Nuprl Lemma : ctt-term-type-is-implies
∀[X:⊢''']. ∀[t:cttTerm(X)]. ∀[T:{X ⊢''' _}]. term(t) ∈ {X ⊢ _:T} supposing type(t)=T
Proof
Definitions occuring in Statement :
ctt-term-term: term(t),
ctt-term-type-is: type(t)=T,
ctt-term-meaning: cttTerm(X),
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,
ctt-term-type-is: type(t)=T,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
cubical-term-eqcd,
ctt-term-term_wf,
subtype_rel_wf,
squash_wf,
true_wf,
istype-universe,
iff_weakening_equal,
subtype_rel_self,
ctt-term-type-is_wf,
cubical-type_wf,
ctt-term-meaning_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalHypSubstitution,
cut,
hypothesis,
thin,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
sqequalRule,
productIsType,
equalityIstype,
hypothesisEquality,
applyLambdaEquality,
setElimination,
rename,
productElimination,
instantiate,
introduction,
extract_by_obid,
isectElimination,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
lambdaEquality_alt,
imageElimination,
universeIsType,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[X:\mvdash{}''']. \mforall{}[t:cttTerm(X)]. \mforall{}[T:\{X \mvdash{}''' \_\}]. term(t) \mmember{} \{X \mvdash{} \_:T\} supposing type(t)=T
Date html generated:
2020_05_20-PM-07_53_06
Last ObjectModification:
2020_05_05-PM-02_10_25
Theory : cubical!type!theory
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