Nuprl Lemma : unsat-omega_start
∀n:ℕ. ∀eqs,ineqs:{L:ℤ List| ||L|| = (n + 1) ∈ ℤ} List. (unsat(omega_start(eqs;ineqs)) ⇒ (¬satisfiable(eqs;ineqs)))
Proof
Definitions occuring in Statement :
omega_start: omega_start(eqs;ineqs),
unsat-int-problem: unsat(p),
satisfiable-integer-problem: satisfiable(eqs;ineqs),
length: ||as||,
list: T List,
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
satisfiable-integer-problem: satisfiable(eqs;ineqs),
exists: ∃x:A. B[x],
unsat-int-problem: unsat(p),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
omega_start: omega_start(eqs;ineqs),
satisfies-integer-problem: satisfies-integer-problem(eqs;ineqs;xs),
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
cons: [a / b],
gcd-reduce-eq-constraints: gcd-reduce-eq-constraints(sat;LL),
accumulate_abort: accumulate_abort(x,sofar.F[x; sofar];s;L),
eager-accum: eager-accum(x,a.f[x; a];y;l),
list_ind: list_ind,
nil: [],
it: ⋅,
gcd-reduce-ineq-constraints: gcd-reduce-ineq-constraints(sat;LL),
satisfies-int-constraint-problem: xs |= p,
cand: A c∧ B,
iff: P ⇐⇒ Q,
satisfies-integer-inequality: xs ⋅ as ≥0,
nat_plus: ℕ+,
le: A ≤ B,
decidable: Dec(P),
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True,
ge: i ≥ j ,
listp: A List+,
guard: {T},
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
satisfies-integer-equality: xs ⋅ as =0
Lemmas referenced :
satisfiable-integer-problem_wf,
subtype_rel_list,
list_wf,
equal-wf-base-T,
unsat-int-problem_wf,
omega_start_wf,
nat_wf,
set_wf,
equal_wf,
length_wf,
list-cases,
product_subtype_list,
l_all_cons,
list_subtype_base,
int_subtype_base,
satisfies-integer-inequality_wf,
satisfies-gcd-reduce-ineq-constraints,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
cons_wf,
nil_wf,
l_all_nil,
gcd-reduce-ineq-constraints_wf,
listp_wf,
subtype_rel_sets,
le_antisymmetry_iff,
add-swap,
unit_wf2,
true_wf,
l_all_wf,
l_member_wf,
satisfies-integer-equality_wf,
satisfies-gcd-reduce-eq-constraints,
gcd-reduce-eq-constraints_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
sqequalHypSubstitution,
productElimination,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
voidElimination,
because_Cache,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
setEquality,
intEquality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
addEquality,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
lambdaEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
independent_pairFormation,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidEquality,
minusEquality,
comment,
unionEquality,
productEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}eqs,ineqs:\{L:\mBbbZ{} List| ||L|| = (n + 1)\} List.
(unsat(omega\_start(eqs;ineqs)) {}\mRightarrow{} (\mneg{}satisfiable(eqs;ineqs)))
Date html generated:
2017_04_14-AM-09_12_32
Last ObjectModification:
2017_02_27-PM-03_50_24
Theory : omega
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