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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