Nuprl Lemma : equiv_path_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. (equiv_path(G;A;B;f) ∈ {G.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,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
equiv_path: equiv_path(G;A;B;f),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
let: let
Lemmas referenced :
equiv-path2_wf,
universe-decode_wf,
universe-comp-fun_wf,
cubical_set_cumulativity-i-j,
comp-fun-to-comp-op_wf,
cube-context-adjoin_wf,
interval-type_wf,
equiv-path1_wf,
composition-structure-cumulativity,
universe-encode_wf,
istype-cubical-term,
cubical-equiv_wf,
istype-cubical-universe-term,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
rename,
dependent_functionElimination,
equalityIstype,
independent_functionElimination,
universeIsType
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(decode(A);decode(B))\}].
(equiv\_path(G;A;B;f) \mmember{} \{G.\mBbbI{} \mvdash{} \_:c\mBbbU{}\})
Date html generated:
2020_05_20-PM-07_29_19
Last ObjectModification:
2020_04_28-PM-07_09_02
Theory : cubical!type!theory
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