i0)(rho)).
((∀J:fset(ℕ). ∀f:I,phi(J). ((a0 (i0)(rho) f) = u((i0) ⋅ f)))
⇒ (∃a1:A((i1)(rho)). ∀J:fset(ℕ). ∀f:I,phi(J). ((a1 (i1)(rho) f) = u((i1) ⋅ f))))⌝
In addition, the operation, comp, that builds a1 from the other inputs
must satisfy composition-uniformity(Gamma;A;comp)
This definition quantifies only over representable cubical sets of the form
I+i,s(phi), and also over the sets rho:Gamma(I+i).
So, when Gamma is a cubical set in universe{n} then this definition of
composition structure is also a type in universe{n}.
A uniform composition operation of this kind can be used to build
a cubical term composition-term that satisfies this typing rule
composition-term_wf
The type of such a construction is composition-function
and it is uniform if uniform-comp-function.
This gives another definition of composition structure:
composition-structure
and the two definitions are equivalent:
composition-op-implies-composition-structure
composition-structure-implies-composition-op
The second definition quantifies over all cubical sets, so when Gamma
is a cubical set in universe{n}, then composition structure for Gamma
is a proposition in universe{n+1}. Nevertheless, the two definitions are equivalent.
We use the composition-op version to show composition for ΠA B, Σ A B, and (Path_A a b):
pi-comp_wf
sigmacomp_wf
path_comp_wf
We will use the second definition to build a composition structure for
Glue [phi ⊢→ (T;w)] A
But, when we define the universes of "fibrant" types we use the original
Gamma ⊢ CompOp(A) definition in order to get cumulativity
(viz. that term in ⌜c𝕌{n}⌝ is also in ⌜c𝕌{n+1}⌝).
⋅
Lemma: comp-nc-0-subset-I_cube
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[phi:𝔽(I)]. ∀J:fset(ℕ). ∀[f:I,phi(J)]. ((i0) ⋅ f ∈ I+i,s(phi)(J))
Lemma: comp-nc-1-subset-I_cube
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[phi:𝔽(I)]. ∀J:fset(ℕ). ∀[f:I,phi(J)]. ((i1) ⋅ f ∈ I+i,s(phi)(J))
Lemma: csm-cubical-subset-type-lemma
∀[Gamma,Delta:j⊢]. ∀[sigma:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Delta(I+i)].
∀[J:fset(ℕ)]. ∀[f:J ⟶ I].
(((A)sigma(f((i1)(rho))) = A(f((i1)((sigma)rho))) ∈ Type) ∧ ((A)sigma(f((i0)(rho))) = A(f((i0)((sigma)rho))) ∈ Type))
Definition: cubical-path-condition
cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0) ==
∀J:fset(ℕ). ∀f:I,phi(J). ((a0 (i0)(rho) f) = u((i0) ⋅ f) ∈ A(f((i0)(rho))))
Lemma: cubical-path-condition_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}]. ∀[a0:A((i0)(rho))].
(cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0) ∈ ℙ')
Lemma: cubical-path-condition-0
∀[I:fset(ℕ)]. ∀[Gamma,A,i,rho,u,a0:Top]. cubical-path-condition(Gamma;A;I;i;rho;0;u;a0)
Lemma: sq_stable__cubical-path-condition
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}]. ∀[a0:A((i0)(rho))].
SqStable(cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0))
Definition: cubical-path-condition'
cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1) ==
∀J:fset(ℕ). ∀f:I,phi(J). ((a1 (i1)(rho) f) = u((i1) ⋅ f) ∈ A(f((i1)(rho))))
Lemma: cubical-path-condition'_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}]. ∀[a1:A((i1)(rho))].
(cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1) ∈ ℙ')
Lemma: sq_stable__cubical-path-condition'
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}]. ∀[a0:A((i1)(rho))].
SqStable(cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a0))
Definition: cubical-path-0
cubical-path-0(Gamma;A;I;i;rho;phi;u) == {a0:A((i0)(rho))| cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0)}
Lemma: cubical-path-0_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}].
(cubical-path-0(Gamma;A;I;i;rho;phi;u) ∈ 𝕌{[i' | j']})
Lemma: cubical-path-0-0
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[u:Top].
cubical-path-0(Gamma;A;I;i;rho;0;u) ≡ A((i0)(rho))
Lemma: member-cubical-path-0-0
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[u:Top]. ∀[x:A((i0)(rho))].
(x ∈ cubical-path-0(Gamma;A;I;i;rho;0;u))
Definition: cubical-path-1
cubical-path-1(Gamma;A;I;i;rho;phi;u) == {a1:A((i1)(rho))| cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1)}
Lemma: cubical-path-1_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}].
(cubical-path-1(Gamma;A;I;i;rho;phi;u) ∈ 𝕌{[i' | j']})
Lemma: cubical-path-1-ap-morph
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}]. ∀[a:cubical-path-1(Gamma;A;I;i;rho;phi;u)]. ∀[J:fset(ℕ)]. ∀[g:J ⟶ I].
∀[j:{j:ℕ| ¬j ∈ J} ].
((a (i1)(rho) g) ∈ cubical-path-1(Gamma;A;J;j;g,i=j(rho);g(phi);(u)subset-trans(I+i;J+j;g,i=j;s(phi))))
Lemma: cubical-path-0-ap-morph
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}]. ∀[a:cubical-path-0(Gamma;A;I;i;rho;phi;u)]. ∀[J:fset(ℕ)]. ∀[g:J ⟶ I].
∀[j:{j:ℕ| ¬j ∈ J} ].
((a (i0)(rho) g) ∈ cubical-path-0(Gamma;A;J;j;g,i=j(rho);g(phi);(u)subset-trans(I+i;J+j;g,i=j;s(phi))))
Lemma: cubical-subset-term-trans
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I,J:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[g:J ⟶ I]. ∀[rho:Gamma(I+i)].
∀[phi:𝔽(I)]. ∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}].
((u)subset-trans(I+i;J+j;g,i=j;s(phi)) ∈ {J+j,s(g(phi)) ⊢ _:(A)<g,i=j(rho)> o iota})
Lemma: csm-cubical-path-0-subtype
∀[Gamma,Delta:j⊢]. ∀[sigma:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Delta(I+i)].
∀[phi:𝔽(I)]. ∀[u:{I+i,s(phi) ⊢ _:((A)sigma)<rho> o iota}].
(cubical-path-0(Delta;(A)sigma;I;i;rho;phi;u) ⊆r cubical-path-0(Gamma;A;I;i;(sigma)rho;phi;u))
Lemma: csm-cubical-path-0-subtype2
∀[Gamma,Delta:j⊢]. ∀[sigma:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Delta(I+i)].
∀[phi:𝔽(I)]. ∀[u1,u2:{I+i,s(phi) ⊢ _:((A)sigma)<rho> o iota}].
cubical-path-0(Delta;(A)sigma;I;i;rho;phi;u1) ⊆r cubical-path-0(Gamma;A;I;i;(sigma)rho;phi;u2)
supposing u1 = u2 ∈ {I+i,s(phi) ⊢ _:((A)sigma)<rho> o iota}
Lemma: csm-cubical-path-1-subtype
∀[Gamma,Delta:j⊢]. ∀[sigma:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Delta(I+i)].
∀[phi:𝔽(I)]. ∀[u:{I+i,s(phi) ⊢ _:((A)sigma)<rho> o iota}].
(cubical-path-1(Gamma;A;I;i;(sigma)rho;phi;u) ⊆r cubical-path-1(Delta;(A)sigma;I;i;rho;phi;u))
Definition: comp-op
comp-op(Gamma;A) ==
I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ cubical-path-1(Gamma;A;I;i;rho;phi;u)
Lemma: comp-op_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢j _}]. (comp-op(Gamma;A) ∈ 𝕌{[i | j]'})
Definition: composition-uniformity
composition-uniformity(Gamma;A;comp) ==
∀I,J:fset(ℕ). ∀i:{i:ℕ| ¬i ∈ I} . ∀j:{j:ℕ| ¬j ∈ J} . ∀g:J ⟶ I. ∀rho:Gamma(I+i). ∀phi:𝔽(I).
∀u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}. ∀a0:cubical-path-0(Gamma;A;I;i;rho;phi;u).
((comp I i rho phi u a0 (i1)(rho) g)
= (comp J j g,i=j(rho) g(phi) (u)subset-trans(I+i;J+j;g,i=j;s(phi)) (a0 (i0)(rho) g))
∈ A(g((i1)(rho))))
Lemma: composition-uniformity_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ cubical-path-1(Gamma;A;I;i;rho;phi;u)].
(composition-uniformity(Gamma;A;comp) ∈ ℙ{[i' | j']})
Lemma: sq_stable__composition-uniformity
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ cubical-path-1(Gamma;A;I;i;rho;phi;u)].
SqStable(composition-uniformity(Gamma;A;comp))
Definition: composition-op
Gamma ⊢ CompOp(A) ==
{comp:I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ cubical-path-1(Gamma;A;I;i;rho;phi;u)|
composition-uniformity(Gamma;A;comp)}
Lemma: composition-op_wf1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢j _}]. (Gamma ⊢ CompOp(A) ∈ 𝕌{[i | j]'})
Lemma: composition-op_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (Gamma ⊢ CompOp(A) ∈ 𝕌{[i' | j']})
Lemma: equal-composition-op
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[c1:Gamma ⊢ CompOp(A)]. ∀[c2:I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ cubical-path-1(Gamma;A;I;i;rho;phi;u)].
c1 = c2 ∈ Gamma ⊢ CompOp(A)
supposing c1
= c2
∈ (I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ cubical-path-1(Gamma;A;I;i;rho;phi;u))
Lemma: equal-composition-op2
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[c1,c2:Gamma ⊢ CompOp(A)].
c1 = c2 ∈ Gamma ⊢ CompOp(A)
supposing ∀I:fset(ℕ). ∀i:{i:ℕ| ¬i ∈ I} . ∀rho:Gamma(I+i). ∀phi:𝔽(I). ∀u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}.
∀a0:cubical-path-0(Gamma;A;I;i;rho;phi;u).
((c1 I i rho phi u a0) = (c2 I i rho phi u a0) ∈ A((i1)(rho)))
Lemma: composition-op-uniformity
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)].
∀I,J:fset(ℕ). ∀i:{i:ℕ| ¬i ∈ I} . ∀j:{j:ℕ| ¬j ∈ J} . ∀g:J ⟶ I. ∀rho:Gamma(I+i). ∀phi:𝔽(I).
∀u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}. ∀a0:cubical-path-0(Gamma;A;I;i;rho;phi;u).
((comp I i rho phi u a0 (i1)(rho) g)
= (comp J j g,i=j(rho) g(phi) (u)subset-trans(I+i;J+j;g,i=j;s(phi)) (a0 (i0)(rho) g))
∈ A(g((i1)(rho))))
Lemma: composition-op-nc-e
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)].
∀I:fset(ℕ). ∀i,j:{j:ℕ| ¬j ∈ I} . ∀rho:Gamma(I+i). ∀phi:𝔽(I). ∀u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}.
∀a0:cubical-path-0(Gamma;A;I;i;rho;phi;u).
((comp I i rho phi u a0) = (comp I j e(i;j)(rho) phi (u)subset-trans(I+i;I+j;e(i;j);s(phi)) a0) ∈ A((i1)(rho)))
Definition: csm-composition
(comp)sigma == λI,i,rho. (comp I i (sigma)rho)
Lemma: csm-composition_wf
∀[Gamma,Delta:j⊢]. ∀[sigma:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)].
((comp)sigma ∈ Delta ⊢ CompOp((A)sigma))
Lemma: csm-composition-exists
∀[Gamma,Delta:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀s:Delta j⟶ Gamma. (Gamma ⊢ CompOp(A) ⇒ Delta ⊢ CompOp((A)s))
Lemma: csm-p-composition-exists
∀[X:j⊢]. ∀[A,T:{X ⊢ _}]. (X ⊢ CompOp(A) ⇒ X.T ⊢ CompOp((A)p))
Lemma: csm-composition-id
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)]. ((comp)1(Gamma) = comp ∈ Gamma ⊢ CompOp(A))
Lemma: csm-composition-comp
∀[X,Y,Z:j⊢]. ∀[s1:Z j⟶ Y]. ∀[s2:Y j⟶ X]. ∀[A:{X ⊢ _}]. ∀[comp:X ⊢ CompOp(A)].
(((comp)s2)s1 = (comp)s2 o s1 ∈ Z ⊢ CompOp((A)s2 o s1))
Lemma: composition-op-1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)].
∀[u:{I+i,s(1) ⊢ _:(A)<rho> o iota}]. ∀[a:cubical-path-0(Gamma;A;I;i;rho;1;u)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I].
((cA I i rho 1 u a (i1)(rho) f) = u((i1) ⋅ f) ∈ A(f((i1)(rho))))
Lemma: composition-op-1-case1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)].
∀[u:{I+i,s(1) ⊢ _:(A)<rho> o iota}]. ∀[a:cubical-path-0(Gamma;A;I;i;rho;1;u)].
((cA I i rho 1 u a) = u((i1)) ∈ A((i1)(rho)))
Definition: filling-uniformity
filling-uniformity(Gamma;A;fill) ==
∀I,J:fset(ℕ). ∀i:{i:ℕ| ¬i ∈ I} . ∀j:{j:ℕ| ¬j ∈ J} . ∀g:J ⟶ I. ∀rho:Gamma(I+i). ∀phi:𝔽(I).
∀u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}. ∀a0:cubical-path-0(Gamma;A;I;i;rho;phi;u).
((fill I i rho phi u a0 rho g,i=j)
= (fill J j g,i=j(rho) g(phi) (u)subset-trans(I+i;J+j;g,i=j;s(phi)) (a0 (i0)(rho) g))
∈ A(g,i=j(rho)))
Lemma: filling-uniformity_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[fill:I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ A(rho)].
(filling-uniformity(Gamma;A;fill) ∈ ℙ{[i' | j']})
Definition: filling-op
filling-op(Gamma;A) ==
{fill:I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ {a:A(rho)|
(section-iota(Gamma;A;I+i;rho;a) = u ∈ {I+i,s(phi) ⊢ _:(A)<rho> o iota}) ∧ ((a rho (i0)) = a0 ∈ A((i0)(rho)))} |
filling-uniformity(Gamma;A;fill)}
Lemma: filling-op_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (filling-op(Gamma;A) ∈ 𝕌{[i' | j']})
Definition: fillterm
fillterm(Gamma;A;I;i;j;rho;a0;u) == λK,f. if isdM0(f i) then (a0 (i0)(rho) s ⋅ f) else u(m(i;j) ⋅ f) fi
Lemma: fillterm_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ I+i} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}]. ∀[a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)].
(fillterm(Gamma;A;I;i;j;rho;a0;u) ∈ {I+i+j,s(fl-join(I+i;s(phi);(i=0))) ⊢ _:(A)<m(i;j)(rho)> o iota})
Lemma: cubical-path-0-fillterm
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ I+i} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
∀[a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)]
((a0 (i0)(rho) s)
∈ cubical-path-0(Gamma;A;I+i;j;m(i;j)(rho);fl-join(I+i;s(phi);(i=0));fillterm(Gamma;A;I;i;j;rho;a0;u)))
Definition: fill_from_comp
fill_from_comp(Gamma;A;comp) ==
λI,i,rho,phi,u,a0. eval j = new-name(I+i) in
comp I+i j m(i;j)(rho) fl-join(I+i;s(phi);(i=0)) fillterm(Gamma;A;I;i;j;rho;a0;u) (a0 (i0)(rho) s)
Lemma: fill_from_comp_wf1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)].
(fill_from_comp(Gamma;A;comp) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ let j = new-name(I+i) in
cubical-path-1(Gamma;A;I+i;j;m(i;j)(rho);fl-join(I+i;s(phi);(i=0));fillterm(Gamma;A;I;i;j;rho;a0;u)))
Lemma: fill_from_comp_property1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)].
∀[phi:𝔽(I)]. ∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}]. ∀[a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)].
((fill_from_comp(Gamma;A;comp) I i rho phi u a0 rho (i0)) = a0 ∈ A((i0)(rho)))
Lemma: fill_from_comp_wf2
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)].
(fill_from_comp(Gamma;A;comp) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ {a:A(rho)|
(section-iota(Gamma;A;I+i;rho;a) = u ∈ {I+i,s(phi) ⊢ _:(A)<rho> o iota}) ∧ ((a rho (i0)) = a0 ∈ A((i0)(rho)))} )
Lemma: fill_from_comp_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)]. (fill_from_comp(Gamma;A;comp) ∈ filling-op(Gamma;A))
Definition: pi-comp-nu
pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u1;j) ==
let v' = fill_from_comp(Gamma;A;cA) J j f,i=1-j(rho) 0 () u1 in
(v' f,i=1-j(rho) r_j)
Lemma: pi-comp-nu_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)].
∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[u1:A(f((i1)(rho)))]. ∀[j:{j:ℕ| ¬j ∈ J} ].
(pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u1;j) ∈ A(r_j(f,i=1-j(rho))))
Lemma: pi-comp-nu-property
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)].
∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[u1:A(f((i1)(rho)))]. ∀[j:{j:ℕ| ¬j ∈ J} ].
((pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u1;j) f,i=j(rho) (j1)) = u1 ∈ A((j1)(f,i=j(rho))))
Lemma: pi-comp-nu-uniformity
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)].
∀[J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. ∀[u:A(f((i1)(rho)))]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[k:{k:ℕ| ¬k ∈ K} ].
((pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u;j) f,i=j(rho) g,j=k)
= pi-comp-nu(Gamma;A;cA;I;i;rho;K;f ⋅ g;(u f((i1)(rho)) g);k)
∈ A(f ⋅ g,i=k(rho)))
Definition: pi-comp-app
pi-comp-app(Gamma;A;I;i;rho;phi;mu;J;f;j;nu) ==
app((mu)subset-trans(I+i;J+j;f,i=j;s(phi)); (canonical-section(Gamma;A;J+j;f,i=j(rho);nu))iota)
Lemma: pi-comp-app_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[mu:{I+i,s(phi) ⊢ _:(ΠA B)<rho> o iota}]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[nu:A(r_j(f,i=1-j(rho)))].
(pi-comp-app(Gamma;A;I;i;rho;phi;mu;J;f;j;nu) ∈ {J+j,s(f(phi)) ⊢ _:(B)<(f,i=j(rho);nu)> o iota})
Definition: pi-comp-lambda
pi-comp-lambda(Gamma;A;I;i;rho;lambda;J;f;j;nu) == lambda J f (nu r_j(f,i=1-j(rho)) (j0))
Lemma: pi-comp-lambda_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[mu:{I+i,s(phi) ⊢ _:(ΠA B)<rho> o iota}]. ∀[lambda:cubical-path-0(Gamma;ΠA B;I;i;rho;phi;mu)]. ∀[J:fset(ℕ)].
∀[f:J ⟶ I]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[nu:A(r_j(f,i=1-j(rho)))].
(pi-comp-lambda(Gamma;A;I;i;rho;lambda;J;f;j;nu)
∈ cubical-path-0(Gamma.A;B;J;j;(f,i=j(rho);nu);f(phi);pi-comp-app(Gamma;A;I;i;rho;phi;mu;J;f;j;nu)))
Definition: pi-comp
pi-comp(Gamma;A;B;cA;cB) ==
λI,i,rho,phi,mu,lambda,J,f,u1. eval j = new-name(J) in
let nu = pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u1;j) in
cB J j (f,i=j(rho);nu) f(phi) pi-comp-app(Gamma;A;I;i;rho;phi;mu;J;f;j;nu)
pi-comp-lambda(Gamma;A;I;i;rho;lambda;J;f;j;nu)
Lemma: pi-comp_wf1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma.A ⊢ CompOp(B)].
(pi-comp(Gamma;A;B;cA;cB) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ mu:{I+i,s(phi) ⊢ _:(ΠA B)<rho> o iota}
⟶ lambda:cubical-path-0(Gamma;ΠA B;I;i;rho;phi;mu)
⟶ J:fset(ℕ)
⟶ f:J ⟶ I
⟶ u1:A(f((i1)(rho)))
⟶ let j = new-name(J) in
let nu = pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u1;j) in
cubical-path-1(Gamma.A;B;J;j;(f,i=j(rho);nu);f(phi);pi-comp-app(Gamma;A;I;i;rho;phi;mu;J;f;j;nu)))
Lemma: pi-comp_wf2
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma.A ⊢ CompOp(B)].
(pi-comp(Gamma;A;B;cA;cB) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ mu:{I+i,s(phi) ⊢ _:(ΠA B)<rho> o iota}
⟶ lambda:cubical-path-0(Gamma;ΠA B;I;i;rho;phi;mu)
⟶ J:fset(ℕ)
⟶ f:J ⟶ I
⟶ u1:A(f((i1)(rho)))
⟶ B((f((i1)(rho));u1)))
Lemma: pi-comp_wf3
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma.A ⊢ CompOp(B)].
(pi-comp(Gamma;A;B;cA;cB) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ mu:{I+i,s(phi) ⊢ _:(ΠA B)<rho> o iota}
⟶ lambda:cubical-path-0(Gamma;ΠA B;I;i;rho;phi;mu)
⟶ ΠA B((i1)(rho)))
Lemma: pi-comp-property
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma.A ⊢ CompOp(B)]. ∀[I:fset(ℕ)].
∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)]. ∀[mu:{I+i,s(phi) ⊢ _:(ΠA B)<rho> o iota}].
∀[lambda:cubical-path-0(Gamma;ΠA B;I;i;rho;phi;mu)]. ∀[J:fset(ℕ)]. ∀[f:I,phi(J)].
((pi-comp(Gamma;A;B;cA;cB) I i rho phi mu lambda (i1)(rho) f) = mu((i1) ⋅ f) ∈ ΠA B(f((i1)(rho))))
Lemma: pi-comp-uniformity
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma.A ⊢ CompOp(B)].
composition-uniformity(Gamma;ΠA B;pi-comp(Gamma;A;B;cA;cB))
Lemma: pi-comp_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma.A ⊢ CompOp(B)].
(pi-comp(Gamma;A;B;cA;cB) ∈ Gamma ⊢ CompOp(ΠA B))
Definition: fun-comp
fun-comp(Gamma; A; B; cA; cB) == pi-comp(Gamma;A;(B)p;cA;(cB)p)
Lemma: fun-comp_wf
∀[Gamma:j⊢]. ∀[A,B:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma ⊢ CompOp(B)].
(fun-comp(Gamma; A; B; cA; cB) ∈ Gamma ⊢ CompOp((A ⟶ B)))
Lemma: fun-comp-exists
∀Gamma:j⊢. ∀A,B:{Gamma ⊢ _}. (Gamma ⊢ CompOp(A) ⇒ Gamma ⊢ CompOp(B) ⇒ Gamma ⊢ CompOp((A ⟶ B)))
Definition: sigmacomp
sigmacomp(Gamma;A;B;cA;cB) ==
λI,i,rho,phi,mu,lambda. let v = fill_from_comp(Gamma;A;cA) I i rho phi mu.1 (fst(lambda)) in
let w = cB I i (rho;v) phi mu.2 (snd(lambda)) in
<(v rho (i1)), w>
Lemma: sigmacomp_wf1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma.A ⊢ CompOp(B)].
(sigmacomp(Gamma;A;B;cA;cB) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ mu:{I+i,s(phi) ⊢ _:(Σ A B)<rho> o iota}
⟶ lambda:cubical-path-0(Gamma;Σ A B;I;i;rho;phi;mu)
⟶ cubical-path-1(Gamma;Σ A B;I;i;rho;phi;mu))
Lemma: sigmacomp_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma.A ⊢ CompOp(B)].
(sigmacomp(Gamma;A;B;cA;cB) ∈ Gamma ⊢ CompOp(Σ A B))
Lemma: sigma-comp-exists
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. ∀B:{Gamma.A ⊢ _}. (Gamma ⊢ CompOp(A) ⇒ Gamma.A ⊢ CompOp(B) ⇒ Gamma ⊢ CompOp(Σ A B))
Lemma: pi-comp-exists
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. ∀B:{Gamma.A ⊢ _}. (Gamma ⊢ CompOp(A) ⇒ Gamma.A ⊢ CompOp(B) ⇒ Gamma ⊢ CompOp(ΠA B))
Lemma: composition-type-lemma1
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)].
((A)[0(𝕀)](rho) = A((new-name(I)0)((s(rho);<new-name(I)>))) ∈ Type)
Lemma: composition-type-lemma2
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)].
(A((new-name(I)1)((s(rho);<new-name(I)>))) = (A)[1(𝕀)](rho) ∈ Type)
Lemma: context-map-lemma1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[i:ℕ]. (<s(rho)> ∈ I+i,s(phi(rho)) j⟶ Gamma, phi)
Lemma: context-map-lemma2
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)].
(<(s(rho);<new-name(I)>)> ∈ I+new-name(I),s(phi(rho)) j⟶ Gamma, phi.𝕀)
Lemma: context-subset-ap-iota
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ((A)iota = A ∈ {Gamma, phi ⊢ _})
Lemma: subset-iota-is-id
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[v:{H, phi ⊢ _}]. (v = (v)iota ∈ {H, phi ⊢ _})
Lemma: context-subset-term-iota
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[v:{Gamma, phi ⊢ _:A}]. ((v)iota = v ∈ {Gamma, phi ⊢ _:A})
Lemma: context-subset-type-iota
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ({Gamma, phi ⊢ _:(A)iota} = {Gamma, phi ⊢ _:A} ∈ 𝕌{[i' | j']})
Lemma: composition-type-lemma3
∀Gamma:j⊢. ∀phi:{Gamma ⊢ _:𝔽}. ∀A:{Gamma.𝕀 ⊢ _}. ∀u:{Gamma, phi.𝕀 ⊢ _:A}. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀a:Gamma(I).
((u)<(s(f(a));<new-name(J)>)> o iota
= ((u)<(s(a);<new-name(I)>)> o iota)subset-trans(I+new-name(I);J+new-name(J);f,new-name(I)=new-name(J);s(phi(a)))
∈ {J+new-name(J),s(phi(f(a))) ⊢ _:(A)<(s(f(a));<new-name(J)>)> o iota})
Lemma: composition-type-lemma4
∀Gamma:j⊢. ∀phi:{Gamma ⊢ _:𝔽}. ∀A:{Gamma.𝕀 ⊢ _}. ∀u:{Gamma, phi.𝕀 ⊢ _:A}. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀rho:Gamma(I).
∀K:fset(ℕ). ∀g:J,phi(f(rho))(K).
((u)<(s(f(rho));<new-name(J)>)> o iota((new-name(J)0) ⋅ g)
= ((u)<(s(rho);<new-name(I)>)> o iota)subset-trans(I+new-name(I);J+new-name(J);f,new-name(I)=new-name(J);
s(phi(rho)))((new-name(J)0) ⋅ g)
∈ A(g((new-name(J)0)((s(f(rho));<new-name(J)>)))))
Lemma: composition-type-lemma5
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[u:{Gamma, phi.𝕀 ⊢ _:A}].
({Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]} ∈ 𝕌{[i | j']})
Lemma: composition-type-lemma6
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[u:{Gamma, phi.𝕀 ⊢ _:A}].
({Gamma ⊢ _:(A)[1(𝕀)][phi |⟶ (u)[1(𝕀)]]} ∈ 𝕌{[i | j']})
Definition: composition-term
comp cA [phi ⊢→ u] a0 ==
λI,rho. (cA I new-name(I) (s(rho);<new-name(I)>) phi(rho) (u)<(s(rho);<new-name(I)>)> o iota a0(rho))
Lemma: composition-term_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma, phi.𝕀 ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}].
(comp cA [phi ⊢→ u] a0 ∈ {Gamma ⊢ _:(A)[1(𝕀)][phi |⟶ (u)[1(𝕀)]]})
Lemma: empty-context-eq-lemma
∀[Gamma:j⊢]. ∀[A,x,y:Top]. (x = y ∈ {Gamma ⊢ _:A}) supposing ∀I:fset(ℕ). (¬Gamma(I))
Lemma: empty-context-lemma
∀[Gamma:j⊢]. ∀[A,x:Top]. (x ∈ {Gamma ⊢ _:A}) supposing ∀I:fset(ℕ). (¬Gamma(I))
Lemma: empty-context-subset-lemma1
∀[Gamma:j⊢]. ∀[A,x:Top]. (x ∈ {Gamma, 0(𝔽).𝕀 ⊢ _:A})
Lemma: empty-context-subset-lemma2
∀[Gamma:j⊢]. ∀[A,x:Top]. (x ∈ {Gamma, 0(𝔽) ⊢ _:A})
Lemma: empty-context-subset-lemma3
∀[Gamma:j⊢]. ∀[A,x,y:Top]. (x = y ∈ {Gamma, 0(𝔽) ⊢ _:A})
Lemma: empty-context-subset-lemma3'
∀[Gamma:j⊢]. ∀[i:{Gamma ⊢ _:𝕀}]. ∀[A,x,y:Top]. (x = y ∈ {Gamma, (i=0), (i=1) ⊢ _:A})
Lemma: empty-context-subset-lemma4
∀[Gamma:j⊢]. ∀[B:{Gamma ⊢ _}]. ∀[A,x,y:Top]. (x = y ∈ {Gamma, 0(𝔽).B ⊢ _:A})
Lemma: empty-context-subset-lemma5
∀[Gamma:j⊢]. ∀[i:{Gamma ⊢ _:𝕀}]. ∀[x,y:Top × Top]. (x = y ∈ {Gamma, (i=0), (i=1) ⊢ _})
Lemma: empty-context-subset-lemma6
∀[Gamma:j⊢]. ∀[x,y:Top × Top]. (x = y ∈ {Gamma, 0(𝔽) ⊢ _})
Lemma: same-cubical-type-trivial_1
∀[X:j⊢]. ∀[i:{X ⊢ _:𝕀}]. ∀[x,y,B:Top]. X, (i=0), (i=1) ⊢ x=y:B
Lemma: case-type-1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}].
