Nuprl Lemma : case-type-0
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}].
∀[A:Top × Top]. ∀[B:{Gamma ⊢ _}]. Gamma ⊢ (if phi then A else B) = B supposing phi = 0(𝔽) ∈ {Gamma ⊢ _:𝔽}
Proof
Definitions occuring in Statement :
case-type: (if phi then A else B),
same-cubical-type: Gamma ⊢ A = B,
face-0: 0(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,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
cubical-type: {X ⊢ _},
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
same-cubical-type: Gamma ⊢ A = B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
case-type-same2,
cubical-type_wf,
istype-top,
face-0_wf,
cubical-term_wf,
face-type_wf,
cubical_set_wf,
empty-context-subset-lemma6,
context-subset_wf,
face-1_wf,
thin-context-subset,
subtype_rel_product,
fset_wf,
nat_wf,
I_cube_wf,
names-hom_wf,
cube-set-restriction_wf,
istype-universe,
top_wf,
subset-cubical-type,
face-and_wf,
face-term-implies-subset,
face-term-implies_wf,
iff_weakening_equal,
face-term-and-implies1,
context-1-subset
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
universeIsType,
productIsType,
because_Cache,
equalityIstype,
inhabitedIsType,
instantiate,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
hyp_replacement,
setElimination,
rename,
functionEquality,
cumulativity,
universeEquality,
functionIsType,
Error :memTop,
lambdaFormation_alt,
productElimination,
independent_functionElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}].
\mforall{}[A:Top \mtimes{} Top]. \mforall{}[B:\{Gamma \mvdash{} \_\}]. Gamma \mvdash{} (if phi then A else B) = B supposing phi = 0(\mBbbF{})
Date html generated:
2020_05_20-PM-04_14_35
Last ObjectModification:
2020_04_10-AM-04_43_35
Theory : cubical!type!theory
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