Nuprl Lemma : fill_from_comp_property1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)]. ∀[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)].
((fill_from_comp(Gamma;A;comp) I i rho phi u a0 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),
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,
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
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],
let: let,
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1),
fillterm: fillterm(Gamma;A;I;i;j;rho;a0;u),
cubical-term-at: u(a),
names: names(I),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {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,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
sq_type: SQType(T),
squash: ↓T,
true: True,
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-1: (i1),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bnot: ¬bb,
assert: ↑b,
nc-0: (i0),
isdM0: isdM0(x),
null: null(as),
empty-fset: {},
nil: [],
nequal: a ≠ b ∈ T ,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
cand: A c∧ B,
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,
cubical-path-0_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
istype-cubical-term,
cubical-subset_wf,
add-name_wf,
cube-set-restriction_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
csm-ap-type_wf,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
I_cube_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,
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,
composition-op_wf,
cubical-type_wf,
cubical_set_wf,
cubical-subset-I_cube,
nc-0_wf,
name-morph-satisfies_wf,
fl-join_wf,
name-morph-satisfies-join,
fl0_wf,
trivial-member-add-name1,
name-morph-satisfies-0,
subtype_rel_self,
new-name_wf,
subtype_base_sq,
not_wf,
set_subtype_base,
int_subtype_base,
nc-1_wf,
nc-m_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type-at_wf,
cube-set-restriction-comp,
iff_weakening_equal,
nh-comp_wf,
names-hom_wf,
nh-comp-assoc,
nc-m-nc-1,
nh-id-right,
cubical-type-ap-morph_wf,
istype-cubical-type-at,
cube-set-restriction-when-id,
subtype_rel-equal,
bool_wf,
bool_subtype_base,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
fset-member-add-name,
dM-lift-inc,
equal-wf-T-base,
isdM0_wf,
btrue_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
f-subset-add-name1,
nh-comp-is-id,
dM_inc_wf,
names-subtype,
names_wf,
nh-comp-sq,
dM-lift-nc-0,
nh-id_wf,
nh-id-left,
cube-set-restriction-id,
cubical-type-ap-morph-id
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
instantiate,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
setElimination,
rename,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
setIsType,
functionIsType,
intEquality,
productElimination,
inrFormation_alt,
promote_hyp,
cumulativity,
setEquality,
hyp_replacement,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
equalityElimination,
inlFormation_alt,
applyLambdaEquality,
sqequalBase,
productIsType
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[comp:Gamma \mvdash{} CompOp(A)]. \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:cubical-path-0(Gamma;A;I;i;rho;phi;u)].
((fill\_from\_comp(Gamma;A;comp) I i rho phi u a0 rho (i0)) = a0)
Date html generated:
2020_05_20-PM-03_54_44
Last ObjectModification:
2020_04_19-PM-08_11_40
Theory : cubical!type!theory
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