Nuprl Lemma : path_term_wf
∀X:j⊢. ∀psi:{X ⊢ _:𝔽}.
∀[r:{X ⊢ _:𝕀}]
∀T:{X.𝕀 ⊢ _}. ∀a,b:{X.𝕀 ⊢ _:T}. ∀w:{X, psi.𝕀 ⊢ _:(Path_T a b)}.
(path_term(psi; w; a; b; r) ∈ {X.𝕀, ((psi)p ∨ (((r=0))p ∨ ((r=1))p)) ⊢ _:T})
Proof
Definitions occuring in Statement :
path_term: path_term(phi; w; a; b; r),
path-type: (Path_A a b),
context-subset: Gamma, phi,
face-zero: (i=0),
face-one: (i=1),
face-or: (a ∨ b),
face-type: 𝔽,
interval-type: 𝕀,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
path_term: path_term(phi; w; a; b; r),
and: P ∧ Q
Lemmas referenced :
path-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
subset-cubical-type,
cubical_set_cumulativity-i-j,
context-subset_wf,
sub_cubical_set_functionality,
context-subset-is-subset,
cubical-term_wf,
cubical-type-cumulativity,
cubical-type-cumulativity2,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
path-term_wf,
csm-ap-term_wf,
csm-face-type,
cc-fst_wf,
csm-interval-type,
subset-cubical-term,
sub_cubical_set_transitivity,
sub_cubical_set_self,
context-adjoin-subset1,
csm-face-zero,
csm-face-one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
universeIsType,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
Error :memTop,
dependent_functionElimination,
independent_pairFormation
Latex:
\mforall{}X:j\mvdash{}. \mforall{}psi:\{X \mvdash{} \_:\mBbbF{}\}.
\mforall{}[r:\{X \mvdash{} \_:\mBbbI{}\}]
\mforall{}T:\{X.\mBbbI{} \mvdash{} \_\}. \mforall{}a,b:\{X.\mBbbI{} \mvdash{} \_:T\}. \mforall{}w:\{X, psi.\mBbbI{} \mvdash{} \_:(Path\_T a b)\}.
(path\_term(psi; w; a; b; r) \mmember{} \{X.\mBbbI{}, ((psi)p \mvee{} (((r=0))p \mvee{} ((r=1))p)) \mvdash{} \_:T\})
Date html generated:
2020_05_20-PM-05_10_31
Last ObjectModification:
2020_04_10-AM-11_42_59
Theory : cubical!type!theory
Home
Index