Nuprl Lemma : mFOL-sequent-freevars-contained
∀s:mFOL-sequent(). ∀L:ℤ List.
(mFOL-sequent-freevars(s) ⊆ L ⇐⇒ mFOL-freevars(snd(s)) ⊆ L ∧ (∀h:mFOL(). ((h ∈ fst(s)) ⇒ mFOL-freevars(h) ⊆ L)))
Proof
Definitions occuring in Statement :
mFOL-sequent-freevars: mFOL-sequent-freevars(s),
mFOL-sequent: mFOL-sequent(),
mFOL-freevars: mFOL-freevars(fmla),
mFOL: mFOL(),
l_contains: A ⊆ B,
l_member: (x ∈ l),
list: T List,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
mFOL-sequent: mFOL-sequent(),
mFOL-sequent-freevars: mFOL-sequent-freevars(s),
pi2: snd(t),
pi1: fst(t),
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
top: Top,
iff: P ⇐⇒ Q,
false: False,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T}
Lemmas referenced :
mFOL-freevars_wf,
list_wf,
list_induction,
mFOL_wf,
all_wf,
iff_wf,
l_contains_wf,
reduce_wf,
l-union_wf,
int-deq_wf,
l_member_wf,
reduce_nil_lemma,
false_wf,
nil_member,
nil_wf,
reduce_cons_lemma,
and_wf,
equal_wf,
or_wf,
l-union-contained,
cons_member,
cons_wf,
mFOL-sequent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
intEquality,
lambdaEquality,
because_Cache,
productEquality,
functionEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
addLevel,
allFunctionality,
impliesFunctionality,
andLevelFunctionality,
allLevelFunctionality,
impliesLevelFunctionality,
rename,
unionElimination,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
equalityTransitivity,
levelHypothesis,
inlFormation,
inrFormation
Latex:
\mforall{}s:mFOL-sequent(). \mforall{}L:\mBbbZ{} List.
(mFOL-sequent-freevars(s) \msubseteq{} L
\mLeftarrow{}{}\mRightarrow{} mFOL-freevars(snd(s)) \msubseteq{} L \mwedge{} (\mforall{}h:mFOL(). ((h \mmember{} fst(s)) {}\mRightarrow{} mFOL-freevars(h) \msubseteq{} L)))
Date html generated:
2018_05_21-PM-10_29_25
Last ObjectModification:
2017_07_26-PM-06_41_35
Theory : minimal-first-order-logic
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