Nuprl Lemma : FOL-abstract_wf
∀[fmla:mFOL()]. (FOL-abstract(fmla) ∈ AbstractFOFormula+(mFOL-freevars(fmla)))
Proof
Definitions occuring in Statement :
FOL-abstract: FOL-abstract(fmla),
mFOL-freevars: mFOL-freevars(fmla),
mFOL: mFOL(),
AbstractFOFormula+: AbstractFOFormula+(vs),
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
mFOatomic: name(vars),
mFOL_size: mFOL_size(p),
spreadn: spread3,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
mFOconnect: mFOconnect(knd;left;right),
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
mFOquant: mFOquant(isall;var;body),
FOL-abstract: FOL-abstract(fmla),
mFOL-freevars: mFOL-freevars(fmla),
mFOL_ind: mFOL_ind,
AbstractFOFormula+: AbstractFOFormula+(vs),
so_lambda: λ2x.t[x],
so_apply: x[s],
l_contains: A ⊆ B,
l_all: (∀x∈L.P[x]),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
le_wf,
mFOL_size_wf,
mFOL_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
mFOL-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
nat_wf,
AbstractFOAtomic+_wf,
FOStruct+_wf,
FOAssignment_wf,
remove-repeats_wf,
int-deq_wf,
subtype_rel_self,
subtype_rel_dep_function,
subtype_rel_FOAssignment,
remove-repeats_property,
select_wf,
length_wf,
select_member,
FOConnective+_wf,
mFOL-freevars_wf,
FOQuantifier+_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
productElimination,
unionElimination,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
promote_hyp,
tokenEquality,
equalityElimination,
instantiate,
cumulativity,
atomEquality,
imageElimination,
addEquality,
universeEquality,
functionEquality
Latex:
\mforall{}[fmla:mFOL()]. (FOL-abstract(fmla) \mmember{} AbstractFOFormula+(mFOL-freevars(fmla)))
Date html generated:
2018_05_21-PM-10_23_04
Last ObjectModification:
2017_07_26-PM-06_38_17
Theory : minimal-first-order-logic
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