Nuprl Lemma : satisfiable-exact-reduce-constraints
∀eqs:ℤ List List. ∀i:ℕ||eqs||. ∀j:ℕ+||eqs[i]||.
∀ineqs:ℤ List List
(satisfiable(eqs;ineqs)
⇒ satisfiable(exact-reduce-constraints(eqs[i];j;eqs);exact-reduce-constraints(eqs[i];j;ineqs)))
supposing exact-eq-constraint(eqs;i;j)
Proof
Definitions occuring in Statement :
exact-reduce-constraints: exact-reduce-constraints(w;j;L),
exact-eq-constraint: exact-eq-constraint(eqs;i;j),
satisfiable-integer-problem: satisfiable(eqs;ineqs),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
exact-eq-constraint: exact-eq-constraint(eqs;i;j),
implies: P ⇒ Q,
satisfiable-integer-problem: satisfiable(eqs;ineqs),
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
sq_type: SQType(T),
prop: ℙ,
l_all: (∀x∈L.P[x]),
cons: [a / b],
list-delete: as\i,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
satisfies-integer-problem: satisfies-integer-problem(eqs;ineqs;xs),
satisfies-integer-equality: xs ⋅ as =0,
satisfies-integer-inequality: xs ⋅ as ≥0,
cand: A c∧ B,
eq_int: (i =z j)
Lemmas referenced :
satisfies-integer-problem-length,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
decidable__le,
length_wf,
select_wf,
list_wf,
istype-false,
not-le-2,
sq_stable__le,
less-iff-le,
condition-implies-le,
minus-add,
istype-void,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-swap,
le-add-cancel,
istype-le,
list-set-type2,
equal-wf-base,
satisfiable-integer-problem_wf,
exact-eq-constraint_wf,
istype-less_than,
int_seg_wf,
int_seg_properties,
less_than_wf,
squash_wf,
true_wf,
length-list-delete,
int_seg_subtype_nat,
subtype_rel_self,
iff_weakening_equal,
decidable__lt,
subtract_wf,
not-lt-2,
minus-minus,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
reduce_hd_cons_lemma,
spread_cons_lemma,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
less_than_transitivity1,
less_than_irreflexivity,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
bool_wf,
assert-bnot,
iff_weakening_uiff,
assert_wf,
list-delete_wf,
satisfies-integer-problem_wf,
exact-reduce-constraints_wf,
list_subtype_base,
select-map,
subtype_rel_list,
top_wf,
length_wf_nat,
exact-reduce-constraints-sqequal,
length-map,
equal_wf,
istype-universe,
length-int-vec-add,
int-vec-mul_wf,
le_weakening,
length-int-vec-mul,
int-dot-reduce-dim,
exact-eq-constraint-implies,
absval_cases,
integer-dot-product_wf,
int-dot-mul-left,
one-mul,
int-vec-mul-mul,
mul-swap,
mul-commutes,
int-vec-add_wf,
ge_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
axiomEquality,
hypothesis,
thin,
rename,
productElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
instantiate,
cumulativity,
intEquality,
Error :lambdaEquality_alt,
closedConclusion,
natural_numberEquality,
setElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
because_Cache,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
applyEquality,
Error :isect_memberEquality_alt,
minusEquality,
Error :dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
Error :inhabitedIsType,
Error :universeIsType,
Error :productIsType,
setEquality,
universeEquality,
promote_hyp,
hypothesis_subsumption,
equalityElimination,
lessCases,
axiomSqEquality,
Error :isectIsTypeImplies,
Error :dependent_pairFormation_alt,
Error :equalityIsType4,
baseApply,
Error :equalityIsType1,
Error :setIsType,
multiplyEquality,
hyp_replacement
Latex:
\mforall{}eqs:\mBbbZ{} List List. \mforall{}i:\mBbbN{}||eqs||. \mforall{}j:\mBbbN{}\msupplus{}||eqs[i]||.
\mforall{}ineqs:\mBbbZ{} List List
(satisfiable(eqs;ineqs)
{}\mRightarrow{} satisfiable(exact-reduce-constraints(eqs[i];j;eqs);
exact-reduce-constraints(eqs[i];j;ineqs)))
supposing exact-eq-constraint(eqs;i;j)
Date html generated:
2019_06_20-PM-00_47_57
Last ObjectModification:
2018_10_18-PM-01_20_39
Theory : omega
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