Nuprl Lemma : case-type-1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}].
∀[A:{Gamma ⊢ _}]. ∀[B:Top × Top]. Gamma ⊢ (if phi then A else B) = A supposing phi = 1(𝔽) ∈ {Gamma ⊢ _:𝔽}
Proof
Definitions occuring in Statement :
case-type: (if phi then A else B),
same-cubical-type: Gamma ⊢ A = B,
face-1: 1(𝔽),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
product: x:A × B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
same-cubical-type: Gamma ⊢ A = B,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
case-type-same1,
thin-context-subset,
face-0_wf,
istype-top,
cubical-type_wf,
face-1_wf,
cubical-term_wf,
face-type_wf,
cubical_set_wf,
empty-context-subset-lemma6,
subset-cubical-type,
context-subset_wf,
face-1-implies-subset,
face-term-implies_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
face-term-implies-same
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productIsType,
because_Cache,
universeIsType,
equalityIstype,
inhabitedIsType,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}].
\mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[B:Top \mtimes{} Top]. Gamma \mvdash{} (if phi then A else B) = A supposing phi = 1(\mBbbF{})
Date html generated:
2020_05_20-PM-04_14_21
Last ObjectModification:
2020_04_10-AM-04_43_07
Theory : cubical!type!theory
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