Nuprl Lemma : csm-ap-term-wf-subset
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[A:{H ⊢ _}]. ∀[t:{H, phi ⊢ _:A}]. ∀[K:j⊢]. ∀[psi:{K ⊢ _:𝔽}]. ∀[B:{K ⊢ _}]. ∀[tau:K j⟶ H].
((t)tau ∈ {K, psi ⊢ _:B}) supposing (K, psi ⊢ B = (A)tau and K ⊢ (psi ⇒ (phi)tau))
Proof
Definitions occuring in Statement :
same-cubical-type: Gamma ⊢ A = B,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
context-subset: Gamma, phi,
face-type: 𝔽,
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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
same-cubical-type: Gamma ⊢ A = B,
and: P ∧ Q
Lemmas referenced :
csm-ap-term_wf,
context-subset_wf,
face-type_wf,
csm-face-type,
thin-context-subset,
context-subset-map,
face-term-implies-subtype,
csm-ap-type_wf,
same-cubical-type_wf,
face-term-implies_wf,
cube_set_map_wf,
cubical-type_wf,
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
because_Cache,
applyEquality,
independent_isectElimination,
universeIsType,
instantiate,
inhabitedIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
applyLambdaEquality,
setElimination,
rename,
productElimination,
lambdaEquality_alt,
hyp_replacement
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[phi:\{H \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{H \mvdash{} \_\}]. \mforall{}[t:\{H, phi \mvdash{} \_:A\}]. \mforall{}[K:j\mvdash{}]. \mforall{}[psi:\{K \mvdash{} \_:\mBbbF{}\}].
\mforall{}[B:\{K \mvdash{} \_\}]. \mforall{}[tau:K j{}\mrightarrow{} H].
((t)tau \mmember{} \{K, psi \mvdash{} \_:B\}) supposing (K, psi \mvdash{} B = (A)tau and K \mvdash{} (psi {}\mRightarrow{} (phi)tau))
Date html generated:
2020_05_20-PM-03_04_05
Last ObjectModification:
2020_04_06-PM-00_02_28
Theory : cubical!type!theory
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