Nuprl Lemma : fun-comp_wf
∀[Gamma:j⊢]. ∀[A,B:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma ⊢ CompOp(B)].
(fun-comp(Gamma; A; B; cA; cB) ∈ Gamma ⊢ CompOp((A ⟶ B)))
Proof
Definitions occuring in Statement :
fun-comp: fun-comp(Gamma; A; B; cA; cB),
composition-op: Gamma ⊢ CompOp(A),
cubical-fun: (A ⟶ B),
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,
fun-comp: fun-comp(Gamma; A; B; cA; cB),
squash: ↓T,
prop: ℙ,
true: True
Lemmas referenced :
pi-comp_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cc-fst_wf,
csm-composition_wf,
composition-op_wf,
squash_wf,
true_wf,
cubical-fun-as-cubical-pi,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A,B:\{Gamma \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} CompOp(A)]. \mforall{}[cB:Gamma \mvdash{} CompOp(B)].
(fun-comp(Gamma; A; B; cA; cB) \mmember{} Gamma \mvdash{} CompOp((A {}\mrightarrow{} B)))
Date html generated:
2020_05_20-PM-04_04_20
Last ObjectModification:
2020_04_16-PM-05_31_59
Theory : cubical!type!theory
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