Nuprl Lemma : cubical-contr_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[p:{Gamma ⊢ _:Contractible(A)}]. ∀[phi:{Gamma ⊢ _:𝔽}].
∀[u:{Gamma, phi ⊢ _:(A)iota}].
(cubical-contr(Gamma; A; cA; p; phi; u) ∈ {Gamma ⊢ _:A[phi |⟶ u]})
Proof
Definitions occuring in Statement :
cubical-contr: cubical-contr(Gamma; A; cA; p; phi; u),
composition-op: Gamma ⊢ CompOp(A),
contractible-type: Contractible(A),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
subset-iota: iota,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-contr: cubical-contr(Gamma; A; cA; p; phi; u),
subtype_rel: A ⊆r B,
contractible-type: Contractible(A),
cubical-type: {X ⊢ _},
cc-snd: q,
cc-fst: p,
csm-ap-type: (AF)s,
csm-id-adjoin: [u],
csm-comp: G o F,
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
compose: f o g,
pi1: fst(t),
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
true: True,
prop: ℙ,
subset-iota: iota,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
istype: istype(T),
csm-ap-term: (t)s,
pi2: snd(t),
sq_type: SQType(T),
interval-type: 𝕀,
constant-cubical-type: (X),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]}
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cc-fst_wf,
path-type_wf,
csm-ap-term_wf,
cc-snd_wf,
csm-adjoin_wf,
csm-comp_wf,
csm-id-adjoin_wf,
cubical-fst_wf,
cubical-pi_wf,
csm-ap-type-iota,
thin-context-subset,
cubical-term-eqcd,
subset-cubical-term,
context-subset_wf,
context-subset-is-subset,
istype-cubical-term,
subset-iota_wf2,
face-type_wf,
contractible-type_wf,
composition-op_wf,
cubical-type_wf,
cubical_set_wf,
cubical-snd_wf,
csm_id_adjoin_fst_type_lemma,
squash_wf,
true_wf,
cubical-app_wf,
csm-context-subset-subtype2,
csm-id_wf,
csm-context-subset-subtype3,
csm-cubical-pi,
cubical-pi-context-subset,
subtype_rel-equal,
equal_wf,
istype-universe,
csm-ap-id-type,
subtype_rel_self,
iff_weakening_equal,
csm-path-type,
cube_set_map_wf,
csm-ap-type-fst-adjoin,
thin-context-subset-adjoin,
context-subset-term-subtype,
csm_ap_term_fst_adjoin_lemma,
subtype_base_sq,
base_wf,
path-type-subset-adjoin,
cubical-path-app_wf,
interval-type_wf,
cubical-term_wf,
cubical-path-app-0,
cubical-path-ap-id-adjoin,
interval-0_wf,
csm-id-adjoin_wf-interval-0,
composition-term_wf,
cc-fst_wf_interval,
csm-composition_wf,
subset-cubical-term2,
sub_cubical_set_self,
interval-1_wf,
cubical-path-app-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
setElimination,
rename,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
cumulativity,
universeIsType,
universeEquality,
hyp_replacement,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
imageElimination,
Error :memTop,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityElimination,
applyLambdaEquality,
baseApply,
closedConclusion,
dependent_set_memberEquality_alt
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} CompOp(A)]. \mforall{}[p:\{Gamma \mvdash{} \_:Contractible(A)\}].
\mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[u:\{Gamma, phi \mvdash{} \_:(A)iota\}].
(cubical-contr(Gamma; A; cA; p; phi; u) \mmember{} \{Gamma \mvdash{} \_:A[phi |{}\mrightarrow{} u]\})
Date html generated:
2020_05_20-PM-04_19_53
Last ObjectModification:
2020_04_19-PM-07_31_27
Theory : cubical!type!theory
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