Nuprl Lemma : satisfies_int_formula_dnf
∀fmla:int_formula(). ∀f:ℤ ⟶ ℤ.
(int_formula_prop(f;fmla) ⇐⇒ (∃X∈int_formula_dnf(fmla). satisfies-poly-constraints(f;X)))
Proof
Definitions occuring in Statement :
int_formula_dnf: int_formula_dnf(fmla),
satisfies-poly-constraints: satisfies-poly-constraints(f;X),
int_formula_prop: int_formula_prop(f;fmla),
int_formula: int_formula(),
l_exists: (∃x∈L. P[x]),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
int_formula_dnf: int_formula_dnf(fmla),
top: Top,
intformless: (left "<" right),
int_formula_ind: int_formula_ind,
satisfies-poly-constraints: satisfies-poly-constraints(f;X),
int_term_to_ipoly: int_term_to_ipoly(t),
itermSubtract: left (-) right,
int_term_ind: int_term_ind,
itermAdd: left (+) right,
itermConstant: "const",
iff: P ⇐⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
iPolynomial: iPolynomial(),
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
polynomial-constraints: polynomial-constraints(),
intformle: left "≤" right,
intformeq: left "=" right,
intformand: left "∧" right,
int_formula_prop: int_formula_prop(f;fmla),
intformor: left "or" right,
intformimplies: left "=>" right,
intformnot: "¬"form,
guard: {T},
iMonomial: iMonomial(),
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
false: False,
sorted: sorted(L),
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
exists: ∃x:A. B[x],
le: A ≤ B,
less_than': less_than'(a;b),
subtract: n - m,
const-poly: const-poly(n),
int_term_value: int_term_value(f;t),
itermMinus: "-"num,
nat_plus: ℕ+,
less_than: a < b,
decidable: Dec(P),
or: P ∨ Q,
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x])
Lemmas referenced :
int_formula-induction,
all_wf,
iff_wf,
int_formula_prop_wf,
l_exists_wf,
polynomial-constraints_wf,
int_formula_dnf_wf,
satisfies-poly-constraints_wf,
l_member_wf,
int_formula_wf,
int_formula_prop_less_lemma,
l_all_nil_iff,
l_all_single,
iPolynomial_wf,
int_term_to_ipoly_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
le_wf,
int_term_value_wf,
ipolynomial-term_wf,
equal_wf,
l_all_wf_nil,
l_all_wf,
cons_wf,
nil_wf,
less_than_wf,
true_wf,
l_exists_single,
subtype_rel_product,
list_wf,
subtype_rel_list,
subtype_rel_self,
int_term_wf,
int_formula_prop_le_lemma,
int_formula_prop_eq_lemma,
equal-wf-T-base,
int_formula_prop_and_lemma,
int_formula_prop_or_lemma,
int_formual_prop_imp_lemma,
int_formula_prop_not_lemma,
add_ipoly-sq,
minus-poly_wf,
add_ipoly_wf,
iMonomial_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
nequal_wf,
le_weakening2,
select_wf,
sq_stable__le,
less_than_transitivity2,
length_wf,
length_of_nil_lemma,
less_than_transitivity1,
less_than_irreflexivity,
int_seg_wf,
sorted_wf,
length_of_cons_lemma,
less-iff-le,
add_functionality_wrt_le,
add-associates,
zero-add,
add-swap,
add-commutes,
le-add-cancel2,
imonomial-less_wf,
minus-one-mul,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
minus-zero,
add-zero,
omega-shadow,
int_seg_properties,
squash_wf,
int_term_value_functionality,
add-ipoly_wf1,
add-ipoly_wf,
const-poly_wf,
add-ipoly-equiv,
iff_weakening_equal,
itermMinus_wf,
add_functionality_wrt_eq,
minus-poly-equiv,
minus_functionality_wrt_eq,
int_term_polynomial,
const-poly-value,
minus-add,
add-is-int-iff,
minus-is-int-iff,
subtract_wf,
le_reflexive,
minus-one-mul-top,
not-le-2,
mul-distributes,
mul-associates,
le-add-cancel,
not-lt-2,
minus-minus,
mul-commutes,
le-add-cancel-alt,
decidable__le,
decidable__lt,
int_term_value_minus_lemma,
itermMinus_functionality,
and-poly-constraints_wf,
satisfies-and-poly-constraints,
or_wf,
l_exists_append,
append_wf,
negate-poly-constraints_wf,
not_wf,
satisfies-negate-poly-constraints,
decidable__exists_int_seg,
decidable__and2,
decidable__all_int_seg,
decidable__equal_int,
not-l_exists
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
functionEquality,
intEquality,
functionExtensionality,
applyEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
setEquality,
independent_functionElimination,
lambdaFormation,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
productElimination,
independent_pairFormation,
impliesFunctionality,
independent_isectElimination,
natural_numberEquality,
andLevelFunctionality,
productEquality,
independent_pairEquality,
because_Cache,
baseClosed,
dependent_set_memberEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
imageElimination,
dependent_pairFormation,
promote_hyp,
addEquality,
universeEquality,
minusEquality,
sqequalIntensionalEquality,
multiplyEquality,
baseApply,
closedConclusion,
levelHypothesis,
unionElimination,
inlFormation,
inrFormation,
impliesLevelFunctionality,
orFunctionality
Latex:
\mforall{}fmla:int\_formula(). \mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbZ{}.
(int\_formula\_prop(f;fmla) \mLeftarrow{}{}\mRightarrow{} (\mexists{}X\mmember{}int\_formula\_dnf(fmla). satisfies-poly-constraints(f;X)))
Date html generated:
2017_09_29-PM-05_56_01
Last ObjectModification:
2017_07_26-PM-01_44_42
Theory : omega
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