∀[A:{Gamma ⊢ _}]. ∀[B:Top × Top]. Gamma ⊢ (if phi then A else B) = A supposing phi = 1(𝔽) ∈ {Gamma ⊢ _:𝔽}
Lemma: case-type-0
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}].
∀[A:Top × Top]. ∀[B:{Gamma ⊢ _}]. Gamma ⊢ (if phi then A else B) = B supposing phi = 0(𝔽) ∈ {Gamma ⊢ _:𝔽}
Lemma: case-term-0
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma ⊢ _}]. ∀[u:Top]. ∀[v:{Gamma ⊢ _:A}].
Gamma ⊢ (u ∨ v)=v:A supposing phi = 0(𝔽) ∈ {Gamma ⊢ _:𝔽}
Lemma: case-term-0'
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma ⊢ _}]. ∀[u:Top]. ∀[v,x:{Gamma ⊢ _:A}].
(Gamma ⊢ (u ∨ v)=x:A) supposing ((x = v ∈ {Gamma ⊢ _:A}) and (phi = 0(𝔽) ∈ {Gamma ⊢ _:𝔽}))
Definition: case-endpoints
[r=0 ⊢→ a; r=1 ⊢→ b] == (a ∨ b)
Lemma: case-endpoints_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[a,b:{G ⊢ _:A}]. ∀[r:{G ⊢ _:𝕀}]. ([r=0 ⊢→ a; r=1 ⊢→ b] ∈ {G, ((r=0) ∨ (r=1)) ⊢ _:A})
Lemma: case-endpoints-0
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[a:{G ⊢ _:A}]. ∀[b:Top]. ([0(𝕀)=0 ⊢→ a; 0(𝕀)=1 ⊢→ b] = a ∈ {G ⊢ _:A})
Lemma: case-endpoints-1
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[a:Top]. ∀[b:{G ⊢ _:A}]. ([1(𝕀)=0 ⊢→ a; 1(𝕀)=1 ⊢→ b] = b ∈ {G ⊢ _:A})
Lemma: csm-case-endpoints
∀[a,b,r,s:Top]. (([r=0 ⊢→ a; r=1 ⊢→ b])s ~ [(r)s=0 ⊢→ (a)s; (r)s=1 ⊢→ (b)s])
Lemma: cubical-term-1-q1
∀[Gamma:j⊢]. ∀[B:{Gamma ⊢ _}]. ∀[z:{Gamma.𝕀 ⊢ _:(B)p}]. (((z)[1(𝕀)])p = z ∈ {Gamma.𝕀, (q=1) ⊢ _:(B)p})
Lemma: cubical-term-0-q0
∀[Gamma:j⊢]. ∀[B:{Gamma ⊢ _}]. ∀[z:{Gamma.𝕀 ⊢ _:(B)p}]. (((z)[0(𝕀)])p = z ∈ {Gamma.𝕀, (q=0) ⊢ _:(B)p})
Definition: rev-type-line
(A)- == (A)(p;1-(q))
Lemma: rev-type-line_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. Gamma.𝕀 ⊢ (A)-
Lemma: csm-rev-type-line
∀[G,K:j⊢]. ∀[A:{G.𝕀 ⊢ _}]. ∀[tau:K j⟶ G]. (((A)-)tau+ = ((A)tau+)- ∈ {K.𝕀 ⊢ _})
Lemma: rev-type-line-0
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. (((A)-)[0(𝕀)] ~ (A)[1(𝕀)])
Lemma: rev-type-line-1
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. (((A)-)[1(𝕀)] ~ (A)[0(𝕀)])
Lemma: rev-rev-type-line
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. (((A)-)- = A ∈ {Gamma.𝕀 ⊢ _})
Definition: rev-type-line-comp
(cA)- == (cA)(p;1-(q))
Lemma: rev-type-line-comp_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ((cA)- ∈ Gamma.𝕀 ⊢ CompOp((A)-))
Lemma: csm-rev-type-line-comp
∀[G,K:j⊢]. ∀[tau:K j⟶ G]. ∀[A:{G.𝕀 ⊢ _}]. ∀[cA:G.𝕀 ⊢ CompOp(A)].
(((cA)-)tau+ = ((cA)tau+)- ∈ K.𝕀 ⊢ CompOp(((A)-)tau+))
Definition: transport
transport(Gamma;a) == comp cA [0(𝔽) ⊢→ discr(⋅)] a
Lemma: transport_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[a:{Gamma ⊢ _:(A)[0(𝕀)]}].
(transport(Gamma;a) ∈ {Gamma ⊢ _:(A)[1(𝕀)]})
Definition: transport-fun
transport-fun(Gamma;A;cA) == (λtransport(Gamma.(A)[0(𝕀)];q))
Lemma: transport-fun_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)].
(transport-fun(Gamma;A;cA) ∈ {Gamma ⊢ _:((A)[0(𝕀)] ⟶ (A)[1(𝕀)])})
Definition: transport-const
transport-const(G;cA;a) == transport(G;a)
Lemma: transport-const_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G ⊢ CompOp(A)]. ∀[a:{G ⊢ _:A}]. (transport-const(G;cA;a) ∈ {G ⊢ _:A})
Definition: const-transport-fun
ConstTrans(A) == cubical-lam(G;transport-const(G.A;(cA)p;q))
Lemma: const-transport-fun_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. (ConstTrans(A) ∈ {Gamma ⊢ _:(A ⟶ A)})
Definition: rev-transport-fun
rev-transport-fun(Gamma;A;cA) == transport-fun(Gamma;(A)-;(cA)(p;1-(q)))
Lemma: rev-transport-fun_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)].
(rev-transport-fun(Gamma;A;cA) ∈ {Gamma ⊢ _:((A)[1(𝕀)] ⟶ (A)[0(𝕀)])})
Definition: cubical-contr
cubical-contr(Gamma; A; cA; p; phi; u) == comp (cA)p [phi ⊢→ (app(p.2; u))p @ q] p.1
Lemma: cubical-contr_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[p:{Gamma ⊢ _:Contractible(A)}]. ∀[phi:{Gamma ⊢ _:𝔽}].
∀[u:{Gamma, phi ⊢ _:(A)iota}].
(cubical-contr(Gamma; A; cA; p; phi; u) ∈ {Gamma ⊢ _:A[phi |⟶ u]})
Lemma: csm-subtype-cubical-subset
∀[Gamma:j⊢]. ∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. (formal-cube(I) j⟶ Gamma ⊆r I,psi j⟶ Gamma)
Lemma: cubical-type-subtype-cubical-subset
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. ({formal-cube(I) ⊢ _} ⊆r {I,psi ⊢ _})
Lemma: cubical-term-subtype-cubical-subset
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. ∀[T:{formal-cube(I) ⊢ _}]. ({formal-cube(I) ⊢ _:T} ⊆r {I,psi ⊢ _:T})
Lemma: context-map_wf_cubical-subset
∀[Gamma:j⊢]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[psi:𝔽(I)]. (<rho> ∈ I,psi j⟶ Gamma)
Definition: composition-function
composition-function{j:l,i:l}(Gamma;A) ==
H:CubicalSet{j}
⟶ sigma:H.𝕀 j⟶ Gamma
⟶ phi:{H ⊢ _:𝔽}
⟶ u:{H, phi.𝕀 ⊢ _:(A)sigma}
⟶ a0:{H ⊢ _:((A)sigma)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}
⟶ {H ⊢ _:((A)sigma)[1(𝕀)][phi |⟶ (u)[1(𝕀)]]}
Lemma: composition-function_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (composition-function{j:l,i:l}(Gamma;A) ∈ 𝕌{[i' | j'']})
Lemma: composition-function-cumulativity
∀Gamma:j⊢. ∀Z:{Gamma ⊢ _}. (composition-function{[i | j]:l, i:l}(Gamma; Z) ⊆r composition-function{j:l,i:l}(Gamma;Z))
Definition: uniform-comp-function
uniform-comp-function{j:l, i:l}(Gamma; A; comp) ==
∀H,K:j⊢. ∀tau:K j⟶ H. ∀sigma:H.𝕀 j⟶ Gamma. ∀phi:{H ⊢ _:𝔽}. ∀u:{H, phi.𝕀 ⊢ _:(A)sigma}.
∀a0:{H ⊢ _:((A)sigma)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}.
((comp H sigma phi u a0)tau
= (comp K sigma o tau+ (phi)tau (u)tau+ (a0)tau)
∈ {K ⊢ _:(((A)sigma)[1(𝕀)])tau[(phi)tau |⟶ ((u)[1(𝕀)])tau]})
Lemma: uniform-comp-function_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:composition-function{j:l,i:l}(Gamma;A)].
(uniform-comp-function{j:l, i:l}(Gamma; A; comp) ∈ ℙ{[i' | j'']})
Lemma: uniform-comp-function-cumulativity
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:composition-function{[i | j]:l, i:l}(Gamma; A)].
(uniform-comp-function{[i | j]:l, i:l}(Gamma; A; comp) ⇒ uniform-comp-function{j:l, i:l}(Gamma; A; comp))
Definition: composition-structure
Gamma ⊢ Compositon(A) ==
{comp:composition-function{j:l,i:l}(Gamma;A)| uniform-comp-function{j:l, i:l}(Gamma; A; comp)}
Lemma: composition-structure_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (Gamma ⊢ Compositon(A) ∈ 𝕌{[i' | j'']})
Lemma: composition-structure-cumulativity
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (Gamma +⊢ Compositon(A) ⊆r Gamma ⊢ Compositon(A))
Lemma: composition-structure-equal
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[c1,c2:Gamma ⊢ Compositon(A)].
c1 = c2 ∈ Gamma ⊢ Compositon(A)
supposing ∀H:j⊢. ∀sigma:H.𝕀 j⟶ Gamma. ∀phi:{H ⊢ _:𝔽}. ∀u:{H, phi.𝕀 ⊢ _:(A)sigma}.
∀a0:{H ⊢ _:((A)sigma)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}.
((c1 H sigma phi u a0) = (c2 H sigma phi u a0) ∈ {H ⊢ _:((A)sigma)[1(𝕀)]})
Lemma: composition-function-subset
∀[Y,X:j⊢].
∀[B:{X ⊢ _}]. (composition-function{j:l,i:l}(X;B) ⊆r composition-function{j:l,i:l}(Y;B))
supposing sub_cubical_set{j:l}(Y; X)
Lemma: composition-structure-subset
∀[Y,X:j⊢]. ∀[B:{X ⊢ _}]. (X ⊢ Compositon(B) ⊆r Y ⊢ Compositon(B)) supposing sub_cubical_set{j:l}(Y; X)
Lemma: composition-in-subset
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G ⊢ Compositon(A)]. ∀[H1,H2:j⊢].
∀[sigma:H1.𝕀 j⟶ G]. ∀[phi:{H1 ⊢ _:𝔽}]. ∀[u:{H1, phi.𝕀 ⊢ _:(A)sigma}].
∀[a0:{H1 ⊢ _:((A)sigma)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}].
((cA H1 sigma phi u a0) = (cA H2 sigma phi u a0) ∈ {H2 ⊢ _:((A)sigma)[1(𝕀)]})
supposing sub_cubical_set{j:l}(H2; H1)
Definition: comp-op-to-comp-fun
cop-to-cfun(cA) == ⌜λH,sigma,phi,u,a0. comp (cA)sigma [phi ⊢→ u] a0⌝
Lemma: comp-op-to-comp-fun_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. (cop-to-cfun(cA) ∈ composition-function{j:l,i:l}(Gamma;A))
Lemma: face-type-comp-at-lemma
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[I,J:fset(ℕ)]. ∀[i:ℕ]. ∀[f:J ⟶ I+i]. ∀[v:H(I)]. (phi(f(s(v))) = f(s(phi(v))) ∈ 𝔽(f))
Lemma: composition-term-uniformity
∀[H,K:j⊢]. ∀[tau:K j⟶ H]. ∀[phi:{H ⊢ _:𝔽}]. ∀[A:{H.𝕀 ⊢ _}]. ∀[u:{H, phi.𝕀 ⊢ _:A}].
∀[a0:{H ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}]. ∀[cA:H.𝕀 ⊢ CompOp(A)].
((comp cA [phi ⊢→ u] a0)tau = comp (cA)tau+ [(phi)tau ⊢→ (u)tau+] (a0)tau ∈ {K ⊢ _:((A)[1(𝕀)])tau})
Lemma: csm-transport
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[a:{Gamma ⊢ _:(A)[0(𝕀)]}]. ∀[H:j⊢]. ∀[s:H j⟶ Gamma].
((transport(Gamma;a))s = transport(H;(a)s) ∈ {H ⊢ _:((A)[1(𝕀)])s})
Lemma: csm-transport-fun
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[H:j⊢]. ∀[s:H j⟶ Gamma].
((transport-fun(Gamma;A;cA))s = transport-fun(H;(A)s+;(cA)s+) ∈ {H ⊢ _:(((A)s+)[0(𝕀)] ⟶ ((A)s+)[1(𝕀)])})
Lemma: csm-comp-op-to-comp-fun
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[H,K:j⊢]. ∀[tau:K j⟶ H]. ∀[sigma:H.𝕀 j⟶ Gamma].
∀[phi:{H ⊢ _:𝔽}]. ∀[u:{H, phi.𝕀 ⊢ _:(A)sigma}]. ∀[a0:{H ⊢ _:((A)sigma)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}].
((cop-to-cfun(cA) H sigma phi u a0)tau
= (cop-to-cfun(cA) K sigma o tau+ (phi)tau (u)tau+ (a0)tau)
∈ {K ⊢ _:(((A)sigma)[1(𝕀)])tau[(phi)tau |⟶ ((u)[1(𝕀)])tau]})
Lemma: comp-op-to-comp-fun_wf2
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. (cop-to-cfun(cA) ∈ Gamma ⊢ Compositon(A))
Lemma: composition-op-implies-composition-structure
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. ∀cA:Gamma ⊢ CompOp(A). Gamma ⊢ Compositon(A)
Lemma: csm-ap-term-cube+
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}].
((u)cube+(I;i) ∈ {formal-cube(I), canonical-section(();𝔽;I;⋅;phi).𝕀 ⊢ _:(A)<rho> o cube+(I;i)})
Lemma: canonical-section-cubical-path-0
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}]. ∀[a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)].
(canonical-section(Gamma;A;I;(i0)(rho);a0)
∈ {formal-cube(I) ⊢ _:((A)<rho> o cube+(I;i))[0(𝕀)][canonical-section(();𝔽;I;⋅;phi) |⟶ ((u)cube+(I;i))[0(𝕀)]]})
Lemma: constrained-cubical-term-to-cubical-path-1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)].
∀phi:𝔽(I)
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}].
∀[v:{formal-cube(I) ⊢ _:((A)<rho> o cube+(I;i))[1(𝕀)][canonical-section(();𝔽;I;⋅;phi) |⟶ ((u)cube+(I;i))[1(𝕀)]]}].
(v(1) ∈ cubical-path-1(Gamma;A;I;i;rho;phi;u))
Definition: comp-fun-to-comp-op1
comp-fun-to-comp-op1(Gamma;A;comp) ==
λI,i,rho,phi,u,a0. (comp formal-cube(I) <rho> o cube+(I;i) canonical-section(();𝔽;I;⋅;phi) (u)cube+(I;i)
canonical-section(Gamma;A;I;(i0)(rho);a0))
Lemma: comp-fun-to-comp-op1_wf
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}.
∀[comp:composition-function{j:l,i:l}(Gamma;A)]
(comp-fun-to-comp-op1(Gamma;A;comp) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ {formal-cube(I) ⊢ _:((A)<rho> o cube+(I;i))[1(𝕀)][canonical-section(();𝔽;I;⋅;phi) |⟶ ((u)cube+(I;i))[1(𝕀)]]})
Definition: comp-fun-to-comp-op
cfun-to-cop(Gamma;A;comp) == λI,i,rho,phi,u,a0. comp-fun-to-comp-op1(Gamma;A;comp) I i rho phi u a0(1)
Lemma: comp-fun-to-comp-op_wf1
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}.
∀[comp:composition-function{j:l,i:l}(Gamma;A)]
(cfun-to-cop(Gamma;A;comp) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ cubical-path-1(Gamma;A;I;i;rho;phi;u))
Lemma: comp-fun-to-comp-op_wf
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. ∀[comp:Gamma ⊢ Compositon(A)]. (cfun-to-cop(Gamma;A;comp) ∈ Gamma ⊢ CompOp(A))
Lemma: composition-structure-implies-composition-op
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. (Gamma ⊢ Compositon(A) ⇒ Gamma ⊢ CompOp(A))
Lemma: comp-op-to-comp-fun-inverse
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. (cfun-to-cop(Gamma;A;cop-to-cfun(cA)) = cA ∈ Gamma ⊢ CompOp(A))
Lemma: comp-fun-to-comp-op-inverse
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)].
(cop-to-cfun(cfun-to-cop(Gamma;A;cA)) = cA ∈ Gamma ⊢ Compositon(A))
Definition: csm-comp-structure
(cA)tau == λH,sigma,phi,u,a0. (cA H tau o sigma phi u a0)
Lemma: csm-comp-structure_wf
∀[Gamma,Delta:j⊢]. ∀[tau:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)].
((cA)tau ∈ Delta ⊢ Compositon((A)tau))
Lemma: csm-comp-structure_wf2
∀[Gamma,Delta:j⊢]. ∀[tau:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma +⊢ Compositon(A)].
((cA)tau ∈ Delta +⊢ Compositon((A)tau))
Lemma: csm-comp-structure-composition-function
∀[Gamma,Delta:j⊢]. ∀[tau:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)].
((cA)tau ∈ composition-function{j:l,i:l}(Delta;(A)tau))
Lemma: csm-comp-structure-subset
∀[Gamma,Delta,tau,cA,phi:Top]. ((cA)tau ~ (cA)tau)
Lemma: csm-comp-structure-id
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)]. ∀[tau:Gamma j⟶ Gamma].
(cA)tau = cA ∈ Gamma ⊢ Compositon(A) supposing tau = 1(Gamma) ∈ Gamma j⟶ Gamma
Lemma: csm-comp-op-to-comp-fun-sq
∀[Gamma,Delta,tau,cA:Top]. ((cop-to-cfun(cA))tau ~ cop-to-cfun((cA)tau))
Lemma: csm-comp-fun-to-comp-op
∀[Gamma,K:j⊢]. ∀[tau:K j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)].
((cfun-to-cop(Gamma;A;cA))tau = cfun-to-cop(K;(A)tau;(cA)tau) ∈ K ⊢ CompOp((A)tau))
Lemma: subset-comp-structure
∀[X,Y:j⊢]. ∀[T:{Y ⊢ _}]. Y ⊢ Compositon(T) ⊆r X ⊢ Compositon(T) supposing sub_cubical_set{j:l}(X; Y)
Definition: rev-type-comp
rev-type-comp(Gamma;cA) == (cA)(p;1-(q))
Lemma: rev-type-comp_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. (rev-type-comp(Gamma;cA) ∈ Gamma.𝕀 ⊢ Compositon((A)-))
Definition: comp_term
comp cA [phi ⊢→ u] a0 == cA Gamma 1(Gamma.𝕀) phi u a0
Lemma: comp_term_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:composition-function{j:l,i:l}(Gamma.𝕀;A)].
∀[u:{Gamma, phi.𝕀 ⊢ _:A}]. ∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}].
(comp cA [phi ⊢→ u] a0 ∈ {Gamma ⊢ _:(A)[1(𝕀)][phi |⟶ (u)[1(𝕀)]]})
Lemma: comp_term-composition-term
∀[Gamma,phi,cA,u,a0:Top]. (comp cop-to-cfun(cA) [phi ⊢→ u] a0 ~ comp cA [phi ⊢→ u] a0)
Lemma: csm-comp_term
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma, phi.𝕀 ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}]. ∀[Delta:j⊢]. ∀[s:Delta j⟶ Gamma].
((comp cA [phi ⊢→ u] a0)s = comp (cA)s+ [(phi)s ⊢→ (u)s+] (a0)s ∈ {Delta ⊢ _:((A)s+)[1(𝕀)][(phi)s |⟶ ((u)s+)[1(𝕀)]]})
Definition: transprt
transprt(G;cA;a0) == comp cA [0(𝔽) ⊢→ discr(⋅)] a0
Lemma: transprt_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[a:{Gamma ⊢ _:(A)[0(𝕀)]}].
(transprt(Gamma;cA;a) ∈ {Gamma ⊢ _:(A)[1(𝕀)]})
Lemma: equals-transprt
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 +⊢ Compositon(A)]. ∀[a:{Gamma ⊢ _:(A)[0(𝕀)]}]. ∀[xx:Top].
(comp cA [0(𝔽) ⊢→ xx] a = transprt(Gamma;cA;a) ∈ {Gamma ⊢ _:(A)[1(𝕀)]})
Definition: transprt-fun
transprt-fun(Gamma;A;cA) == (λtransprt(Gamma.(A)[0(𝕀)];(cA)p+;q))
Lemma: transprt-fun_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 +⊢ Compositon(A)].
(transprt-fun(Gamma;A;cA) ∈ {Gamma ⊢ _:((A)[0(𝕀)] ⟶ (A)[1(𝕀)])})
Lemma: csm-transprt
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[a:{Gamma ⊢ _:(A)[0(𝕀)]}]. ∀[H:j⊢]. ∀[s:H j⟶ Gamma].
((transprt(Gamma;cA;a))s = transprt(H;(cA)s+;(a)s) ∈ {H ⊢ _:((A)[1(𝕀)])s})
Lemma: csm-transprt-fun
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 +⊢ Compositon(A)]. ∀[H:j⊢]. ∀[s:H j⟶ Gamma].
((transprt-fun(Gamma;A;cA))s = transprt-fun(H;(A)s+;(cA)s+) ∈ {H ⊢ _:(((A)s+)[0(𝕀)] ⟶ ((A)s+)[1(𝕀)])})
Definition: transprt-const
transprt-const(G;cA;a) == transprt(G;(cA)p;a)
Lemma: transprt-const_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[a:{G ⊢ _:A}]. (transprt-const(G;cA;a) ∈ {G ⊢ _:A})
Lemma: csm-transprt-const
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[a:{G ⊢ _:A}]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((transprt-const(G;cA;a))s = transprt-const(H;(cA)s;(a)s) ∈ {H ⊢ _:(A)s})
Definition: comp_trm
comp_trm(Gamma;cA;phi;u;a0) == comp cA [phi ⊢→ u] a0
Lemma: comp_trm_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:composition-function{j:l,i:l}(Gamma.𝕀;A)].
∀[u:{Gamma, phi.𝕀 ⊢ _:A}]. ∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}].
(comp_trm(Gamma;
cA;
phi;
u;
a0) ∈ {Gamma ⊢ _:(A)[1(𝕀)]})
Lemma: comp_term-subset
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma, phi.𝕀 ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}]. ∀[psi:{Gamma ⊢ _:𝔽}].
(comp cA [phi ⊢→ u] a0 = comp cA [phi ⊢→ u] a0 ∈ {Gamma, psi ⊢ _:(A)[1(𝕀)]})
Definition: filling-function
filling-function{j:l, i:l}(Gamma;A) ==
H:CubicalSet{j}
⟶ sigma:H.𝕀 j⟶ Gamma
⟶ phi:{H ⊢ _:𝔽}
⟶ u:{H.𝕀, (phi)p ⊢ _:(A)sigma}
⟶ a0:{H ⊢ _:((A)sigma)[0(𝕀)][phi |⟶ u[0]]}
⟶ {H.𝕀 ⊢ _:(A)sigma[(phi)p |⟶ u]}
Lemma: filling-function_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (filling-function{j:l, i:l}(Gamma;A) ∈ 𝕌{[i' | j'']})
Definition: uniform-filling-function
uniform-filling-function{j:l, i:l}(Gamma;A;fill) ==
∀H,K:j⊢. ∀tau:K j⟶ H. ∀sigma:H.𝕀 j⟶ Gamma. ∀phi:{H ⊢ _:𝔽}. ∀u:{H.𝕀, (phi)p ⊢ _:(A)sigma}.
∀a0:{H ⊢ _:((A)sigma)[0(𝕀)][phi |⟶ u[0]]}.
((fill H sigma phi u a0)tau+ = (fill K sigma o tau+ (phi)tau (u)tau+ (a0)tau) ∈ {K.𝕀 ⊢ _:((A)sigma)tau+})
Lemma: uniform-filling-function_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:filling-function{j:l, i:l}(Gamma;A)].
(uniform-filling-function{j:l, i:l}(Gamma;A;comp) ∈ ℙ{[i' | j'']})
Definition: filling-structure
Gamma ⊢ Filling(A) == {fill:filling-function{j:l, i:l}(Gamma;A)| uniform-filling-function{j:l, i:l}(Gamma;A;fill)}
Lemma: filling-structure_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (Gamma ⊢ Filling(A) ∈ 𝕌{[i' | j'']})
Definition: csm-m
m == λI,c. let a,s = c in let rho,r = a in (rho;r ∧ s)
Lemma: csm-m_wf
∀[Gamma:j⊢]. (m ∈ Gamma.𝕀.𝕀 j⟶ Gamma.𝕀)
Lemma: cc-fst-comp-csm-m-term
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. (((phi)p)m = ((phi)p)p ∈ {H.𝕀.𝕀 ⊢ _:𝔽})
Lemma: 0-comp-cc-fst-comp-m
∀[H:j⊢]. ([0(𝕀)] o p o m = m ∈ H.𝕀.𝕀, ((q=0))p j⟶ H.𝕀)
Lemma: csm-m-comp-0
∀[H:j⊢]. ([0(𝕀)] o p = m o [0(𝕀)] ∈ H.𝕀 ij⟶ H.𝕀)
Lemma: csm-m-comp-1
∀[H:j⊢]. (m o [1(𝕀)] = 1(H.𝕀) ∈ H.𝕀 ij⟶ H.𝕀)
Lemma: csm+-comp-m
∀[H,K:j⊢]. ∀[tau:K j⟶ H]. (m o tau++ = tau+ o m ∈ K.𝕀.𝕀 ij⟶ H.𝕀)
Lemma: cc-fst+-comp-0
∀[G:j⊢]. (p+ o [0(𝕀)] = [0(𝕀)] o p ∈ G.𝕀 ij⟶ G.𝕀)
Lemma: cc-fst+-0-type
∀[G:j⊢]. ∀[A:{G.𝕀 ⊢ _}]. (((A)[0(𝕀)])p = ((A)p+)[0(𝕀)] ∈ {G.𝕀 ⊢ _})
Lemma: cc-fst+-comp-1
∀[G:j⊢]. (p+ o [1(𝕀)] = [1(𝕀)] o p ∈ G.𝕀 ij⟶ G.𝕀)
Lemma: cc-fst+-1-type
∀[G:j⊢]. ∀[A:{G.𝕀 ⊢ _}]. (((A)[1(𝕀)])p = ((A)p+)[1(𝕀)] ∈ {G.𝕀 ⊢ _})
Definition: comp-to-fill
comp-to-fill(Gamma;cA) == λH,sigma,phi,u,a0. (cA H.𝕀 sigma o m ((phi)p ∨ (q=0)) ((u)m ∨ ((a0)p)m) (a0)p)
Lemma: comp-to-fill_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:composition-function{j:l,i:l}(Gamma;A)].
(comp-to-fill(Gamma;cA) ∈ filling-function{j:l, i:l}(Gamma;A))
Lemma: comp-to-fill_wf2
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)]. (comp-to-fill(Gamma;cA) ∈ Gamma ⊢ Filling(A))
Lemma: composition-implies-filling-structure
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. (Gamma ⊢ Compositon(A) ⇒ Gamma ⊢ Filling(A))
Definition: fill_term
fill cA [phi ⊢→ u] a0 == comp-to-fill(Gamma.𝕀;cA) Gamma 1(Gamma.𝕀) phi u a0
Lemma: fill_term_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:composition-function{j:l,i:l}(Gamma.𝕀;A)].
∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}]. ∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ u[0]]}].
(fill cA [phi ⊢→ u] a0 ∈ {Gamma.𝕀 ⊢ _:A[(phi)p |⟶ u]})
Lemma: csm-fill_term
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ u[0]]}]. ∀[Delta:j⊢]. ∀[s:Delta j⟶ Gamma].
