Nuprl Lemma : csm-equivU
∀[G,K:j⊢]. ∀[tau:K j⟶ G]. ∀[E:{G.𝕀 ⊢ _}]. ∀[cE:G.𝕀 ⊢ CompOp(E)].
((equivU(G;E;cE))tau = equivU(K;(E)tau+;(cE)tau+) ∈ {K ⊢ _:Equiv(((E)tau+)[0(𝕀)];((E)tau+)[1(𝕀)])})
Proof
Definitions occuring in Statement :
equivU: equivU(G;E;cE),
csm-composition: (comp)sigma,
composition-op: Gamma ⊢ CompOp(A),
cubical-equiv: Equiv(T;A),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm+: tau+,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _: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,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap-type: (AF)s,
cc-fst: p,
interval-1: 1(𝕀),
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
pi1: fst(t),
equivU: equivU(G;E;cE),
squash: ↓T,
prop: ℙ,
csm+: tau+,
csm-comp: G o F,
all: ∀x:A. B[x],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
interval-type: 𝕀,
cc-snd: q,
constant-cubical-type: (X),
pi2: snd(t),
compose: f o g,
csm-composition: (comp)sigma
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf,
interval-0_wf,
cc-fst_wf_interval,
transport_wf,
cubical-equiv_wf,
csm-cubical-equiv,
subset-cubical-term2,
sub_cubical_set_self,
istype-cubical-term,
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm+_wf_interval,
interval-1_wf,
cubical-term-eqcd,
csm-transport,
equiv-comp_wf,
csm-composition_wf,
cubical-id-equiv_wf,
subtype_rel_self,
iff_weakening_equal,
csm_id_adjoin_fst_type_lemma,
csm-ap-id-type,
equivU_wf,
member_wf,
csm-id_wf,
csm-equiv-comp,
csm-cubical-id-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
hypothesis,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyEquality,
independent_isectElimination,
sqequalRule,
universeIsType,
setElimination,
rename,
productElimination,
inhabitedIsType,
lambdaEquality_alt,
imageElimination,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
Error :memTop,
hyp_replacement,
cumulativity,
functionExtensionality,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
applyLambdaEquality,
functionIsType
Latex:
\mforall{}[G,K:j\mvdash{}]. \mforall{}[tau:K j{}\mrightarrow{} G]. \mforall{}[E:\{G.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cE:G.\mBbbI{} \mvdash{} CompOp(E)].
((equivU(G;E;cE))tau = equivU(K;(E)tau+;(cE)tau+))
Date html generated:
2020_05_20-PM-07_21_47
Last ObjectModification:
2020_04_28-PM-00_51_34
Theory : cubical!type!theory
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