Nuprl Lemma : fill-path_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[x:{Gamma ⊢ _:(A)[1(𝕀)]}]. ∀[y:{Gamma ⊢ _:(A)[0(𝕀)]}].
∀[z:{Gamma.𝕀 ⊢ _:((A)[0(𝕀)])p}].
(fill-path(Gamma;A;cA;x;y;z) ∈ {Gamma ⊢ _:(Path_(A)[1(𝕀)] x app(transport-fun(Gamma;A;cA); y))}) supposing
(((z)[0(𝕀)] = app(rev-transport-fun(Gamma;A;cA); x) ∈ {Gamma ⊢ _:(A)[0(𝕀)]}) and
((z)[1(𝕀)] = y ∈ {Gamma ⊢ _:(A)[0(𝕀)]}))
Proof
Definitions occuring in Statement :
fill-path: fill-path(Gamma;A;cA;x;y;z),
rev-transport-fun: rev-transport-fun(Gamma;A;cA),
transport-fun: transport-fun(Gamma;A;cA),
composition-op: Gamma ⊢ CompOp(A),
path-type: (Path_A a b),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-app: app(w; u),
csm-id-adjoin: [u],
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
squash: ↓T,
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap-type: (AF)s,
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
true: True,
uimplies: b supposing a,
fill-path: fill-path(Gamma;A;cA;x;y;z),
interval-1: 1(𝕀),
and: P ∧ Q,
prop: ℙ
Lemmas referenced :
fillpath_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf-interval-0,
cc-fst_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-1,
csm_id_adjoin_fst_type_lemma,
cubical-term_wf,
cubical-type-cumulativity,
cubical-type-cumulativity2,
csm-ap-term_wf,
cubical-app_wf_fun,
rev-transport-fun_wf,
composition-op_wf,
cubical-type_wf,
cubical_set_wf,
equal_wf,
transport-fun_wf,
squash_wf,
true_wf,
path-type_wf,
term-to-path_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
imageElimination,
setElimination,
rename,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
universeIsType,
independent_isectElimination,
axiomEquality,
equalityIstype,
isect_memberEquality_alt,
isectIsTypeImplies,
setEquality,
productEquality,
applyLambdaEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)]. \mforall{}[x:\{Gamma \mvdash{} \_:(A)[1(\mBbbI{})]\}].
\mforall{}[y:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})]\}]. \mforall{}[z:\{Gamma.\mBbbI{} \mvdash{} \_:((A)[0(\mBbbI{})])p\}].
(fill-path(Gamma;A;cA;x;y;z)
\mmember{} \{Gamma \mvdash{} \_:(Path\_(A)[1(\mBbbI{})] x app(transport-fun(Gamma;A;cA); y))\}) supposing
(((z)[0(\mBbbI{})] = app(rev-transport-fun(Gamma;A;cA); x)) and
((z)[1(\mBbbI{})] = y))
Date html generated:
2020_05_20-PM-04_56_44
Last ObjectModification:
2020_04_13-PM-02_20_54
Theory : cubical!type!theory
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