Nuprl 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 ⊢ _})
Proof
Definitions occuring in Statement :
glue-type: Glue [phi ⊢→ (T;w)] A,
context-subset: Gamma, phi,
face-type: 𝔽,
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,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
glue-cube: glue-cube(Gamma;A;phi;T;w;I;rho),
csm-ap-term: (t)s,
cubical-term-at: u(a),
cubical-type-at: A(a),
pi1: fst(t),
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,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
context-subset: Gamma, phi,
squash: ↓T,
true: True,
glue-equations: glue-equations(Gamma;A;phi;T;w;I;rho;t;a),
cubical-type: {X ⊢ _},
subset-iota: iota,
csm-ap-type: (AF)s,
csm-ap: (s)x,
cubical-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
glue-type: Glue [phi ⊢→ (T;w)] A,
glue-morph: glue-morph(Gamma;A;phi;T;w;I;rho;J;f;u)
Lemmas referenced :
csm-ap-term_wf,
face-type_wf,
csm-face-type,
context-subset_wf,
cube_set_map_wf,
istype-cubical-term,
cubical-fun_wf,
thin-context-subset,
cubical-type_wf,
cubical_set_wf,
context-subset-map,
cubical-type-equal2,
csm-ap-type_wf,
glue-type_wf,
cubical-term-eqcd,
csm-cubical-fun,
I_cube_wf,
fset_wf,
nat_wf,
fl-eq_wf,
cubical-term-at_wf,
subtype_rel_self,
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,
equal_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,
csm-ap-type-at,
csm-ap_wf,
I_cube_pair_redex_lemma,
cubical-type-at_wf,
names-hom_wf,
squash_wf,
true_wf,
istype-universe,
cube-set-restriction_wf,
csm-ap-restriction,
iff_weakening_equal,
csm-ap-term-at,
cube_set_restriction_pair_lemma,
nh-comp_wf,
cubical-term-at-comp-is-1,
csm-cubical-type-ap-morph,
cube-set-restriction-comp,
face-term-at-restriction-eq-1,
cubical-type-ap-morph_wf,
istype-cubical-type-at,
cubical_type_at_pair_lemma,
cube-set-restriction-id,
nh-id_wf,
glue-cube_wf,
equal-glue-cube,
glue-morph_wf,
subtype_rel-equal,
btrue_wf,
iff_imp_equal_bool,
iff_functionality_wrt_iff,
istype-true
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
hyp_replacement,
dependent_functionElimination,
lambdaFormation_alt,
rename,
because_Cache,
unionElimination,
equalityElimination,
productElimination,
productEquality,
cumulativity,
isectEquality,
setElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
independent_functionElimination,
voidElimination,
dependent_set_memberEquality_alt,
setEquality,
functionEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_pairEquality_alt,
functionExtensionality,
functionIsType,
independent_pairFormation,
independent_pairEquality
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{}[Z:j\mvdash{}]. \mforall{}[s:Z j{}\mrightarrow{} Gamma].
((Glue [phi \mvdash{}\mrightarrow{} (T;w)] A)s = Z \mvdash{} Glue [(phi)s \mvdash{}\mrightarrow{} ((T)s;(w)s)] (A)s)
Date html generated:
2020_05_20-PM-05_41_44
Last ObjectModification:
2020_04_21-PM-06_56_24
Theory : cubical!type!theory
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