Nuprl 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
Proof
Definitions occuring in Statement :
glue-type: Glue [phi ⊢→ (T;w)] A,
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _: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,
glue-type: Glue [phi ⊢→ (T;w)] A,
cubical-type: {X ⊢ _},
and: P ∧ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
istype-cubical-term,
context-subset_wf,
cubical-fun_wf,
thin-context-subset,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
glue-cube_wf,
I_cube_wf,
fset_wf,
nat_wf,
glue-morph_wf,
names-hom_wf,
cube-set-restriction_wf,
glue-morph-id,
glue-morph-comp,
nh-id_wf,
subtype_rel-equal,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube-set-restriction-id,
subtype_rel_self,
iff_weakening_equal,
nh-comp_wf,
cube-set-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
instantiate,
dependent_pairEquality_alt,
lambdaEquality_alt,
because_Cache,
functionIsType,
applyEquality,
independent_pairFormation,
lambdaFormation_alt,
dependent_functionElimination,
productElimination,
productIsType,
equalityIstype,
independent_isectElimination,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
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)\}].
Gamma \mvdash{} Glue [phi \mvdash{}\mrightarrow{} (T;w)] A
Date html generated:
2020_05_20-PM-05_40_45
Last ObjectModification:
2020_04_21-PM-06_01_57
Theory : cubical!type!theory
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