Nuprl Lemma : comp-to-fill_wf2
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)]. (comp-to-fill(Gamma;cA) ∈ Gamma ⊢ Filling(A))
Proof
Definitions occuring in Statement :
comp-to-fill: comp-to-fill(Gamma;cA),
filling-structure: Gamma ⊢ Filling(A),
composition-structure: Gamma ⊢ Compositon(A),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
composition-structure: Gamma ⊢ Compositon(A),
filling-structure: Gamma ⊢ Filling(A),
prop: ℙ,
uniform-filling-function: uniform-filling-function{j:l, i:l}(Gamma;A;fill),
all: ∀x:A. B[x],
comp-to-fill: comp-to-fill(Gamma;cA),
uimplies: b supposing a,
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
csm+: tau+,
csm-comp: G o F,
guard: {T},
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
true: True,
squash: ↓T,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
same-cubical-type: Gamma ⊢ A = B,
partial-term-0: u[0],
csm-ap: (s)x,
csm-adjoin: (s;u),
csm-id: 1(X),
compose: f o g,
cc-adjoin-cube: (v;u),
csm-m: m,
csm-id-adjoin: [u],
interval-0: 0(𝕀),
same-cubical-term: X ⊢ u=v:A,
context-subset: Gamma, phi,
pi1: fst(t),
csm-ap-term: (t)s,
cubical-term-at: u(a),
case-term: (u ∨ v),
cube-context-adjoin: X.A,
pi2: snd(t),
face-zero: (i=0),
not: ¬A,
false: False,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
bdd-distributive-lattice: BoundedDistributiveLattice,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
btrue: tt,
bfalse: ff,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
record-update: r[x := v],
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
face-lattice: face-lattice(T;eq),
face_lattice: face_lattice(I),
record-select: r.x,
lattice-point: Point(l),
face-presheaf: 𝔽,
functor-ob: ob(F),
I_cube: A(I),
face-type: 𝔽,
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
dM0: 0,
interval-presheaf: 𝕀,
free-dist-lattice: free-dist-lattice(T; eq),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
dM: dM(I),
DeMorgan-algebra: DeMorganAlgebra,
respects-equality: respects-equality(S;T),
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
interval-1: 1(𝕀),
face-term-implies: Gamma ⊢ (phi ⇒ psi),
composition-function: composition-function{j:l,i:l}(Gamma;A),
face-or: (a ∨ b)
Lemmas referenced :
comp-to-fill_wf,
uniform-filling-function_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf,
interval-0_wf,
partial-term-0_wf,
constrained-cubical-term-eqcd,
istype-cubical-term,
context-subset_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf_interval,
thin-context-subset,
cube_set_map_wf,
csm+_wf_interval,
csm-comp_wf,
csm-m_wf,
face-or_wf,
face-zero_wf,
cc-snd_wf,
context-subset-map,
csm-id-adjoin_wf-interval-0,
cubical_set_cumulativity-i-j,
cc-fst_wf,
sub_cubical_set_self,
subset-cubical-term,
context-subset-is-subset,
true_wf,
squash_wf,
cubical-term_wf,
csm-comp-type,
cubical-type-cumulativity2,
0-comp-cc-fst-comp-m,
interval-1_wf,
iff_weakening_equal,
subtype_rel_self,
csm-ap-id-type,
istype-universe,
equal_wf,
csm-m-comp-1,
csm-comp-term,
face-and_wf,
csm-ap-term-wf-subset,
face-term-and-implies1,
face-term-and-implies2,
face-term-implies-subset,
sub_cubical_set-cumulativity1,
csm-subset-domain,
csm-context-subset-subtype2,
context-iterated-subset,
case-term_wf,
csm-m-comp-0,
nat_wf,
fset_wf,
I_cube_wf,
I_cube_pair_redex_lemma,
face-or-eq-1,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
lattice-join_wf,
lattice-meet_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
assert-fl-eq,
eqtt_to_assert,
lattice-1_wf,
face_lattice_wf,
lattice-point_wf,
cubical-term-at_wf,
fl-eq_wf,
cubical_type_at_pair_lemma,
cubical-type-at_wf,
istype-cubical-type-at,
bdd-lattice_wf,
bdd-distributive-lattice_wf,
DeMorgan-algebra_wf,
subtype_rel_transitivity,
DeMorgan-algebra-subtype,
bdd-distributive-lattice-subtype-bdd-lattice,
dM_wf,
lattice-0-meet,
interval-type-at,
csm-ap-term-at,
dM0_wf,
DeMorgan-algebra-axioms_wf,
DeMorgan-algebra-structure-subtype,
DeMorgan-algebra-structure_wf,
cc-adjoin-cube_wf,
subtype-respects-equality,
cubical-term-equal,
cubical-term-eqcd,
csm-face-or,
csm-comp-assoc,
csm-ap-id-term,
context-adjoin-subset2,
cc-fst-comp-csm-m-term,
context-adjoin-subset4,
csm+_wf,
csm-interval-type,
face-term-implies-same,
csm-id-adjoin_wf-interval-1,
face-or-at,
csm-face-zero,
free-DeMorgan-lattice_wf,
subtype_rel-equal,
names-deq_wf,
names_wf,
dm-neg_wf,
dM-to-FL_wf,
csm+-comp-m,
csm-case-term,
sub_cubical_set_wf,
csm-constrained-cubical-term,
face-type-at,
cube_set_map_cumulativity-i-j,
subset-cubical-type,
csm-id-comp,
csm-adjoin_wf,
constrained-cubical-term_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
universeIsType,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
lambdaFormation_alt,
independent_isectElimination,
Error :memTop,
dependent_functionElimination,
because_Cache,
applyEquality,
independent_functionElimination,
equalityIstype,
hyp_replacement,
lambdaEquality_alt,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
applyLambdaEquality,
productElimination,
universeEquality,
productIsType,
independent_pairFormation,
sqequalBase,
cumulativity,
functionExtensionality,
voidElimination,
promote_hyp,
dependent_pairFormation_alt,
isectEquality,
productEquality,
equalityElimination,
unionElimination,
dependent_pairEquality_alt
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} Compositon(A)].
(comp-to-fill(Gamma;cA) \mmember{} Gamma \mvdash{} Filling(A))
Date html generated:
2020_05_20-PM-04_47_33
Last ObjectModification:
2020_05_02-AM-10_49_40
Theory : cubical!type!theory
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