Nuprl Lemma : satisfiable-elim-eq-constraints
∀eqs,ineqs:ℤ List List. ∀xs:ℤ List.
(satisfies-integer-problem(eqs;ineqs;xs)
⇒ satisfies-integer-problem([];(eager-map(λeq.eager-map(λx.(-x);eq);eqs) @ eqs) @ ineqs;xs))
Proof
Definitions occuring in Statement :
satisfies-integer-problem: satisfies-integer-problem(eqs;ineqs;xs),
eager-map: eager-map(f;as),
append: as @ bs,
nil: [],
list: T List,
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
minus: -n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
satisfies-integer-problem: satisfies-integer-problem(eqs;ineqs;xs),
and: P ∧ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
top: Top,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
nat: ℕ,
le: A ≤ B,
less_than: a < b,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
subtract: n - m,
sq_type: SQType(T),
guard: {T},
satisfies-integer-equality: xs ⋅ as =0,
satisfies-integer-inequality: xs ⋅ as ≥0,
int-vec-mul: a * as,
true: True,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
ge: i ≥ j
Lemmas referenced :
l_all_nil,
satisfies-integer-problem_wf,
list_wf,
list-value-type,
eager-map_wf,
int-value-type,
l_all_append,
satisfies-integer-inequality_wf,
append_wf,
map_wf,
map-length,
length_wf_nat,
and_wf,
equal_wf,
nat_wf,
less_than_wf,
lelt_wf,
length_wf,
int_seg_wf,
eager-map-is-map,
select-map,
subtype_rel_list,
top_wf,
select_wf,
sq_stable__le,
non_neg_length,
map_length,
set_subtype_base,
le_wf,
int_subtype_base,
subtract_wf,
minus-one-mul,
add-swap,
add-commutes,
add-associates,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
zero-add,
one-mul,
subtype_base_sq,
int_seg_properties,
nat_properties,
length-map,
list_subtype_base,
minus-one-mul-top,
squash_wf,
true_wf,
int-dot-mul-left,
zero_ann_a,
decidable__equal_int,
integer-dot-product_wf,
false_wf,
not-equal-2,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
condition-implies-le,
minus-add,
minus-zero,
or_wf,
iff_weakening_equal,
le_reflexive,
equal-wf-base,
le_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
hypothesisEquality,
intEquality,
because_Cache,
lambdaEquality,
independent_isectElimination,
minusEquality,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality,
addLevel,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
levelHypothesis,
natural_numberEquality,
applyEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
sqequalIntensionalEquality,
promote_hyp,
addEquality,
multiplyEquality,
instantiate,
cumulativity,
functionExtensionality,
functionEquality,
universeEquality,
unionElimination,
inlFormation,
inrFormation,
orFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}eqs,ineqs:\mBbbZ{} List List. \mforall{}xs:\mBbbZ{} List.
(satisfies-integer-problem(eqs;ineqs;xs)
{}\mRightarrow{} satisfies-integer-problem([];(eager-map(\mlambda{}eq.eager-map(\mlambda{}x.(-x);eq);eqs) @ eqs) @ ineqs;xs))
Date html generated:
2017_04_14-AM-09_05_38
Last ObjectModification:
2017_02_27-PM-03_45_15
Theory : omega
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