((fill cA [phi ⊢→ u] a0)s+ = fill (cA)s+ [(phi)s ⊢→ (u)s+] (a0)s ∈ {Delta.𝕀 ⊢ _:(A)s+[((phi)s)p |⟶ (u)s+]})
Lemma: fill_term_1
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[T:{H.𝕀 ⊢ _}]. ∀[u:{H.𝕀, (phi)p ⊢ _:T}]. ∀[a0:{H ⊢ _:(T)[0(𝕀)][phi |⟶ u[0]]}].
∀[cT:H.𝕀 ⊢ Compositon(T)].
((fill cT [phi ⊢→ u] a0)[1(𝕀)] = comp cT [phi ⊢→ u] a0 ∈ {H ⊢ _:(T)[1(𝕀)]})
Lemma: fill_term_0
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[T:{H.𝕀 ⊢ _}]. ∀[u:{H.𝕀, (phi)p ⊢ _:T}]. ∀[a0:{H ⊢ _:(T)[0(𝕀)][phi |⟶ u[0]]}].
∀[cT:H.𝕀 ⊢ Compositon(T)].
((fill cT [phi ⊢→ u] a0)[0(𝕀)] = a0 ∈ {H ⊢ _:(T)[0(𝕀)]})
Definition: rev_fill_term
rev_fill_term(Gamma;cA;phi;u;a1) == (fill rev-type-comp(Gamma;cA) [phi ⊢→ (u)(p;1-(q))] a1)(p;1-(q))
Lemma: rev_fill_term_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a1:{Gamma ⊢ _:(A)[1(𝕀)][phi |⟶ u[1]]}].
(rev_fill_term(Gamma;cA;phi;u;a1) ∈ {Gamma.𝕀 ⊢ _:A[(phi)p |⟶ u]})
Lemma: rev_fill_term_1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a1:{Gamma ⊢ _:(A)[1(𝕀)][phi |⟶ u[1]]}].
((rev_fill_term(Gamma;cA;phi;u;a1))[1(𝕀)] = a1 ∈ {Gamma ⊢ _:(A)[1(𝕀)]})
Lemma: rev_fill_term_0
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a1:{Gamma ⊢ _:(A)[1(𝕀)][phi |⟶ u[1]]}].
((rev_fill_term(Gamma;cA;phi;u;a1))[0(𝕀)]
= comp rev-type-comp(Gamma;cA) [phi ⊢→ (u)(p;1-(q))] a1
∈ {Gamma ⊢ _:(A)[0(𝕀)]})
Lemma: csm-rev_fill_term
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a1:{Gamma ⊢ _:(A)[1(𝕀)][phi |⟶ u[1]]}]. ∀[Delta:j⊢]. ∀[s:Delta j⟶ Gamma].
((rev_fill_term(Gamma;cA;phi;u;a1))s+
= rev_fill_term(Delta;(cA)s+;(phi)s;(u)s+;(a1)s)
∈ {Delta.𝕀 ⊢ _:(A)s+[((phi)s)p |⟶ (u)s+]})
Lemma: trivial-constrained-term
∀[Gamma:j⊢]. ∀[B:{Gamma ⊢ _}]. ∀[xx:Top]. ({Gamma ⊢ _:B} ⊆r {Gamma ⊢ _:B[0(𝔽) |⟶ xx]})
Definition: revfill
revfill(Gamma;cA;a1) == rev_fill_term(Gamma;cA;0(𝔽);discr(⋅);a1)
Lemma: revfill_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[a1:{Gamma ⊢ _:(A)[1(𝕀)]}].
(revfill(Gamma;cA;a1) ∈ {Gamma.𝕀 ⊢ _:A})
Lemma: csm-revfill
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[a1:{Gamma ⊢ _:(A)[1(𝕀)]}]. ∀[Delta:j⊢].
∀[s:Delta j⟶ Gamma].
((revfill(Gamma;cA;a1))s+ = revfill(Delta;(cA)s+;(a1)s) ∈ {Delta.𝕀 ⊢ _:(A)s+})
Lemma: revfill-1
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[a1:{Gamma ⊢ _:(A)[1(𝕀)]}].
((revfill(Gamma;cA;a1))[1(𝕀)] = a1 ∈ {Gamma ⊢ _:(A)[1(𝕀)]})
Definition: filling_term
fill cA [phi ⊢→ u] a0 == fill cop-to-cfun(cA) [phi ⊢→ u] a0
Lemma: filling_term_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ u[0]]}].
(fill cA [phi ⊢→ u] a0 ∈ {Gamma.𝕀 ⊢ _:A[(phi)p |⟶ u]})
Lemma: csm-filling_term
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ u[0]]}]. ∀[Delta:j⊢]. ∀[s:Delta j⟶ Gamma].
((fill cA [phi ⊢→ u] a0)s+ = fill (cA)s+ [(phi)s ⊢→ (u)s+] (a0)s ∈ {Delta.𝕀 ⊢ _:(A)s+[((phi)s)p |⟶ (u)s+]})
Lemma: filling_term_0
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[T:{H.𝕀 ⊢ _}]. ∀[u:{H.𝕀, (phi)p ⊢ _:T}]. ∀[a0:{H ⊢ _:(T)[0(𝕀)][phi |⟶ u[0]]}].
∀[cT:H.𝕀 ⊢ CompOp(T)].
((fill cT [phi ⊢→ u] a0)[0(𝕀)] = a0 ∈ {H ⊢ _:(T)[0(𝕀)]})
Lemma: filling_term_1
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[T:{H.𝕀 ⊢ _}]. ∀[u:{H.𝕀, (phi)p ⊢ _:T}]. ∀[a0:{H ⊢ _:(T)[0(𝕀)][phi |⟶ u[0]]}].
∀[cT:H.𝕀 ⊢ CompOp(T)].
((fill cT [phi ⊢→ u] a0)[1(𝕀)] = comp cT [phi ⊢→ u] a0 ∈ {H ⊢ _:(T)[1(𝕀)]})
Definition: fill-type-up
fill-type-up(Gamma;A;cA) == (λ(fill (cA)p+ [0(𝔽) ⊢→ discr(⋅)] q)swap-interval(Gamma;(A)[0(𝕀)]))
Lemma: fill-type-up_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)].
(fill-type-up(Gamma;A;cA) ∈ {Gamma.𝕀 ⊢ _:(((A)[0(𝕀)])p ⟶ A)})
Lemma: fill-type-up-0
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma ⊢ _:(A)[0(𝕀)]}].
((app(fill-type-up(Gamma;A;cA); (u)p))[0(𝕀)] = u ∈ {Gamma ⊢ _:(A)[0(𝕀)]})
Lemma: fill-type-up-1
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma ⊢ _:(A)[0(𝕀)]}].
((app(fill-type-up(Gamma;A;cA); (u)p))[1(𝕀)] = app(transport-fun(Gamma;A;cA); u) ∈ {Gamma ⊢ _:(A)[1(𝕀)]})
Definition: fill-type-down
fill-type-down(Gamma;A;cA) == (fill-type-up(Gamma;(A)-;(cA)(p;1-(q))))(p;1-(q))
Lemma: fill-type-down_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)].
(fill-type-down(Gamma;A;cA) ∈ {Gamma.𝕀 ⊢ _:(((A)[1(𝕀)])p ⟶ A)})
Lemma: fill-type-down-1
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma ⊢ _:(A)[1(𝕀)]}].
((app(fill-type-down(Gamma;A;cA); (u)p))[1(𝕀)] = u ∈ {Gamma ⊢ _:(A)[1(𝕀)]})
Lemma: fill-type-down-0
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma ⊢ _:(A)[1(𝕀)]}].
((app(fill-type-down(Gamma;A;cA); (u)p))[0(𝕀)] = app(rev-transport-fun(Gamma;A;cA); u) ∈ {Gamma ⊢ _:(A)[0(𝕀)]})
Definition: fillpath
fillpath(Gamma;A;cA;x;y;z) ==
comp (cA)p+ [((q=0) ∨ (q=1)) ⊢→ [(q)p=0 ⊢→ (app(fill-type-down(Gamma;A;cA); (x)p))p+;
(q)p=1 ⊢→ (app(fill-type-up(Gamma;A;cA); (y)p))p+]] z
Lemma: fillpath_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[x:{Gamma ⊢ _:(A)[1(𝕀)]}]. ∀[y:{Gamma ⊢ _:(A)[0(𝕀)]}].
∀[z:{Gamma.𝕀 ⊢ _:((A)[0(𝕀)])p}].
(fillpath(Gamma;A;cA;x;y;z) ∈ {t:{Gamma.𝕀 ⊢ _:((A)[1(𝕀)])p}|
((t)[0(𝕀)] = x ∈ {Gamma ⊢ _:(A)[1(𝕀)]})
∧ ((t)[1(𝕀)] = app(transport-fun(Gamma;A;cA); y) ∈ {Gamma ⊢ _:(A)[1(𝕀)]})} ) supposing
(((z)[0(𝕀)] = app(rev-transport-fun(Gamma;A;cA); x) ∈ {Gamma ⊢ _:(A)[0(𝕀)]}) and
((z)[1(𝕀)] = y ∈ {Gamma ⊢ _:(A)[0(𝕀)]}))
Definition: fill-path
fill-path(Gamma;A;cA;x;y;z) == <>(fillpath(Gamma;A;cA;x;y;z))
Lemma: fill-path_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[x:{Gamma ⊢ _:(A)[1(𝕀)]}]. ∀[y:{Gamma ⊢ _:(A)[0(𝕀)]}].
∀[z:{Gamma.𝕀 ⊢ _:((A)[0(𝕀)])p}].
(fill-path(Gamma;A;cA;x;y;z) ∈ {Gamma ⊢ _:(Path_(A)[1(𝕀)] x app(transport-fun(Gamma;A;cA); y))}) supposing
(((z)[0(𝕀)] = app(rev-transport-fun(Gamma;A;cA); x) ∈ {Gamma ⊢ _:(A)[0(𝕀)]}) and
((z)[1(𝕀)] = y ∈ {Gamma ⊢ _:(A)[0(𝕀)]}))
Lemma: fill-path_wf_const
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[x,y:{Gamma ⊢ _:A}]. ∀[z:{Gamma.𝕀 ⊢ _:(A)p}].
(fill-path(Gamma;(A)p;(cA)p;x;y;z) ∈ {Gamma ⊢ _:(Path_A x app(transport-fun(Gamma;(A)p;(cA)p); y))}) supposing
(((z)[0(𝕀)] = app(rev-transport-fun(Gamma;(A)p;(cA)p); x) ∈ {Gamma ⊢ _:((A)p)[0(𝕀)]}) and
((z)[1(𝕀)] = y ∈ {Gamma ⊢ _:((A)p)[0(𝕀)]}))
Definition: trans-const-path
trans-const-path(G;cA;a) == <>(rev_fill_term(G;(cA)p;0(𝔽);discr(⋅);a))
Lemma: trans-const-path_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[a:{G ⊢ _:A}].
(trans-const-path(G;cA;a) ∈ {G ⊢ _:(Path_A transprt-const(G;cA;a) a)})
Definition: comp-path
pth_a_b + pth_b_c ==
<>(comp ((cA)p)p [((q=0) ∨ (q=1)) ⊢→ ((path-eta(refl(a)))p+ ∨ (path-eta(pth_b_c))p+)] path-eta(pth_a_b))
Lemma: comp-path_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G ⊢ Compositon(A)]. ∀[a,b,c:{G ⊢ _:A}]. ∀[pth_a_b:{G ⊢ _:(Path_A a b)}].
∀[pth_b_c:{G ⊢ _:(Path_A b c)}].
(pth_a_b + pth_b_c ∈ {G ⊢ _:(Path_A a c)})
Definition: comp_path
pth_a_b + pth_b_c == pth_a_b + pth_b_c
Lemma: comp_path_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G ⊢ Compositon(A)]. ∀[a,b,c:{G ⊢ _:A}]. ∀[pth_a_b:{G ⊢ _:(Path_A a b)}].
∀[pth_b_c:{G ⊢ _:(Path_A b c)}].
(pth_a_b + pth_b_c ∈ {G ⊢ _:(Path_A a c)})
Definition: contractible-to-prop
contractible-to-prop(X;A;cA;c) == (λ(λrev-path(X.A.(A)p;contr-path(((c)p)p;(q)p)) + contr-path(((c)p)p;q)))
Lemma: contractible-to-prop_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[cA:X +⊢ Compositon(A)]. ∀[c:{X ⊢ _:Contractible(A)}].
(contractible-to-prop(X;A;cA;c) ∈ {X ⊢ _:isProp(A)})
Lemma: contractible-iff-inhabited-prop
∀X:j⊢. ∀A:{X ⊢ _}. ∀cA:X +⊢ Compositon(A). ({X ⊢ _:Contractible(A)} ⇐⇒ {X ⊢ _:isProp(A)} ∧ {X ⊢ _:A})
Definition: sigma_comp
sigma_comp(cA;cB) ==
λH,sigma,phi,u,a0. let a = fill (cA)sigma [phi ⊢→ u.1] a0.1 in
let b = comp ((cB)sigma+)[a] [phi ⊢→ u.2] a0.2 in
cubical-pair((a)[1(𝕀)];b)
Lemma: sigma_comp_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[cA:X ⊢ Compositon(A)]. ∀[cB:X.A +⊢ Compositon(B)].
(sigma_comp(cA;cB) ∈ composition-function{j:l,i:l}(X;Σ A B))
Lemma: sigma_comp-sq
∀[X,A,cA,cB:Top].
(sigma_comp(cA;cB) ~ λH,sigma,phi,u,a0. let a = cA H.𝕀 (λx,x@0. (sigma x (m x x@0)))
(λI,rho. phi I (fst(rho)) ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. if (phi I (fst((m I rho)))==1)
then fst((u I (m I rho)))
else fst((a0 I (fst((m I rho)))))
fi )
(λI,a. (fst((a0 I (fst(a)))))) in
let b = cB H (λx,x@0. <sigma x x@0, a x x@0>) phi (λI,a. (snd((u I a))))
(λI,a. (snd((a0 I a)))) in
λI,a@0. <a I <a@0, 1>, b I a@0>)
Lemma: sigma_comp_wf2
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[cA:X ⊢ Compositon(A)]. ∀[cB:X.A +⊢ Compositon(B)].
(sigma_comp(cA;cB) ∈ X ⊢ Compositon(Σ A B))
Lemma: csm-sigma_comp
∀[X,A,cA,cB,H,tau:Top]. ((sigma_comp(cA;cB))tau ~ sigma_comp((cA)tau;(cB)tau+))
Lemma: csm-sigma_comp2
∀[X,H,cA,cB,A,tau:Top]. ((sigma_comp(cA;cB))tau ~ sigma_comp((cA)tau;(cB)tau+))
Lemma: csm-sigma_comp3
∀[H,A',X,cA,cB,A,tau:Top]. ((sigma_comp(cA;cB))tau ~ sigma_comp((cA)tau;(cB)tau+))
Lemma: fst-transprt-sigma
∀[X:j⊢]. ∀[A:{X.𝕀 ⊢ _}]. ∀[B:{X.𝕀.A ⊢ _}]. ∀[cA:X.𝕀 +⊢ Compositon(A)]. ∀[cB:X.𝕀.A +⊢ Compositon(B)].
∀[pr:{X ⊢ _:(Σ A B)[0(𝕀)]}].
(transprt(X;sigma_comp(cA;cB);pr).1 = transprt(X;cA;pr.1) ∈ {X ⊢ _:(A)[1(𝕀)]})
Lemma: fst-transprt-const-sigma
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[cA:X +⊢ Compositon(A)]. ∀[cB:X.A +⊢ Compositon(B)]. ∀[pr:{X ⊢ _:Σ A B}].
(transprt-const(X;sigma_comp(cA;cB);pr).1 = transprt-const(X;cA;pr.1) ∈ {X ⊢ _:A})
Definition: pi_comp
pi_comp(X;A;cA;cB) ==
λH,sigma,phi,u,a0. let v = revfill(H.((A)sigma)[1(𝕀)];((cA)sigma)p+;q) in
(λcomp (cB)(sigma o p+;v) [(phi)p ⊢→ app((u)p+; v)] app((a0)p; (v)[0(𝕀)]))
Lemma: pi_comp_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[cA:X +⊢ Compositon(A)]. ∀[cB:X.A +⊢ Compositon(B)].
(pi_comp(X;A;cA;cB) ∈ composition-function{j:l,i:l}(X;ΠA B))
Lemma: pi_comp-sq
∀[X,A,cA,cB:Top].
(pi_comp(X;A;cA;cB) ~ λH,sigma,phi,u,a0. let v = λI,a. (cA H.((A)sigma)[1(𝕀)].𝕀
(λx,x@0. (sigma x <fst(fst((m x x@0))), ¬(snd((m x x@0)))>))
(λI,rho. 0 ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. (snd(fst((m I rho)))))
(λI,a. (snd(fst(a))))
I
<fst(a), ¬(snd(a))>) in
(λcB H.((A)sigma)[1(𝕀)] (λx,x@0. <sigma x <fst(fst(x@0)), snd(x@0)>, v x \000Cx@0>)
(λI,a. (phi I (fst(a))))
(λI,a. (u I <fst(fst(a)), snd(a)> I 1 (v I a)))
(λI,a. (a0 I (fst(a)) I 1 (v I <a, 0>)))))
Lemma: csm-pi_comp
∀[X,Y,tau,A,cA,cB:Top]. ((pi_comp(X;A;cA;cB))tau ~ pi_comp(Y;(A)tau;(cA)tau;(cB)tau+))
Lemma: pi_comp_wf2
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[cA:X +⊢ Compositon(A)]. ∀[cB:X.A +⊢ Compositon(B)].
(pi_comp(X;A;cA;cB) ∈ X ⊢ Compositon(ΠA B))
Lemma: pi_comp_wf_fun
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[cA:X +⊢ Compositon(A)]. ∀[cB:X +⊢ Compositon(B)].
(pi_comp(X;A;cA;(cB)p) ∈ X ⊢ Compositon((A ⟶ B)))
Definition: pathtype-comp
pathtype-comp(G;A;cA) == λH,sigma,phi,u,a0. <>comp ((cA)sigma)p+ [(phi)p ⊢→ (u)p+ @ (q)p] (a0)p @ q
Lemma: pathtype-comp_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G ⊢ Compositon(A)]. (pathtype-comp(G;A;cA) ∈ G ⊢ Compositon(Path(A)))
Lemma: csm-pathtype-comp
∀[G,A,cA,H,tau:Top]. ((pathtype-comp(G;A;cA))tau ~ pathtype-comp(H;(A)tau;(cA)tau))
Definition: path-term
path-term(phi;w;a;b;r) == (w @ r ∨ (a ∨ b))
Lemma: path-term_wf
∀X:j⊢. ∀psi:{X ⊢ _:𝔽}.
∀[r:{X ⊢ _:𝕀}]
∀T:{X ⊢ _}. ∀a,b:{X ⊢ _:T}. ∀w:{X, psi ⊢ _:(Path_T a b)}.
(path-term(psi;w;a;b;r) ∈ {X, (psi ∨ ((r=0) ∨ (r=1))) ⊢ _:T})
Lemma: path-term-case1
∀X:j⊢. ∀psi:{X ⊢ _:𝔽}.
∀[r:{X ⊢ _:𝕀}]
∀T:{X ⊢ _}. ∀a,b:{X ⊢ _:T}. ∀w:{X, psi ⊢ _:(Path_T a b)}. (path-term(psi;w;a;b;r) = w @ r ∈ {X, psi ⊢ _:T})
Lemma: path-term-equal
∀X:j⊢. ∀psi:{X ⊢ _:𝔽}.
∀[r:{X ⊢ _:𝕀}]
∀T:{X ⊢ _}. ∀a,b:{X ⊢ _:T}. ∀w:{X, psi ⊢ _:(Path_T a b)}.
∀[z:{X ⊢ _:T}]
path-term(psi;w;a;b;r) = z ∈ {X, (psi ∨ ((r=0) ∨ (r=1))) ⊢ _:T}
supposing (w @ r = z ∈ {X, psi ⊢ _:T}) ∧ (a = z ∈ {X, (r=0) ⊢ _:T}) ∧ (b = z ∈ {X, (r=1) ⊢ _:T})
Lemma: path-term-0
∀X:j⊢. ∀psi:{X ⊢ _:𝔽}. ∀T:{X ⊢ _}. ∀a,b:{X ⊢ _:T}. ∀w:{X, psi ⊢ _:(Path_T a b)}.
(path-term(psi;w;a;b;0(𝕀)) = a ∈ {X ⊢ _:T})
Lemma: path-term-1
∀X:j⊢. ∀psi:{X ⊢ _:𝔽}. ∀T:{X ⊢ _}. ∀a,b:{X ⊢ _:T}. ∀w:{X, psi ⊢ _:(Path_T a b)}.
(path-term(psi;w;a;b;1(𝕀)) = b ∈ {X ⊢ _:T})
Lemma: csm-path-term
∀[phi,r,s,a,b,w:Top]. ((path-term(phi;w;a;b;r))s ~ path-term((phi)s;(w)s;(a)s;(b)s;(r)s))
Definition: path_term
path_term(phi; w; a; b; r) == path-term((phi)p;w;a;b;(r)p)
Lemma: path_term_wf
∀X:j⊢. ∀psi:{X ⊢ _:𝔽}.
∀[r:{X ⊢ _:𝕀}]
∀T:{X.𝕀 ⊢ _}. ∀a,b:{X.𝕀 ⊢ _:T}. ∀w:{X, psi.𝕀 ⊢ _:(Path_T a b)}.
(path_term(psi; w; a; b; r) ∈ {X.𝕀, ((psi)p ∨ (((r=0))p ∨ ((r=1))p)) ⊢ _:T})
Lemma: csm-path_term
∀[psi,r,s,a,b,w:Top]. ((path_term(psi; w; a; b; r))s ~ path-term(((psi)p)s;(w)s;(a)s;(b)s;((r)p)s))
Lemma: path_term-0
∀[H,A,psi,r,a,b,w:Top].
((path_term(psi; w; a; b; r))[0(𝕀)] ~ path-term((psi)1(H.A);(w)[0(𝕀)];(a)[0(𝕀)];(b)[0(𝕀)];(r)1(H.A)))
Lemma: path_term-1
∀[H,A,psi,r,a,b,w:Top].
((path_term(psi; w; a; b; r))[1(𝕀)] ~ path-term((psi)1(H.A);(w)[1(𝕀)];(a)[1(𝕀)];(b)[1(𝕀)];(r)1(H.A)))
Definition: path_comp
path_comp(G;A;a;b;
cA) ==
λH,sigma,phi,u,a0. <>(comp ((cA)sigma)p+ [((phi)p ∨ ((q=0) ∨ (q=1))) ⊢→ path_term((phi)p;
(u)p+;
((a)sigma)p+;
((b)sigma)p+;
q)] (a0)p @ q)
Lemma: path_comp_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[a,b:{G ⊢ _:A}]. ∀[cA:G ⊢ Compositon(A)].
(path_comp(G;A;a;b;
cA) ∈ G ⊢ Compositon((Path_A a b)))
Lemma: path-comp-exists
∀G:j⊢. ∀A:{G ⊢ _}. ∀a,b:{G ⊢ _:A}. (G ⊢ CompOp(A) ⇒ G ⊢ CompOp((Path_A a b)))
Lemma: csm-path_comp
∀[G,H,A,a,b,cA,tau:Top]. ((path_comp(G;A;a;b;cA))tau ~ path_comp(H;(A)tau;(a)tau;(b)tau;(cA)tau))
Lemma: cubical-pi-comp-structure
∀X:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. (X ⊢ Compositon(A) ⇒ X.A +⊢ Compositon(B) ⇒ X ⊢ Compositon(ΠA B))
Lemma: transform-comp-structure
∀Gamma,Delta:j⊢. ∀tau:Delta j⟶ Gamma. ∀A:{Gamma ⊢ _}. (Gamma ⊢ Compositon(A) ⇒ Delta ⊢ Compositon((A)tau))
Definition: fiber-comp
fiber-comp(X;T;A;w;a;cT;cA) == sigma_comp(cT;path_comp(X.T;(A)p;(a)p;app((w)p; q);(cA)p))
Lemma: fiber-comp_wf
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[cT:X ⊢ Compositon(T)]. ∀[cA:X +⊢ Compositon(A)].
(fiber-comp(X;T;A;w;a;cT;cA) ∈ X ⊢ Compositon(Fiber(w;a)))
Lemma: fiber-comp-exists
∀X:j⊢. ∀T,A:{X ⊢ _}. ∀w:{X ⊢ _:(T ⟶ A)}. ∀a:{X ⊢ _:A}. (X ⊢ CompOp(T) ⇒ X ⊢ CompOp(A) ⇒ X ⊢ CompOp(Fiber(w;a)))
Lemma: csm-fiber-comp-sq
∀[G,A,T,a,cA,cT,H,s,f:Top]. ((fiber-comp(G;T;A;f;a;cT;cA))s ~ fiber-comp(H;(T)s;(A)s;(f)s;(a)s;(cT)s;(cA)s))
Lemma: fiber-comp-subset
∀[X,T,A,w,a,cT,cA,phi:Top]. (fiber-comp(X, phi;T;A;w;a;cT;cA) ~ fiber-comp(X;T;A;w;a;cT;cA))
Lemma: csm-fiber-comp
∀[G:j⊢]. ∀[A,T:{G ⊢ _}]. ∀[a:{G ⊢ _:A}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cT:G ⊢ Compositon(T)]. ∀[H:j⊢]. ∀[s:H j⟶ G].
∀[f:{G ⊢ _:(T ⟶ A)}].
((fiber-comp(G;T;A;f;a;cT;cA))s = fiber-comp(H;(T)s;(A)s;(f)s;(a)s;(cT)s;(cA)s) ∈ H ⊢ Compositon(Fiber((f)s;(a)s)))
Lemma: fiber-member-transprt-const-fiber-comp
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[pr:{X ⊢ _:Fiber(w;a)}]. ∀[cT:X +⊢ Compositon(T)].
∀[cA:X +⊢ Compositon(A)].
(fiber-member(transprt-const(X;fiber-comp(X;T;A;w;a;cT;cA);pr)) = transprt-const(X;cT;pr.1) ∈ {X ⊢ _:T})
Definition: contractible_comp
contractible_comp(X;A;cA) == sigma_comp(cA;pi_comp(X.A;(A)p;(cA)p;path_comp(X.A.(A)p;((A)p)p;(q)p;q;((cA)p)p)))
Lemma: contractible_comp_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[cA:X +⊢ Compositon(A)]. (contractible_comp(X;A;cA) ∈ X ⊢ Compositon(Contractible(A)))
Lemma: contractible_comp-sq
∀[X,A,cA:Top].
(contractible_comp(X;A;cA)
~ λH,sigma,phi,u,a0.
let a = cA H.𝕀 (λx,x@0. (sigma x (m x x@0))) (λI,rho. phi I (fst(rho)) ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. if (phi I (fst((m I rho)))==1)
then fst((u I (m I rho)))
else fst((a0 I (fst((m I rho)))))
fi )
(λI,a. (fst((a0 I (fst(a)))))) in
let b = let v = λI,a@0. (cA H.((A)sigma)[1(𝕀)].𝕀
(λx,x@0. (sigma x <fst(fst((m x x@0))), ¬(snd((m x x@0)))>))
(λI,rho. 0 ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. if (0==1) then ⋅ else snd(fst((m I rho))) fi )
(λI,a. (snd(fst(a))))
I
<fst(a@0), ¬(snd(a@0))>) in
λI,a@0,J,f,u@0,J@0,f@0,u@1.
(cA H.((A)sigma)[1(𝕀)].𝕀
(λx,x@0. (sigma x <fst(fst(fst(x@0))), snd(x@0)>))
(λI,rho.
phi I (fst(fst(rho))) ∨ dM-to-FL(I;¬(...)) ∨ ...)\000C
(λI,rho. if (phi I (fst(fst(fst(rho))))==1)
then (snd((u I <fst(fst(fst(rho))), snd(rho)>)\000C))
I
1
(v I <fst(fst(rho)), snd(rho)>)
I
1
(snd(fst(rho)))
if (dM-to-FL(I;¬(snd(fst(rho))))==1)
then a I <fst(fst(fst(rho))), snd(rho)>
else v I <fst(fst(rho)), snd(rho)>
fi )
(λI,a. ((snd((a0 I (fst(fst(a)))))) I 1 (v I <fst(a), 0>)\000C I
1
(snd(a))))
J@0
<f@0((f(a@0);u@0)), u@1>) in
cubical-pair(λI,a@0. (a I <a@0, 1(𝕀) I a@0>);b))
Lemma: contractible_comp-apply-sq
∀[X,A,cA,B,a,b,c,d,J,x:Top].
