Nuprl Lemma : glue-type-1'
∀[Gamma:j⊢]. ∀[A,T:{Gamma ⊢ _}]. ∀[w:{Gamma ⊢ _:(T ⟶ A)}]. ∀[phi:{Gamma ⊢ _:𝔽}].
Gamma ⊢ Glue [phi ⊢→ (T;w)] A = T ∈ {Gamma ⊢ _} supposing phi = 1(𝔽) ∈ {Gamma ⊢ _:𝔽}
Proof
Definitions occuring in Statement :
glue-type: Glue [phi ⊢→ (T;w)] A,
face-1: 1(𝔽),
face-type: 𝔽,
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
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,
prop: ℙ,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
glue-type-1,
thin-context-subset,
face-1_wf,
subset-cubical-term2,
context-subset_wf,
subset-context-1,
cubical-fun_wf,
subset-cubical-type,
context-1-subset,
cubical-fun-subset,
istype-cubical-term,
cubical-type_wf,
cubical_set_wf,
face-type_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
glue-type_wf,
cubical-term-eqcd,
subset-cubical-term,
context-subset-is-subset,
subtype_rel_wf,
cubical-term_wf,
iff_weakening_equal,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
applyEquality,
independent_isectElimination,
sqequalRule,
Error :memTop,
inhabitedIsType,
universeIsType,
instantiate,
equalityIstype,
hyp_replacement,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A,T:\{Gamma \mvdash{} \_\}]. \mforall{}[w:\{Gamma \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}].
Gamma \mvdash{} Glue [phi \mvdash{}\mrightarrow{} (T;w)] A = T supposing phi = 1(\mBbbF{})
Date html generated:
2020_05_20-PM-05_42_59
Last ObjectModification:
2020_04_21-PM-07_28_47
Theory : cubical!type!theory
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