Nuprl Lemma : satisfiable-full-omega-tt
∀[fmla:int_formula()]. ↑isl(full-omega(fmla)) supposing 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(),
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
isl: isl(x),
true: True,
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False,
not: ¬A
Lemmas referenced :
full-omega_wf,
set_wf,
bool_wf,
equal-wf-T-base,
not_wf,
satisfiable_int_formula_wf,
eqtt_to_assert,
assert_witness,
isl_wf,
unit_wf2,
btrue_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bfalse_wf,
int_formula_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
baseClosed,
lambdaFormation,
setElimination,
rename,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
natural_numberEquality,
applyEquality,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
because_Cache,
voidElimination,
isect_memberEquality,
setEquality
Latex:
\mforall{}[fmla:int\_formula()]. \muparrow{}isl(full-omega(fmla)) supposing satisfiable\_int\_formula(fmla)
Date html generated:
2017_09_29-PM-05_56_22
Last ObjectModification:
2017_07_26-PM-01_47_04
Theory : omega
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