(contractible_comp(X;A;cA) B (λx,y. a[x;y]) (λx,y. b[x;y]) (λx,y. c[x;y]) (λx,y. d[x;y]) J x
~ <cA <λI.(alpha:B(I) × 𝕀(alpha)), λI,J,f,p. <f(fst(p)), (snd(p) fst(p) f)>> (λx,x@0. a[x;m x x@0])
(λI,rho. b[I;fst(rho)] ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. if (b[I;fst((m I rho))]==1) then fst(c[I;m I rho]) else fst(d[I;fst((m I rho))]) fi )
(λI,a. (fst(d[I;fst(a)])))
J
<x, 1>
, λJ@0,f,u@0,J@1,f@0,u@1. (cA
<λI.(alpha:alpha:B(I) × ((A)λx,y. a[x;y])[λI,rho. 1](alpha) × 𝕀(alpha))
, λI,J,f,p. <<f(fst(fst(p))), (snd(fst(p)) fst(fst(p)) f)>, (snd(p) fst(p) f)>
>
(λx,x@0. a[x;<fst(fst(fst(x@0))), snd(x@0)>])
(λI,rho. b[I;fst(fst(rho))] ∨ dM-to-FL(I;¬(snd(rho))) ∨ dM-to-FL(I;snd(rho)))
(λI,rho. if (b[I;fst(fst(fst(rho)))]==1)
then (snd(c[I;<fst(fst(fst(rho))), snd(rho)>])) I 1
(cA
<λI.(alpha:alpha:B(I) × ((A)λx,y. a[x;y])[λI,rho. 1](alpha) × 𝕀(alpha))
, λI,J,f,p. <<f(fst(fst(p))), (snd(fst(p)) fst(fst(p)) f)>
, (snd(p) fst(p) f)
>
>
(λx,x@0. a[x;<fst(fst((m x x@0))), ¬(snd((m x x@0)))>])
(λI,rho. 0 ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. (snd(fst((m I rho)))))
(λI,a. (snd(fst(a))))
I
<fst(fst(rho)), ¬(snd(rho))>)
I
1
(snd(fst(rho)))
if (dM-to-FL(I;¬(snd(fst(rho))))==1)
then cA <λI.(alpha:B(I) × 𝕀(alpha)), λI,J,f,p. <f(fst(p)), (snd(p) fst(p) f)>>
(λx,x@0. a[x;m x x@0])
(λI,rho. b[I;fst(rho)] ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. if (b[I;fst((m I rho))]==1)
then fst(c[I;m I rho])
else fst(d[I;fst((m I rho))])
fi )
(λI,a. (fst(d[I;fst(a)])))
I
<fst(fst(fst(rho))), snd(rho)>
else cA
<λI.(alpha:alpha:B(I) × ((A)λx,y. a[x;y])[λI,rho. 1](alpha) × 𝕀(alpha))
, λI,J,f,p. <<f(fst(fst(p))), (snd(fst(p)) fst(fst(p)) f)>
, (snd(p) fst(p) f)
>
>
(λx,x@0. a[x;<fst(fst((m x x@0))), ¬(snd((m x x@0)))>])
(λI,rho. 0 ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. (snd(fst((m I rho)))))
(λI,a. (snd(fst(a))))
I
<fst(fst(rho)), ¬(snd(rho))>
fi )
(λI,a@0. ((snd(d[I;fst(fst(a@0))])) I 1
(cA
<λI.(alpha:alpha:B(I) × ((A)λx,y. a[x;y])[λI,rho. 1](alpha) × 𝕀(alpha))
, λI,J,f,p. <<f(fst(fst(p))), (snd(fst(p)) fst(fst(p)) f)>, (snd(p) fst(p) f)>
>
(λx,x@0. a[x;<fst(fst((m x x@0))), ¬(snd((m x x@0)))>])
(λI,rho. 0 ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. (snd(fst((m I rho)))))
(λI,a. (snd(fst(a))))
I
<fst(a@0), ¬(0)>)
I
1
(snd(a@0))))
J@1
<<f@0(f(x)), (u@0 f(x) f@0)>, u@1>)
>)
Lemma: csm-contractible_comp
∀[X,H,cA,A,tau:Top]. ((contractible_comp(X;A;cA))tau ~ contractible_comp(H;(A)tau;(cA)tau))
Definition: contractible-comp
contractible-comp(X;A;cA) == cfun-to-cop(X;Contractible(A);contractible_comp(X;A;cop-to-cfun(cA)))
Lemma: contractible-comp_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[cA:X ⊢ CompOp(A)]. (contractible-comp(X;A;cA) ∈ X ⊢ CompOp(Contractible(A)))
Lemma: contractible-comp-exists
∀X:j⊢. ∀A:{X ⊢ _}. (X ⊢ CompOp(A) ⇒ X ⊢ CompOp(Contractible(A)))
Definition: compatible-composition
compatible-composition{j:l, i:l}(Gamma; phi; psi; A; B; cA; cB) ==
∀H:j⊢. ∀sigma:H.𝕀 j⟶ Gamma, (phi ∧ psi). ∀chi:{H ⊢ _:𝔽}. ∀u:{H, chi.𝕀 ⊢ _:(A)sigma}.
∀a0:{H ⊢ _:((A)sigma)[0(𝕀)][chi |⟶ (u)[0(𝕀)]]}.
((cB H sigma chi u a0) = (cA H sigma chi u a0) ∈ {H ⊢ _:((A)sigma)[1(𝕀)]})
Lemma: compatible-composition_wf
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[cA:Gamma, phi ⊢ Compositon(A)].
∀[cB:Gamma, psi ⊢ Compositon(B)].
compatible-composition{j:l, i:l}(Gamma; phi; psi; A; B; cA; cB) ∈ ℙ{[i | j'']} supposing Gamma, (phi ∧ psi) ⊢ A = B
Lemma: compatible-composition-disjoint
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[cA:Gamma, phi ⊢ Compositon(A)].
∀[cB:Gamma, psi ⊢ Compositon(B)].
compatible-composition{j:l, i:l}(Gamma; phi; psi; A; B; cA; cB) supposing Gamma ⊢ ((phi ∧ psi) ⇒ 0(𝔽))
Definition: case-type-comp
case-type-comp(G; phi; psi; A; B; cA; cB) ==
λH,sigma,chi,u,a0. (cA H, (∀ (phi)sigma) sigma chi u a0 ∨ cB H, (∀ (psi)sigma) sigma chi u a0)
Lemma: case-type-comp_wf1
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[cA:Gamma, phi ⊢ Compositon(A)].
∀[cB:Gamma, psi ⊢ Compositon(B)].
(case-type-comp(Gamma; phi; psi; A; B; cA; cB)
∈ composition-function{j:l,i:l}(Gamma, (phi ∨ psi);(if phi then A else B))) supposing
(compatible-composition{j:l, i:l}(Gamma; phi; psi; A; B; cA; cB) and
Gamma, (phi ∧ psi) ⊢ A = B)
Lemma: case-type-comp_wf
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[cA:Gamma, phi ⊢ Compositon(A)].
∀[cB:Gamma, psi ⊢ Compositon(B)].
(case-type-comp(Gamma; phi; psi; A; B; cA; cB) ∈ Gamma, (phi ∨ psi) ⊢ Compositon((if phi then A else B))) supposing
(compatible-composition{j:l, i:l}(Gamma; phi; psi; A; B; cA; cB) and
Gamma, (phi ∧ psi) ⊢ A = B)
Lemma: case-type-comp-true-false
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}].
(∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)]. ∀[B:{Gamma, psi ⊢ _}]. ∀[cB:Gamma, psi ⊢ Compositon(B)].
(case-type-comp(Gamma; phi; psi; A; B; cA; cB) = cA ∈ Gamma ⊢ Compositon(A))) supposing
(Gamma ⊢ (1(𝔽) ⇒ phi) and
Gamma ⊢ (psi ⇒ 0(𝔽)))
Lemma: case-type-comp-false-true
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}].
(∀[B:{Gamma ⊢ _}]. ∀[cB:Gamma ⊢ Compositon(B)]. ∀[A:{Gamma, phi ⊢ _}]. ∀[cA:Gamma, phi ⊢ Compositon(A)].
(case-type-comp(Gamma; phi; psi; A; B; cA; cB) = cB ∈ Gamma ⊢ Compositon(B))) supposing
(Gamma ⊢ (1(𝔽) ⇒ psi) and
(phi = 0(𝔽) ∈ {Gamma ⊢ _:𝔽}))
Lemma: csm-case-type-comp
∀[H,Gamma,tau,phi,psi,A,B,cA,cB:Top].
((case-type-comp(Gamma; phi; psi; A; B; cA; cB))tau ~ case-type-comp(H;
(phi)tau;
(psi)tau;
(A)tau;
(B)tau;
(cA)tau;
(cB)tau))
Lemma: case-type-comp-disjoint
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[cA:Gamma, phi ⊢ Compositon(A)].
∀[cB:Gamma, psi ⊢ Compositon(B)].
case-type-comp(Gamma; phi; psi; A; B; cA; cB) ∈ Gamma, (phi ∨ psi) ⊢ Compositon((if phi then A else B))
supposing Gamma ⊢ ((phi ∧ psi) ⇒ 0(𝔽))
Lemma: case-type-comp-partition
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[cA:Gamma, phi ⊢ Compositon(A)].
∀[cB:Gamma, psi ⊢ Compositon(B)].
case-type-comp(Gamma; phi; psi; A; B; cA; cB) ∈ Gamma ⊢ Compositon((if phi then A else B))
supposing Gamma ⊢ ((phi ∧ psi) ⇒ 0(𝔽)) ∧ Gamma ⊢ (1(𝔽) ⇒ (phi ∨ psi))
Definition: fibrant-type
FibrantType(X) == A:{X ⊢ _} × X ⊢ CompOp(A)
Lemma: fibrant-type_wf
∀[X:j⊢]. (FibrantType(X) ∈ 𝕌{[i' | j']})
Lemma: fibrant-type_wf_formal-cube
∀[I:fset(ℕ)]. (FibrantType(formal-cube(I)) ∈ 𝕌')
Lemma: fibrant-type-cumulativity
∀[X:j⊢]. (FibrantType(X) ⊆r fibrant-type{i':l}(X))
Definition: csm-fibrant-type
csm-fibrant-type(G;H;s;FT) == let A,cA = FT in <(A)s, (cA)s>
Lemma: csm-fibrant-type_wf
∀[G,H:j⊢]. ∀[s:H j⟶ G]. ∀[FT:FibrantType(G)]. (csm-fibrant-type(G;H;s;FT) ∈ FibrantType(H))
Lemma: csm-fibrant-type-id
∀[G:j⊢]. ∀[FT:FibrantType(G)]. ∀[tau:G j⟶ G].
csm-fibrant-type(G;G;tau;FT) = FT ∈ FibrantType(G) supposing tau = 1(G) ∈ G j⟶ G
Lemma: csm-fibrant-comp
∀[X,Y,Z:j⊢]. ∀[g:Z j⟶ Y]. ∀[f:Y j⟶ X]. ∀[FT:FibrantType(X)].
(csm-fibrant-type(X;Z;f o g;FT) = csm-fibrant-type(Y;Z;g;csm-fibrant-type(X;Y;f;FT)) ∈ FibrantType(Z))
Definition: discrete-comp
discrete-comp(G;T) == λI,i,rho,phi,u,a0. a0
Lemma: discrete-comp_wf
∀[G:j⊢]. ∀[T:Type]. (discrete-comp(G;T) ∈ G ⊢ CompOp(discr(T)))
Definition: discrete_comp
discrete_comp(G;T) == λH,sigma,phi,u,a0. a0
Lemma: discrete_comp_wf
∀[G:j⊢]. ∀[T:Type]. (discrete_comp(G;T) ∈ G ⊢ Compositon(discr(T)))
Definition: extension-fun
extension-fun{i:l}(Gamma;A) ==
H:CubicalSet ⟶ sigma:H ⟶ Gamma ⟶ phi:{H ⊢ _:𝔽} ⟶ u:{H, phi ⊢ _:(A)sigma} ⟶ {H ⊢ _:(A)sigma[phi |⟶ u]}
Lemma: extension-fun_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (extension-fun{j:l}(Gamma; A) ∈ 𝕌{[i | j'']})
Definition: uniform-extension-fun
uniform-extension-fun{i:l}(Gamma;A;ext) ==
∀H,K:⊢. ∀tau:K ⟶ H. ∀sigma:H ⟶ Gamma. ∀phi:{H ⊢ _:𝔽}. ∀u:{H, phi ⊢ _:(A)sigma}.
((ext H sigma phi u)tau = (ext K sigma o tau (phi)tau (u)tau) ∈ {K ⊢ _:((A)sigma)tau[(phi)tau |⟶ (u)tau]})
Lemma: uniform-extension-fun_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[ext:extension-fun{j:l}(Gamma; A)].
(uniform-extension-fun{j:l}(Gamma; A; ext) ∈ ℙ{[i | j'']})
Definition: uniform-extend
uniform-extend{i:l}(Gamma; A) == {ext:extension-fun{i:l}(Gamma;A)| uniform-extension-fun{i:l}(Gamma;A;ext)}
Lemma: uniform-extend_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (Gamma ⊢ Extension(A) ∈ 𝕌{[i | j'']})
Definition: extend-to-contraction
extend-to-contraction(Gamma;A;ext) ==
contr-witness(Gamma;ext Gamma 1(Gamma) 0(𝔽) discr(⋅);<>(ext Gamma.A.𝕀 p o p (q=1) (q)p))
Lemma: extend-to-contraction_wf
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. ∀ext:Gamma +⊢ Extension(A).
(extend-to-contraction(Gamma;A;ext) ∈ {Gamma ⊢ _:Contractible(A)})
Definition: contraction-to-extend
contraction-to-extend(Gamma;A;cA;ctr) ==
λH,sigma,phi,u. comp ((cA)sigma)p [phi ⊢→ path-eta(contr-path((ctr)sigma;u))] contr-center((ctr)sigma)
Lemma: contraction-to-extend_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)]. ∀[ctr:{Gamma ⊢ _:Contractible(A)}].
(contraction-to-extend(Gamma;A;cA;ctr) ∈ Gamma ⊢ Extension(A))
Lemma: uniform-extend-contractible
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. ∀ext:Gamma +⊢ Extension(A). {Gamma ⊢ _:Contractible(A)}
Lemma: contractible-uniform-extend
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. ∀cA:Gamma ⊢ Compositon(A). ∀ctr:{Gamma ⊢ _:Contractible(A)}. Gamma ⊢ Extension(A)
Definition: extend-face-term
This construction extends a partial term u ∈ {I,phi ⊢ _:𝔽} to a
total formula in 𝔽(I).
For each face, fc = (irr_face(I;as;bs)),
in the components of phi, there is a "free-est" morphism,
g=(irr-face-morph(I;as;bs)), that makes (fc)g = 1.
This makes (fc ∧ u(g)) a well-defined formula in 𝔽(I).
The join of all of these formulas is the extension.
It satisfies these additional properties:
extend-face-term-property
extend-face-term-le
extend-face-term-unique
extend-face-term-morph
We use it to define the composition operation for ⌜𝔽⌝
face_comp.
face_comp_wf⋅
extend-face-term(I;phi;u) ==
\/(λpr.let as,bs = pr
in irr_face(I;as;bs) ∧ u(irr-face-morph(I;as;bs))"(face_lattice_components(I;phi)))
Lemma: extend-face-term_wf
∀[I:fset(ℕ)]. ∀[phi:𝔽(I)]. ∀[u:{I,phi ⊢ _:𝔽}]. (extend-face-term(I;phi;u) ∈ 𝔽(I))
Lemma: extend-face-term-property
∀[I:fset(ℕ)]. ∀[phi:𝔽(I)]. ∀[u:{I,phi ⊢ _:𝔽}]. ∀[J:fset(ℕ)]. ∀[g:I,phi(J)].
((extend-face-term(I;phi;u))<g> = u(g) ∈ Point(face_lattice(J)))
Lemma: extend-face-term-le
∀[I:fset(ℕ)]. ∀[phi:𝔽(I)]. ∀[u:{I,phi ⊢ _:𝔽}]. extend-face-term(I;phi;u) ≤ phi
Lemma: extend-face-term-uniqueness
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))]. ∀[u:{I,phi ⊢ _:𝔽}]. ∀[a,b:Point(face_lattice(I))].
a = b ∈ Point(face_lattice(I))
supposing a ≤ phi
∧ b ≤ phi
∧ (∀[g:{f:I ⟶ I| (phi f) = 1} ]. ((a)<g> = u(g) ∈ Point(face_lattice(I))))
∧ (∀[g:{f:I ⟶ I| (phi f) = 1} ]. ((b)<g> = u(g) ∈ Point(face_lattice(I))))
Lemma: extend-face-term-unique
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))]. ∀[u:{I,phi ⊢ _:𝔽}]. ∀[a:Point(face_lattice(I))].
a = extend-face-term(I;phi;u) ∈ Point(face_lattice(I))
supposing a ≤ phi ∧ (∀[g:{f:I ⟶ I| (phi f) = 1} ]. ((a)<g> = u(g) ∈ Point(face_lattice(I))))
Lemma: extend-face-term-morph
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))]. ∀[u:{I,phi ⊢ _:𝔽}]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I].
((extend-face-term(I;phi;u))<f> = extend-face-term(J;(phi)<f>;(u)subset-trans(I;J;f;phi)) ∈ Point(face_lattice(J)))
Definition: face_comp
face_comp() == λI,i,rho,phi,u,a0. (i1)(extend-face-term(I+i;s(phi);u))
Lemma: face_comp_wf
∀[G:j⊢]. (face_comp() ∈ G ⊢ CompOp(𝔽))
Definition: face-comp
face-comp() == cop-to-cfun(face_comp())
Lemma: face-comp_wf
∀[G:j⊢]. (face-comp() ∈ G ⊢ Compositon(𝔽))
Definition: bij-contr
bij-contr(X; A; f; g; cA; b; c) ==
contr-witness(X;app(g; contr-center(c));map-path(X.A;(g)p;contr-path((c)p;app((f)p; q))) + b)
Lemma: bij-contr_wf
∀X:j⊢. ∀A,B:{X ⊢ _}. ∀f:{X ⊢ _:(A ⟶ B)}. ∀g:{X ⊢ _:(B ⟶ A)}. ∀cA:X +⊢ Compositon(A).
∀b:{X.A ⊢ _:(Path_(A)p app((g)p; app((f)p; q)) q)}. ∀c:{X ⊢ _:Contractible(B)}.
(bij-contr(X; A; f; g; cA; b; c) ∈ {X ⊢ _:Contractible(A)})
Lemma: bijection-preserves-contractible
∀X:j⊢. ∀A,B:{X ⊢ _}. ∀f:{X ⊢ _:(A ⟶ B)}. ∀g:{X ⊢ _:(B ⟶ A)}. ∀cA:X +⊢ Compositon(A).
∀b:{X.A ⊢ _:(Path_(A)p app((g)p; app((f)p; q)) q)}. ∀c:{X ⊢ _:Contractible(B)}.
{X ⊢ _:Contractible(A)}
Comment: pres doc
================================= pres ======================================
The cubical term pres f [phi ⊢→ t] t0 is a witness to the fact that
composition is `preserved by` function application.⋅
Definition: pres-a0
pres-a0(G;f;t0) == app((f)[0(𝕀)]; t0)
Lemma: pres-a0_wf
∀[G:j⊢]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t0:{G ⊢ _:(T)[0(𝕀)]}]. (pres-a0(G;f;t0) ∈ {G ⊢ _:(A)[0(𝕀)]})
Definition: pres-c1
pres-c1(G;phi;f;t;t0;cA) == comp cA [phi ⊢→ app(f; t)] app((f)[0(𝕀)]; t0)
Lemma: pres-c1_wf
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cA:composition-function{j:l,i:l}(G.𝕀;A)].
(pres-c1(G;phi;f;t;t0;cA) ∈ {G ⊢ _:(A)[1(𝕀)][phi |⟶ app(f; t)[1]]})
Lemma: csm-pres-c1
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cA:G.𝕀 ⊢ Compositon(A)]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((pres-c1(G;phi;f;t;t0;cA))s
= pres-c1(H;(phi)s;(f)s+;(t)s+;(t0)s;(cA)s+)
∈ {H ⊢ _:((A)s+)[1(𝕀)][(phi)s |⟶ app((f)s+; (t)s+)[1]]})
Definition: pres-c2
pres-c2(G;phi;f;t;t0;cT) == app((f)[1(𝕀)]; comp cT [phi ⊢→ t] t0)
Lemma: pres-c2_wf
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:composition-function{j:l,i:l}(G.𝕀;T)].
(pres-c2(G;phi;f;t;t0;cT) ∈ {G ⊢ _:(A)[1(𝕀)][phi |⟶ app(f; t)[1]]})
Lemma: csm-pres-c2
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 ⊢ Compositon(T)]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((pres-c2(G;phi;f;t;t0;cT))s
= pres-c2(H;(phi)s;(f)s+;(t)s+;(t0)s;(cT)s+)
∈ {H ⊢ _:((A)s+)[1(𝕀)][(phi)s |⟶ app((f)s+; (t)s+)[1]]})
Definition: pres-v
pres-v(G;phi;t;t0;cT) == fill cT [phi ⊢→ t] t0
Lemma: pres-v_wf
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[T:{G.𝕀 ⊢ _}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}]. ∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}].
∀[cT:composition-function{j:l,i:l}(G.𝕀;T)].
(pres-v(G;phi;t;t0;cT) ∈ {G.𝕀 ⊢ _:T[(phi)p |⟶ t]})
Lemma: csm-pres-v
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[T:{G.𝕀 ⊢ _}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}]. ∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}].
∀[cT:G.𝕀 ⊢ Compositon(T)]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((pres-v(G;phi;t;t0;cT))s+ = pres-v(H;(phi)s;(t)s+;(t0)s;(cT)s+) ∈ {H.𝕀 ⊢ _:(T)s+[((phi)s)p |⟶ (t)s+]})
Definition: presw
presw(G;phi;f;t;t0;cT) == app(f; pres-v(G;phi;t;t0;cT))
Lemma: presw_wf
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:composition-function{j:l,i:l}(G.𝕀;T)].
(presw(G;phi;f;t;t0;cT) ∈ {G.𝕀 ⊢ _:A})
Lemma: csm-presw
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 ⊢ Compositon(T)]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((presw(G;phi;f;t;t0;cT))s+ = presw(H;(phi)s;(f)s+;(t)s+;(t0)s;(cT)s+) ∈ {H.𝕀 ⊢ _:(A)s+})
Lemma: pres-a0-constraint
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 +⊢ Compositon(T)].
((pres-a0(G;f;t0))p ∈ {G.𝕀 ⊢ _:((A)p+)[0(𝕀)][((phi)p ∨ (q=1)) |⟶ ((presw(G;phi;f;t;t0;cT))p+)[0(𝕀)]]})
Lemma: presw-pres-c2
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 ⊢ Compositon(T)].
((((presw(G;phi;f;t;t0;cT))p+)[1(𝕀)])[1(𝕀)] = pres-c2(G;phi;f;t;t0;cT) ∈ {G ⊢ _:(A)[1(𝕀)]})
Lemma: presw-pres-c1
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 +⊢ Compositon(T)]. ∀[cA:G.𝕀 ⊢ Compositon(A)].
G ⊢ (comp (cA)p+ [((phi)p ∨ (q=1)) ⊢→ (presw(G;phi;f;t;t0;cT))p+] (pres-a0(G;f;t0))p)[0(𝕀)]=pres-c1(G;phi;f;t;t0;cA):
(A)[1(𝕀)]
Definition: pres
pres f [phi ⊢→ t] t0 == <>(comp (cA)p+ [((phi)p ∨ (q=1)) ⊢→ (presw(G;phi;f;t;t0;cT))p+] (pres-a0(G;f;t0))p)
Lemma: pres_wf
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 +⊢ Compositon(T)]. ∀[cA:G.𝕀 +⊢ Compositon(A)].
(pres f [phi ⊢→ t] t0 ∈ {G ⊢ _:(Path_(A)[1(𝕀)] pres-c1(G;phi;f;t;t0;cA) pres-c2(G;phi;f;t;t0;cT))})
Lemma: csm-pres
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 +⊢ Compositon(T)]. ∀[cA:G.𝕀 +⊢ Compositon(A)]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((pres f [phi ⊢→ t] t0)s
= pres (f)s+ [(phi)s ⊢→ (t)s+] (t0)s
∈ {H ⊢ _:(Path_((A)s+)[1(𝕀)] pres-c1(H;(phi)s;(f)s+;(t)s+;(t0)s;(cA)s+) pres-c2(H;(phi)s;(f)s+;(t)s+;(t0)s;(cT)s+))})
Lemma: pres-invariant
∀[G,H:j⊢].
∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 +⊢ Compositon(T)]. ∀[cA:G.𝕀 +⊢ Compositon(A)].
(pres f [phi ⊢→ t] t0
= pres f [phi ⊢→ t] t0
∈ {G ⊢ _:(Path_(A)[1(𝕀)] pres-c1(G;phi;f;t;t0;cA) pres-c2(G;phi;f;t;t0;cT))})
supposing H = G ∈ CubicalSet{j}
Lemma: pres-constraint
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 +⊢ Compositon(T)]. ∀[cA:G.𝕀 +⊢ Compositon(A)].
(pres f [phi ⊢→ t] t0
= G ⊢ <>((app(f; t)[1])p)
∈ {G, phi ⊢ _:(Path_(A)[1(𝕀)] pres-c1(G;phi;f;t;t0;cA) pres-c2(G;phi;f;t;t0;cT))})
Lemma: pres_wf2
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 +⊢ Compositon(T)]. ∀[cA:G.𝕀 +⊢ Compositon(A)].
(pres f [phi ⊢→ t] t0 ∈ {G ⊢ _:(Path_(A)[1(𝕀)] pres-c1(G;phi;f;t;t0;cA) pres-c2(G;phi;f;t;t0;cT))[phi
|⟶ <>((app(f; t)[1])p)]})
Definition: equiv-term
equiv f [phi ⊢→ (t, c)] a ==
let e = equiv-contr(f;a) in
let u = (contr-path(e;fiber-point(t;c)))p @ q in
comp (cF)p [phi ⊢→ u] contr-center(e)
Lemma: equiv-term_wf
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[t:{G, phi ⊢ _:T}]. ∀[a:{G ⊢ _:A}].
∀[c:{G, phi ⊢ _:(Path_A a app(equiv-fun(f); t))}]. ∀[cF:G ⊢ Compositon(Fiber(equiv-fun(f);a))].
(equiv f [phi ⊢→ (t, c)] a ∈ {G ⊢ _:Fiber(equiv-fun(f);a)[phi |⟶ fiber-point(t;c)]})
Lemma: equiv-term-subset
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[t:{G, phi ⊢ _:T}]. ∀[a:{G ⊢ _:A}].
∀[c:{G, phi ⊢ _:(Path_A a app(equiv-fun(f); t))}]. ∀[cF:G ⊢ Compositon(Fiber(equiv-fun(f);a))]. ∀[psi:{G ⊢ _:𝔽}].
(equiv f [phi ⊢→ (t, c)] a = equiv f [phi ⊢→ (t, c)] a ∈ {G, psi ⊢ _:Fiber(equiv-fun(f);a)})
Lemma: equiv-term-0
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}].
∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[a:{G ⊢ _:A}]. ∀[t,c:Top]. ∀[cF:G +⊢ Compositon(Fiber(equiv-fun(f);a))].
(equiv f [phi ⊢→ (t, c)] a = transprt(G;(cF)p;contr-center(equiv-contr(f;a))) ∈ {G ⊢ _:Fiber(equiv-fun(f);a)})
supposing phi = 0(𝔽) ∈ {G ⊢ _:𝔽}
Lemma: equiv-term-0-subset-1
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}].
∀[psi:{G ⊢ _:𝔽}]
∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[a:{G ⊢ _:A}]. ∀[t,c:Top]. ∀[cF:G +⊢ Compositon(Fiber(equiv-fun(f);a))].
(equiv f [phi ⊢→ (t, c)] a = transprt(G;(cF)p;contr-center(equiv-contr(f;a))) ∈ {G ⊢ _:Fiber(equiv-fun(f);a)})
supposing psi = 1(𝔽) ∈ {G ⊢ _:𝔽}
supposing phi = 0(𝔽) ∈ {G ⊢ _:𝔽}
Definition: equiv_term
equiv f [phi ⊢→ (t,c)] a == equiv f [phi ⊢→ (t, c)] a
Lemma: equiv_term_wf
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[t:{G, phi ⊢ _:T}]. ∀[a:{G ⊢ _:A}].
∀[c:{G, phi ⊢ _:(Path_A a app(equiv-fun(f); t))}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cT:G +⊢ Compositon(T)].
(equiv f [phi ⊢→ (t,c)] a ∈ {G ⊢ _:Fiber(equiv-fun(f);a)[phi |⟶ fiber-point(t;c)]})
Lemma: equiv_term-0
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}].
∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[a:{G ⊢ _:A}]. ∀[t,c:Top]. ∀[cA:G +⊢ Compositon(A)].
∀[cT:G +⊢ Compositon(T)].
