Nuprl Lemma : case-term-wf3
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, (phi ∨ psi) ⊢ _}]. ∀[v:{Gamma, psi ⊢ _:A}].
∀[u:{Gamma, phi ⊢ _:A[psi |⟶ v]}].
((u ∨ v) ∈ {t:{Gamma, (phi ∨ psi) ⊢ _:A}| Gamma, psi ⊢ t=v:A} )
Proof
Definitions occuring in Statement :
case-term: (u ∨ v),
same-cubical-term: X ⊢ u=v:A,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-or: (a ∨ b),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
same-cubical-term: X ⊢ u=v:A,
and: P ∧ Q
Lemmas referenced :
context-subset-subtype-or,
constrained-cubical-term_wf,
cubical-type-cumulativity2,
context-subset_wf,
subset-cubical-term,
context-subset-is-subset,
face-type_wf,
subset-cubical-term2,
context-iterated-subset1,
context-subset-subtype-or2,
cubical-term_wf,
cubical_set_cumulativity-i-j,
cubical-type_wf,
face-or_wf,
cubical_set_wf,
case-term_wf2,
case-term-equal-right,
face-and_wf,
sub_cubical_set_transitivity,
context-iterated-subset,
context-subset-swap,
subset-cubical-type,
sub_cubical_set_functionality2,
face-term-implies-subset,
face-term-implies-or2,
same-cubical-term_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
dependent_set_memberEquality_alt,
universeIsType,
cut,
hypothesisEquality,
applyEquality,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
hypothesis,
sqequalRule,
instantiate,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
setElimination,
rename,
independent_pairFormation,
productElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma, (phi \mvee{} psi) \mvdash{} \_\}]. \mforall{}[v:\{Gamma, psi \mvdash{} \_:A\}].
\mforall{}[u:\{Gamma, phi \mvdash{} \_:A[psi |{}\mrightarrow{} v]\}].
((u \mvee{} v) \mmember{} \{t:\{Gamma, (phi \mvee{} psi) \mvdash{} \_:A\}| Gamma, psi \mvdash{} t=v:A\} )
Date html generated:
2020_05_20-PM-03_12_33
Last ObjectModification:
2020_04_07-PM-00_52_07
Theory : cubical!type!theory
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