Nuprl Lemma : FOAssigment-rename_wf
∀[Dom:Type]. ∀[x,y:ℤ]. ∀[fmla:mFOL()]. ∀[a:FOAssignment(mFOL-freevars(mFOL-rename(fmla;y;x)),Dom)].
FOAssigment-rename(a;fmla;x;y) ∈ FOAssignment(mFOL-freevars(fmla),Dom) supposing ¬(x ∈ mFOL-boundvars(fmla))
Proof
Definitions occuring in Statement :
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(),
FOAssignment: FOAssignment(vs,Dom),
l_member: (x ∈ l),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
member: t ∈ T,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
FOAssigment-rename: FOAssigment-rename(a;fmla;x;y),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
FOAssignment: FOAssignment(vs,Dom),
l_contains: A ⊆ B,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
istype: istype(T),
squash: ↓T
Lemmas referenced :
deq-member_wf,
int-deq_wf,
mFOL-freevars_wf,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
l_member_wf,
mFOL-boundvars_wf,
istype-void,
FOAssignment_wf,
mFOL-rename_wf,
mFOL_wf,
istype-int,
istype-universe,
update-assignment_wf,
subtype_rel_FOAssignment,
int_subtype_base,
assert_wf,
bnot_wf,
eq_int_wf,
not_wf,
equal-wf-base,
istype-assert,
filter_wf5,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert_of_eq_int,
member_filter,
mFOL-freevars-of-rename,
l_all_iff,
subtype_rel_wf,
squash_wf,
true_wf,
list_wf,
trivial-mFOL-rename,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesis,
hypothesisEquality,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
because_Cache,
voidElimination,
universeIsType,
axiomEquality,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality,
applyEquality,
inlFormation_alt,
independent_pairFormation,
productIsType,
sqequalBase,
lambdaEquality_alt,
closedConclusion,
setElimination,
rename,
setIsType,
dependent_set_memberEquality_alt,
inrFormation_alt,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[Dom:Type]. \mforall{}[x,y:\mBbbZ{}]. \mforall{}[fmla:mFOL()]. \mforall{}[a:FOAssignment(mFOL-freevars(mFOL-rename(fmla;y;x)),Dom)].
FOAssigment-rename(a;fmla;x;y) \mmember{} FOAssignment(mFOL-freevars(fmla),Dom)
supposing \mneg{}(x \mmember{} mFOL-boundvars(fmla))
Date html generated:
2019_10_16-AM-11_39_07
Last ObjectModification:
2018_12_01-AM-00_03_45
Theory : minimal-first-order-logic
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