Nuprl Lemma : exact-eq-constraint-implies
∀[eqs:ℤ List List]. ∀[i:ℕ||eqs||]. ∀[j:ℕ||eqs[i]||].
∀ineqs:ℤ List List. ∀xs:ℤ List.
(satisfies-integer-problem(eqs;ineqs;xs)
⇒ (xs[j] = if (eqs[i][j] =z 1) then -1 * eqs[i]\j ⋅ xs\j else eqs[i]\j ⋅ xs\j fi ∈ ℤ))
supposing exact-eq-constraint(eqs;i;j)
Proof
Definitions occuring in Statement :
exact-eq-constraint: exact-eq-constraint(eqs;i;j),
satisfies-integer-problem: satisfies-integer-problem(eqs;ineqs;xs),
int-vec-mul: a * as,
list-delete: as\i,
integer-dot-product: as ⋅ bs,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
satisfies-integer-problem: satisfies-integer-problem(eqs;ineqs;xs),
and: P ∧ Q,
l_all: (∀x∈L.P[x]),
satisfies-integer-equality: xs ⋅ as =0,
prop: ℙ,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
cand: A c∧ B,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
top: Top,
subtract: n - m,
exact-eq-constraint: exact-eq-constraint(eqs;i;j),
less_than: a < b,
true: True,
nequal: a ≠ b ∈ T
Lemmas referenced :
satisfies-integer-problem_wf,
list_wf,
exact-eq-constraint_wf,
int_seg_wf,
length_wf,
select_wf,
sq_stable__le,
int-dot-select,
int_seg_subtype_nat,
false_wf,
less_than_transitivity1,
le_weakening,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_subtype_base,
int-dot-mul-left,
list-delete_wf,
subtract_wf,
integer-dot-product_wf,
add-associates,
minus-zero,
one-mul,
minus-one-mul-top,
add-zero,
zero-add,
add-swap,
add-commutes,
add-mul-special,
zero-mul,
absval_unfold,
lt_int_wf,
assert_of_lt_int,
top_wf,
less_than_wf,
equal-wf-T-base,
minus-one-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
extract_by_obid,
isectElimination,
intEquality,
sqequalRule,
lambdaEquality,
axiomEquality,
because_Cache,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
independent_pairFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
minusEquality,
addEquality,
multiplyEquality,
voidEquality,
lessCases,
sqequalAxiom
Latex:
\mforall{}[eqs:\mBbbZ{} List List]. \mforall{}[i:\mBbbN{}||eqs||]. \mforall{}[j:\mBbbN{}||eqs[i]||].
\mforall{}ineqs:\mBbbZ{} List List. \mforall{}xs:\mBbbZ{} List.
(satisfies-integer-problem(eqs;ineqs;xs)
{}\mRightarrow{} (xs[j] = if (eqs[i][j] =\msubz{} 1) then -1 * eqs[i]\mbackslash{}j \mcdot{} xs\mbackslash{}j else eqs[i]\mbackslash{}j \mcdot{} xs\mbackslash{}j fi ))
supposing exact-eq-constraint(eqs;i;j)
Date html generated:
2017_04_14-AM-09_04_59
Last ObjectModification:
2017_02_27-PM-03_45_03
Theory : omega
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