Nuprl Lemma : fill_from_comp_wf2
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)].
(fill_from_comp(Gamma;A;comp) ∈ 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:cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ {a:A(rho)|
(section-iota(Gamma;A;I+i;rho;a) = u ∈ {I+i,s(phi) ⊢ _:(A)<rho> o iota}) ∧ ((a rho (i0)) = a0 ∈ A((i0)(rho)))} )
Proof
Definitions occuring in Statement :
fill_from_comp: fill_from_comp(Gamma;A;comp),
composition-op: Gamma ⊢ CompOp(A),
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
section-iota: section-iota(Gamma;A;I;rho;a),
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type-ap-morph: (u a f),
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-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],
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
face-presheaf: 𝔽,
I_cube: A(I),
all: ∀x:A. B[x],
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,
let: let,
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
cubical-type: {X ⊢ _},
subset-iota: iota,
csm-comp: G o F,
csm-ap-type: (AF)s,
compose: f o g,
csm-ap: (s)x,
section-iota: section-iota(Gamma;A;I;rho;a),
canonical-section: canonical-section(Gamma;A;I;rho;a),
csm-ap-term: (t)s,
cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1),
fl-join: fl-join(I;x;y),
name-morph-satisfies: (psi f) = 1,
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,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
formal-cube: formal-cube(I),
names-hom: I ⟶ J,
context-map: <rho>,
functor-arrow: arrow(F),
cube-set-restriction: f(s),
fillterm: fillterm(Gamma;A;I;i;j;rho;a0;u),
cubical-term-at: u(a),
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
fl-morph: <f>,
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
face-lattice-property,
free-dist-lattice-with-constraints-property,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
pi2: snd(t),
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
dM: dM(I),
dM-lift: dM-lift(I;J;f),
nc-1: (i1),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
DeMorgan-algebra: DeMorganAlgebra,
nc-0: (i0),
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
free-DeMorgan-algebra-property,
free-dist-lattice-property,
empty-fset: {},
nil: [],
lattice-0: 0,
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
dM0: 0,
nc-s: s,
dM_inc: <x>,
dminc: <i>,
free-dl-inc: free-dl-inc(x),
fset-singleton: {x},
cons: [a / b]
Lemmas referenced :
fill_from_comp_wf1,
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,
subtype_rel_set,
cubical-type-at_wf,
cube-set-restriction_wf,
nc-m_wf,
subtype_rel_universe1,
cubical-path-condition'_wf,
new-name_wf,
fl-join_wf,
nc-s_wf,
f-subset-add-name,
fillterm_wf,
istype-cubical-type-at,
subtype_rel-equal,
fill_from_comp_property1,
nc-0_wf,
cubical-type-ap-morph_wf,
cubical-path-0_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-term_wf,
cubical-subset_wf,
face-presheaf_wf2,
csm-ap-type_wf,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
I_cube_wf,
not_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
composition-op_wf,
cubical-type_wf,
cubical_set_wf,
cubical-path-1_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube-set-restriction-comp,
nc-1_wf,
subtype_rel_self,
iff_weakening_equal,
cube-set-restriction-when-id,
nh-comp_wf,
nc-m-nc-1,
cubical-term-equal2,
csm-ap-type-at,
section-iota_wf,
cubical-subset-I_cube-member,
member-cubical-subset-I_cube,
iff_transitivity,
lattice-point_wf,
face_lattice_wf,
fl-morph_wf,
lattice-join_wf,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-1_wf,
or_wf,
fl-morph-join,
face_lattice-1-join-irreducible,
cat-ob_wf,
op-cat_wf,
cube-cat_wf,
s-comp-if-lemma1,
names-hom_wf,
nh-comp-assoc,
istype-void,
nh-id-right,
fl-morph-restriction,
names_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
dM-lift-inc,
int_subtype_base,
isdM0_wf,
assert-isdM0,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
nc-0-comp-s,
cubical-subset-I_cube,
name-morph-satisfies_wf,
cubical-term-at_wf,
csm-ap_wf,
f-subset-add-name1,
decidable__equal_int,
nh-comp-sq,
dM0_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
dM-lift-0,
dM-lift_wf2,
not-added-name,
dM_inc_wf,
names-subtype,
set_subtype_base,
istype-nat,
fset-member-add-name,
equal_functionality_wrt_subtype_rel2,
istype-cubical-term,
face-lattice-property,
free-dist-lattice-with-constraints-property,
free-DeMorgan-algebra-property,
free-dist-lattice-property
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
functionExtensionality,
sqequalRule,
dependent_functionElimination,
Error :memTop,
because_Cache,
dependent_set_memberEquality_alt,
universeIsType,
applyEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
voidElimination,
instantiate,
cumulativity,
productIsType,
equalityIstype,
inhabitedIsType,
setEquality,
intEquality,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
hyp_replacement,
productEquality,
isectEquality,
unionIsType,
inlFormation_alt,
functionIsType,
equalityElimination,
promote_hyp,
applyLambdaEquality,
functionEquality,
sqequalBase,
inrFormation_alt
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[comp:Gamma \mvdash{} CompOp(A)].
(fill\_from\_comp(Gamma;A;comp) \mmember{} I:fset(\mBbbN{})
{}\mrightarrow{} i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\}
{}\mrightarrow{} rho:Gamma(I+i)
{}\mrightarrow{} phi:\mBbbF{}(I)
{}\mrightarrow{} u:\{I+i,s(phi) \mvdash{} \_:(A)<rho> o iota\}
{}\mrightarrow{} a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)
{}\mrightarrow{} \{a:A(rho)| (section-iota(Gamma;A;I+i;rho;a) = u) \mwedge{} ((a rho (i0)) = a0)\} )
Date html generated:
2020_05_20-PM-03_55_17
Last ObjectModification:
2020_04_21-AM-00_50_55
Theory : cubical!type!theory
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