Nuprl Lemma : cubical-path-0-fillterm
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ I+i} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
∀[a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)]
((a0 (i0)(rho) s)
∈ cubical-path-0(Gamma;A;I+i;j;m(i;j)(rho);fl-join(I+i;s(phi);(i=0));fillterm(Gamma;A;I;i;j;rho;a0;u)))
Proof
Definitions occuring in Statement :
fillterm: fillterm(Gamma;A;I;i;j;rho;a0;u),
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type-ap-morph: (u a f),
cubical-type: {X ⊢ _},
subset-iota: iota,
cubical-subset: I,psi,
fl-join: fl-join(I;x;y),
face-presheaf: 𝔽,
fl0: (x=0),
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-m: m(i;j),
nc-0: (i0),
nc-s: s,
add-name: I+i,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
face-presheaf: 𝔽,
I_cube: A(I),
names: names(I),
subtype_rel: A ⊆r B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
cube-set-restriction: f(s),
fl-join: fl-join(I;x;y),
name-morph-satisfies: (psi f) = 1,
pi2: snd(t),
functor-ob: ob(F),
pi1: fst(t),
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
bdd-distributive-lattice: BoundedDistributiveLattice,
true: True,
fillterm: fillterm(Gamma;A;I;i;j;rho;a0;u),
cubical-term-at: u(a),
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
compose: f o g,
dM: dM(I),
dM-lift: dM-lift(I;J;f),
nc-0: (i0),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
label: ...$L... t,
context-map: <rho>,
subset-iota: iota,
csm-comp: G o F,
csm-ap: (s)x,
functor-arrow: arrow(F),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
ob_pair_lemma,
fl0_wf,
trivial-member-add-name1,
fset-member_wf,
nat_wf,
int-deq_wf,
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,
f-subset-add-name1,
f-subset-add-name,
cubical-path-0_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-term_wf,
cubical-subset_wf,
cube-set-restriction_wf,
face-presheaf_wf2,
nc-s_wf,
csm-ap-type_wf,
cubical-type-cumulativity,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
I_cube_wf,
istype-nat,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
istype-void,
fset_wf,
cubical-type_wf,
cubical_set_wf,
cubical-type-ap-morph_wf,
nc-0_wf,
subtype_rel-equal,
cubical-type-at_wf,
nc-m_wf,
equal_wf,
cube-set-restriction-comp,
iff_weakening_equal,
nc-0-s-commute,
nh-comp_wf,
nc-m-nc-0,
cubical-path-condition_wf,
fl-join_wf,
fillterm_wf,
cubical-subset-I_cube-member,
iff_transitivity,
lattice-point_wf,
face_lattice_wf,
lattice-join_wf,
fl-morph_wf,
subtype_rel_self,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
or_wf,
squash_wf,
true_wf,
istype-universe,
fl-morph-join,
face_lattice-1-join-irreducible,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
subtype_base_sq,
int_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
dM-lift-inc,
isdM0_wf,
names-hom_wf,
nh-comp-assoc,
assert-isdM0,
cubical-type-ap-morph-comp,
nh-comp-nc-m-eq2,
fset-member-add-name,
set_subtype_base,
not_wf,
s-comp-nc-0',
istype-cubical-type-at,
cubical-type-ap-morph-comp-eq,
iff_weakening_uiff,
assert_wf,
dM_wf,
dM0_wf,
member-cubical-subset-I_cube,
fl-morph-comp2,
fl-morph-fl0-is-1,
csm-ap-type-at,
cubical-term-at_wf,
cubical-subset-I_cube,
name-morph-satisfies_wf,
name-morph-satisfies-comp,
nh-id-right,
s-comp-nc-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
isectElimination,
because_Cache,
dependent_set_memberEquality_alt,
universeIsType,
applyEquality,
hypothesisEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
voidElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
setIsType,
functionIsType,
intEquality,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
productEquality,
cumulativity,
isectEquality,
universeEquality,
equalityIstype,
unionIsType,
equalityElimination,
promote_hyp,
productIsType,
applyLambdaEquality,
inrFormation_alt,
sqequalBase,
hyp_replacement
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} I+i\} ].
\mforall{}[rho:Gamma(I+i)]. \mforall{}[phi:\mBbbF{}(I)].
\mforall{}u:\{I+i,s(phi) \mvdash{} \_:(A)<rho> o iota\}
\mforall{}[a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)]
((a0 (i0)(rho) s)
\mmember{} cubical-path-0(Gamma;A;I+i;j;m(i;j)(rho);fl-join(I+i;s(phi);(i=0));...))
Date html generated:
2020_05_20-PM-03_54_07
Last ObjectModification:
2020_04_09-PM-03_36_02
Theory : cubical!type!theory
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