Nuprl Lemma : case-term-wf4
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, phi ⊢ _}]. ∀[B:{Gamma, psi ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}].
∀[v:{Gamma, psi ⊢ _:B}].
((u ∨ v) ∈ {Gamma, (phi ∨ psi) ⊢ _:(if phi then A else B)}) supposing
(Gamma, (phi ∧ psi) ⊢ u=v:A and
Gamma, (phi ∧ psi) ⊢ A = B)
Proof
Definitions occuring in Statement :
case-term: (u ∨ v),
case-type: (if phi then A else B),
same-cubical-term: X ⊢ u=v:A,
same-cubical-type: Gamma ⊢ A = B,
context-subset: Gamma, phi,
face-or: (a ∨ b),
face-and: (a ∧ b),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
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,
squash: ↓T,
prop: ℙ,
true: True,
same-cubical-type: Gamma ⊢ A = B,
guard: {T},
implies: P ⇒ Q,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
same-cubical-term: X ⊢ u=v:A,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
same-cubical-type_wf,
context-subset_wf,
face-and_wf,
subset-cubical-type,
face-term-implies-subset,
face-term-and-implies1,
face-term-and-implies2,
istype-cubical-term,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
case-term_wf2,
case-type_wf,
cubical-term_wf,
squash_wf,
true_wf,
case-type-same2,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
equal_functionality_wrt_subtype_rel2,
case-type-same1,
subset-cubical-term,
context-subset-is-subset,
context-subset-subtype-and,
context-iterated-subset,
sub_cubical_set_wf,
face-and-com,
iff_weakening_equal,
sub_cubical_set_transitivity,
context-subset-swap,
sub_cubical_set_functionality2,
context-iterated-subset2,
face-or_wf,
face-term-implies-or1,
subset-cubical-term2,
cubical-term-eqcd,
same-cubical-term_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
independent_isectElimination,
instantiate,
lambdaEquality_alt,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
cumulativity,
independent_functionElimination,
dependent_set_memberEquality_alt,
productElimination,
inhabitedIsType,
independent_pairFormation,
equalityIstype,
universeEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma, phi \mvdash{} \_\}]. \mforall{}[B:\{Gamma, psi \mvdash{} \_\}].
\mforall{}[u:\{Gamma, phi \mvdash{} \_:A\}]. \mforall{}[v:\{Gamma, psi \mvdash{} \_:B\}].
((u \mvee{} v) \mmember{} \{Gamma, (phi \mvee{} psi) \mvdash{} \_:(if phi then A else B)\}) supposing
(Gamma, (phi \mwedge{} psi) \mvdash{} u=v:A and
Gamma, (phi \mwedge{} psi) \mvdash{} A = B)
Date html generated:
2020_05_20-PM-03_12_47
Last ObjectModification:
2020_04_19-PM-01_55_45
Theory : cubical!type!theory
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