Nuprl Lemma : mFOL-freevars-of-rename
∀fmla:mFOL(). ∀old,new:ℤ.
∀x:ℤ
((x ∈ mFOL-freevars(mFOL-rename(fmla;old;new)))
⇐⇒ ((¬(x = old ∈ ℤ)) ∧ (x ∈ mFOL-freevars(fmla))) ∨ ((x = new ∈ ℤ) ∧ (old ∈ mFOL-freevars(fmla))))
supposing ¬(new ∈ mFOL-boundvars(fmla))
Proof
Definitions occuring in Statement :
mFOL-rename: mFOL-rename(fmla;old;new),
mFOL-boundvars: mFOL-boundvars(fmla),
mFOL-freevars: mFOL-freevars(fmla),
mFOL: mFOL(),
l_member: (x ∈ l),
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
mFOquant: mFOquant(isall;var;body),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
mFOconnect: mFOconnect(knd;left;right),
false: False,
not: ¬A,
mFOL_ind: mFOL_ind,
mFOatomic: name(vars),
mFOL-boundvars: mFOL-boundvars(fmla),
mFOL-rename: mFOL-rename(fmla;old;new),
mFOL-freevars: mFOL-freevars(fmla),
implies: P ⇒ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
or: P ∨ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
cand: A c∧ B,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
sq_type: SQType(T),
exists: ∃x:A. B[x]
Lemmas referenced :
bool_wf,
insert_wf,
int-deq_wf,
l-union_wf,
istype-atom,
list_wf,
istype-int,
istype-void,
nil_wf,
mFOL_wf,
int_subtype_base,
equal-wf-base,
mFOL-rename_wf,
mFOL-freevars_wf,
iff_wf,
mFOL-boundvars_wf,
l_member_wf,
not_wf,
mFOL-induction,
member-map,
member-remove-repeats,
remove-repeats_wf,
ifthenelse_wf,
map_wf,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
subtype_base_sq,
val-union-l-union,
int-valueall-type,
member-union,
member-insert,
member_filter,
bnot_wf,
filter_wf5,
istype-assert,
assert_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
equalityIstype,
unionIsType,
productIsType,
isectIsType,
functionIsType,
rename,
inhabitedIsType,
functionIsTypeImplies,
voidElimination,
dependent_functionElimination,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_functionElimination,
universeIsType,
because_Cache,
applyEquality,
productEquality,
unionEquality,
hypothesis,
hypothesisEquality,
isectEquality,
intEquality,
functionEquality,
lambdaEquality_alt,
sqequalRule,
thin,
isectElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
natural_numberEquality,
sqequalBase,
baseClosed,
closedConclusion,
baseApply,
inlFormation_alt,
promote_hyp,
dependent_pairFormation_alt,
inrFormation_alt,
equalityElimination,
unionElimination,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
cumulativity,
instantiate,
productElimination,
independent_pairFormation,
setElimination,
setIsType,
Error :memTop
Latex:
\mforall{}fmla:mFOL(). \mforall{}old,new:\mBbbZ{}.
\mforall{}x:\mBbbZ{}
((x \mmember{} mFOL-freevars(mFOL-rename(fmla;old;new)))
\mLeftarrow{}{}\mRightarrow{} ((\mneg{}(x = old)) \mwedge{} (x \mmember{} mFOL-freevars(fmla))) \mvee{} ((x = new) \mwedge{} (old \mmember{} mFOL-freevars(fmla))))
supposing \mneg{}(new \mmember{} mFOL-boundvars(fmla))
Date html generated:
2020_05_20-AM-09_08_23
Last ObjectModification:
2020_02_06-PM-01_55_55
Theory : minimal-first-order-logic
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