Nuprl Lemma : equiv-comp_wf
∀[H:j⊢]. ∀[A,E:{H ⊢ _}]. ∀[cA:H ⊢ CompOp(A)]. ∀[cE:H ⊢ CompOp(E)]. (equiv-comp(H;A;E;cA;cE) ∈ H ⊢ CompOp(Equiv(A;E)))
Proof
Definitions occuring in Statement :
equiv-comp: equiv-comp(H;A;E;cA;cE),
composition-op: Gamma ⊢ CompOp(A),
cubical-equiv: Equiv(T;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,
equiv-comp: equiv-comp(H;A;E;cA;cE),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
comp-fun-to-comp-op_wf,
cubical-equiv_wf,
equiv_comp_wf,
comp-op-to-comp-fun_wf2,
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[A,E:\{H \mvdash{} \_\}]. \mforall{}[cA:H \mvdash{} CompOp(A)]. \mforall{}[cE:H \mvdash{} CompOp(E)].
(equiv-comp(H;A;E;cA;cE) \mmember{} H \mvdash{} CompOp(Equiv(A;E)))
Date html generated:
2020_05_20-PM-07_20_18
Last ObjectModification:
2020_04_25-PM-09_49_50
Theory : cubical!type!theory
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