(equiv f [phi ⊢→ (t,c)] a
= transprt(G;(fiber-comp(G;T;A;equiv-fun(f);a;cT;cA))p;contr-center(equiv-contr(f;a)))
∈ {G ⊢ _:Fiber(equiv-fun(f);a)})
supposing phi = 0(𝔽) ∈ {G ⊢ _:𝔽}
Definition: csm-path-ap-q
csm-path-ap-q(H;G;s;t) == ((t)p)s+ @ q
Lemma: csm-path-ap-q_wf
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[H:j⊢]. ∀[s:H j⟶ G]. ∀[B:{G ⊢ _}]. ∀[a,b:{G, phi ⊢ _:B}]. ∀[t:{G, phi ⊢ _:(Path_B a b)}].
(csm-path-ap-q(H;G;s;t) ∈ {H.𝕀, ((phi)p)s+ ⊢ _:((B)p)s+})
Lemma: csm-equiv-term
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[t:{G, phi ⊢ _:T}]. ∀[a:{G ⊢ _:A}].
∀[c:{G, phi ⊢ _:(Path_A a app(equiv-fun(f); t))}]. ∀[cF:G ⊢ Compositon(Fiber(equiv-fun(f);a))]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((equiv f [phi ⊢→ (t, c)] a)s
= equiv (f)s [(phi)s ⊢→ ((t)s, (c)s)] (a)s
∈ {H ⊢ _:Fiber(equiv-fun((f)s);(a)s)[(phi)s |⟶ fiber-point((t)s;(c)s)]})
Lemma: csm-equiv_term
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[t:{G, phi ⊢ _:T}]. ∀[a:{G ⊢ _:A}].
∀[c:{G, phi ⊢ _:(Path_A a app(equiv-fun(f); t))}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cT:G +⊢ Compositon(T)]. ∀[H:j⊢].
∀[s:H j⟶ G].
((equiv f [phi ⊢→ (t,c)] a)s
= equiv (f)s [(phi)s ⊢→ ((t)s,(c)s)] (a)s
∈ {H ⊢ _:Fiber(equiv-fun((f)s);(a)s)[(phi)s |⟶ fiber-point((t)s;(c)s)]})
Comment: glueing doc
================================= glueing ===================================
The hardest theorems of the theory are in this section where we define the
cubical type Glue [phi ⊢→ (T;w)] A and show that it has a composition structure.
⋅
Definition: glue-equations
glue-equations(Gamma;A;phi;T;w;I;rho;t;a) ==
(∀J:fset(ℕ). ∀f:{f:J ⟶ I| phi(f(rho)) = 1 ∈ Point(face_lattice(J))} . ∀K:fset(ℕ). ∀g:K ⟶ J.
((t J f f(rho) g) = (t K f ⋅ g) ∈ T(f ⋅ g(rho))))
∧ (∀J:fset(ℕ). ∀f:{f:J ⟶ I| phi(f(rho)) = 1 ∈ Point(face_lattice(J))} .
((w(f(rho)) J 1 (t J f)) = (a rho f) ∈ A(f(rho))))
Lemma: glue-equations_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[t:J:fset(ℕ) ⟶ f:{f:J ⟶ I| phi(f(rho)) = 1 ∈ Point(face_lattice(J))} ⟶ T(f(rho))].
∀[a:A(rho)].
(glue-equations(Gamma;A;phi;T;w;I;rho;t;a) ∈ ℙ)
Definition: glue-cube
glue-cube(Gamma;A;phi;T;w;I;rho) ==
if (phi(rho)==1)
then T(rho)
else {ta:J:fset(ℕ) ⟶ f:{f:J ⟶ I| phi(f(rho)) = 1 ∈ Point(face_lattice(J))} ⟶ T(f(rho)) × A(rho)|
let t,a = ta
in glue-equations(Gamma;A;phi;T;w;I;rho;t;a)}
fi
Lemma: glue-cube_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[I:fset(ℕ)]. ∀[rho:Gamma(I)].
(glue-cube(Gamma;A;phi;T;w;I;rho) ∈ Type)
Lemma: equal-glue-cube
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}].
∀phi:{Gamma ⊢ _:𝔽}
∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀I:fset(ℕ). ∀rho:Gamma(I). ∀u,v:glue-cube(Gamma;A;phi;T;w;I;rho).
u = v ∈ glue-cube(Gamma;A;phi;T;w;I;rho)
supposing if (phi(rho)==1)
then u = v ∈ T(rho)
else u = v ∈ (J:fset(ℕ) ⟶ f:{f:J ⟶ I| phi(f(rho)) = 1 ∈ Point(face_lattice(J))} ⟶ T(f(rho)) × A(rho))
fi
Definition: glue-morph
glue-morph(Gamma;A;phi;T;w;I;rho;J;f;u) ==
if (phi(rho)==1)
then (u rho f)
else let t,a = u
in if (phi(f(rho))==1) then t J f else <λK,g. (t K f ⋅ g), (a rho f)> fi
fi
Lemma: glue-morph_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[u:glue-cube(Gamma;A;phi;T;w;I;rho)].
(glue-morph(Gamma;A;phi;T;w;I;rho;J;f;u) ∈ glue-cube(Gamma;A;phi;T;w;J;f(rho)))
Lemma: glue-morph-comp
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. ∀[u:glue-cube(Gamma;A;phi;T;w;I;rho)].
(glue-morph(Gamma;A;phi;T;w;J;f(rho);K;g;glue-morph(Gamma;A;phi;T;w;I;rho;J;f;u))
= glue-morph(Gamma;A;phi;T;w;I;rho;K;f ⋅ g;u)
∈ glue-cube(Gamma;A;phi;T;w;K;f ⋅ g(rho)))
Lemma: glue-morph-id
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. ∀phi:{Gamma ⊢ _:𝔽}. ∀T:{Gamma, phi ⊢ _}. ∀w:{Gamma, phi ⊢ _:(T ⟶ A)}. ∀I:fset(ℕ).
∀a:Gamma(I). ∀u:glue-cube(Gamma;A;phi;T;w;I;a).
(glue-morph(Gamma;A;phi;T;w;I;a;I;1;u) = u ∈ glue-cube(Gamma;A;phi;T;w;I;a))
Definition: glue-type
Glue [phi ⊢→ (T;w)] A == <λI,rho. glue-cube(Gamma;A;phi;T;w;I;rho), λI,J,f,rho,u. glue-morph(Gamma;A;phi;T;w;I;rho;J;f;\000Cu)>
Lemma: glue-type_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
Gamma ⊢ Glue [phi ⊢→ (T;w)] A
Lemma: glue-type-subset
∀[Gamma,A,phi,T,w,xx:Top]. (Gamma, xx ⊢ Glue [phi ⊢→ (T;w)] A ~ Gamma ⊢ Glue [phi ⊢→ (T;w)] A)
Lemma: csm-glue-type
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}]. ∀[Z:j⊢].
∀[s:Z j⟶ Gamma].
((Glue [phi ⊢→ (T;w)] A)s = Z ⊢ Glue [(phi)s ⊢→ ((T)s;(w)s)] (A)s ∈ {Z ⊢ _})
Lemma: glue-type-constraint
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
Gamma, phi ⊢ Glue [phi ⊢→ (T;w)] A = T
Lemma: glue-type-term-subtype
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
({Gamma ⊢ _:Glue [phi ⊢→ (T;w)] A} ⊆r {Gamma, phi ⊢ _:T})
Lemma: glue-type-term-subtype2
∀[Gamma:j⊢]. ∀[psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, psi ⊢ _}]. ∀[phi:{Gamma, psi ⊢ _:𝔽}]. ∀[T:{Gamma, psi, phi ⊢ _}].
∀[w:{Gamma, psi, phi ⊢ _:(T ⟶ A)}].
({Gamma, psi ⊢ _:Glue [phi ⊢→ (T;w)] A} ⊆r {Gamma, psi, phi ⊢ _:T})
Lemma: glue-type-1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[T:{Gamma, 1(𝔽) ⊢ _}]. ∀[w:{Gamma, 1(𝔽) ⊢ _:(T ⟶ A)}].
(Gamma ⊢ Glue [1(𝔽) ⊢→ (T;w)] A = T ∈ {Gamma ⊢ _})
Lemma: glue-type-1'
∀[Gamma:j⊢]. ∀[A,T:{Gamma ⊢ _}]. ∀[w:{Gamma ⊢ _:(T ⟶ A)}]. ∀[phi:{Gamma ⊢ _:𝔽}].
Gamma ⊢ Glue [phi ⊢→ (T;w)] A = T ∈ {Gamma ⊢ _} supposing phi = 1(𝔽) ∈ {Gamma ⊢ _:𝔽}
Definition: glue-term
glue [phi ⊢→ t] a == λI,rho. if (phi(rho)==1) then t(rho) else <λJ,f. t(f(rho)), a(rho)> fi
Lemma: glue-term_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[t:{Gamma, phi ⊢ _:T}]. ∀[a:{Gamma ⊢ _:A[phi |⟶ app(w; t)]}].
(glue [phi ⊢→ t] a ∈ {Gamma ⊢ _:Glue [phi ⊢→ (T;w)] A})
Lemma: csm-glue-term
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[t:{Gamma, phi ⊢ _:T}]. ∀[a:{Gamma ⊢ _:A[phi |⟶ app(w; t)]}]. ∀[H:j⊢]. ∀[s:H j⟶ Gamma].
((glue [phi ⊢→ t] a)s = H ⊢ glue [(phi)s ⊢→ (t)s] (a)s ∈ {H ⊢ _:(Glue [phi ⊢→ (T;w)] A)s})
Lemma: glue-term-constraint
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[t:{Gamma, phi ⊢ _:T}]. ∀[A,a:Top].
Gamma, phi ⊢ glue [phi ⊢→ t] a=t:T
Lemma: glue-term-subset
∀[Gamma:j⊢]. ∀[A,phi,T,t,a,xx:Top]. (Gamma, xx ⊢ glue [phi ⊢→ t] a ~ Gamma ⊢ glue [phi ⊢→ t] a)
Lemma: glue-term-1
∀[Gamma:j⊢]. ∀[T:{Gamma ⊢ _}]. ∀[t:{Gamma ⊢ _:T}]. ∀[A,a:Top]. (Gamma ⊢ glue [1(𝔽) ⊢→ t] a = t ∈ {Gamma ⊢ _:T})
Lemma: glue-term-1'
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma ⊢ _}]. ∀[t:{Gamma ⊢ _:T}]. ∀[A,a:Top].
Gamma ⊢ glue [phi ⊢→ t] a = t ∈ {Gamma ⊢ _:T} supposing phi = 1(𝔽) ∈ {Gamma ⊢ _:𝔽}
Lemma: glue-term_wf2
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[t:{Gamma, phi ⊢ _:T}]. ∀[a:{Gamma ⊢ _:A[phi |⟶ app(w; t)]}].
(glue [phi ⊢→ t] a ∈ {Gamma ⊢ _:Glue [phi ⊢→ (T;w)] A[phi |⟶ t]})
Definition: unglue-term
unglue(b) == λI,rho. if (phi(rho)==1) then w I rho I 1 b(rho) else snd(b(rho)) fi
Lemma: unglue-term_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[b:{Gamma ⊢ _:Glue [phi ⊢→ (T;w)] A}].
(unglue(b) ∈ {Gamma ⊢ _:A})
Lemma: unglue-term_wf2
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[b:{Gamma ⊢ _:Glue [phi ⊢→ (T;w)] A}].
(unglue(b) ∈ {Gamma ⊢ _:A[phi |⟶ app(w; b)]})
Lemma: unglue-term-1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[b:{Gamma ⊢ _:Glue [phi ⊢→ (T;w)] A}].
unglue(b) = app(w; b) ∈ {Gamma ⊢ _:A} supposing phi = 1(𝔽) ∈ {Gamma ⊢ _:𝔽}
Lemma: csm-unglue
∀[phi,w,s,b:Top]. ((unglue(b))s ~ unglue((b)s))
Lemma: unglue-glue
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[t:{Gamma, phi ⊢ _:T}]. ∀[a:{Gamma ⊢ _:A[phi |⟶ app(w; t)]}].
(unglue(glue [phi ⊢→ t] a) = a ∈ {Gamma ⊢ _:A})
Lemma: glue-unglue
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[b:{Gamma ⊢ _:Glue [phi ⊢→ (T;w)] A}].
(Gamma ⊢ glue [phi ⊢→ b] unglue(b) = b ∈ {Gamma ⊢ _:Glue [phi ⊢→ (T;w)] A})
Definition: glue-comp
comp(Glue [phi ⊢→ (T, f)] A) ==
λH,sigma,psi,b,b0. let a = unglue(b) in
let a0 = unglue(b0) in
let a'1 = comp (cA)sigma [psi ⊢→ a] a0 in
let t'1 = comp (cT)sigma [psi ⊢→ b] b0 in
let w = pres (equiv-fun(f))sigma [psi ⊢→ b] b0 in
let st = (t'1 ∨ (b)[1(𝕀)]) in
let sw = (w ∨ <>((app((equiv-fun(f))sigma; b)[1])p)) in
let cF = fiber-comp(H, ((phi)sigma)[1(𝕀)];((T)sigma)[1(𝕀)];((A)sigma)[1(𝕀)];
equiv-fun(((f)sigma)[1(𝕀)]);a'1;(cT)sigma o [1(𝕀)];(cA)sigma o [1(𝕀)]) in
let z = equiv ((f)sigma)[1(𝕀)] [((∀ (phi)sigma) ∨ psi) ⊢→ (st, sw)] a'1 in
let t1 = fiber-member(z) in
let alpha = fiber-path(z) in
let a1 = comp (cA)sigma o [1(𝕀)] o p [(((phi)sigma)[1(𝕀)] ∨ psi) ⊢→ ((alpha)p @ q ∨ (a[1])p)]
a'1 in
glue [((phi)sigma)[1(𝕀)] ⊢→ t1] a1
Lemma: glue-comp_wf
∀G:j⊢. ∀A:{G ⊢ _}. ∀cA:G +⊢ Compositon(A). ∀psi:{G ⊢ _:𝔽}. ∀T:{G, psi ⊢ _}. ∀cT:G, psi +⊢ Compositon(T).
∀f:{G, psi ⊢ _:Equiv(T;A)}.
(comp(Glue [psi ⊢→ (T, f)] A) ∈ composition-function{j:l,i:l}(G;Glue [psi ⊢→ (T;equiv-fun(f))] A))
Comment: glue-comp-wf2-step2-defn
let addr = 121121222121311213211112 in let step2 = make_sub_prf
(make_proof_address_term_aux (ioid Error :glue-comp_wf2) addr) in
(view_show o ioid_term) step2
Comment: glue-comp-wf2-step3-def
let addr = 221121222111211222 in (view_show o ioid_term)(make_sub_prf
(make_proof_address_term_aux (ioid Error :glue-comp_wf2-step2) addr))
⋅
Comment: glue-comp-wf2-step4-def
let addr = 211722222121 in (view_show o ioid_term)(make_sub_prf
(make_proof_address_term_aux (ioid Error :glue-comp_wf2-step3) addr))
⋅
Lemma: glue-comp_wf2
∀G:j⊢. ∀A:{G ⊢ _}. ∀cA:G +⊢ Compositon(A). ∀psi:{G ⊢ _:𝔽}. ∀T:{G, psi ⊢ _}. ∀cT:G, psi +⊢ Compositon(T).
∀f:{G, psi ⊢ _:Equiv(T;A)}.
(comp(Glue [psi ⊢→ (T, f)] A) ∈ G ⊢ Compositon(Glue [psi ⊢→ (T;equiv-fun(f))] A))
Lemma: glue-comp-agrees
The type (not displayed) of the equality in this lemma
is composition-structure{i:l}(G, psi; T)
This means that in "extent" psi, when ⌜G ⊢ Glue [psi ⊢→ (T;f)] A = T ∈ {G, psi ⊢ _}⌝, the
composition for ⌜Glue [psi ⊢→ (T;f)] A⌝ is the same as the composition for T.
This property of the compostion for Glue is used in construction of the
composition for c𝕌 (the cubiucal universe type).⋅
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[psi:{G ⊢ _:𝔽}]. ∀[T:{G, psi ⊢ _}]. ∀[cT:G, psi +⊢ Compositon(T)].
∀[f:{G, psi ⊢ _:Equiv(T;A)}].
(comp(Glue [psi ⊢→ (T, f)] A) = cT ∈ G, psi ⊢ Compositon(T))
Lemma: glue-comp-agrees2
The type (not displayed) of the equality in this lemma
is composition-structure{i:l}(G, psi; T)
This means that in "extent" psi, when ⌜G ⊢ Glue [psi ⊢→ (T;f)] A = T ∈ {G, psi ⊢ _}⌝, the
composition for ⌜Glue [psi ⊢→ (T;f)] A⌝ is the same as the composition for T.
This property of the compostion for Glue is used in construction of the
composition for c𝕌 (the cubiucal universe type).⋅
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[psi:{G ⊢ _:𝔽}]. ∀[T:{G, psi ⊢ _}]. ∀[cT:G, psi +⊢ Compositon(T)].
∀[f:{G, psi ⊢ _:Equiv(T;A)}].
comp(Glue [psi ⊢→ (T, f)] A) = cT ∈ G ⊢ Compositon(T) supposing G ⊢ (1(𝔽) ⇒ psi)
Lemma: csm-glue-comp
∀[G:j⊢]. ∀[K:Top]. ∀[A:{G ⊢ _}]. ∀[psi:{G ⊢ _:𝔽}]. ∀[T:{G, psi ⊢ _}]. ∀[cA,cT,f,tau:Top].
((comp(Glue [psi ⊢→ (T, f)] A) )tau ~ comp(Glue [(psi)tau ⊢→ ((T)tau, (f)tau)] (A)tau) )
Lemma: csm-glue-comp-agrees
The type (not displayed) of the equality in this lemma
is composition-structure{i:l}(G, psi; T)
This means that in "extent" psi, when ⌜G ⊢ Glue [psi ⊢→ (T;f)] A = T ∈ {G, psi ⊢ _}⌝, the
composition for ⌜Glue [psi ⊢→ (T;f)] A⌝ is the same as the composition for T.
This property of the compostion for Glue is used in construction of the
composition for c𝕌 (the cubiucal universe type).⋅
∀[H,G:j⊢]. ∀[tau:H j⟶ G]. ∀[A:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[psi:{G ⊢ _:𝔽}]. ∀[T:{G, psi ⊢ _}].
∀[cT:G, psi +⊢ Compositon(T)]. ∀[f:{G, psi ⊢ _:Equiv(T;A)}].
(comp(Glue [psi ⊢→ (T, f)] A) )tau = (cT)tau ∈ H ⊢ Compositon((T)tau) supposing H ⊢ (1(𝔽) ⇒ (psi)tau)
Definition: gluetype
gluetype(Gamma;A;phi;T;f) == Glue [phi ⊢→ (T;equiv-fun(f))] A
Lemma: gluetype_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[f:{Gamma, phi ⊢ _:Equiv(T;A)}].
Gamma ⊢ gluetype(Gamma;A;phi;T;f)
Lemma: glue-comp_wf3
∀G:j⊢. ∀A:{G ⊢ _}. ∀cA:G +⊢ Compositon(A). ∀psi:{G ⊢ _:𝔽}. ∀T:{G, psi ⊢ _}. ∀cT:G, psi +⊢ Compositon(T).
∀f:{G, psi ⊢ _:Equiv(T;A)}.
(comp(Glue [psi ⊢→ (T, f)] A) ∈ G ⊢ Compositon(gluetype(G;A;psi;T;f)))
Comment: cubical universe doc
============================= cubical universe ================================
We can now define the cubical universe c𝕌 of fibrant types.
To get cumulativity cubical-universe-cumulativity we must use
the definition of fibrant that quantifies only over representable cubical sets.
compU_wf shows that the Universe has a composition structure.⋅
Definition: closed-cubical-universe
cc𝕌 == <λI.FibrantType(formal-cube(I)), λI,J,f,FT. csm-fibrant-type(formal-cube(I);formal-cube(J);<f>;FT)>
Lemma: closed-cubical-universe_wf
cc𝕌 ∈ { * ⊢' _}
Lemma: closed-cubical-universe-cumulativity
closed-cubical-term(cc𝕌) ⊆r closed-cubical-term(cc𝕌')
Definition: cubical-universe
c𝕌 == closed-type-to-type(cc𝕌)
Lemma: cubical-universe_wf
∀[X:j⊢]. X ⊢' c𝕌
Lemma: cubical-universe-ap-morph
∀[I,b,J,f,a:Top]. ((b a f) ~ csm-fibrant-type(formal-cube(I);formal-cube(J);<f>;b))
Lemma: cubical-term_wf-universe
∀X:j⊢. ({X ⊢ _:c𝕌} ∈ 𝕌{[i' | j']})
Lemma: istype-cubical-universe-term
∀X:j⊢. istype({X ⊢ _:c𝕌})
Lemma: cubical-universe-at
∀[I,a:Top]. (c𝕌(a) ~ A:{formal-cube(I) ⊢ _} × formal-cube(I) ⊢ CompOp(A))
Lemma: cube-context-adjoin_wf_universe
∀[Gamma:⊢]. Gamma.c𝕌 ⊢
Lemma: cubical-universe-at-equal
∀[X:j⊢]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[x,y:c𝕌(a)].
x = y ∈ c𝕌(a)
supposing ((fst(x)) = (fst(y)) ∈ {formal-cube(I) ⊢ _}) ∧ ((snd(x)) = (snd(y)) ∈ formal-cube(I) ⊢ CompOp(fst(x)))
Definition: universe-type
universe-type(t;I;a) == fst(t(a))
Lemma: universe-type_wf
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. formal-cube(I) ⊢ universe-type(t;I;a)
Lemma: csm-cubical-universe
∀[s:Top]. ((c𝕌)s ~ c𝕌)
Lemma: csm-ap-term-universe
∀[X,H:j⊢]. ∀[s:H j⟶ X]. ∀[t:{X ⊢ _:c𝕌}]. ((t)s ∈ {H ⊢ _:c𝕌})
Lemma: cubical-universe-p
∀[G:j⊢]. ∀[A:{G ⊢ _:c𝕌}]. ∀[T:{G ⊢ _}]. ((A)p ∈ {G.T ⊢ _:c𝕌})
Lemma: cubical-universe-at-cumulativity
∀[I:fset(ℕ)]. ∀[a:Top]. (c𝕌(a) ⊆r c𝕌'(a))
Lemma: cubical-universe-cumulativity2
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. (t ∈ {X ⊢ _:c𝕌'})
Lemma: cubical-universe-cumulativity
∀[X:j⊢]. ({X ⊢ _:c𝕌} ⊆r {X ⊢ _:c𝕌'})
Definition: universe-encode
encode(T;cT) == λI,a. <(T)<a>, (cT)<a>>
Lemma: universe-encode_wf
∀[G:j⊢]. ∀[T:{G ⊢ _}]. ∀[cT:G ⊢ CompOp(T)]. (encode(T;cT) ∈ {G ⊢ _:c𝕌})
Lemma: csm-universe-encode
∀[G:j⊢]. ∀[T:{G ⊢ _}]. ∀[cT:G ⊢ CompOp(T)]. ∀[H:j⊢]. ∀[s:H j⟶ G]. ((encode(T;cT))s = encode((T)s;(cT)s) ∈ {H ⊢ _:c𝕌})
Definition: universe-decode
decode(t) == <λI,a. fst(t(a))(1), λI,J,f,a,x. (x 1 f)>
Lemma: universe-decode_wf
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. X ⊢ decode(t)
Lemma: csm-universe-decode
∀[t,s:Top]. ((decode(t))s ~ decode((t)s))
Lemma: universe-decode-type
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. ∀[I:fset(ℕ)]. ∀[rho:X(I)]. (decode((t)<rho>) = universe-type(t;I;rho) ∈ {formal-cube(I) ⊢ _})
Lemma: universe-decode-restriction
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. ∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[rho:X(J)].
(decode(t)(f(rho)) = universe-type(t;J;rho)(f(1)) ∈ Type)
Lemma: csm-universe-type
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[K:fset(ℕ)]. ∀[f:K ⟶ I].
((universe-type(t;I;a))<f> = universe-type(t;K;f(a)) ∈ {formal-cube(K) ⊢ _})
Lemma: universe-type-at
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[K:fset(ℕ)]. ∀[f:K ⟶ I].
(universe-type(t;I;a)(f) = decode(t)(f(a)) ∈ Type)
Definition: universe-comp-op
compOp(t) == λI,i,rho. ((snd(t(rho))) I i 1)
Lemma: universe-comp-op_wf
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. (compOp(t) ∈ X ⊢ CompOp(decode(t)))
Lemma: csm-universe-comp-op
∀[E,s:Top]. ((compOp(E))s ~ compOp((E)s))
Lemma: universe-encode-decode
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. (encode(decode(t);compOp(t)) = t ∈ {X ⊢ _:c𝕌})
Lemma: universe-decode-encode
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[cT:X ⊢ CompOp(T)]. (decode(encode(T;cT)) = T ∈ {X ⊢ _})
Lemma: universe-comp-op-encode
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[cT:X ⊢ CompOp(T)]. (compOp(encode(T;cT)) = cT ∈ X ⊢ CompOp(T))
Definition: universe-comp-fun
CompFun(A) == cop-to-cfun(compOp(A))
Lemma: universe-comp-fun_wf
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. (CompFun(t) ∈ X +⊢ Compositon(decode(t)))
Lemma: csm-universe-comp-fun
∀[s,A,G,H:Top]. ((CompFun(A))s ~ CompFun((A)s))
Definition: equiv_comp
equiv_comp(H;A;E;cA;cE) ==
sigma_comp(pi_comp(H;A;cA;(cE)p);pi_comp(H.(A ⟶ E);(E)p;(cE)p;contractible_comp(H.(A ⟶ E).(E)p;Fiber((q)p;q);
fiber-comp(H.(A ⟶ E).(E)p;((A)p)p;
((E)p)p;(q)p;q;((cA)p)p;
((cE)p)p))))
Lemma: equiv_comp_wf
∀[H:j⊢]. ∀[A,E:{H ⊢ _}]. ∀[cA:H +⊢ Compositon(A)]. ∀[cE:H +⊢ Compositon(E)].
(equiv_comp(H;A;E;cA;cE) ∈ H +⊢ Compositon(Equiv(A;E)))
Lemma: equiv_comp-sq
∀[H,A,E,cA,cE:Top].
(equiv_comp(H;A;E;cA;cE)
~ sigma_comp(pi_comp(H;A;cA;λH@0,sigma,phi,u,a0. (cE H@0 (λx,x@0. (fst((sigma x x@0)))) phi u a0));
pi_comp(H.(A ⟶ E);(E)λI,p. (fst(p));λH@0,sigma,phi,u,a0. (cE H@0 (λx,x@0. (fst((sigma x x@0)))) phi u a0\000C);
contractible_comp(H.(A ⟶ E).(E)λI,p. (fst(p));Fiber(λI,a. (snd(fst(a)));λI,p. (snd(p)));
λH@0,sigma,phi,u,a0,I,a@0.
<cA H@0.𝕀 (λx,x@0. (fst(fst((sigma x (m x x@0))))))
(λI,rho. phi I (fst(rho)) ∨ dM-to-FL(I;¬(snd(rho))))\000C
(λI,rho. if (phi I (fst((m I rho)))==1)
then fst((u I (m I rho)))
else fst((a0 I (fst((m I rho)))))
fi )
(λI,a. (fst((a0 I (fst(a))))))
I
<a@0, 1>
, λJ,f,u@0.
(cE H@0.𝕀
(λx,x@0. (fst(fst((sigma x
<fst(fst(x@0))
, snd(x@0)
>)))))
(λI,rho.
phi I (fst(rho)) ∨ ... ∨ (...=1)
I
rho)
(λI,rho.
if (phi I (fst(fst(rho)))==1)
then (snd((u I
<fst(fst(rho))
, snd(rho)
>)))
I
1
(snd(fst(rho)))
if (dM-to-FL(I;¬(snd(...)))==1)
then snd((sigma I
<fst(fst(rho))
, snd(rho)
>))
else (snd(fst((sigma I
<fst(fst(rho))
, snd(rho)
>))))
I
1
(cA H@0.𝕀
(λx,x@0.
(fst(fst((sigma x
(m x
x@0)))))\000C)
(λI,a.
