Nuprl Lemma : full-omega_wf
∀[fmla:int_formula()]. (full-omega(fmla) ∈ {b:𝔹| b = ff ⇒ (¬satisfiable_int_formula(fmla))} )
Proof
Definitions occuring in Statement :
full-omega: full-omega(fmla),
satisfiable_int_formula: satisfiable_int_formula(fmla),
int_formula: int_formula(),
bfalse: ff,
bool: 𝔹,
uall: ∀[x:A]. B[x],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
full-omega: full-omega(fmla),
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
polynomial-constraints: polynomial-constraints(),
so_lambda: λ2x.t[x],
so_apply: x[s],
iPolynomial: iPolynomial(),
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
and: P ∧ Q,
squash: ↓T,
guard: {T},
iMonomial: iMonomial(),
subtype_rel: A ⊆r B,
istype: istype(T),
prop: ℙ,
int_nzero: ℤ-o,
has-value: (a)↓,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
tunion: ⋃x:A.B[x],
so_apply: x[s1;s2],
bool: 𝔹,
unit: Unit,
not: ¬A,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
top: Top,
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
bfalse: ff,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
cand: A c∧ B,
false: False,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
true: True,
pi2: snd(t),
int-constraint-problem: IntConstraints,
l_all: (∀x∈L.P[x]),
pi1: fst(t),
nat: ℕ,
isr: isr(x)
Lemmas referenced :
satisfiable_int_formula_dnf,
int_formula_dnf_wf,
evalall-reduce,
list_wf,
polynomial-constraints_wf,
int_formula_wf,
list-valueall-type,
product-valueall-type,
iPolynomial_wf,
set-valueall-type,
iMonomial_wf,
all_wf,
int_seg_wf,
length_wf,
imonomial-less_wf,
select_wf,
sq_stable__le,
less_than_transitivity2,
le_weakening2,
int_nzero_wf,
sorted_wf,
istype-int,
nequal_wf,
int-valueall-type,
value-type-has-value,
list-value-type,
satisfiable_int_formula_wf,
l_exists_wf,
satisfiable_polynomial_constraints_wf,
l_member_wf,
eager-accum-list_accum,
bool_wf,
bfalse_wf,
bor_wf,
union-valueall-type,
unit_wf2,
equal-valueall-type,
list_accum_wf,
list_induction,
equal-wf-T-base,
l_all_wf,
list_accum_nil_lemma,
istype-void,
l_all_nil,
list_accum_cons_lemma,
l_all_cons,
eqtt_to_assert,
btrue_neq_bfalse,
btrue_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal_wf,
squash_wf,
true_wf,
equal-wf-base-T,
testxxx_lemma,
pcs-to-integer-problem_wf,
omega_wf,
satisfiable-pcs-to-integer-problem,
sq_stable__all,
satisfiable-integer-problem_wf,
subtype_rel_list,
list_subtype_base,
int_subtype_base,
false_wf,
sq_stable_from_decidable,
decidable__false,
isr-omega,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
sqequalRule,
independent_isectElimination,
Error :equalityIsType1,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
axiomEquality,
Error :universeIsType,
Error :lambdaEquality_alt,
because_Cache,
natural_numberEquality,
setElimination,
rename,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
setEquality,
intEquality,
callbyvalueReduce,
Error :productIsType,
Error :functionIsType,
Error :setIsType,
Error :dependent_set_memberEquality_alt,
Error :equalityIsType3,
functionEquality,
applyEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :equalityIsType4,
functionExtensionality,
unionElimination,
equalityElimination,
independent_pairFormation,
Error :dependent_pairFormation_alt,
promote_hyp,
instantiate,
cumulativity,
hyp_replacement,
universeEquality,
Error :equalityIsType2,
baseApply,
closedConclusion,
addEquality
Latex:
\mforall{}[fmla:int\_formula()]. (full-omega(fmla) \mmember{} \{b:\mBbbB{}| b = ff {}\mRightarrow{} (\mneg{}satisfiable\_int\_formula(fmla))\} )
Date html generated:
2019_06_20-PM-00_51_40
Last ObjectModification:
2018_10_03-PM-03_00_23
Theory : omega
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