Nuprl Lemma : trivial-mFOL-rename
∀x,y:ℤ. ∀fmla:mFOL().  ((¬(y ∈ mFOL-freevars(fmla))) 
⇒ (fmla = mFOL-rename(fmla;y;x) ∈ mFOL()))
Proof
Definitions occuring in Statement : 
mFOL-rename: mFOL-rename(fmla;old;new)
, 
mFOL-freevars: mFOL-freevars(fmla)
, 
mFOL: mFOL()
, 
l_member: (x ∈ l)
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
int: ℤ
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
so_lambda: λ2x.t[x]
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
so_apply: x[s]
, 
mFOL-rename: mFOL-rename(fmla;old;new)
, 
mFOL-freevars: mFOL-freevars(fmla)
, 
mFOatomic: name(vars)
, 
mFOL_ind: mFOL_ind, 
squash: ↓T
, 
true: True
, 
mFOconnect: mFOconnect(knd;left;right)
, 
uimplies: b supposing a
, 
mFOquant: mFOquant(isall;var;body)
, 
guard: {T}
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
exists: ∃x:A. B[x]
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
bnot: ¬bb
, 
assert: ↑b
, 
false: False
, 
subtype_rel: A ⊆r B
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
not: ¬A
, 
cand: A c∧ B
, 
nequal: a ≠ b ∈ T 
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
Lemmas referenced : 
mFOL-induction, 
not_wf, 
l_member_wf, 
mFOL-freevars_wf, 
equal_wf, 
mFOL_wf, 
mFOL-rename_wf, 
mFOatomic_wf, 
remove-repeats_wf, 
int-deq_wf, 
list_wf, 
mFOconnect_wf, 
val-union_wf, 
int-valueall-type, 
mFOquant_wf, 
filter_wf5, 
bnot_wf, 
eq_int_wf, 
bool_wf, 
ifthenelse_wf, 
eqtt_to_assert, 
assert_of_eq_int, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot, 
neg_assert_of_eq_int, 
squash_wf, 
true_wf, 
trivial_map, 
iff_weakening_equal, 
strong-subtype-l_member, 
strong-subtype-self, 
remove-repeats_property, 
int_subtype_base, 
val-union-l-union, 
member-union, 
member_filter, 
iff_transitivity, 
assert_wf, 
equal-wf-base, 
iff_weakening_uiff, 
assert_of_bnot, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformeq_wf, 
itermVar_wf, 
intformnot_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_eq_lemma, 
int_term_value_var_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
intEquality, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
sqequalRule, 
lambdaEquality, 
functionEquality, 
hypothesisEquality, 
hypothesis, 
independent_functionElimination, 
applyEquality, 
imageElimination, 
because_Cache, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
atomEquality, 
independent_isectElimination, 
setElimination, 
rename, 
setEquality, 
unionElimination, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
productElimination, 
dependent_pairFormation, 
promote_hyp, 
dependent_functionElimination, 
instantiate, 
cumulativity, 
voidElimination, 
universeEquality, 
inlFormation, 
inrFormation, 
independent_pairFormation, 
impliesFunctionality, 
int_eqEquality, 
isect_memberEquality, 
voidEquality, 
computeAll
Latex:
\mforall{}x,y:\mBbbZ{}.  \mforall{}fmla:mFOL().    ((\mneg{}(y  \mmember{}  mFOL-freevars(fmla)))  {}\mRightarrow{}  (fmla  =  mFOL-rename(fmla;y;x)))
Date html generated:
2018_05_21-PM-10_22_00
Last ObjectModification:
2017_07_26-PM-06_38_05
Theory : minimal-first-order-logic
Home
Index