Nuprl Lemma : equiv-path1-constraint
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}].
G.𝕀, ((q=0) ∨ (q=1)) ⊢ equiv-path1(G;A;B;f) = (if (q=0) then (A)p else (B)p)
Proof
Definitions occuring in Statement :
equiv-path1: equiv-path1(G;A;B;f),
cubical-equiv: Equiv(T;A),
case-type: (if phi then A else B),
same-cubical-type: Gamma ⊢ A = B,
context-subset: Gamma, phi,
face-zero: (i=0),
face-one: (i=1),
face-or: (a ∨ b),
interval-type: 𝕀,
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x]
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,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
guard: {T},
equiv-path1: equiv-path1(G;A;B;f)
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,
glue-type-constraint,
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,
cubical-equiv_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{}[f:\{G \mvdash{} \_:Equiv(A;B)\}].
G.\mBbbI{}, ((q=0) \mvee{} (q=1)) \mvdash{} equiv-path1(G;A;B;f) = (if (q=0) then (A)p else (B)p)
Date html generated:
2020_05_20-PM-07_26_49
Last ObjectModification:
2020_04_25-PM-10_13_05
Theory : cubical!type!theory
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