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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