(phi I
(fst(a))) ∨ (λI,p. .\000C..=0))
(λI,rho.
if (...==1)
then fst((u I
(m I
rho)))
else fst((a0 I
...))
fi )
(λI,a. (fst((a0 I
(fst(a))))))
I
<fst(fst(rho)), snd(rho)>)
fi )
(λI,a. ((snd((a0 I (fst(a))))) I 1
(snd(a))))
J
(f(a@0);u@0))
>))))
Definition: equiv_comp_apply
equiv_comp_apply(H;A;E;cA;cE;I;a;b;c;d) ==
<λJ,f,u@0. (cE <λJ.J ⟶ I, λI,J,f,g. g ⋅ f>.𝕀.((A)λx,x@0. <a[x;m x x@0], b[x;m x x@0]>)[1(𝕀)]
(λx,x@0. <a[x;m x <fst(fst(x@0)), snd(x@0)>], b[x;m x <fst(fst(x@0)), snd(x@0)>]>)
(λI,a. c[I;fst(fst(a))] ∨ dM-to-FL(I;¬(snd(fst(a)))))
(λI@0,a@0. (if (c[I@0;fst((m I@0 <fst(fst(a@0)), snd(a@0)>))]==1)
then fst(⋅)
else fst(d[I@0;fst((m I@0 <fst(fst(a@0)), snd(a@0)>))])
fi
I@0
1
(cA <λJ.J ⟶ I, λI,J,f,g. g ⋅ f>.𝕀.((A)λx,x@0. <a[x;m x x@0], b[x;m x x@0]>)[1(𝕀)].𝕀
(λx,x@0. <a[x;m x <fst(fst((m x x@0))), ¬(snd((m x x@0)))>]
, b[x;m x <fst(fst((m x x@0))), ¬(snd((m x x@0)))>]
>)
(λI,rho. 0 ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. (snd(fst((m I rho)))))
(λI,a. (snd(fst(a))))
I@0
<fst(a@0), ¬(snd(a@0))>)))
(λI@0,a@0. ((fst(d[I@0;fst(fst(a@0))])) I@0 1
(cA <λJ.J ⟶ I, λI,J,f,g. g ⋅ f>.𝕀.((A)λx,x@0. <a[x;m x x@0], b[x;m x x@0]>)[1(𝕀)].𝕀
(λx,x@0. <a[x;m x <fst(fst((m x x@0))), ¬(snd((m x x@0)))>]
, b[x;m x <fst(fst((m x x@0))), ¬(snd((m x x@0)))>]
>)
(λI,rho. 0 ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. (snd(fst((m I rho)))))
(λI,a. (snd(fst(a))))
I@0
<a@0, ¬(0)>)))
J
(f(<1, 1>);u@0))
, λJ,f,u@0.
(cubical-pair(λI@0,a@1.
<cA
<λJ.J ⟶ I, λI,J,f,g. g ⋅ f>.(let x,F = E
in <λJ,c. (x J <a[J;c], b[J;c]>)
, λK,J,f,c,u. (F K J f <a[K;c], b[K;c]> u)
>)[1(𝕀)].𝕀.𝕀
(λx,x@0. <a[x;<fst(fst((m x (m x x@0)))), snd((m x (m x x@0)))>]
, b[x;<fst(fst((m x (m x x@0)))), snd((m x (m x x@0)))>]
>)
(λI,rho.
c[I;
fst(fst(fst(rho)))] ∨ dM-to-FL(I;¬(snd(fst(rho)))) ∨ dM-to-FL(I;¬(...)))
(λI@0,rho.
if (c[I@0;fst(fst(fst((m I@0
rho))))] ∨ dM-to-FL(I@0;¬(snd(fst((m I@0
rho)))))==1)
then fst(if (c[I@0;fst(fst((m I@0 (m I@0 rho))))]==1)
then fst(((snd(⋅)) I@0 1
(cE
<λJ.J ⟶ I, λI,J,f,g. g ⋅ f>.(let x,F = E
in <λJ,c. (x J <a[J;c], b[J;c]>)
, λK,J,f,c,u. (F K J f
<a[K;c], b[K;c]>
u)
>)[1(𝕀)].𝕀
(λx,x@0. <a[x;<fst(fst((m x x@0))), ¬(snd((m x x@0)))>]
, b[x;<fst(fst((m x x@0))), ¬(snd((m x x@0)))>]
>)
(λI,rho. 0 ∨ dM-to-FL(I;¬(snd(rho))))
(λI,rho. (snd(fst((m I rho)))))
(λI,a. (snd(fst(a))))
I@0
<fst((m I@0 (m I@0 rho)))
, ¬(snd((m I@0 (m I@0 rho))))
>)))
else fst(...)
fi )
else ...
fi )
...
...
...
, ...
>;...)
...
...)
>
Lemma: equiv_comp-apply-sq
∀[H,A,E,cA,cE,I,a,b,c,d:Top].
(equiv_comp(H;A;E;cA;cE) formal-cube(I) (λx,y. <a[x;y], b[x;y]>) (λK,f. c[K;f]) (λI@0,a. ⋅) (λK,f. d[K;f]) I 1
~ equiv_comp_apply(H;A;E;cA;cE;I;a;b;c;d))
Lemma: csm-equiv_comp-sq
∀[H,K,tau,A,E,cA,cE:Top].
((equiv_comp(H;A;E;cA;cE))tau ~ sigma_comp(pi_comp(K;(A)tau;(cA)tau;((cE)tau)p);
pi_comp(K.((A)tau ⟶ (E)tau);((E)tau)p;((cE)tau)p;
contractible_comp(K.((A)tau ⟶ (E)tau).((E)tau)p;
(Fiber((q)p;q))tau++;
fiber-comp(K.((A)tau ⟶ (E)tau).((E)tau)p;
(((A)tau)p)p;(((E)tau)p)p;(q)p;q;
(((cA)tau)p)p;(((cE)tau)p)p)))))
Lemma: csm-equiv_comp
∀[H,K:j⊢]. ∀[tau:K j⟶ H]. ∀[A,E:{H ⊢ _}]. ∀[cA:H +⊢ Compositon(A)]. ∀[cE:H +⊢ Compositon(E)].
((equiv_comp(H;A;E;cA;cE))tau = equiv_comp(K;(A)tau;(E)tau;(cA)tau;(cE)tau) ∈ K ⊢ Compositon(Equiv((A)tau;(E)tau)))
Definition: equiv-comp
equiv-comp(H;A;E;cA;cE) == cfun-to-cop(H;Equiv(A;E);equiv_comp(H;A;E;cop-to-cfun(cA);cop-to-cfun(cE)))
Lemma: equiv-comp_wf
∀[H:j⊢]. ∀[A,E:{H ⊢ _}]. ∀[cA:H ⊢ CompOp(A)]. ∀[cE:H ⊢ CompOp(E)]. (equiv-comp(H;A;E;cA;cE) ∈ H ⊢ CompOp(Equiv(A;E)))
Lemma: equiv-comp-exists
∀H:j⊢. ∀A,E:{H ⊢ _}. (H ⊢ CompOp(A) ⇒ H ⊢ CompOp(E) ⇒ H ⊢ CompOp(Equiv(A;E)))
Lemma: csm-equiv-comp
∀[H,K:j⊢]. ∀[tau:K j⟶ H]. ∀[A,E:{H ⊢ _}]. ∀[cA:H ⊢ CompOp(A)]. ∀[cE:H ⊢ CompOp(E)].
((equiv-comp(H;A;E;cA;cE))tau = equiv-comp(K;(A)tau;(E)tau;(cA)tau;(cE)tau) ∈ K ⊢ CompOp(Equiv((A)tau;(E)tau)))
Definition: equivU
equivU(G;E;cE) == transport(G;IdEquiv(G;(E)[0(𝕀)]))
Lemma: equivU_wf
∀[G:j⊢]. ∀[E:{G.𝕀 ⊢ _}]. ∀[cE:G.𝕀 ⊢ CompOp(E)]. (equivU(G;E;cE) ∈ {G ⊢ _:Equiv((E)[0(𝕀)];(E)[1(𝕀)])})
Definition: comp-U
comp-U() ==
λG,phi,E,A. encode(Glue [phi ⊢→ (decode((E)[1(𝕀)]);equiv-fun(equivU(G, phi;(decode(E))-;(compOp(E))-)))] decode(A);
cfun-to-cop(G;Glue [phi ⊢→ (decode((E)[1(𝕀)]);equiv-fun(equivU(G, phi;(decode(E))-;(compOp(E))-)))]
decode(A)
;comp(Glue [phi ⊢→ (decode((E)[1(𝕀)]), equivU(G, phi;(decode(E))-;(compOp(E))-))] decode(A)) ))
Lemma: csm-equivU
∀[G,K:j⊢]. ∀[tau:K j⟶ G]. ∀[E:{G.𝕀 ⊢ _}]. ∀[cE:G.𝕀 ⊢ CompOp(E)].
((equivU(G;E;cE))tau = equivU(K;(E)tau+;(cE)tau+) ∈ {K ⊢ _:Equiv(((E)tau+)[0(𝕀)];((E)tau+)[1(𝕀)])})
Definition: compU
compU() ==
λG,s,phi,E,A. encode(Glue [phi ⊢→ (decode((E)[1(𝕀)]);equiv-fun(equivU(G, phi;(decode(E))-;(compOp(E))-)))] decode(A);
cfun-to-cop(G;Glue [phi ⊢→ (decode((E)[1(𝕀)]);equiv-fun(equivU(G, phi;(decode(E))-;
(compOp(E))-)))] decode(A)
;comp(Glue [phi ⊢→ (decode((E)[1(𝕀)]), equivU(G, phi;(decode(E))-;(compOp(E))-))]
decode(A)) ))
Lemma: compU-wf-lemma1
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}].
(λE,A. encode(Glue [phi ⊢→ (decode((E)[1(𝕀)]);equiv-fun(equivU(G, phi;(decode(E))-;(compOp(E))-)))] decode(A);
cfun-to-cop(G;Glue [phi ⊢→ (decode((E)[1(𝕀)]);equiv-fun(equivU(G, phi;(decode(E))-;(compOp(E))-)))]
decode(A)
;comp(Glue [phi ⊢→ (decode((E)[1(𝕀)]), equivU(G, phi;(decode(E))-;(compOp(E))-))] decode(A)) ))
∈ u:{G, phi.𝕀 ⊢ _:c𝕌} ⟶ a0:{G ⊢ _:c𝕌[phi |⟶ (u)[0(𝕀)]]} ⟶ {G ⊢ _:c𝕌[phi |⟶ (u)[1(𝕀)]]})
Lemma: compU_wf1
∀[G:j⊢]. (compU() ∈ composition-function{j:l, i':l}(G; c𝕌))
Lemma: compU_wf
∀[G:j⊢]. (compU() ∈ G ⊢ Compositon'(c𝕌))
Definition: univ-comp
univ-comp{i:l}() == cfun-to-cop(();c𝕌;compU())
Lemma: univ-comp-sq
∀[G:Top]. (univ-comp{i:l}() ~ cfun-to-cop(G;c𝕌;compU()))
Lemma: univ-comp_wf
∀[G:j⊢]. (univ-comp{i:l}() ∈ G ⊢ CompOp(c𝕌))
Lemma: comp-op-to-comp-fun-univ-comp
∀[G:j⊢]. (cop-to-cfun(univ-comp{i:l}()) ∈ composition-structure{[i' | j]:l, i':l}(G; c𝕌))
Lemma: csm-univ-comp
∀[s:Top]. ((univ-comp{i:l}())s ~ univ-comp{i:l}())
Lemma: composition-op-universe-sq
∀[G:Top]
(G ⊢ CompOp(c𝕌) ~ {comp:I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:G(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:c𝕌}
⟶ {a0:FibrantType(formal-cube(I))|
∀J:fset(ℕ). ∀f:I,phi(J).
(csm-fibrant-type(formal-cube(I);formal-cube(J);<f>;a0)
= u((i0) ⋅ f)
∈ FibrantType(formal-cube(J)))}
⟶ {a1:FibrantType(formal-cube(I))|
∀J:fset(ℕ). ∀f:I,phi(J).
(csm-fibrant-type(formal-cube(I);formal-cube(J);<f>;a1)
= u((i1) ⋅ f)
∈ FibrantType(formal-cube(J)))} |
∀I,J:fset(ℕ). ∀i:{i:ℕ| ¬i ∈ I} . ∀j:{j:ℕ| ¬j ∈ J} . ∀g:J ⟶ I. ∀rho:G(I+i). ∀phi:𝔽(I).
∀u:{I+i,s(phi) ⊢ _:c𝕌}. ∀a0:{a0:FibrantType(formal-cube(I))|
∀J:fset(ℕ). ∀f:I,phi(J).
(csm-fibrant-type(formal-cube(I);formal-cube(J);<f>;a0)
= u((i0) ⋅ f)
∈ FibrantType(formal-cube(J)))} .
(csm-fibrant-type(formal-cube(I);formal-cube(J);<g>;comp I i rho phi u a0)
= (comp J j g,i=j(rho) g(phi) (u)subset-trans(I+i;J+j;g,i=j;s(phi))
csm-fibrant-type(formal-cube(I);formal-cube(J);<g>;a0))
∈ FibrantType(formal-cube(J)))} )
Definition: universe-term
t𝕌 == encode(c𝕌;univ-comp{i:l}())
Lemma: universe-term_wf
∀[G:j⊢]. (t𝕌 ∈ {G ⊢ _:c𝕌'})
Lemma: decode-universe-term
∀[G:j⊢]. (decode(t𝕌) = c𝕌 ∈ {G ⊢' _})
Lemma: comp-universe-term
∀[G:j⊢]. (compOp(t𝕌) = univ-comp{i:l}() ∈ G ⊢ CompOp(c𝕌))
Definition: cubical-equiv-by-cases
cubical-equiv-by-cases(G;B;f) == ((f)p ∨ IdEquiv(G.𝕀, (q=1);(B)p))
Lemma: cubical-equiv-by-cases_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}].
(cubical-equiv-by-cases(G;B;f) ∈ {G.𝕀, ((q=0) ∨ (q=1)) ⊢ _:Equiv((if (q=0) then (A)p else (B)p);(B)p)})
Lemma: csm-cubical-equiv-by-cases
∀G:j⊢. ∀A,B:{G ⊢ _}. ∀f:{G ⊢ _:Equiv(A;B)}. ∀H:j⊢. ∀s:H j⟶ G.
((cubical-equiv-by-cases(G;B;f))s+
= cubical-equiv-by-cases(H;(B)s;(f)s)
∈ {H.𝕀, ((q=0) ∨ (q=1)) ⊢ _:Equiv((if (q=0) then ((A)s)p else ((B)s)p);((B)s)p)})
Definition: equiv-path1
equiv-path1(G;A;B;f) ==
Glue [((q=0) ∨ (q=1)) ⊢→ ((if (q=0) then (A)p else (B)p);equiv-fun(cubical-equiv-by-cases(G;B;f)))] (B)p
Lemma: equiv-path1_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}]. G.𝕀 ⊢ equiv-path1(G;A;B;f)
Lemma: csm-equiv-path1
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((equiv-path1(G;A;B;f))s+ = equiv-path1(H;(A)s;(B)s;(f)s) ∈ {H.𝕀 ⊢ _})
Lemma: equiv-path1-constraint
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}].
G.𝕀, ((q=0) ∨ (q=1)) ⊢ equiv-path1(G;A;B;f) = (if (q=0) then (A)p else (B)p)
Lemma: equiv-path1-0
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}]. ((equiv-path1(G;A;B;f))[0(𝕀)] = A ∈ {G ⊢ _})
Lemma: equiv-path1-1
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}]. ((equiv-path1(G;A;B;f))[1(𝕀)] = B ∈ {G ⊢ _})
Lemma: face-term-0-and-1
∀G:j⊢. G.𝕀 ⊢ (((q=0) ∧ (q=1)) ⇒ 0(𝔽))
Definition: equiv-path2
equiv-path2(G;A;B;cA;cB;f) ==
comp(Glue [((q=0) ∨ (q=1)) ⊢→ ((if (q=0) then (A)p else (B)p), cubical-equiv-by-cases(G;B;f))] (B)p)
Lemma: equiv-path2_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cB:G +⊢ Compositon(B)]. ∀[f:{G ⊢ _:Equiv(A;B)}].
(equiv-path2(G;A;B;cA;cB;f) ∈ G.𝕀 +⊢ Compositon(equiv-path1(G;A;B;f)))
Lemma: csm-equiv-path2
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cB:G +⊢ Compositon(B)]. ∀[f:{G ⊢ _:Equiv(A;B)}]. ∀[H:j⊢].
∀[s:H j⟶ G].
((equiv-path2(G;A;B;cA;cB;f))s+
= equiv-path2(H;(A)s;(B)s;(cA)s;(cB)s;(f)s)
∈ H.𝕀 +⊢ Compositon(equiv-path1(H;(A)s;(B)s;(f)s)))
Lemma: equiv-path2-0
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cB:G +⊢ Compositon(B)].
((equiv-path2(G;A;B;cA;cB;f))[0(𝕀)] = cA ∈ G +⊢ Compositon(A))
Lemma: equiv-path2-1
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cB:G +⊢ Compositon(B)].
((equiv-path2(G;A;B;cA;cB;f))[1(𝕀)] = cB ∈ G +⊢ Compositon(B))
Definition: equiv_path
equiv_path(G;A;B;f) ==
let T = equiv-path1(G;decode(A);decode(B);f) in
let C = equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f) in
encode(T;cfun-to-cop(G.𝕀;T;C))
Lemma: equiv_path_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. (equiv_path(G;A;B;f) ∈ {G.𝕀 ⊢ _:c𝕌})
Lemma: csm-equiv_path
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((equiv_path(G;A;B;f))s+ = equiv_path(H;(A)s;(B)s;(f)s) ∈ {H.𝕀 ⊢ _:c𝕌})
Lemma: equiv_path-0
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. G ⊢ (equiv_path(G;A;B;f))[0(𝕀)]=A:c𝕌
Lemma: equiv_path-1
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. G ⊢ (equiv_path(G;A;B;f))[1(𝕀)]=B:c𝕌
Definition: equiv-path
EquivPath(G;A;B;f) == <>(equiv_path(G;A;B;f))
Lemma: equiv-path_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. (EquivPath(G;A;B;f) ∈ {G ⊢ _:(Path_c𝕌 A B)})
Definition: univ-a
UA == cubical-lam(G;EquivPath(G.Equiv(decode(A);decode(B));(A)p;(B)p;q))
Lemma: univ-a_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. (UA ∈ {G ⊢ _:(Equiv(decode(A);decode(B)) ⟶ (Path_c𝕌 A B))})
Lemma: univ-a_wf2
∀[G:j⊢]. (UA ∈ {G.c𝕌.c𝕌 ⊢ _:(Equiv(decode(q);decode((q)p)) ⟶ (Path_c𝕌 q (q)p))})
Lemma: app-univ-a-1
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}].
(app(UA; f) = (EquivPath(G.Equiv(decode(A);decode(B));(A)p;(B)p;q))[f] ∈ {G ⊢ _:(Path_c𝕌 A B)})
Lemma: app-univ-a
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}].
(app(UA; f) = EquivPath(G;A;B;f) ∈ {G ⊢ _:(Path_c𝕌 A B)})
Definition: univ-trans
univ-trans(G;T) == transprt-fun(G;decode(T);cop-to-cfun(compOp(T)))
Lemma: univ-trans_wf
∀[G:j⊢]. ∀[T:{G.𝕀 ⊢ _:c𝕌}]. (univ-trans(G;T) ∈ {G ⊢ _:((decode(T))[0(𝕀)] ⟶ (decode(T))[1(𝕀)])})
Lemma: csm-univ-trans
∀[G:j⊢]. ∀[T:{G.𝕀 ⊢ _:c𝕌}]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((univ-trans(G;T))s = univ-trans(H;(T)s+) ∈ {H ⊢ _:(((decode(T))s+)[0(𝕀)] ⟶ ((decode(T))s+)[1(𝕀)])})
Definition: path-trans
PathTransport(p) == univ-trans(G;path-eta(p))
Lemma: path-trans_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[p:{G ⊢ _:(Path_c𝕌 A B)}]. (PathTransport(p) ∈ {G ⊢ _:(decode(A) ⟶ decode(B))})
Lemma: path-trans-sq
∀[G,pth,I,a,J,h,u:Top]. (PathTransport(pth) I a J h u ~ compOp(path-eta(pth)) J new-name(J) <s(h(a)), <new-name(J)>> 0 \000Cdiscr(⋅) u)
Lemma: path-trans-sq2
∀[G,pth,I,a,J,h,u:Top].
(PathTransport(pth) I a J h u ~ (snd((pth(s(h(a))) J+new-name(J) 1 <new-name(J)>))) J new-name(J) 1 0 discr(⋅) u)
Definition: cubical-subst
cubical-subst(G;f;pth;x) == app(PathTransport(map-path(G;f;pth)); x)
Lemma: cubical-subst_wf
∀[G:j⊢]. ∀[A:{G ⊢' _}]. ∀[a,b:{G ⊢ _:A}]. ∀[p:{G ⊢ _:(Path_A a b)}]. ∀[f:{G ⊢ _:(A ⟶ c𝕌)}].
∀[x:{G ⊢ _:decode(app(f; a))}].
(cubical-subst(G;f;p;x) ∈ {G ⊢ _:decode(app(f; b))})
Definition: equivTerm
equivTerm(G;A;B) == encode(Equiv(decode(A);decode(B));equiv-comp(G;decode(A);decode(B);compOp(A);compOp(B)))
Lemma: equivTerm_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. (equivTerm(G;A;B) ∈ {G ⊢ _:c𝕌})
Lemma: csm-equivTerm
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[H:j⊢]. ∀[s:H j⟶ G]. ((equivTerm(G;A;B))s = equivTerm(H;(A)s;(B)s) ∈ {H ⊢ _:c𝕌})
Lemma: decode-equivTerm
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. (decode(equivTerm(G;A;B)) = Equiv(decode(A);decode(B)) ∈ {G ⊢ _})
Definition: transEquiv
transEquiv{i:l}(G;A;p) == cubical-subst(G;cubical-lam(G;equivTerm(G.c𝕌;(A)p;q));p;IdEquiv(G;decode(A)))
Lemma: transEquiv_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[p:{G ⊢ _:(Path_c𝕌 A B)}]. (transEquiv{i:l}(G;A;p) ∈ {G ⊢ _:Equiv(decode(A);decode(B))})
Definition: transEquiv-trans
transEquivFun(p) == equiv-fun(transEquiv{i:l}(G;A;p))
Lemma: transEquiv-trans_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[p:{G ⊢ _:(Path_c𝕌 A B)}]. (transEquivFun(p) ∈ {G ⊢ _:(decode(A) ⟶ decode(B))})
Lemma: transEquiv-trans-sq
∀[G,A,p,I,a,J,f,u:Top].
(transEquivFun(p) I a J f u ~ let x,cA = p I+new-name(I) 1(s(1(a))) I+new-name(I) 1 <new-name(I)>
in cA J new-name(J)
1 ⋅ cube+(I;new-name(I)) J+new-name(J) <1 ⋅ f ⋅ s, 1 ∧ <new-name(J)>> ⋅ 1
(0)<1 ⋅ f> ∨ 0
(λI@1,a@0. (if ((0)<1 ⋅ f ⋅ s ⋅ a@0>==1) then (fst(⋅)) I@1 1 else λu.u fi
((snd((A I@1+new-name(I@1)
cube+(I;new-name(I)) I@1+new-name(I@1)
<1 ⋅ f ⋅ s ⋅ a@0 ⋅ s
, 1 ∧ ¬(dM-lift(I@1+new-name(I@1);I@1;s)
¬(dM-lift(I@1;J+new-name(J);a@0)
<new-name(J)>) ∧ <new-name(I@1)>)
>(1(1(1(s(1(a)))))))))
I@1
new-name(I@1)
1
0 ∨ dM-to-FL(I@1;¬(¬(dM-lift(I@1;J+new-name(J);a@0) <new-name(J)>)))
(λI@0,a@1. ((((u 1 s) 1 a@0) 1 s) 1 a@1))
((u 1 s) 1 a@0))))
((snd((A J+new-name(J)
cube+(I;new-name(I)) J+new-name(J) <1 ⋅ f ⋅ s, 1 ∧ ¬(1 ∧ <new-name(J)>)>(1(1(1\000C(s(1(a)))))))))
J
new-name(J)
1
0
(λI@1,a@0. ((u 1 s) 1 a@0))
u))
Lemma: universe-path-type-lemma-0
∀G:j⊢. ∀A,B:{G ⊢ _:c𝕌}. ∀p:{G ⊢ _:(Path_c𝕌 A B)}. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀v:G(I+new-name(I)).
(universe-type(A;I+new-name(I);v)((new-name(I)0) ⋅ f)
= fst((p(v) I+new-name(I) 1 <new-name(I)>))((new-name(I)0) ⋅ f)
∈ Type)
Lemma: universe-path-type-lemma-1
∀G:j⊢. ∀A,B:{G ⊢ _:c𝕌}. ∀p:{G ⊢ _:(Path_c𝕌 A B)}. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀v:G(I+new-name(I)).
(universe-type(B;I+new-name(I);v)((new-name(I)1) ⋅ f)
= fst((p(v) I+new-name(I) 1 <new-name(I)>))((new-name(I)1) ⋅ f)
∈ Type)
Lemma: transEquiv-trans-eq
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[p:{G ⊢ _:(Path_c𝕌 A B)}].
(transEquivFun(p)
= (λI,a,J,f,u. ((snd((p(s(a)) I+new-name(I) 1 <new-name(I)>))) J new-name(J) f,new-name(I)=new-name(J) 0 ⋅
(compOp(A) J new-name(J) s(f(a)) 0 ⋅ u)))
∈ {G ⊢ _:(decode(A) ⟶ decode(B))})
Lemma: transEquiv-trans-eq2
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[p:{G ⊢ _:(Path_c𝕌 A B)}].
(transEquivFun(p)
= (λI,a,J,h,u. ((snd((p(s(h(a))) J+new-name(J) 1 <new-name(J)>))) J new-name(J) 1 0 discr(⋅)
(compOp(A) J new-name(J) s(h(a)) 0 ⋅ u)))
∈ {G ⊢ _:(decode(A) ⟶ decode(B))})
Lemma: transEquiv-trans-eq-path-trans
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[p:{G ⊢ _:(Path_c𝕌 A B)}].
(transEquivFun(p) = (PathTransport(p) o ConstTrans(decode(A))) ∈ {G ⊢ _:(decode(A) ⟶ decode(B))})
Definition: trans-equiv-path
trans-equiv-path(G;A;B;f) ==
let tr = λb.transprt-const(G.decode(A);(CompFun(B))p;b) in
let b = app(equiv-fun((f)p); q) in
cubical-lam(G;tr (tr b))
Lemma: trans-equiv-path_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}].
(trans-equiv-path(G;A;B;f) ∈ {G ⊢ _:(decode(A) ⟶ decode(B))})
Lemma: csm-trans-equiv-path
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((trans-equiv-path(G;A;B;f))s = trans-equiv-path(H;(A)s;(B)s;(f)s) ∈ {H ⊢ _:(decode((A)s) ⟶ decode((B)s))})
Lemma: app-trans-equiv-path
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. ∀[a:{G ⊢ _:decode(A)}].
(app(trans-equiv-path(G;A;B;f); a)
= transprt-const(G;CompFun(B);transprt-const(G;CompFun(B);app(equiv-fun(f); a)))
∈ {G ⊢ _:decode(B)})
Lemma: univ-trans-equiv_path
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}].
(univ-trans(G;equiv_path(G;A;B;f)) = trans-equiv-path(G;A;B;f) ∈ {G ⊢ _:(decode(A) ⟶ decode(B))})
Definition: univalence
Univalence == Πc𝕌 Contractible(Σ c𝕌 Equiv(decode((q)p);decode(q)))
Lemma: univalence_wf
∀[G:j⊢]. G ⊢' Univalence
Definition: uabeta_aux
uabeta_aux(G;A;B;f) ==
let b = app(equiv-fun((f)p); q) in
let b' = transprt-const(G.decode(A);(CompFun(B))p;b) in
let pth1 = trans-const-path(G.decode(A);(CompFun(B))p;b') in
let pth2 = trans-const-path(G.decode(A);(CompFun(B))p;b) in
pth1 + pth2
Lemma: uabeta_aux_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}].
