Nuprl Lemma : unsat-omega_step
∀p:IntConstraints. (unsat(omega_step(p)) ⇒ unsat(p))
Proof
Definitions occuring in Statement :
omega_step: omega_step(p),
unsat-int-problem: unsat(p),
int-constraint-problem: IntConstraints,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
omega_step: omega_step(p),
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
int-constraint-problem: IntConstraints,
tunion: ⋃x:A.B[x],
pi2: snd(t),
int-problem-dimension: dim(p),
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
nil: [],
it: ⋅,
first-success: first-success(f;L),
list_ind: list_ind,
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),
exact-reduce-constraints: exact-reduce-constraints(w;j;L),
evalall: evalall(t),
map: map(f;as),
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
subtract: n - m,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
le: A ≤ B,
unsat-int-problem: unsat(p),
satisfies-int-constraint-problem: xs |= p,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
unit: Unit,
exact-eq-constraint: exact-eq-constraint(eqs;i;j),
satisfiable-integer-problem: satisfiable(eqs;ineqs),
exists: ∃x:A. B[x],
satisfies-integer-problem: satisfies-integer-problem(eqs;ineqs;xs),
gcd-reduce-ineq-constraints: gcd-reduce-ineq-constraints(sat;LL),
cand: A c∧ B,
satisfies-integer-inequality: xs ⋅ as ≥0,
nat_plus: ℕ+,
listp: A List+,
isl: isl(x),
outl: outl(x),
assert: ↑b,
satisfies-integer-equality: xs ⋅ as =0,
bool: 𝔹,
bnot: ¬bb,
int_upper: {i...}
Lemmas referenced :
decidable__lt,
int-problem-dimension_wf,
nat_wf,
top_wf,
less_than_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
set_wf,
list_wf,
equal_wf,
length_wf,
list-cases,
null_nil_lemma,
product_subtype_list,
reduce_hd_cons_lemma,
subtract_wf,
null_cons_lemma,
false_wf,
not-lt-2,
le_antisymmetry_iff,
condition-implies-le,
add-associates,
add-commutes,
add-swap,
add_functionality_wrt_le,
zero-add,
le-add-cancel,
first-success_wf,
equal-wf-base-T,
equal-wf-base,
list_subtype_base,
set_subtype_base,
int_seg_wf,
equal-wf-T-base,
absval_wf,
select_wf,
decidable__le,
not-le-2,
sq_stable__le,
minus-add,
minus-one-mul,
minus-one-mul-top,
find-exact-eq-constraint_wf,
add-zero,
unit_wf2,
l_all_wf,
l_member_wf,
satisfies-integer-equality_wf,
satisfies-integer-inequality_wf,
unsat-int-problem_wf,
omega_step_wf,
int-constraint-problem_wf,
satisfiable-exact-reduce-constraints,
subtype_rel_list,
lelt_wf,
satisfies-integer-problem_wf,
exact-reduce-constraints_wf2,
satisfiable-integer-problem_wf,
nil_wf,
l_all_nil,
all_wf,
not_wf,
l_all_wf_nil,
l_all_cons,
satisfies-gcd-reduce-ineq-constraints,
not-equal-2,
minus-zero,
cons_wf,
gcd-reduce-ineq-constraints_wf,
listp_wf,
subtype_rel_sets,
minus-minus,
true_wf,
satisfies-gcd-reduce-eq-constraints,
gcd-reduce-eq-constraints_wf,
null_wf,
bool_wf,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
satisfies-shadow_inequalities,
le-add-cancel2,
le_wf,
shadow_inequalities_wf,
exists_wf,
satisfiable-elim-eq-constraints,
append_wf,
eager-map_wf,
set-value-type,
list-value-type,
int-value-type,
map-length,
map_wf,
eager-map-is-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
natural_numberEquality,
unionElimination,
because_Cache,
lessCases,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
independent_functionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
addEquality,
promote_hyp,
hypothesis_subsumption,
minusEquality,
setEquality,
baseApply,
closedConclusion,
productEquality,
unionEquality,
equalityElimination,
dependent_set_memberEquality,
addLevel,
levelHypothesis,
dependent_pairFormation,
inlEquality,
dependent_pairEquality,
independent_pairEquality
Latex:
\mforall{}p:IntConstraints. (unsat(omega\_step(p)) {}\mRightarrow{} unsat(p))
Date html generated:
2018_05_21-PM-00_24_35
Last ObjectModification:
2018_05_19-AM-06_58_23
Theory : omega
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