Nuprl Lemma : csm-ap-type_wf
∀[Gamma,Delta:CubicalSet]. ∀[s:Delta ⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. Delta ⊢ (A)s
Proof
Definitions occuring in Statement :
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 ⊢ _},
csm-ap-type: (AF)s,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
all: ∀x:A. B[x],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
list_wf,
coordinate_name_wf,
csm-ap_wf,
I-cube_wf,
subtype_rel-equal,
cube-set-restriction_wf,
equal_wf,
csm-ap-restriction,
iff_weakening_equal,
name-morph_wf,
name-comp_wf,
cube-set-restriction-comp,
squash_wf,
true_wf,
cube-set-map_wf,
all_wf,
id-morph_wf,
cube-set-restriction-id,
cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
dependent_set_memberEquality,
sqequalRule,
dependent_pairEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
extract_by_obid,
isectElimination,
hypothesis,
because_Cache,
independent_isectElimination,
instantiate,
imageElimination,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionEquality,
lambdaFormation,
independent_pairFormation,
spreadEquality,
productEquality,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[Gamma,Delta:CubicalSet]. \mforall{}[s:Delta {}\mrightarrow{} Gamma]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. Delta \mvdash{} (A)s
Date html generated:
2018_05_23-PM-06_28_45
Last ObjectModification:
2018_05_16-PM-02_30_19
Theory : cubical!sets
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