Nuprl Lemma : csm-equiv-path1
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((equiv-path1(G;A;B;f))s+ = equiv-path1(H;(A)s;(B)s;(f)s) ∈ {H.𝕀 ⊢ _})
Proof
Definitions occuring in Statement :
equiv-path1: equiv-path1(G;A;B;f),
cubical-equiv: Equiv(T;A),
interval-type: 𝕀,
csm+: tau+,
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],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
equiv-path1: equiv-path1(G;A;B;f),
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
guard: {T},
csm+: tau+,
csm-comp: G o F,
cubical-type: {X ⊢ _},
csm-ap: (s)x,
csm-adjoin: (s;u),
pi1: fst(t),
compose: f o g,
squash: ↓T,
prop: ℙ,
true: True,
csm-ap-term: (t)s,
pi2: snd(t)
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cc-fst_wf_interval,
subset-cubical-type,
context-subset_wf,
context-subset-is-subset,
istype-cubical-term,
face-type_wf,
csm-glue-type,
face-or_wf,
face-zero_wf,
cc-snd_wf,
face-one_wf,
case-type_wf,
same-cubical-type-zero-and-one,
face-0_wf,
equiv-fun_wf,
cubical-equiv-by-cases_wf,
csm+_wf_interval,
cube_set_map_wf,
cubical-equiv_wf,
cubical-type_wf,
cubical_set_wf,
csm-equiv-fun,
glue-type_wf,
squash_wf,
true_wf,
cubical-fun_wf,
thin-context-subset,
q-csm+,
csm-face-or,
csm-face-zero,
csm-face-one,
csm-case-type,
csm-cubical-equiv-by-cases
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
hypothesis,
applyEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
universeIsType,
inhabitedIsType,
setElimination,
rename,
productElimination,
Error :memTop,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(A;B)\}]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} G].
((equiv-path1(G;A;B;f))s+ = equiv-path1(H;(A)s;(B)s;(f)s))
Date html generated:
2020_05_20-PM-07_26_34
Last ObjectModification:
2020_04_25-PM-10_12_28
Theory : cubical!type!theory
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