Nuprl Lemma : cubical-pi_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. X ⊢ ΠA B
Proof
Definitions occuring in Statement :
cubical-pi: ΠA B,
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-pi: ΠA B,
cubical-type: {X ⊢ _},
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
so_apply: x[s],
uimplies: b supposing a,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
top: Top,
prop: ℙ,
cand: A c∧ B
Lemmas referenced :
cubical-type_wf,
cube-context-adjoin_wf,
cubical-set_wf,
cubical-pi-family_wf,
I-cube_wf,
list_wf,
coordinate_name_wf,
name-comp_wf,
subtype_rel_dep_function,
cubical-type-at_wf,
cube-set-restriction_wf,
cc-adjoin-cube_wf,
subtype_rel-equal,
equal_wf,
cube-set-restriction-comp,
iff_weakening_equal,
name-morph_wf,
all_wf,
cubical-type-ap-morph_wf,
cc-adjoin-cube-restriction,
squash_wf,
true_wf,
name-comp-assoc,
name-comp-id-left,
id-morph_wf,
cube-set-restriction-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
because_Cache,
dependent_pairEquality,
lambdaEquality,
setElimination,
rename,
applyEquality,
functionExtensionality,
dependent_functionElimination,
independent_isectElimination,
instantiate,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
lambdaFormation,
voidElimination,
voidEquality,
functionEquality,
hyp_replacement,
independent_pairFormation,
productEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. X \mvdash{} \mPi{}A B
Date html generated:
2017_10_05-AM-10_14_17
Last ObjectModification:
2017_07_28-AM-11_19_16
Theory : cubical!sets
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