Nuprl Lemma : member-mFOL-sequent-freevars
∀s:mFOL-sequent(). ∀v:ℤ.
((v ∈ mFOL-sequent-freevars(s)) ⇐⇒ (v ∈ mFOL-freevars(snd(s))) ∨ (∃h∈fst(s). (v ∈ mFOL-freevars(h))))
Proof
Definitions occuring in Statement :
mFOL-sequent-freevars: mFOL-sequent-freevars(s),
mFOL-sequent: mFOL-sequent(),
mFOL-freevars: mFOL-freevars(fmla),
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
mFOL-sequent: mFOL-sequent(),
pi2: snd(t),
pi1: fst(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
mFOL-sequent-freevars: mFOL-sequent-freevars(s),
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
cand: A c∧ B,
l_all: (∀x∈L.P[x]),
guard: {T},
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b,
squash: ↓T,
l_exists: (∃x∈L. P[x])
Lemmas referenced :
length_wf,
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
select_wf,
not-member-mFOL-sequent-freevars,
decidable__l_exists_better-extract,
decidable__equal_int,
decidable__l_member,
decidable__or,
mFOL-sequent_wf,
mFOL_wf,
l_exists_wf,
mFOL-freevars_wf,
or_wf,
mFOL-sequent-freevars_wf,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
lambdaEquality,
setElimination,
rename,
because_Cache,
setEquality,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
voidElimination,
inlFormation,
inrFormation,
independent_isectElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageElimination
Latex:
\mforall{}s:mFOL-sequent(). \mforall{}v:\mBbbZ{}.
((v \mmember{} mFOL-sequent-freevars(s))
\mLeftarrow{}{}\mRightarrow{} (v \mmember{} mFOL-freevars(snd(s))) \mvee{} (\mexists{}h\mmember{}fst(s). (v \mmember{} mFOL-freevars(h))))
Date html generated:
2016_05_15-PM-10_26_17
Last ObjectModification:
2016_01_16-PM-04_32_41
Theory : minimal-first-order-logic
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