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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