Nuprl Lemma : csm-equiv-path2
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cB:G +⊢ Compositon(B)]. ∀[f:{G ⊢ _:Equiv(A;B)}]. ∀[H:j⊢].
∀[s:H j⟶ G].
((equiv-path2(G;A;B;cA;cB;f))s+
= equiv-path2(H;(A)s;(B)s;(cA)s;(cB)s;(f)s)
∈ H.𝕀 +⊢ Compositon(equiv-path1(H;(A)s;(B)s;(f)s)))
Proof
Definitions occuring in Statement :
equiv-path2: equiv-path2(G;A;B;cA;cB;f),
equiv-path1: equiv-path1(G;A;B;f),
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
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],
equiv-path2: equiv-path2(G;A;B;cA;cB;f),
member: t ∈ T,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
uimplies: b supposing a,
guard: {T},
cubical-type: {X ⊢ _},
csm+: tau+,
csm-ap: (s)x,
csm-comp: G o F,
csm-adjoin: (s;u),
pi1: fst(t),
compose: f o g,
csm-ap-term: (t)s,
pi2: snd(t),
csm-comp-structure: (cA)tau,
equiv-path1: equiv-path1(G;A;B;f),
and: P ∧ Q
Lemmas referenced :
csm-glue-comp,
cube-context-adjoin_wf,
interval-type_wf,
csm-ap-type_wf,
cc-fst_wf_interval,
face-or_wf,
face-zero_wf,
cc-snd_wf,
face-one_wf,
cube_set_map_wf,
istype-cubical-term,
cubical-equiv_wf,
composition-structure_wf,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf,
subset-cubical-type,
context-subset_wf,
context-subset-is-subset,
face-type_wf,
case-type_wf,
same-cubical-type-zero-and-one,
face-0_wf,
csm-case-type,
csm-face-or,
csm-face-zero,
csm-face-one,
csm-case-type-comp,
case-type-comp-disjoint,
csm-comp-structure_wf,
cube_set_map_cumulativity-i-j,
composition-structure-subset,
face-term-0-and-1,
csm-cubical-equiv-by-cases,
q-csm+,
glue-comp_wf2,
csm-comp-structure_wf2,
glue-type_wf,
equiv-fun_wf,
sub_cubical_set_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
hypothesis,
hypothesisEquality,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
because_Cache,
universeIsType,
inhabitedIsType,
applyEquality,
lambdaFormation_alt,
independent_isectElimination,
dependent_functionElimination,
setElimination,
rename,
productElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
applyLambdaEquality,
lambdaEquality_alt,
hyp_replacement
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_\}]. \mforall{}[cA:G +\mvdash{} Compositon(A)]. \mforall{}[cB:G +\mvdash{} Compositon(B)].
\mforall{}[f:\{G \mvdash{} \_:Equiv(A;B)\}]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} G].
((equiv-path2(G;A;B;cA;cB;f))s+ = equiv-path2(H;(A)s;(B)s;(cA)s;(cB)s;(f)s))
Date html generated:
2020_05_20-PM-07_28_18
Last ObjectModification:
2020_04_28-PM-10_05_07
Theory : cubical!type!theory
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