Nuprl Lemma : csm-fiber-member
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[p:{X ⊢ _:Fiber(w;a)}]. ∀[H:j⊢]. ∀[s:H j⟶ X].
((fiber-member(p))s = fiber-member((p)s) ∈ {H ⊢ _:(T)s})
Proof
Definitions occuring in Statement :
fiber-member: fiber-member(p),
cubical-fiber: Fiber(w;a),
cubical-fun: (A ⟶ B),
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
cubical-fiber: Fiber(w;a),
fiber-member: fiber-member(p),
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
and: P ∧ Q
Lemmas referenced :
csm-ap-cubical-fst,
cube_set_map_wf,
cubical-term_wf,
cubical-fiber_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-fun_wf,
cubical-type_wf,
cubical_set_wf,
cc-snd_wf,
csm-ap-term_wf,
cube-context-adjoin_wf,
cc-fst_wf,
csm-cubical-fun,
cubical-app_wf_fun,
csm-ap-type_wf,
path-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
universeIsType,
inhabitedIsType,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
applyLambdaEquality,
setElimination,
rename,
productElimination,
lambdaEquality_alt,
hyp_replacement
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T,A:\{X \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[a:\{X \mvdash{} \_:A\}]. \mforall{}[p:\{X \mvdash{} \_:Fiber(w;a)\}]. \mforall{}[H:j\mvdash{}].
\mforall{}[s:H j{}\mrightarrow{} X].
((fiber-member(p))s = fiber-member((p)s))
Date html generated:
2020_05_20-PM-03_24_30
Last ObjectModification:
2020_04_07-PM-04_10_58
Theory : cubical!type!theory
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