Nuprl Lemma : FOL-abstract-rename
∀[Dom:Type]. ∀[S:FOStruct+{i:l}(Dom)].
∀x,y:ℤ. ∀fmla:mFOL().
((¬(x ∈ mFOL-boundvars(fmla)))
⇒ (∀a1:FOAssignment(mFOL-freevars(mFOL-rename(fmla;y;x)),Dom). ∀a2:FOAssignment(mFOL-freevars(fmla),Dom).
((a2 = FOAssigment-rename(a1;fmla;x;y) ∈ FOAssignment(mFOL-freevars(fmla),Dom))
⇒ (Dom,S,a2 +|= FOL-abstract(fmla) = Dom,S,a1 +|= FOL-abstract(mFOL-rename(fmla;y;x)) ∈ ℙ))))
Proof
Definitions occuring in Statement :
FOL-abstract: FOL-abstract(fmla),
FOAssigment-rename: FOAssigment-rename(a;fmla;x;y),
mFOL-rename: mFOL-rename(fmla;old;new),
mFOL-boundvars: mFOL-boundvars(fmla),
mFOL-freevars: mFOL-freevars(fmla),
mFOL: mFOL(),
FOSatWith+: Dom,S,a +|= fmla,
FOStruct+: FOStruct+{i:l}(Dom),
FOAssignment: FOAssignment(vs,Dom),
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
uimplies: b supposing a,
so_apply: x[s],
not: ¬A,
mFOL-boundvars: mFOL-boundvars(fmla),
mFOL_ind: mFOL_ind,
mFOatomic: name(vars),
nil: [],
it: ⋅,
false: False,
guard: {T},
mFOL-rename: mFOL-rename(fmla;old;new),
FOL-abstract: FOL-abstract(fmla),
AbstractFOAtomic+: AbstractFOAtomic+(n;L),
FOSatWith+: Dom,S,a +|= fmla,
squash: ↓T,
FOStruct+: FOStruct+{i:l}(Dom),
FOStruct: FOStruct(Dom),
top: Top,
compose: f o g,
bool: 𝔹,
unit: Unit,
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,
subtype_rel: A ⊆r B,
true: True,
FOAssigment-rename: FOAssigment-rename(a;fmla;x;y),
mFOL-freevars: mFOL-freevars(fmla),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
update-assignment: a[x := v],
FOAssignment: FOAssignment(vs,Dom),
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
mFOconnect: mFOconnect(knd;left;right),
FOConnective+: FOConnective+(knd),
let: let,
cand: A c∧ B,
mFOquant: mFOquant(isall;var;body),
FOQuantifier+: FOQuantifier+(isall),
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind
Lemmas referenced :
mFOL-induction,
not_wf,
l_member_wf,
mFOL-boundvars_wf,
all_wf,
FOAssignment_wf,
mFOL-freevars_wf,
mFOL-rename_wf,
equal_wf,
FOAssigment-rename_wf,
FOSatWith+_wf,
FOL-abstract_wf,
mFOL_wf,
mFOatomic_wf,
nil_wf,
istype-void,
list_wf,
istype-atom,
mFOconnect_wf,
mFOquant_wf,
bool_wf,
istype-int,
FOStruct+_wf,
istype-universe,
b-union_wf,
squash_wf,
true_wf,
map-map,
list-subtype,
map_wf,
eq_int_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,
subtype_rel_self,
deq-member_wf,
int-deq_wf,
remove-repeats_wf,
assert_wf,
bnot_wf,
istype-assert,
bool_cases,
assert-deq-member,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
subtype_rel_sets_simple,
member-remove-repeats,
equal-wf-base,
eq_int_eq_true,
btrue_wf,
iff_weakening_equal,
int_subtype_base,
set_subtype_base,
bfalse_wf,
assert_elim,
btrue_neq_bfalse,
strong-subtype-l_member,
strong-subtype-self,
remove-repeats_property,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
eq_int_eq_false,
member-union,
val-union-l-union,
int-valueall-type,
ifthenelse_wf,
eq_atom_wf,
subtype_rel_FOAssignment,
union-contains,
equal_functionality_wrt_subtype_rel2,
union-contains2,
AbstractFOFormula+_wf,
trivial-mFOL-rename,
member-insert,
update-assignment_wf,
member_filter,
filter_wf5,
intformand_wf,
int_formula_prop_and_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
lambdaEquality_alt,
functionEquality,
closedConclusion,
intEquality,
hypothesisEquality,
because_Cache,
cumulativity,
independent_isectElimination,
universeEquality,
universeIsType,
independent_functionElimination,
equalityIstype,
inhabitedIsType,
functionIsType,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
isect_memberEquality_alt,
voidElimination,
setEquality,
functionExtensionality,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation_alt,
promote_hyp,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
tokenEquality,
natural_numberEquality,
independent_pairFormation,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
baseApply,
sqequalBase,
approximateComputation,
int_eqEquality,
inlFormation_alt,
inrFormation_alt,
productEquality,
unionEquality,
setIsType
Latex:
\mforall{}[Dom:Type]. \mforall{}[S:FOStruct+\{i:l\}(Dom)].
\mforall{}x,y:\mBbbZ{}. \mforall{}fmla:mFOL().
((\mneg{}(x \mmember{} mFOL-boundvars(fmla)))
{}\mRightarrow{} (\mforall{}a1:FOAssignment(mFOL-freevars(mFOL-rename(fmla;y;x)),Dom).
\mforall{}a2:FOAssignment(mFOL-freevars(fmla),Dom).
((a2 = FOAssigment-rename(a1;fmla;x;y))
{}\mRightarrow{} (Dom,S,a2 +|= FOL-abstract(fmla) = Dom,S,a1 +|= FOL-abstract(mFOL-rename(fmla;y;x))))))
Date html generated:
2019_10_16-AM-11_40_06
Last ObjectModification:
2018_12_01-PM-05_07_41
Theory : minimal-first-order-logic
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