Nuprl Lemma : path-term-case1
∀X:j⊢. ∀psi:{X ⊢ _:𝔽}.
∀[r:{X ⊢ _:𝕀}]
∀T:{X ⊢ _}. ∀a,b:{X ⊢ _:T}. ∀w:{X, psi ⊢ _:(Path_T a b)}. (path-term(psi;w;a;b;r) = w @ r ∈ {X, psi ⊢ _:T})
Proof
Definitions occuring in Statement :
path-term: path-term(phi;w;a;b;r),
cubical-path-app: pth @ r,
path-type: (Path_A a b),
context-subset: Gamma, phi,
face-type: 𝔽,
interval-type: 𝕀,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
path-term: path-term(phi;w;a;b;r),
same-cubical-term: X ⊢ u=v:A,
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
case-term-equal-left,
thin-context-subset,
path-type_wf,
cubical-term_wf,
context-subset_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
interval-type_wf,
face-type_wf,
cubical_set_wf,
cubical-path-app_wf,
context-subset-term-subtype,
subset-cubical-term2,
sub_cubical_set_self,
subset-cubical-term,
context-subset-is-subset,
path-type-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :memTop,
universeIsType,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}X:j\mvdash{}. \mforall{}psi:\{X \mvdash{} \_:\mBbbF{}\}.
\mforall{}[r:\{X \mvdash{} \_:\mBbbI{}\}]
\mforall{}T:\{X \mvdash{} \_\}. \mforall{}a,b:\{X \mvdash{} \_:T\}. \mforall{}w:\{X, psi \mvdash{} \_:(Path\_T a b)\}. (path-term(psi;w;a;b;r) = w @ r)
Date html generated:
2020_05_20-PM-05_09_30
Last ObjectModification:
2020_04_10-AM-11_40_51
Theory : cubical!type!theory
Home
Index