Nuprl Lemma : filling_term_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ u[0]]}].
(fill cA [phi ⊢→ u] a0 ∈ {Gamma.𝕀 ⊢ _:A[(phi)p |⟶ u]})
Proof
Definitions occuring in Statement :
filling_term: fill cA [phi ⊢→ u] a0,
composition-op: Gamma ⊢ CompOp(A),
partial-term-0: u[0],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
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,
filling_term: fill cA [phi ⊢→ u] a0,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
fill_term_wf,
cube-context-adjoin_wf,
interval-type_wf,
constrained-cubical-term_wf,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm-id-adjoin_wf-interval-0,
partial-term-0_wf,
cubical-term_wf,
context-subset_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf,
thin-context-subset,
composition-op_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
because_Cache,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
Error :memTop
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)].
\mforall{}[u:\{Gamma.\mBbbI{}, (phi)p \mvdash{} \_:A\}]. \mforall{}[a0:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})][phi |{}\mrightarrow{} u[0]]\}].
(fill cA [phi \mvdash{}\mrightarrow{} u] a0 \mmember{} \{Gamma.\mBbbI{} \mvdash{} \_:A[(phi)p |{}\mrightarrow{} u]\})
Date html generated:
2020_05_20-PM-04_53_33
Last ObjectModification:
2020_04_10-AM-11_32_09
Theory : cubical!type!theory
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