Nuprl Lemma : csm-cubical-pi
∀X,Delta:CubicalSet. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta ⟶ X. ((ΠA B)s = Delta ⊢ Π(A)s (B)(s o p;q) ∈ {Delta ⊢ _})
Proof
Definitions occuring in Statement :
cubical-pi: ΠA B,
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-comp: G o F,
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
csm-ap-type: (AF)s,
cubical-pi: ΠA B,
cubical-pi-family: cubical-pi-family(X;A;B;I;a)
Lemmas referenced :
cubical-type-equal,
csm-ap-type_wf,
cubical-pi_wf,
cube-set-map_wf,
cubical-type_wf,
cube-context-adjoin_wf,
cubical-set_wf,
I-cube_wf,
list_wf,
coordinate_name_wf,
csm-cubical-pi-family,
cubical-pi-family_wf,
csm-ap_wf,
name-morph_wf,
cubical-pi-family-comp,
cube-set-restriction_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
sqequalRule,
dependent_pairEquality,
lambdaEquality,
dependent_functionElimination,
applyEquality,
because_Cache,
functionEquality
Latex:
\mforall{}X,Delta:CubicalSet. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}. \mforall{}s:Delta {}\mrightarrow{} X.
((\mPi{}A B)s = Delta \mvdash{} \mPi{}(A)s (B)(s o p;q))
Date html generated:
2018_05_23-PM-06_29_44
Last ObjectModification:
2018_05_20-PM-04_11_47
Theory : cubical!sets
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