Nuprl 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})
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,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
same-cubical-type: Gamma ⊢ A = B,
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
glue-type-constraint,
istype-cubical-term,
context-subset_wf,
cubical-fun_wf,
thin-context-subset,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
glue-type_wf,
same-cubical-type_wf,
context-subset-term-subtype,
subtype_rel_wf,
squash_wf,
true_wf,
istype-universe,
cubical-term-eqcd,
iff_weakening_equal,
subset-cubical-term,
sub_cubical_set_self
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
universeIsType,
instantiate,
lambdaEquality_alt,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
sqequalRule,
equalityElimination,
imageElimination,
universeEquality,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
because_Cache
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\} \msubseteq{}r \{Gamma, phi \mvdash{} \_:T\})
Date html generated:
2020_05_20-PM-05_42_15
Last ObjectModification:
2020_04_21-PM-06_57_48
Theory : cubical!type!theory
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