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
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