Nuprl Lemma : full-omega-unsat
∀[fmla:int_formula()]. ¬satisfiable_int_formula(fmla) supposing inr Ax ≤ full-omega(fmla)
Proof
Definitions occuring in Statement :
full-omega: full-omega(fmla),
satisfiable_int_formula: satisfiable_int_formula(fmla),
int_formula: int_formula(),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
inr: inr x ,
sqle: s ≤ t,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
bool: 𝔹,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff
Lemmas referenced :
satisfiable-full-omega-tt,
satisfiable_int_formula_wf,
sqle_wf_base,
int_formula-subtype-base,
int_formula_wf,
full-omega_wf,
set_wf,
bool_wf,
equal-wf-T-base,
not_wf,
equal_wf,
true_wf,
unit_subtype_base,
false_wf,
has-value_wf_base,
is-exception_wf,
not-bfalse-sqle-btrue
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
independent_isectElimination,
hypothesis,
because_Cache,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
baseClosed,
baseApply,
closedConclusion,
applyEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
setElimination,
rename,
unionElimination,
divergentSqle,
sqleRule,
sqleReflexivity
Latex:
\mforall{}[fmla:int\_formula()]. \mneg{}satisfiable\_int\_formula(fmla) supposing inr Ax \mleq{} full-omega(fmla)
Date html generated:
2017_09_29-PM-05_56_26
Last ObjectModification:
2017_06_01-AM-10_01_56
Theory : omega
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