Nuprl Lemma : csm-ap-term_wf
∀[Delta,Gamma:CubicalSet]. ∀[s:Delta ⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[t:{Gamma ⊢ _:A}]. ((t)s ∈ {Delta ⊢ _:(A)s})
Proof
Definitions occuring in Statement :
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:AF},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-type: {X ⊢ _},
cubical-term: {X ⊢ _:AF},
pi1: fst(t),
csm-ap-type: (AF)s,
csm-ap-term: (t)s,
and: P ∧ Q,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
csm-ap_wf,
I-cube_wf,
list_wf,
coordinate_name_wf,
equal_wf,
squash_wf,
true_wf,
cube-set-restriction_wf,
subtype_rel-equal,
csm-ap-restriction,
subtype_rel_self,
iff_weakening_equal,
name-morph_wf,
all_wf,
cubical-term_wf,
cubical-type_wf,
cube-set-map_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
dependent_set_memberEquality,
lambdaEquality,
applyEquality,
hypothesisEquality,
extract_by_obid,
isectElimination,
hypothesis,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
independent_isectElimination,
instantiate,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[Delta,Gamma:CubicalSet]. \mforall{}[s:Delta {}\mrightarrow{} Gamma]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[t:\{Gamma \mvdash{} \_:A\}].
((t)s \mmember{} \{Delta \mvdash{} \_:(A)s\})
Date html generated:
2018_05_23-PM-06_28_54
Last ObjectModification:
2018_05_16-PM-02_30_25
Theory : cubical!sets
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