Nuprl 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})
Proof
Definitions occuring in Statement :
glue-term: glue [phi ⊢→ t] a,
glue-type: Glue [phi ⊢→ (T;w)] A,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-app: app(w; u),
cubical-fun: (A ⟶ B),
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
glue-type: Glue [phi ⊢→ (T;w)] A,
csm-ap-type: (AF)s,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
pi1: fst(t),
true: True,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
glue-term: glue [phi ⊢→ t] a,
csm-ap-term: (t)s,
face-type: 𝔽,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
face-presheaf: 𝔽,
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
context-subset: Gamma, phi,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]}
Lemmas referenced :
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
csm-ap-type_wf,
glue-type_wf,
csm-ap-term_wf,
glue-term_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
context-subset_wf,
cube_set_map_wf,
constrained-cubical-term_wf,
cubical-app_wf_fun,
thin-context-subset,
istype-cubical-term,
cubical-fun_wf,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
csm-face-type,
context-subset-map,
cubical-term-eqcd,
csm-cubical-fun,
csm-cubical-app,
csm-constrained-cubical-term,
cubical_type_at_pair_lemma,
equal-glue-cube,
csm-ap_wf,
subtype_rel_self,
glue-cube_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type-at_wf,
csm-glue-type,
iff_weakening_equal,
fl-eq_wf,
cubical-term-at_wf,
lattice-point_wf,
face_lattice_wf,
lattice-1_wf,
eqtt_to_assert,
assert-fl-eq,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
I_cube_pair_redex_lemma,
csm-ap-term-at,
csm-context-subset-subtype2,
cube-set-restriction_wf,
csm-ap-restriction,
names-hom_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
functionExtensionality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
universeIsType,
inhabitedIsType,
Error :memTop,
lambdaEquality_alt,
hyp_replacement,
dependent_functionElimination,
setElimination,
rename,
universeEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productEquality,
cumulativity,
isectEquality,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
voidElimination,
dependent_set_memberEquality_alt,
independent_pairEquality,
setEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[T:\{Gamma, phi \mvdash{} \_\}].
\mforall{}[w:\{Gamma, phi \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[t:\{Gamma, phi \mvdash{} \_:T\}]. \mforall{}[a:\{Gamma \mvdash{} \_:A[phi |{}\mrightarrow{} app(w; t)]\}].
\mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} Gamma].
((glue [phi \mvdash{}\mrightarrow{} t] a)s = H \mvdash{} glue [(phi)s \mvdash{}\mrightarrow{} (t)s] (a)s)
Date html generated:
2020_05_20-PM-05_43_37
Last ObjectModification:
2020_04_21-PM-07_35_10
Theory : cubical!type!theory
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