Nuprl Lemma : contraction-to-extend_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)]. ∀[ctr:{Gamma ⊢ _:Contractible(A)}].
(contraction-to-extend(Gamma;A;cA;ctr) ∈ Gamma ⊢ Extension(A))
Proof
Definitions occuring in Statement :
contraction-to-extend: contraction-to-extend(Gamma;A;cA;ctr),
uniform-extend: uniform-extend{i:l}(Gamma; A),
composition-structure: Gamma ⊢ Compositon(A),
contractible-type: Contractible(A),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
contraction-to-extend: contraction-to-extend(Gamma;A;cA;ctr),
extension-fun: extension-fun{i:l}(Gamma;A),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
guard: {T},
csm-comp-structure: (cA)tau,
interval-type: 𝕀,
csm-comp: G o F,
compose: f o g,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
csm-id: 1(X),
csm-ap: (s)x,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-adjoin: (s;u),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
uniform-extend: uniform-extend{i:l}(Gamma; A),
uniform-extension-fun: uniform-extension-fun{i:l}(Gamma;A;ext),
prop: ℙ,
cubical-path-app: pth @ r,
cand: A c∧ B,
composition-structure: Gamma ⊢ Compositon(A),
squash: ↓T,
true: True,
cc-fst: p,
csm+: tau+,
interval-1: 1(𝕀),
cc-snd: q,
constant-cubical-type: (X),
pi1: fst(t),
contr-path: contr-path(c;x),
path-eta: path-eta(pth),
csm-ap-term: (t)s,
pi2: snd(t)
Lemmas referenced :
contr-center_wf,
csm-ap-type_wf,
csm-ap-term_wf,
contractible-type_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-contractible-type,
contr-path_wf,
context-subset_wf,
thin-context-subset,
contractible-type-subset,
context-subset-is-subset,
path-eta_wf,
path-type-subtype,
context-subset-term-subtype,
csm-comp-structure_wf,
cube-context-adjoin_wf,
interval-type_wf,
cc-fst_wf_interval,
csm-comp_term,
csm_id_adjoin_fst_type_lemma,
subset-cubical-term,
istype-cubical-term,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm-context-subset-subtype2,
face-type_wf,
cube_set_map_wf,
uniform-extension-fun_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
cubical-path-app-0,
cubical-path-app-1,
path-eta-0,
path-eta-1,
comp_term_wf,
constrained-cubical-term_wf,
squash_wf,
true_wf,
csm-id-adjoin-subset,
equal_wf,
istype-universe,
cubical-term_wf,
csm-path-eta,
csm-cubical-app,
csm-cubical-snd,
csm-contr-center,
csm-face-type,
context-subset-map,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
applyEquality,
independent_isectElimination,
sqequalRule,
Error :memTop,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
independent_pairFormation,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
productElimination,
instantiate,
setElimination,
rename,
dependent_set_memberEquality_alt,
universeIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
natural_numberEquality,
hyp_replacement,
universeEquality,
functionEquality,
cumulativity
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} Compositon(A)]. \mforall{}[ctr:\{Gamma \mvdash{} \_:Contractible(A)\}].
(contraction-to-extend(Gamma;A;cA;ctr) \mmember{} Gamma \mvdash{} Extension(A))
Date html generated:
2020_05_20-PM-05_23_07
Last ObjectModification:
2020_04_17-PM-00_39_36
Theory : cubical!type!theory
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