(uabeta_aux(G;A;B;f) ∈ {G.decode(A) ⊢ _:let b = app(equiv-fun((f)p); q) in
let b' = transprt-const(G.decode(A);(CompFun(B))p;b) in
let b'' = transprt-const(G.decode(A);(CompFun(B))p;b') in
(Path_(decode(B))p b'' b)})
Definition: transport-type
TransportType(A) == ∀[B:{G ⊢ _:c𝕌}]. ({G ⊢ _:(Path_c𝕌 A B)} ⟶ {G ⊢ _:(decode(A) ⟶ decode(B))})
Lemma: transport-type_wf
∀[G:j⊢]. ∀[A:{G ⊢ _:c𝕌}]. (TransportType(A) ∈ 𝕌{[i'' | j'']})
Definition: uabetatype
uabetatype(G;A;B;f) ==
ΠEquiv(decode(A);decode(B)) Π(decode(A))p (Path_((decode(B))p)p app(f G.Equiv(decode(A);decode(B)).(decode(A))p
((A)p)p
app(UA; (q)p); q) app(equiv-fun((q)p); q))
Lemma: uabetatype_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:G:CubicalSet{i|j} ⟶ A:{G ⊢ _:c𝕌} ⟶ TransportType(A)]. G ⊢ uabetatype(G;A;B;f)
Definition: uabeta-type
uabeta-type(G;A;B) ==
ΠEquiv(decode(A);decode(B)) Π(decode(A))p (Path_((decode(B))p)p app(PathTransport(app(UA; (q)p)); q)
app(equiv-fun((q)p); q))
Lemma: uabeta-type_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. G ⊢ uabeta-type(G;A;B)
Definition: uabeta
uabeta(G;A;B) == (λ(λuabeta_aux(G.Equiv(decode(A);decode(B));(A)p;(B)p;q)))
Lemma: uabeta_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. (uabeta(G;A;B) ∈ {G ⊢ _:uabeta-type(G;A;B)})
Definition: transEquivbeta-type
transEquivbeta-type{i:l}(G;A;B) ==
ΠEquiv(decode(A);decode(B)) Π(decode(A))p (Path_((decode(B))p)p app(transEquivFun(app(UA; (q)p)); q)
app(equiv-fun((q)p); q))
Lemma: transEquivbeta-type_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. G ⊢ transEquivbeta-type{i:l}(G;A;B)
Definition: univalence_t
univalence_t{i:l}(G) ==
(λbij-contr(G.c𝕌; Σ c𝕌 Equiv(decode((q)p);decode(q)); cubical-sigma-fun(G.c𝕌;c𝕌;Equiv(decode((q)p);decode(q));UA);
cubical-sigma-fun(G.c𝕌;c𝕌;(Path_c𝕌 (q)p q);44); 55; 66; singleton-contr(G.c𝕌;c𝕌;q)))
Definition: univalence-term
univalence-term{i:l}(G; A; B) == equiv-witness(UA;is-equiv-witness(G;(Path_c𝕌 A B);fiber-point(33;44);55))
Definition: cubical-bool
Bool == encode(discr(𝔹);discrete-comp(();𝔹))
Lemma: cubical-bool_wf
Bool ∈ {() ⊢ _:c𝕌}
Lemma: csm-cubical-bool
∀[H:j⊢]. ∀[s:H j⟶ ()]. ((Bool)s = Bool ∈ {H ⊢ _:c𝕌})
Definition: bool-negation-equiv
bool-negation-equiv(X) == bijection-equiv(X;𝔹;𝔹;λb.(¬bb);λb.(¬bb))
Lemma: bool-negation-equiv_wf
bool-negation-equiv(()) ∈ {() ⊢ _:Equiv(decode(Bool);decode(Bool))}
Definition: neg-bool-trans
neg-bool-trans() == PathTransport(EquivPath(();Bool;Bool;bool-negation-equiv(())))
Lemma: neg-bool-trans_wf
neg-bool-trans() ∈ {() ⊢ _:(discr(𝔹) ⟶ discr(𝔹))}
Definition: ctt-level-type
{X ⊢lvl _} == if (lvl =z 0) then {X ⊢ _} if (lvl =z 1) then {X ⊢' _} if (lvl =z 2) then {X ⊢'' _} else {X ⊢''' _} fi
Lemma: ctt-level-type_wf
∀[X:⊢''']. ∀[lvl:ℕ]. (X ⊢lvl ∈ 𝕌{i''''})
Lemma: ctt-level-type-cumulativity
∀[X:⊢''']. ∀[lvl:ℕ4]. ({X ⊢lvl _} ⊆r ctt-level-type{i':l}(X; lvl))
Lemma: ctt-level-type-cumulativity2
∀[X:⊢''']. ∀[a,b:ℕ4]. {X ⊢a _} ⊆r {X ⊢b _} supposing a ≤ b
Lemma: ctt-level-type-subtype
∀[X:⊢''']. ∀[lvl:ℕ4]. ({X ⊢lvl _} ⊆r {X ⊢''' _})
Lemma: ctt-level-type-subtype2
∀[X:⊢''']. ∀[lvl:ℕ4]. ({X ⊢ _} ⊆r {X ⊢lvl _})
Definition: ctt-level-comp
Comp(X;lvl;T) ==
if (lvl =z 0) then composition-structure{i''':l, i:l}(X; T)
if (lvl =z 1) then composition-structure{i''':l, i':l}(X; T)
if (lvl =z 2) then composition-structure{i''':l, i'':l}(X; T)
else composition-structure{i''':l, i''':l}(X; T)
fi
Lemma: ctt-level-comp_wf
∀[X:⊢''']. ∀[lvl:ℕ4]. ∀[T:{X ⊢lvl _}]. (Comp(X;lvl;T) ∈ 𝕌{i'''''})
Lemma: ctt-level-comp-cumulativity
∀[X:⊢''']. ∀[a:ℕ4]. ∀[T:{X ⊢a _}]. ∀[b:ℕ4]. Comp(X;a;T) ⊆r Comp(X;b;T) supposing a ≤ b
Definition: levelsup
levelsup(x;y) == imax(x;y)
Lemma: levelsup_wf
∀[x,y:ℕ4]. (levelsup(x;y) ∈ ℕ4)
Lemma: csm-ap-type_wf-level-type
∀[X,Y:⊢''']. ∀[s:cube_set_map{i''':l}(X; Y)]. ∀[lvl:ℕ4]. ∀[T:{Y ⊢lvl _}]. ((T)s ∈ X ⊢lvl )
Lemma: cubical-term_wf-level-type
∀[X:⊢''']. ∀[lvl:ℕ4]. ∀[T:{X ⊢lvl _}]. ({X ⊢ _:T} ∈ 𝕌{i''''})
Lemma: cube-context-adjoin_wf-level-type
∀[X:⊢''']. ∀[lvl:ℕ4]. ∀[T:{X ⊢lvl _}]. X.T ⊢'''
Lemma: csm-ap-term_wf-level-type
∀[X,Y:⊢''']. ∀[s:cube_set_map{i''':l}(X; Y)]. ∀[lvl:ℕ4]. ∀[T:{Y ⊢lvl _}]. ∀[t:{Y ⊢ _:T}]. ((t)s ∈ {X ⊢ _:(T)s})
Lemma: cubical-sigma_wf-level-type
∀[K:⊢''']. ∀[a,b:ℕ4]. ∀[A:{K ⊢a _}]. ∀[B:{K.A ⊢b _}]. (Σ A B ∈ K ⊢levelsup(a;b) )
Lemma: cubical-pi_wf-level-type
∀[K:⊢''']. ∀[a,b:ℕ4]. ∀[A:{K ⊢a _}]. ∀[B:{K.A ⊢b _}]. (ΠA B ∈ K ⊢levelsup(a;b) )
Definition: ctt-term-meaning
cttTerm(X) == lvl:ℕ4 × T:{X ⊢lvl _} × {X ⊢ _:T}
Lemma: ctt-term-meaning_wf
∀[X:⊢''']. (cttTerm(X) ∈ 𝕌{i''''})
Definition: ctt-term-level
level(t) == fst(t)
Lemma: ctt-term-level_wf
∀[X:⊢''']. ∀[t:cttTerm(X)]. (level(t) ∈ ℕ4)
Definition: ctt-term-type
type(t) == fst(snd(t))
Lemma: ctt-term-type_wf
∀[X:⊢''']. ∀[t:cttTerm(X)]. (type(t) ∈ X ⊢level(t) )
Definition: ctt-term-type-is
type(t)=T == type(t) = T ∈ {X ⊢''' _}
Lemma: ctt-term-type-is_wf
∀[X:⊢''']. ∀[t:cttTerm(X)]. ∀[T:{X ⊢''' _}]. (type(t)=T ∈ ℙ{i''''})
Definition: ctt-term-term
term(t) == snd(snd(t))
Lemma: ctt-term-term_wf
∀[X:⊢''']. ∀[t:cttTerm(X)]. (term(t) ∈ {X ⊢ _:type(t)})
Lemma: ctt-term-type-is-implies
∀[X:⊢''']. ∀[t:cttTerm(X)]. ∀[T:{X ⊢''' _}]. term(t) ∈ {X ⊢ _:T} supposing type(t)=T
Lemma: ctt-term-meaning-cumulativity
∀[X:⊢''']. (cttTerm(X) ⊆r ctt-term-meaning{i':l}(X))
Lemma: ctt-term-meaning-subtype
∀[X,Y:⊢''']. cttTerm(X) ⊆r cttTerm(Y) supposing sub_cubical_set{i''':l}(Y; X)
Definition: mk-ctt-term-mng
mk-ctt-term-mng(lvl; T; t) == <lvl, T, t>
Lemma: mk-ctt-term-mng_wf
∀[X:⊢''']. ∀[lvl:ℕ4]. ∀[T:{X ⊢lvl _}]. ∀[t:{X ⊢ _:T}]. (mk-ctt-term-mng(lvl; T; t) ∈ cttTerm(X))
Definition: csm-ap-term-meaning
(t)s == let lvl,A,a = t in <lvl, (A)s, (a)s>
Lemma: csm-ap-term-meaning_wf
∀[X,Y:⊢''']. ∀[s:cube_set_map{i''':l}(X; Y)]. ∀[t:cttTerm(Y)]. ((t)s ∈ cttTerm(X))
Definition: ctt-type-meaning1
ctt-type-meaning1{i:l}(X) == lvl:ℕ4 × {X ⊢lvl _} × Top
Lemma: ctt-type-meaning1_wf
∀[X:⊢''']. (ctt-type-meaning1{i:l}(X) ∈ 𝕌{i''''})
Definition: ctt-type-meaning
cttType(X) == lvl:ℕ4 × T:{X ⊢lvl _} × Comp(X;lvl;T)
Lemma: ctt-type-meaning_wf
∀[X:⊢''']. (cttType(X) ∈ 𝕌{i'''''})
Definition: ctt-type-level
level(T) == fst(T)
Lemma: ctt-type-level_wf
∀[X:⊢''']. ∀[T:cttType(X)]. (level(T) ∈ ℕ4)
Definition: ctt-type-type
type(T) == fst(snd(T))
Lemma: ctt-type-type_wf
∀[X:⊢''']. ∀[T:cttType(X)]. (type(T) ∈ X ⊢level(T) )
Definition: ctt-type-comp
comp(T) == snd(snd(T))
Lemma: ctt-type-comp_wf
∀[X:⊢''']. ∀[T:cttType(X)]. (comp(T) ∈ Comp(X;level(T);type(T)))
Definition: mk_ctt-type-mng
cttType(levl= lvltype= Tcomp= cT) == <lvl, T, cT>
Lemma: mk_ctt-type-mng_wf
∀[X:⊢''']. ∀[lvl:ℕ4]. ∀[T:{X ⊢lvl _}]. ∀[cT:if (lvl =z 0) then composition-structure{i''':l, i:l}(X; T)
if (lvl =z 1) then composition-structure{i''':l, i':l}(X; T)
if (lvl =z 2) then composition-structure{i''':l, i'':l}(X; T)
else composition-structure{i''':l, i''':l}(X; T)
fi ].
(cttType(levl= lvl
type= T
comp= cT) ∈ cttType(X))
Lemma: ctt-type-meaning-subtype
∀[X:⊢''']. (cttType(X) ⊆r ctt-type-meaning1{i:l}(X))
Lemma: ctt-term-meaning-subtype2
∀[X:⊢''']. (cttTerm(X) ⊆r ctt-type-meaning1{i:l}(X))
Lemma: cc-snd_wf-level-type
∀[X:⊢''']. ∀[lvl:ℕ4]. ∀[T:{X ⊢lvl _}]. (q ∈ {X.T ⊢ _:(T)p})
Definition: var-term-meaning
var-term-meaning(lvl;T) == mk-ctt-term-mng(lvl; (T)p; q)
Lemma: var-term-meaning_wf
∀[X:⊢''']. ∀[lvl:ℕ4]. ∀[T:{X ⊢lvl _}]. (var-term-meaning(lvl;T) ∈ cttTerm(X.T))
Definition: cubical_context
CubicalContext == X:CubicalSet''' × vs:varname() List × ({v:varname()| (v ∈ vs)} ⟶ cttTerm(X))
Lemma: cubical_context_wf
CubicalContext ∈ 𝕌{i'''''}
Lemma: cubical_context_cumulativity
CubicalContext ⊆r cubical_context{i':l}
Definition: in-context-dom
in-context-dom(ctxt;v) == let X,vs,f = ctxt in (v ∈ vs)
Lemma: in-context-dom_wf
∀[ctxt:CubicalContext]. ∀[v:varname()]. (in-context-dom(ctxt;v) ∈ ℙ)
Definition: cubical_context_val
cubical_context_val(ctxt;v) == let X,vs,f = ctxt in f v
Lemma: cubical_context_val_wf
∀[ctxt:CubicalContext]. ∀[v:varname()].
cubical_context_val(ctxt;v) ∈ cttTerm(fst(ctxt)) supposing in-context-dom(ctxt;v)
Definition: cubical-context
?CubicalContext == Provisional''''(CubicalContext)
Lemma: cubical-context_wf
?CubicalContext ∈ 𝕌{i'''''}
Definition: contextof
contextof(X) == allow(X)
Lemma: cubical-context_cumulativity
?CubicalContext ⊆r ?CubicalContext'
Lemma: bind-provision-cubical-context-equations
(∀[x:CubicalContext]. ∀[f:CubicalContext ⟶ ?CubicalContext]. (bind-provision(OK(x);u.f u) = (f x) ∈ ?CubicalContext))
∧ (∀[m:?CubicalContext]. (bind-provision(m;u.OK(u)) = m ∈ ?CubicalContext))
∧ (∀[m:?CubicalContext]. ∀[f,g:CubicalContext ⟶ ?CubicalContext].
(bind-provision(bind-provision(m;u.f u);u.g u) = bind-provision(m;u.bind-provision(f u;u.g u)) ∈ ?CubicalContext))
Lemma: trivial-same-cubical-context
∀X,Y:?CubicalContext. ((Y = bind-provision(X;ctxt.OK(ctxt)) ∈ ?CubicalContext) ⇒ (X = Y ∈ ?CubicalContext))
Definition: context-ok
context-ok(ctxt) == allowed(ctxt)
Lemma: context-ok_wf
∀[ctxt:?CubicalContext]. (context-ok(ctxt) ∈ ℙ{i''''})
Lemma: contextof_wf
∀[X:?CubicalContext]. contextof(X) ∈ CubicalContext supposing context-ok(X)
Definition: context-set
context-set(ctxt) == fst(allow(ctxt))
Lemma: context-set_wf
∀[ctxt:?CubicalContext]. context-set(ctxt) ⊢''' supposing context-ok(ctxt)
Lemma: bind-provision_wf_context
∀[x:?CubicalContext]. ∀[f:CubicalContext ⟶ ?CubicalContext]. (bind-provision(x;t.f[t]) ∈ ?CubicalContext)
Lemma: trivial-same-context-set
∀X,Y:?CubicalContext.
((Y = bind-provision(X;ctxt.OK(ctxt)) ∈ ?CubicalContext)
⇒ context-ok(X)
⇒ {(X = Y ∈ ?CubicalContext) ∧ (context-set(X) = context-set(Y) ∈ CubicalSet''')})
Definition: update-context-lvl
update-context-lvl(ctxt;lvl;T;v) ==
let X,vs,f = ctxt in
<X.T, [v / vs], λx.if eq_var(x;v) then var-term-meaning(lvl;T) else (f x)p fi >
Lemma: update-context-lvl_wf
∀[ctxt:CubicalContext]. ∀[lvl:ℕ4]. ∀[T:{fst(ctxt) ⊢lvl _}]. ∀[v:varname()].
(update-context-lvl(ctxt;lvl;T;v) ∈ CubicalContext)
Definition: update-cubical-context-I
ctxt,v:I == update-context-lvl(ctxt;0;𝕀;v)
Lemma: update-cubical-context-I_wf
∀[ctxt:CubicalContext]. ∀[v:varname()]. (ctxt,v:I ∈ CubicalContext)
Definition: update-provisional-context-I
X,v:I == provision(allowed(X); allow(X),v:I)
Lemma: update-provisional-context-I_wf
∀[ctxt:?CubicalContext]. ∀[v:varname()]. (ctxt,v:I ∈ ?CubicalContext)
Definition: update-cubical-context
ctxt; v:T == provision(allowed(T); let lvl,A = allow(T) in update-context-lvl(ctxt;lvl;A;v))
Lemma: update-cubical-context_wf
∀[ctxt:CubicalContext]. ∀[v:varname()]. ∀[T:Provisional''''(lvl:ℕ4 × {fst(ctxt) ⊢lvl _})].
(ctxt; v:T ∈ ?CubicalContext)
Definition: update-cubical-context2
ctxt; v:fst(T) == provision(allowed(T); let lvl,A,_ = allow(T) in update-context-lvl(ctxt;lvl;A;v))
Lemma: update-cubical-context2_wf
∀[ctxt:CubicalContext]. ∀[v:varname()]. ∀[T:Provisional''''(ctt-type-meaning1{i:l}(fst(ctxt)))].
(ctxt; v:fst(T) ∈ ?CubicalContext)
Definition: restrict-cubical-context
restrict-cubical-context{i:l}(ctxt;trm) ==
provision(allowed(trm) ∧ type(allow(trm))=𝔽; let X,vs,f = ctxt in
<X, term(allow(trm)), vs, f>)
Lemma: restrict-cubical-context_wf
∀[ctxt:CubicalContext]. ∀[trm:Provisional''''(cttTerm(fst(ctxt)))].
(restrict-cubical-context{i:l}(ctxt;trm) ∈ ?CubicalContext)
Definition: cubical-context-lookup
cubical-context-lookup(ctxt;v) == provision(in-context-dom(allow(ctxt);v); cubical_context_val(allow(ctxt);v))
Lemma: cubical-context-lookup_wf
∀[ctxt:?CubicalContext]. ∀[v:varname()].
(cubical-context-lookup(ctxt;v) ∈ Provisional''''(cttTerm(context-set(ctxt))) supposing context-ok(ctxt))
Definition: update-context2
update-context2(X;v;t) == bind-provision(X;ctxt.ctxt; v:fst(t))
Lemma: update-context2_wf
∀[X:?CubicalContext]
∀[v:varname()]. ∀[t:Provisional''''(cttType(context-set(X)))]. (update-context2(X;v;t) ∈ ?CubicalContext)
supposing context-ok(X)
Lemma: context-set-update-context2
∀X:?CubicalContext
(context-ok(X)
⇒ (∀[v:varname()]. ∀[t:Provisional''''(cttType(context-set(X)))].
∀X':?CubicalContext
((X' = update-context2(X;v;t) ∈ ?CubicalContext)
⇒ {allowed(t) ⇒ (context-ok(X') ∧ (context-set(X') = context-set(X).type(allow(t)) ∈ CubicalSet'''))})))
Lemma: update-context2-ok
∀X:?CubicalContext
(context-ok(X)
⇒ (∀[v:varname()]. ∀[t:Provisional''''(cttType(context-set(X)))].
∀X':?CubicalContext. ((X' = update-context2(X;v;t) ∈ ?CubicalContext) ⇒ {allowed(t) ⇒ context-ok(X')})))
Definition: update-context3
update-context3(X;v;t) == bind-provision(X;ctxt.ctxt; v:fst(t))
Lemma: update-context3_wf
∀[X:?CubicalContext]
∀[v:varname()]. ∀[t:Provisional''''(cttTerm(context-set(X)))]. (update-context3(X;v;t) ∈ ?CubicalContext)
supposing context-ok(X)
Lemma: context-set-update-context3
∀X:?CubicalContext
(context-ok(X)
⇒ (∀[v:varname()]. ∀[t:Provisional''''(cttTerm(context-set(X)))].
∀X':?CubicalContext
((X' = update-context3(X;v;t) ∈ ?CubicalContext)
⇒ {allowed(t) ⇒ (context-ok(X') ∧ (context-set(X') = context-set(X).type(allow(t)) ∈ CubicalSet'''))})))
Definition: update-context4
update-context4{i:l}(X;phi) == bind-provision(X;ctxt.restrict-cubical-context{i:l}(ctxt;phi))
Lemma: update-context4_wf
∀[X:?CubicalContext]
∀[phi:Provisional''''(cttTerm(context-set(X)))]. (update-context4{i:l}(X;phi) ∈ ?CubicalContext)
supposing context-ok(X)
Lemma: context-set-update-context4
∀X:?CubicalContext
(context-ok(X)
⇒ (∀[phi:Provisional''''(cttTerm(context-set(X)))]
∀X':?CubicalContext
((X' = update-context4{i:l}(X;phi) ∈ ?CubicalContext)
⇒ {allowed(phi)
⇒ type(allow(phi))=𝔽
⇒ (context-ok(X') ∧ (context-set(X') = context-set(X), term(allow(phi)) ∈ CubicalSet'''))})))
Definition: ctt-tokens
ctt-tokens() ==
``Glue case sigma path pi F I decode1 decode2 decode3 encode1 encode2 encode3 pathabs pathap lambda apply pair fst ...
Lemma: ctt-tokens_wf
ctt-tokens() ∈ Atom List
Definition: ctt-op
CttOp ==
k:{k:Atom| (k ∈ ``opid nType nterm univ``)} × if k =a "opid" then {f:Atom| (f ∈ ctt-tokens())}
if k =a "nType" then Type
if k =a "nterm" then T:Type × T
else ℕ
fi
Lemma: ctt-op_wf
CttOp ∈ 𝕌'
Definition: ctt-op-sort
ctt-op-sort(f) ==
let k,v = f
in if k =a "opid"
then if v ∈b ``Glue decode1 decode2 decode3 case sigma path pi F`` then "fibrant"
if v =a "I" then "type"
else "term"
fi
if k =a "nType" then "fibrant"
if k =a "nterm" then "term"
else "fibrant"
fi
Lemma: ctt-op-sort_wf
∀[f:CttOp]. (ctt-op-sort(f) ∈ {x:Atom| (x ∈ ``fibrant type term``)} )
Definition: ctt-kind
ctt-kind(t) ==
if isvarterm(t) then 0
if ctt-op-sort(term-opr(t)) =a "fibrant" then 1
if ctt-op-sort(term-opr(t)) =a "type" then 2
else 0
fi
Lemma: ctt-kind_wf
∀[t:term(CttOp)]. (ctt-kind(t) ∈ ℕ)
Definition: ctt-opid-arity
ctt-opid-arity(t) ==
if t =a "inv" then [<0, 0>]
if t =a "max" then [<0, 0>; <0, 0>]
if t =a "min" then [<0, 0>; <0, 0>]
if t =a "comp" then [<0, 0>; <1, 1>; <1, 0>; <0, 0>]
if t =a "glue" then [<0, 0>; <0, 0>; <0, 0>; <0, 0>]
if t =a "unglue" then [<0, 1>; <0, 0>; <0, 1>; <0, 0>; <0, 0>]
if t =a "meet" then [<0, 0>; <0, 0>]
if t =a "join" then [<0, 0>; <0, 0>]
if t =a "eq0" then [<0, 0>]
if t =a "eq1" then [<0, 0>]
if t =a "0" then []
if t =a "1" then []
if t =a "fst" then [<0, 1>; <1, 1>; <0, 0>]
if t =a "snd" then [<0, 1>; <1, 1>; <0, 0>]
if t =a "pair" then [<0, 0>; <1, 1>; <0, 0>]
if t =a "lambda" then [<0, 1>; <1, 0>]
if t =a "apply" then [<0, 0>; <1, 1>; <0, 0>]
if t =a "pathabs" then [<0, 1>; <1, 0>]
if t =a "pathap" then [<0, 1>; <0, 0>; <0, 0>]
if t =a "decode1" then [<0, 0>]
if t =a "decode2" then [<0, 0>]
if t =a "decode3" then [<0, 0>]
if t =a "encode1" then [<0, 1>]
if t =a "encode2" then [<0, 1>]
if t =a "encode3" then [<0, 1>]
if t =a "I" then []
if t =a "F" then []
if t =a "pi" then [<0, 1>; <1, 1>]
if t =a "sigma" then [<0, 1>; <1, 1>]
if t =a "path" then [<0, 1>; <0, 0>; <0, 0>]
if t =a "case" then [<0, 0>; <0, 0>; <0, 1>; <0, 1>]
if t =a "Glue" then [<0, 1>; <0, 0>; <0, 1>; <0, 0>]
else []
fi
Lemma: ctt-opid-arity_wf
∀[t:Atom]. (ctt-opid-arity(t) ∈ (ℕ × ℕ) List)
Definition: ctt-arity
ctt-arity(x) ==
let k,v = x
in if k =a "opid" then ctt-opid-arity(v)
if k =a "nType" then []
if k =a "nterm" then []
else []
fi
Lemma: ctt-arity_wf
∀[x:CttOp]. (ctt-arity(x) ∈ (ℕ × ℕ) List)
Definition: ctt-term
CttTerm == wfterm(CttOp;λt.ctt-kind(t);λx.ctt-arity(x))
Lemma: ctt-term_wf
CttTerm ∈ 𝕌'
Lemma: ctt-term-induction
∀[P:CttTerm ⟶ ℙ{i'''''}]
((∀v:{v:varname()| ¬(v = nullvar() ∈ varname())} . P[varterm(v)])
⇒ (∀f:CttOp. ∀bts:{bts:(varname() List × CttTerm) List|
(||bts|| = ||ctt-arity(f)|| ∈ ℤ)
∧ (∀i:ℕ||bts||
((||fst(bts[i])|| = (fst(ctt-arity(f)[i])) ∈ ℤ)
∧ (ctt-kind(snd(bts[i])) = (snd(ctt-arity(f)[i])) ∈ ℤ)))} .
((∀i:ℕ||bts||. P[snd(bts[i])]) ⇒ P[mkwfterm(f;bts)]))
⇒ {∀t:CttTerm. P[t]})
Definition: ctt-opr-is
ctt-opr-is(f;s) == let k,v = f in k =a "opid" ∧b v =a s
Lemma: ctt-opr-is_wf
∀[f:CttOp]. ∀[s:Atom]. (ctt-opr-is(f;s) ∈ 𝔹)
Lemma: assert-ctt-opr-is
∀[f:CttOp]. ∀[s:Atom]. uiff(↑ctt-opr-is(f;s);f = <"opid", s> ∈ CttOp)
Lemma: ctt-opr-is-implies
∀[f:CttOp]. ∀[s:Atom]. f ~ <"opid", s> supposing ↑ctt-opr-is(f;s)
Definition: ctt-term-is
ctt-term-is(s;t) == (¬bisvarterm(t)) ∧b let k,v = term-opr(t) in k =a "opid" ∧b v =a s
Lemma: ctt-term-is_wf
∀[s:Atom]. ∀[t:CttTerm]. (ctt-term-is(s;t) ∈ 𝔹)
Definition: ctt-eq
ctt-eq{i:l}(a;b) == alpha-eq-terms(CttOp;a;b)
Lemma: ctt-eq_wf
∀[a,b:CttTerm]. (ctt-eq{i:l}(a;b) ∈ ℙ')
Lemma: assert-ctt-term-is
∀s:Atom. ∀t:CttTerm.
((↑ctt-term-is(s;t))
⇒ {(¬↑isvarterm(t))
∧ (term-opr(t) = <"opid", s> ∈ CttOp)
∧ (||term-bts(t)|| = ||ctt-opid-arity(s)|| ∈ ℤ)
∧ (∀i:ℕ||term-bts(t)||
((||fst(term-bts(t)[i])|| = (fst(ctt-opid-arity(s)[i])) ∈ ℤ)
∧ (ctt-kind(snd(term-bts(t)[i])) = (snd(ctt-opid-arity(s)[i])) ∈ ℤ)))
∧ (t ~ mkwfterm(term-opr(t);term-bts(t)))})
Lemma: term-accum_wf_ctt1
∀[R:?CubicalContext ⟶ CttTerm ⟶ ℙ{i'''''}]. ∀[Q:?CubicalContext
⟶ f:CttOp
⟶ vs:(varname() List)
⟶ {L:(t:CttTerm × p:?CubicalContext × R[p;t]) List|
||L|| < ||ctt-arity(f)||
∧ (||vs|| = (fst(ctt-arity(f)[||L||])) ∈ ℤ)
∧ (∀i:ℕ||L||
(ctt-kind(fst(L[i])) = (snd(ctt-arity(f)[i])) ∈ ℤ))}
⟶ ?CubicalContext]. ∀[varcase:p:?CubicalContext
⟶ v:{v:varname()|
¬(v = nullvar() ∈ varname())}
⟶ R[p;varterm(v)]].
∀[mktermcase:p:?CubicalContext
⟶ f:CttOp
⟶ bts:wf-bound-terms(CttOp;λt.ctt-kind(t);λf.ctt-arity(f);f)
⟶ L:{L:(t:CttTerm × p:?CubicalContext × R[p;t]) List|
(||L|| = ||bts|| ∈ ℤ)
∧ (∀i:ℕ||L||. ((fst(L[i])) = (snd(bts[i])) ∈ term(CttOp)))
∧ (∀i:ℕ||L||. ((fst(snd(L[i]))) = Q[p;f;fst(bts[i]);firstn(i;L)] ∈ ?CubicalContext))}
⟶ R[p;mkwfterm(f;bts)]]. ∀[t:CttTerm]. ∀[p:?CubicalContext].
