Nuprl Lemma : member-all-vars-mkterm
∀[opr:Type]
  ∀f:opr. ∀v:varname(). ∀bts:bound-term(opr) List.
    ((v ∈ all-vars(mkterm(f;bts))) 
⇐⇒ ∃bt:bound-term(opr). ((bt ∈ bts) ∧ ((v ∈ fst(bt)) ∨ (v ∈ all-vars(snd(bt))))))
Proof
Definitions occuring in Statement : 
all-vars: all-vars(t)
, 
bound-term: bound-term(opr)
, 
mkterm: mkterm(opr;bts)
, 
varname: varname()
, 
l_member: (x ∈ l)
, 
list: T List
, 
uall: ∀[x:A]. B[x]
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
or: P ∨ Q
, 
and: P ∧ Q
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
mkterm: mkterm(opr;bts)
, 
all-vars: all-vars(t)
, 
member: t ∈ T
, 
so_lambda: λ2x.t[x]
, 
so_lambda: λ2x y.t[x; y]
, 
bound-term: bound-term(opr)
, 
so_apply: x[s1;s2]
, 
prop: ℙ
, 
and: P ∧ Q
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
or: P ∨ Q
, 
so_apply: x[s]
, 
implies: P 
⇒ Q
, 
iff: P 
⇐⇒ Q
, 
exists: ∃x:A. B[x]
, 
rev_implies: P 
⇐ Q
, 
uimplies: b supposing a
, 
not: ¬A
, 
false: False
, 
cand: A c∧ B
, 
l_member: (x ∈ l)
, 
nat: ℕ
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
select: L[n]
, 
cons: [a / b]
, 
nat_plus: ℕ+
, 
decidable: Dec(P)
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
guard: {T}
, 
uiff: uiff(P;Q)
, 
ge: i ≥ j 
, 
subtype_rel: A ⊆r B
, 
label: ...$L... t
, 
squash: ↓T
, 
true: True
Lemmas referenced : 
list_induction, 
bound-term_wf, 
all_wf, 
list_wf, 
varname_wf, 
iff_wf, 
l_member_wf, 
list_accum_wf, 
l-union_wf, 
var-deq_wf, 
all-vars_wf, 
or_wf, 
exists_wf, 
list_accum_nil_lemma, 
nil_wf, 
null_nil_lemma, 
btrue_wf, 
member-implies-null-eq-bfalse, 
btrue_neq_bfalse, 
list_accum_cons_lemma, 
cons_wf, 
istype-universe, 
member-union, 
istype-void, 
istype-le, 
length_of_cons_lemma, 
add_nat_plus, 
length_wf_nat, 
decidable__lt, 
full-omega-unsat, 
intformnot_wf, 
intformless_wf, 
itermConstant_wf, 
istype-int, 
int_formula_prop_not_lemma, 
int_formula_prop_less_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_wf, 
istype-less_than, 
nat_plus_properties, 
add-is-int-iff, 
intformand_wf, 
itermVar_wf, 
itermAdd_wf, 
intformeq_wf, 
int_formula_prop_and_lemma, 
int_term_value_var_lemma, 
int_term_value_add_lemma, 
int_formula_prop_eq_lemma, 
false_wf, 
length_wf, 
select_wf, 
nat_properties, 
decidable__le, 
intformle_wf, 
int_formula_prop_le_lemma, 
cons_member, 
subtype_rel_self, 
equal_wf, 
squash_wf, 
true_wf, 
iff_weakening_equal, 
term_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
lambdaFormation_alt, 
sqequalRule, 
cut, 
thin, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
hypothesis, 
lambdaEquality_alt, 
because_Cache, 
productElimination, 
universeIsType, 
inhabitedIsType, 
productEquality, 
independent_functionElimination, 
dependent_functionElimination, 
Error :memTop, 
independent_pairFormation, 
inlFormation_alt, 
productIsType, 
unionIsType, 
unionElimination, 
independent_isectElimination, 
equalityTransitivity, 
equalitySymmetry, 
voidElimination, 
rename, 
spreadEquality, 
independent_pairEquality, 
promote_hyp, 
functionIsType, 
inrFormation_alt, 
instantiate, 
universeEquality, 
dependent_pairFormation_alt, 
dependent_set_memberEquality_alt, 
natural_numberEquality, 
approximateComputation, 
applyLambdaEquality, 
setElimination, 
pointwiseFunctionality, 
baseApply, 
closedConclusion, 
baseClosed, 
int_eqEquality, 
equalityIstype, 
applyEquality, 
imageElimination, 
imageMemberEquality
Latex:
\mforall{}[opr:Type]
    \mforall{}f:opr.  \mforall{}v:varname().  \mforall{}bts:bound-term(opr)  List.
        ((v  \mmember{}  all-vars(mkterm(f;bts)))
        \mLeftarrow{}{}\mRightarrow{}  \mexists{}bt:bound-term(opr).  ((bt  \mmember{}  bts)  \mwedge{}  ((v  \mmember{}  fst(bt))  \mvee{}  (v  \mmember{}  all-vars(snd(bt))))))
Date html generated:
2020_05_19-PM-09_56_25
Last ObjectModification:
2020_03_09-PM-04_09_21
Theory : terms
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