Nuprl Lemma : equiv-path2_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cB:G +⊢ Compositon(B)]. ∀[f:{G ⊢ _:Equiv(A;B)}].
(equiv-path2(G;A;B;cA;cB;f) ∈ G.𝕀 +⊢ Compositon(equiv-path1(G;A;B;f)))
Proof
Definitions occuring in Statement :
equiv-path2: equiv-path2(G;A;B;cA;cB;f),
equiv-path1: equiv-path1(G;A;B;f),
composition-structure: Gamma ⊢ Compositon(A),
cubical-equiv: Equiv(T;A),
interval-type: 𝕀,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: 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-path2: equiv-path2(G;A;B;cA;cB;f),
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}
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,
case-type-comp-disjoint,
face-zero_wf,
cc-snd_wf,
face-one_wf,
csm-comp-structure_wf,
composition-structure-subset,
cubical_set_cumulativity-i-j,
face-term-0-and-1,
glue-comp_wf2,
csm-comp-structure_wf2,
face-or_wf,
case-type_wf,
same-cubical-type-zero-and-one,
face-0_wf,
cubical-equiv-by-cases_wf,
cubical-equiv_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf
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
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)\}].
(equiv-path2(G;A;B;cA;cB;f) \mmember{} G.\mBbbI{} +\mvdash{} Compositon(equiv-path1(G;A;B;f)))
Date html generated:
2020_05_20-PM-07_27_57
Last ObjectModification:
2020_04_28-PM-04_28_42
Theory : cubical!type!theory
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