(term-accum(t with p)
p,f,vs,tr.Q[p;f;vs;tr]
varterm(x) with p ⇒ varcase[p;x]
mkterm(f,bts) with p ⇒ trs.mktermcase[p;f;bts;trs] ∈ R[p;t])
Definition: ctt-meaning-type
ctt-meaning-type{i:l}(X;t) ==
if isvarterm(t) then cttTerm(X)
if ctt-op-sort(term-opr(t)) =a "fibrant" then cttType(X)
if ctt-op-sort(term-opr(t)) =a "type" then lvl:ℕ4 × {X ⊢lvl _}
else cttTerm(X)
fi
Lemma: ctt-meaning-type_wf
∀[X:⊢''']. ∀[t:CttTerm]. (ctt-meaning-type{i:l}(X;t) ∈ 𝕌{i'''''})
Definition: ctt_meaning
[[ctxt;t]] == Provisional''''(ctt-meaning-type{i:l}(context-set(ctxt);t)) supposing context-ok(ctxt)
Lemma: ctt_meaning_wf
∀[ctxt:?CubicalContext]. ∀[t:CttTerm]. ([[ctxt;t]] ∈ 𝕌{i'''''})
Lemma: ctt_meaning_functionality
∀[ctxt:?CubicalContext]. ∀[t,t':CttTerm]. [[ctxt;t]] = [[ctxt;t']] ∈ 𝕌{i'''''} supposing ctt-eq{i:l}(t;t')
Definition: ctt-meaning-triple
CttMeaningTriple == X:?CubicalContext × bt:varname() List × CttTerm × [[X;snd(bt)]]
Lemma: ctt-meaning-triple_wf
CttMeaningTriple ∈ 𝕌{i'''''}
Definition: ctt-is-term
ctt-is-term(t) == isvarterm(t) ∨bctt-op-sort(term-opr(t)) =a "term"
Lemma: ctt-is-term_wf
∀[t:term(CttOp)]. (ctt-is-term(t) ∈ 𝔹)
Lemma: assert-ctt-is-term
∀[t:CttTerm]
∀X:?CubicalContext. [[X;t]] ⊆r Provisional''''(cttTerm(context-set(X))) supposing context-ok(X)
supposing ↑ctt-is-term(t)
Lemma: ctt-kind-0
∀[t:CttTerm]
∀X:?CubicalContext. [[X;t]] ⊆r Provisional''''(cttTerm(context-set(X))) supposing context-ok(X)
supposing ctt-kind(t) = 0 ∈ ℤ
Definition: ctt-is-fibrant
ctt-is-fibrant(t) == (¬bisvarterm(t)) ∧b ctt-op-sort(term-opr(t)) =a "fibrant"
Lemma: ctt-is-fibrant_wf
∀[t:term(CttOp)]. (ctt-is-fibrant(t) ∈ 𝔹)
Lemma: assert-ctt-is-fibrant
∀[t:CttTerm]
∀X:?CubicalContext. [[X;t]] ⊆r Provisional''''(cttType(context-set(X))) supposing context-ok(X)
supposing ↑ctt-is-fibrant(t)
Lemma: ctt-kind-1
∀[t:CttTerm]
∀X:?CubicalContext. [[X;t]] ⊆r Provisional''''(cttType(context-set(X))) supposing context-ok(X)
supposing ctt-kind(t) = 1 ∈ ℤ
Definition: ctt-subterm-context
ctt-subterm-context{i:l}(ctxt;f;n;vs;m) ==
if (ctt-opr-is(f;"pi") ∨bctt-opr-is(f;"sigma") ∨bctt-opr-is(f;"fst") ∨bctt-opr-is(f;"snd") ∨bctt-opr-is(f;"lambda"))
∧b (n =z 1)
then ctxt; hd(vs):fst(m)
if ctt-opr-is(f;"case") ∧b ((n =z 2) ∨b(n =z 3)) then restrict-cubical-context{i:l}(ctxt;m)
if ctt-opr-is(f;"Glue") then if (n =z 2) ∨b(n =z 3) then restrict-cubical-context{i:l}(ctxt;m) else OK(ctxt) fi
if (ctt-opr-is(f;"pair") ∨bctt-opr-is(f;"apply")) ∧b (n =z 1) then ctxt; hd(vs):fst(m)
if ctt-opr-is(f;"pathabs") ∧b (n =z 1) then ctxt; hd(vs):OK(<0, 𝕀>)
if ctt-opr-is(f;"comp")
then if (n =z 1) then ctxt; hd(vs):OK(<0, 𝕀>)
if (n =z 2) then restrict-cubical-context{i:l}(ctxt;m),hd(vs):I
else OK(ctxt)
fi
if ctt-opr-is(f;"glue") ∧b ((n =z 2) ∨b(n =z 3)) then restrict-cubical-context{i:l}(ctxt;m)
if ctt-opr-is(f;"unglue") ∧b ((n =z 2) ∨b(n =z 3)) then restrict-cubical-context{i:l}(ctxt;m)
else OK(ctxt)
fi
Lemma: ctt-subterm-context_wf
∀[ctxt:CubicalContext]. ∀[f:CttOp]. ∀[n:ℕ]. ∀[vs:varname() List].
∀[m:Provisional''''(cttTerm(fst(ctxt)))
supposing ↑(((ctt-opr-is(f;"Glue") ∨bctt-opr-is(f;"case") ∨bctt-opr-is(f;"unglue")) ∧b ((n =z 2) ∨b(n =z 3)))
∨b(ctt-opr-is(f;"comp") ∧b (n =z 2))
∨b(ctt-opr-is(f;"glue") ∧b ((n =z 2) ∨b(n =z 3) ∨b(n =z 4)))) ⋂ Provisional''''(ctt-type-meaning1{i:l}(fst(ctxt)))
supposing ↑((ctt-opr-is(f;"pi")
∨bctt-opr-is(f;"sigma")
∨bctt-opr-is(f;"lambda")
∨bctt-opr-is(f;"apply")
∨bctt-opr-is(f;"pair")
∨bctt-opr-is(f;"fst")
∨bctt-opr-is(f;"snd"))
∧b (n =z 1))].
ctt-subterm-context{i:l}(ctxt;f;n;vs;m) ∈ ?CubicalContext
supposing (↑(((ctt-opr-is(f;"pi")
∨bctt-opr-is(f;"sigma")
∨bctt-opr-is(f;"lambda")
∨bctt-opr-is(f;"apply")
∨bctt-opr-is(f;"pair")
∨bctt-opr-is(f;"fst")
∨bctt-opr-is(f;"snd")
∨bctt-opr-is(f;"pathabs")
∨bctt-opr-is(f;"comp"))
∧b (n =z 1))
∨b(ctt-opr-is(f;"comp") ∧b (n =z 2))))
⇒ 0 < ||vs||
Definition: provisional-subterm-context
provisional-subterm-context{i:l}(X;f;vs;L) ==
bind-provision(X;ctxt.ctt-subterm-context{i:l}(ctxt;f;||L||;vs;if (ctt-opr-is(f;"Glue")
∧b ((||L|| =z 2) ∨b(||L|| =z 3)))
∨b(ctt-opr-is(f;"case") ∧b (||L|| =z 3))
∨b(ctt-opr-is(f;"unglue")
∧b ((||L|| =z 2) ∨b(||L|| =z 3)))
then snd(snd(L[1]))
else snd(snd(L[0]))
fi ))
Lemma: provisional-subterm-context_wf
∀[X:?CubicalContext]. ∀[f:CttOp]. ∀[vs:varname() List]. ∀[L:{L:(a:?CubicalContext
× bt:varname() List × CttTerm
× [[a;snd(bt)]]) List|
||L|| < ||ctt-arity(f)||
∧ (||vs|| = (fst(ctt-arity(f)[||L||])) ∈ ℤ)
∧ (∀i:ℕ||L||
(ctt-kind(snd(fst(snd(L[i]))))
= (snd(ctt-arity(f)[i]))
∈ ℤ))
∧ ((||L|| = 1 ∈ ℤ)
⇒ ((↑ctt-opr-is(f;"pi"))
∨ (↑ctt-opr-is(f;"sigma"))
∨ (↑ctt-opr-is(f;"lambda"))
∨ (↑ctt-opr-is(f;"apply"))
∨ (↑ctt-opr-is(f;"pair"))
∨ (↑ctt-opr-is(f;"fst"))
∨ (↑ctt-opr-is(f;"snd")))
⇒ ((fst(L[0])) = X ∈ ?CubicalContext))
∧ ((||L|| = 2 ∈ ℤ)
⇒ ((↑ctt-opr-is(f;"Glue"))
∨ (↑ctt-opr-is(f;"case"))
∨ (↑ctt-opr-is(f;"comp"))
∨ (↑ctt-opr-is(f;"unglue"))
∨ (↑ctt-opr-is(f;"glue")))
⇒ ((fst(L[0])) = X ∈ ?CubicalContext))
∧ ((||L|| = 2 ∈ ℤ)
⇒ ((↑ctt-opr-is(f;"Glue")) ∨ (↑ctt-opr-is(f;"unglue")))
⇒ ((fst(L[1])) = X ∈ ?CubicalContext))
∧ ((||L|| = 3 ∈ ℤ)
⇒ ((↑ctt-opr-is(f;"Glue"))
∨ (↑ctt-opr-is(f;"case"))
∨ (↑ctt-opr-is(f;"unglue")))
⇒ ((fst(L[1])) = X ∈ ?CubicalContext))
∧ ((||L|| = 3 ∈ ℤ)
⇒ (↑ctt-opr-is(f;"glue"))
⇒ ((fst(L[0])) = X ∈ ?CubicalContext))
∧ ((||L|| = 4 ∈ ℤ)
⇒ (↑ctt-opr-is(f;"glue"))
⇒ ((fst(L[0])) = X ∈ ?CubicalContext))} ].
(provisional-subterm-context{i:l}(X;f;vs;L) ∈ ?CubicalContext)
Definition: ctt-binder-context
ctt-binder-context ==
λX,vs,f,trs,bt. if ||trs|| <z ||ctt-arity(f)||
∧b (||fst(bt)|| =z fst(ctt-arity(f)[||trs||]))
∧b bdd-all(||trs||;i.(ctt-kind(snd(fst(snd(trs[i])))) =z snd(ctt-arity(f)[i])))
then provisional-subterm-context{i:l}(X;f;fst(bt);trs)
else X
fi
Lemma: ctt-binder-context_wf
ctt-binder-context ∈ ?CubicalContext
⟶ (varname() List)
⟶ CttOp
⟶ very-dep-fun(?CubicalContext;varname() List × CttTerm;a,bt.[[a;snd(bt)]])
Lemma: wfterm-accum_wf_ctt1
∀[m:X:?CubicalContext
⟶ vs:(varname() List)
⟶ f:CttOp
⟶ L:{L:CttMeaningTriple List|
vdf-eq(?CubicalContext;ctt-binder-context X vs f;L)
∧ (↑wf-term(λf.ctt-arity(f);λt.ctt-kind(t);mkterm(f;map(λx.(fst(snd(x)));L))))}
⟶ [[X;mkterm(f;map(λx.(fst(snd(x)));L))]]]. ∀[varcase:X:?CubicalContext
⟶ vs:(varname() List)
⟶ v:{v:varname()| ¬(v = nullvar() ∈ varname())}
⟶ [[X;varterm(v)]]]. ∀[X:?CubicalContext]. ∀[t:CttTerm].
(wfterm-accum(X;t)
p,vs,v.varcase[p;vs;v]
prm,vs,f,L.m[prm;vs;f;L]
p0,ws,op,sofar,bt.ctt-binder-context[p0;ws;op;sofar;bt] ∈ [[X;t]])
Definition: eq0-meaning
eq0-meaning{i:l}(Z) ==
let X,t,m = Z in
provision(allowed(m) ∧ type(allow(m))=𝕀; mk-ctt-term-mng(level(allow(m)); 𝔽; (term(allow(m))=0)))
Definition: eq1-meaning
eq1-meaning{i:l}(Z) ==
let X,t,m = Z in
provision(allowed(m) ∧ type(allow(m))=𝕀; mk-ctt-term-mng(level(allow(m)); 𝔽; (term(allow(m))=1)))
Definition: meet-meaning
meet-meaning{i:l}(A; B) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
provision(allowed(ma) ∧ allowed(mb) ∧ type(allow(ma))=𝔽 ∧ type(allow(mb))=𝔽;
mk-ctt-term-mng(levelsup(level(allow(ma));level(allow(mb))); 𝔽; (term(allow(ma)) ∧ term(allow(mb)))))
Definition: join-meaning
join-meaning{i:l}(A; B) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
provision(allowed(ma) ∧ allowed(mb) ∧ type(allow(ma))=𝔽 ∧ type(allow(mb))=𝔽;
mk-ctt-term-mng(levelsup(level(allow(ma));level(allow(mb))); 𝔽; (term(allow(ma)) ∨ term(allow(mb)))))
Definition: max-meaning
max-meaning{i:l}(A; B) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
provision(allowed(ma) ∧ allowed(mb) ∧ type(allow(ma))=𝕀 ∧ type(allow(mb))=𝕀;
mk-ctt-term-mng(levelsup(level(allow(ma));level(allow(mb))); 𝕀; term(allow(ma)) ∨ term(allow(mb))))
Definition: min-meaning
min-meaning{i:l}(A; B) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
provision(allowed(ma) ∧ allowed(mb) ∧ type(allow(ma))=𝕀 ∧ type(allow(mb))=𝕀;
mk-ctt-term-mng(levelsup(level(allow(ma));level(allow(mb))); 𝕀; term(allow(ma)) ∧ term(allow(mb))))
Definition: inv-meaning
inv-meaning{i:l}(A) ==
let Xa,a,ma = A in
provision(allowed(ma) ∧ type(allow(ma))=𝕀; mk-ctt-term-mng(level(allow(ma)); 𝕀; 1-(term(allow(ma)))))
Definition: pi-meaning
pi-meaning{i:l}(A; B) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
provision(allowed(ma) ∧ allowed(mb); cttType(levl= levelsup(level(allow(ma));level(allow(mb)))
type= Πtype(allow(ma)) type(allow(mb))
comp= pi_comp(context-set(Xa);type(allow(ma));comp(allow(ma));
comp(allow(mb)))))
Definition: sigma-meaning
sigma-meaning{i:l}(A; B) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
provision(allowed(ma) ∧ allowed(mb); cttType(levl= levelsup(level(allow(ma));level(allow(mb)))
type= Σ type(allow(ma)) type(allow(mb))
comp= sigma_comp(comp(allow(ma));comp(allow(mb)))))
Definition: path-meaning
path-meaning{i:l}(A; B; C) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
let Xc,c,mc = C in
provision(allowed(ma) ∧ allowed(mb) ∧ allowed(mc) ∧ type(allow(mb))=type(allow(ma)) ∧ type(allow(mc))=type(allow(ma));
cttType(levl= level(allow(ma))
type= (Path_type(allow(ma)) term(allow(mb)) term(allow(mc)))
comp= path_comp(context-set(Xa);type(allow(ma));term(allow(mb));term(allow(mc));
comp(allow(ma)))))
Definition: pathapp-meaning
pathapp-meaning{i:l}(A; B; C) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
let Xc,c,mc = C in
provision(allowed(ma)
∧ allowed(mb)
∧ allowed(mc)
∧ type(allow(mc))=𝕀
∧ ({context-set(Xa) ⊢ _:type(allow(mb))} ⊆r {context-set(Xa) ⊢ _:Path(type(allow(ma)))});
mk-ctt-term-mng(level(allow(ma)); type(allow(ma)); term(allow(mb)) @ term(allow(mc))))
Definition: pathabs-meaning
pathabs-meaning{i:l}(A; B) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
provision(allowed(ma) ∧ allowed(mb) ∧ type(allow(mb))=(type(allow(ma)))p; let b = term(allow(mb)) in
mk-ctt-term-mng(level(allow(ma));
(Path_type(allow(ma))
(b)[0(𝕀)]
(b)[1(𝕀)]); <>(b))\000C)
Definition: apply-meaning
apply-meaning{i:l}(A; B; C) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
let Xc,c,mc = C in
provision(allowed(ma) ∧ allowed(mb) ∧ allowed(mc) ∧ type(allow(mc))=Πtype(allow(ma)) type(allow(mb));
mk-ctt-term-mng(level(allow(mb)); (type(allow(mb)))[term(allow(ma))];
cubical-apply(term(allow(mc));term(allow(ma)))))
Definition: lambda-meaning
lambda-meaning{i:l}(A; B) ==
let Xa,dXa,ma = A in
let Xb,b,mb = B in
provision(allowed(ma) ∧ allowed(mb); mk-ctt-term-mng(levelsup(level(allow(ma));level(allow(mb)));
Πtype(allow(ma)) type(allow(mb)); (λterm(allow(mb)))))
Definition: pair-meaning
pair-meaning{i:l}(A; B; C) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
let Xc,c,mc = C in
provision(allowed(ma) ∧ allowed(mb) ∧ allowed(mc) ∧ type(allow(mc))=(type(allow(mb)))[term(allow(ma))];
mk-ctt-term-mng(levelsup(level(allow(ma));level(allow(mb))); Σ type(allow(ma)) type(allow(mb));
cubical-pair(term(allow(ma));term(allow(mc)))))
Definition: fst-meaning
fst-meaning{i:l}(A; B; C) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
let Xc,c,mc = C in
provision(allowed(ma) ∧ allowed(mb) ∧ allowed(mc) ∧ type(allow(mc))=Σ type(allow(ma)) type(allow(mb));
mk-ctt-term-mng(level(allow(ma)); type(allow(ma)); term(allow(mc)).1))
Definition: snd-meaning
snd-meaning{i:l}(A; B; C) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
let Xc,c,mc = C in
provision(allowed(ma) ∧ allowed(mb) ∧ allowed(mc) ∧ type(allow(mc))=Σ type(allow(ma)) type(allow(mb));
mk-ctt-term-mng(level(allow(mb)); (type(allow(mb)))[term(allow(mc)).1]; term(allow(mc)).2))
Definition: comp-meaning
comp-meaning{i:l}(A; B; C; D) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
let Xc,c,mc = C in
let Xd,d,md = D in
provision(allowed(ma)
∧ type(allow(ma))=𝔽
∧ allowed(mb)
∧ allowed(mc)
∧ allowed(md)
∧ type(allow(mc))=type(allow(mb))
∧ type(allow(md))=(type(allow(mb)))[0(𝕀)]
∧ ((term(allow(mc)))[0(𝕀)]
= term(allow(md))
∈ {context-set(Xa), term(allow(ma)) ⊢ _:(type(allow(mb)))[0(𝕀)]});
mk-ctt-term-mng(level(allow(mb)); (type(allow(mb)))[1(𝕀)];
comp comp(allow(mb)) [term(allow(ma)) ⊢→ term(allow(mc))] term(allow(md))))
Definition: case-meaning
case-meaning{i:l}(A; B; C; D) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
let Xc,c,mc = C in
let Xd,d,md = D in
provision((allowed(ma) ∧ allowed(mb))
∧ (type(allow(ma))=𝔽
∧ type(allow(mb))=𝔽
∧ context-set(Xa) ⊢ ((term(allow(ma)) ∧ term(allow(mb))) ⇒ 0(𝔽))
∧ context-set(Xa) ⊢ (1(𝔽) ⇒ (term(allow(ma)) ∨ term(allow(mb)))))
∧ allowed(mc)
∧ allowed(md); cttType(levl= levelsup(level(allow(mc));level(allow(md)))
type= (if term(allow(ma)) then type(allow(mc)) else type(allow(md)))
comp= case-type-comp(context-set(Xa);
term(allow(ma));
term(allow(mb));
type(allow(mc));
type(allow(md));
comp(allow(mc));
comp(allow(md)))))
Definition: Glue-meaning
Glue-meaning{i:l}(A; B; C; D) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
let Xc,c,mc = C in
let Xd,d,md = D in
provision((allowed(ma) ∧ allowed(mb))
∧ type(allow(mb))=𝔽
∧ (allowed(mc) ∧ allowed(md))
∧ let A = type(allow(ma)) in
let phi = term(allow(mb)) in
let T = type(allow(mc)) in
type(allow(md))=Equiv(T;A); let A = type(allow(ma)) in
let cA = comp(allow(ma)) in
let phi = term(allow(mb)) in
let T = type(allow(mc)) in
let cT = comp(allow(mc)) in
let f = term(allow(md)) in
cttType(levl= levelsup(level(allow(ma));level(allow(mc)))
type= gluetype(context-set(Xa);A;phi;T;f)
comp= comp(Glue [phi ⊢→ (T, f)] A) ))
Definition: glue-meaning
glue-meaning{i:l}(A; B; C; D) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
let Xc,c,mc = C in
let Xd,d,md = D in
provision((allowed(ma) ∧ allowed(mb))
∧ type(allow(ma))=𝔽
∧ (allowed(mc) ∧ allowed(md))
∧ let phi = term(allow(ma)) in
type(allow(md))=(type(allow(mc)) ⟶ type(allow(mb)))
∧ (app(term(allow(md)); term(allow(mc)))
= term(allow(mb))
∈ {context-set(Xa), phi ⊢ _:type(allow(mb))});
let phi = term(allow(ma)) in
mk-ctt-term-mng(levelsup(level(allow(mb));level(allow(mc)));
Glue [phi ⊢→ (type(allow(mc));term(allow(md)))] type(allow(mb));
glue [phi ⊢→ term(allow(mc))] term(allow(mb))))
Definition: unglue-meaning
unglue-meaning{i:l}(A; B; C; D; E) ==
let Xa,a,ma = A in
let Xb,b,mb = B in
let Xc,c,mc = C in
let Xd,d,md = D in
let Xe,e,me = E in
provision((allowed(ma) ∧ allowed(mb))
∧ type(allow(mb))=𝔽
∧ (allowed(mc) ∧ allowed(md) ∧ allowed(me))
∧ let A = type(allow(ma)) in
let phi = term(allow(mb)) in
let T = type(allow(mc)) in
let G = fst(allow(me)) in
type(allow(md))=(T ⟶ A) ∧ type(allow(me))=Glue [phi ⊢→ (T;term(allow(md)))] A;
mk-ctt-term-mng(level(allow(ma)); type(allow(ma)); unglue(term(allow(me)))))
Definition: decode-meaning
decode-meaning{i:l}(n; A) ==
let Xa,a,ma = A in
provision(allowed(ma) ∧ type(allow(ma))=if (n =z 0) then c𝕌 if (n =z 1) then c𝕌' else c𝕌'' fi ;
let t = term(allow(ma)) in
cttType(levl= n
type= decode(t)
comp= CompFun(t)))
Definition: encode-meaning
encode-meaning{i:l}(n; A) ==
let Xa,a,ma = A in
provision(allowed(ma) ∧ level(allow(ma)) < n; mk-ctt-term-mng(n; if (n =z 1) then c𝕌
if (n =z 2) then c𝕌'
else c𝕌''
fi ; encode(type(allow(ma));
cfun-to-cop(context-set(Xa);type(allow(ma))
;comp(allow(ma))))))
Definition: univ-meaning
univ-meaning{i:l}(n) ==
OK(if (n =z 0) then cttType(levl= 1type= c𝕌comp= compU())
if (n =z 1) then cttType(levl= 2type= c𝕌'comp= compU())
else cttType(levl= 3
type= c𝕌''
comp= compU())
fi )
Definition: ctt-op-meaning
ctt-op-meaning{i:l}(X; vs; f; L) ==
let k,v = f
in if k =a "nType" then OK(<0, discr(v), discrete_comp(context-set(X);v)>)
if k =a "nterm" then OK(let T,t = v in mk-ctt-term-mng(0; discr(T); discr(t)))
if k =a "opid"
then if v =a "0" then OK(mk-ctt-term-mng(0; 𝕀; 0(𝕀)))
if v =a "1" then OK(mk-ctt-term-mng(0; 𝕀; 1(𝕀)))
if v =a "eq0" then eq0-meaning{i:l}(L[0])
if v =a "eq1" then eq1-meaning{i:l}(L[0])
if v =a "meet" then meet-meaning{i:l}(L[0]; L[1])
if v =a "join" then join-meaning{i:l}(L[0]; L[1])
if v =a "max" then max-meaning{i:l}(L[0]; L[1])
if v =a "min" then min-meaning{i:l}(L[0]; L[1])
if v =a "I" then OK(<0, 𝕀>)
if v =a "F" then OK(cttType(levl= 0type= 𝔽comp= face-comp()))
if v =a "inv" then inv-meaning{i:l}(L[0])
if v =a "sigma" then sigma-meaning{i:l}(L[0]; L[1])
if v =a "pi" then pi-meaning{i:l}(L[0]; L[1])
if v =a "Glue" then Glue-meaning{i:l}(L[0]; L[1]; L[2]; L[3])
if v =a "case" then case-meaning{i:l}(L[0]; L[1]; L[2]; L[3])
if v =a "path" then path-meaning{i:l}(L[0]; L[1]; L[2])
if v =a "decode1" then decode-meaning{i:l}(0; L[0])
if v =a "decode2" then decode-meaning{i:l}(1; L[0])
if v =a "decode3" then decode-meaning{i:l}(2; L[0])
if v =a "encode1" then encode-meaning{i:l}(1; L[0])
if v =a "encode2" then encode-meaning{i:l}(2; L[0])
if v =a "encode3" then encode-meaning{i:l}(3; L[0])
if v =a "pathabs" then pathabs-meaning{i:l}(L[0]; L[1])
if v =a "pathap" then pathapp-meaning{i:l}(L[0]; L[1]; L[2])
if v =a "lambda" then lambda-meaning{i:l}(L[0]; L[1])
if v =a "apply" then apply-meaning{i:l}(L[0]; L[1]; L[2])
if v =a "pair" then pair-meaning{i:l}(L[0]; L[1]; L[2])
if v =a "fst" then fst-meaning{i:l}(L[0]; L[1]; L[2])
if v =a "snd" then snd-meaning{i:l}(L[0]; L[1]; L[2])
if v =a "glue" then glue-meaning{i:l}(L[0]; L[1]; L[2]; L[3])
if v =a "unglue" then unglue-meaning{i:l}(L[0]; L[1]; L[2]; L[3]; L[4])
if v =a "comp" then comp-meaning{i:l}(L[0]; L[1]; L[2]; L[3])
else provision(False; ⋅)
fi
else univ-meaning{i:l}(v)
fi
Lemma: ctt-op-meaning_wf
∀[X:?CubicalContext]. ∀[vs:varname() List]. ∀[f:CttOp].
∀[L:{L:CttMeaningTriple List|
vdf-eq(?CubicalContext;ctt-binder-context X vs f;L)
∧ (↑wf-term(λf.ctt-arity(f);λt.ctt-kind(t);mkterm(f;map(λx.(fst(snd(x)));L))))} ].
(ctt-op-meaning{i:l}(X; vs; f; L) ∈ [[X;mkterm(f;map(λx.(fst(snd(x)));L))]])
Definition: ctt-context-term-mng
ctt-context-term-mng{i:l}(X;t) ==
wfterm-accum(X;t)
ctxt,vs,v.cubical-context-lookup(ctxt;v)
ctxt,vs,f,L.ctt-op-meaning{i:l}(ctxt; vs; f; L)
p0,ws,op,sofar,bt.ctt-binder-context[p0;ws;op;sofar;bt]
Lemma: ctt-context-term-mng_wf
∀[X:?CubicalContext]. ∀[t:CttTerm]. (ctt-context-term-mng{i:l}(X;t) ∈ [[X;t]])
Comment: intersection type doc
We can define a cubical (higher dimensional) analogue ⋂A B
of the intersection type.⋅
Definition: cubical-isect-family
cubical-isect-family(X;A;B;I;a) ==
{w:J:fset(ℕ) ⟶ f:J ⟶ I ⟶ (⋂u:A(f(a)). B((f(a);u)))|
∀J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀u:A(f(a)). ((w J f (f(a);u) g) = (w K f ⋅ g) ∈ B(g((f(a);u))))}
Lemma: cubical-isect-family_wf
∀[X:⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. (cubical-isect-family(X;A;B;I;a) ∈ Type)
Lemma: cubical-isect-family-comp
∀X,Delta:⊢. ∀s:Delta ⟶ X. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀a:Delta(I). ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}.
∀w:cubical-isect-family(X;A;B;I;(s)a).
(λK,g. (w K f ⋅ g) ∈ cubical-isect-family(X;A;B;J;(s)f(a)))
Lemma: csm-cubical-isect-family
∀X,Delta:⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta ⟶ X. ∀I:fset(ℕ). ∀a:Delta(I).
(cubical-isect-family(X;A;B;I;(s)a) = cubical-isect-family(Delta;(A)s;(B)(s o p;q);I;a) ∈ Type)
Definition: cubical-isect
⋂A B == <λI,a. cubical-isect-family(X;A;B;I;a), λI,J,f,a,w,K,g. (w K f ⋅ g)>
Lemma: cubical-isect_wf
∀[X:⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. X ⊢ ⋂A B
Lemma: csm-cubical-isect
∀X,Delta:⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta ⟶ X. ((⋂A B)s = ⋂(A)s (B)(s o p;q) ∈ {Delta ⊢ _})
Lemma: cubical-isect-subset-adjoin2
∀[X,phi,A,B,C,D:Top]. (⋂A B ~ ⋂A B)
Definition: cubical-isect-elim
cubical-isect-elim(t) == λI,a. (t I (fst(a)) I 1)
Lemma: cubical-isect-elim_wf
∀[X:⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[t:{X ⊢ _:⋂A B}]. (cubical-isect-elim(t) ∈ {X.A ⊢ _:B})
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