Nuprl Lemma : csm-equiv-comp
∀[H,K:j⊢]. ∀[tau:K j⟶ H]. ∀[A,E:{H ⊢ _}]. ∀[cA:H ⊢ CompOp(A)]. ∀[cE:H ⊢ CompOp(E)].
((equiv-comp(H;A;E;cA;cE))tau = equiv-comp(K;(A)tau;(E)tau;(cA)tau;(cE)tau) ∈ K ⊢ CompOp(Equiv((A)tau;(E)tau)))
Proof
Definitions occuring in Statement :
equiv-comp: equiv-comp(H;A;E;cA;cE),
csm-composition: (comp)sigma,
composition-op: Gamma ⊢ CompOp(A),
cubical-equiv: Equiv(T;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],
member: t ∈ T,
equiv-comp: equiv-comp(H;A;E;cA;cE),
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
all: ∀x:A. B[x],
true: True,
uimplies: b supposing a,
guard: {T},
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
csm-comp-fun-to-comp-op,
cubical-equiv_wf,
equiv_comp_wf,
comp-op-to-comp-fun_wf2,
composition-structure-cumulativity,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-composition_wf,
comp-fun-to-comp-op_wf,
csm-cubical-equiv,
equal_functionality_wrt_subtype_rel2,
composition-structure_wf,
csm-ap-type_wf,
csm-equiv_comp,
cube_set_map_cumulativity-i-j,
subtype_rel_self,
iff_weakening_equal,
csm-comp-op-to-comp-fun-sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
universeIsType,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
hyp_replacement,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
cumulativity,
independent_isectElimination,
independent_functionElimination,
productElimination,
Error :memTop
Latex:
\mforall{}[H,K:j\mvdash{}]. \mforall{}[tau:K j{}\mrightarrow{} H]. \mforall{}[A,E:\{H \mvdash{} \_\}]. \mforall{}[cA:H \mvdash{} CompOp(A)]. \mforall{}[cE:H \mvdash{} CompOp(E)].
((equiv-comp(H;A;E;cA;cE))tau = equiv-comp(K;(A)tau;(E)tau;(cA)tau;(cE)tau))
Date html generated:
2020_05_20-PM-07_20_51
Last ObjectModification:
2020_04_27-PM-02_04_50
Theory : cubical!type!theory
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