Nuprl Lemma : csm-equiv_path
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((equiv_path(G;A;B;f))s+ = equiv_path(H;(A)s;(B)s;(f)s) ∈ {H.𝕀 ⊢ _:c𝕌})
Proof
Definitions occuring in Statement :
equiv_path: equiv_path(G;A;B;f),
universe-decode: decode(t),
cubical-universe: c𝕌,
cubical-equiv: Equiv(T;A),
interval-type: 𝕀,
csm+: tau+,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
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_path: equiv_path(G;A;B;f),
let: let,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
csm+: tau+,
csm-comp: G o F,
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equiv-path2_wf,
universe-decode_wf,
universe-comp-fun_wf,
cubical_set_cumulativity-i-j,
csm-universe-encode,
cube-context-adjoin_wf,
interval-type_wf,
equiv-path1_wf,
comp-fun-to-comp-op_wf,
csm+_wf_interval,
cube_set_map_wf,
istype-cubical-term,
cubical-equiv_wf,
istype-cubical-universe-term,
cubical_set_wf,
universe-encode_wf,
squash_wf,
true_wf,
composition-op_wf,
cubical-type-cumulativity2,
cubical-type_wf,
csm-equiv-path1,
csm-universe-decode,
csm-ap-term_wf,
cubical-term-eqcd,
csm-cubical-equiv,
cubical-universe_wf,
csm-cubical-universe,
cubical-term_wf,
composition-structure-cumulativity,
subtype_rel-equal,
equal_wf,
istype-universe,
csm-comp-fun-to-comp-op,
cube_set_map_cumulativity-i-j,
subtype_rel_self,
iff_weakening_equal,
composition-structure_wf,
csm-equiv-path2,
csm-ap-type_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-universe-comp-fun
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesis,
hypothesisEquality,
instantiate,
applyEquality,
dependent_functionElimination,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
equalityTransitivity,
lambdaEquality_alt,
imageElimination,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
Error :memTop,
independent_isectElimination,
cumulativity,
universeEquality,
hyp_replacement,
lambdaFormation_alt,
equalityIstype,
independent_functionElimination,
rename,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
applyLambdaEquality,
setElimination,
productElimination
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(decode(A);decode(B))\}]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} G].
((equiv\_path(G;A;B;f))s+ = equiv\_path(H;(A)s;(B)s;(f)s))
Date html generated:
2020_05_20-PM-07_29_41
Last ObjectModification:
2020_04_30-AM-09_45_33
Theory : cubical!type!theory
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