Nuprl Lemma : fiber-comp_wf
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[cT:X ⊢ Compositon(T)]. ∀[cA:X +⊢ Compositon(A)].
(fiber-comp(X;T;A;w;a;cT;cA) ∈ X ⊢ Compositon(Fiber(w;a)))
Proof
Definitions occuring in Statement :
fiber-comp: fiber-comp(X;T;A;w;a;cT;cA),
composition-structure: Gamma ⊢ Compositon(A),
cubical-fiber: Fiber(w;a),
cubical-fun: (A ⟶ B),
cubical-term: {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,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
and: P ∧ Q,
cubical-fiber: Fiber(w;a),
fiber-comp: fiber-comp(X;T;A;w;a;cT;cA)
Lemmas referenced :
cc-snd_wf,
csm-ap-term_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-fun_wf,
cc-fst_wf,
csm-cubical-fun,
cubical-term_wf,
sigma_comp_wf2,
path-type_wf,
csm-ap-type_wf,
cubical-app_wf_fun,
path_comp_wf,
csm-comp-structure_wf,
composition-structure_wf,
istype-cubical-term,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
productElimination,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T,A:\{X \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[a:\{X \mvdash{} \_:A\}]. \mforall{}[cT:X \mvdash{} Compositon(T)].
\mforall{}[cA:X +\mvdash{} Compositon(A)].
(fiber-comp(X;T;A;w;a;cT;cA) \mmember{} X \mvdash{} Compositon(Fiber(w;a)))
Date html generated:
2020_05_20-PM-05_13_07
Last ObjectModification:
2020_04_17-AM-00_18_57
Theory : cubical!type!theory
Home
Index