Nuprl Lemma : sq_stable__cubical-path-condition'
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}]. ∀[a0:A((i1)(rho))].
SqStable(cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a0))
Proof
Definitions occuring in Statement :
cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1),
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
subset-iota: iota,
cubical-subset: I,psi,
face-presheaf: 𝔽,
csm-comp: G o F,
context-map: <rho>,
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-1: (i1),
nc-s: s,
add-name: I+i,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1),
all: ∀x:A. B[x],
and: P ∧ Q,
sq_stable: SqStable(P),
implies: P ⇒ Q,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
context-map: <rho>,
subset-iota: iota,
csm-comp: G o F,
csm-ap: (s)x,
compose: f o g,
functor-arrow: arrow(F),
cube-set-restriction: f(s),
true: True,
guard: {T}
Lemmas referenced :
cubical-type-ap-morph_wf,
cubical-subset-I_cube-member,
cube-set-restriction_wf,
nc-1_wf,
I_cube_wf,
cubical-subset_wf,
istype-cubical-type-at,
add-name_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
cubical-term_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
istype-nat,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
istype-void,
fset_wf,
cubical-type_wf,
cubical_set_wf,
cubical-term-at_wf,
comp-nc-1-subset-I_cube,
csm-ap-type-at,
cubical-type-at_wf,
cube-set-restriction-comp,
sq_stable__all,
equal_wf,
sq_stable__equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
productElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
instantiate,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
dependent_set_memberEquality_alt,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
applyEquality,
setIsType,
functionIsType,
intEquality,
imageElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
cumulativity,
functionEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[rho:Gamma(I+i)]. \mforall{}[phi:\mBbbF{}(I)].
\mforall{}[u:\{I+i,s(phi) \mvdash{} \_:(A)<rho> o iota\}]. \mforall{}[a0:A((i1)(rho))].
SqStable(cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a0))
Date html generated:
2020_05_20-PM-03_45_43
Last ObjectModification:
2020_04_09-AM-10_59_41
Theory : cubical!type!theory
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