Nuprl Lemma : l_member-first
∀[A:Type]. ∀d:A List. ∀x:A. ∀eq:EqDecider(A). ((x ∈ d) ⇒ (∃i:ℕ||d||. ((∀j:ℕi. (¬(d[j] = x ∈ A))) ∧ (d[i] = x ∈ A))))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
deq: EqDecider(T),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
eqof: eqof(d),
deq: EqDecider(T),
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
select: L[n],
cons: [a / b],
cand: A c∧ B,
ge: i ≥ j ,
iff: P ⇐⇒ Q,
subtract: n - m,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
all_wf,
deq_wf,
l_member_wf,
exists_wf,
int_seg_wf,
length_wf,
not_wf,
equal_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
list_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
eqof_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
length_of_cons_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cons_wf,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
lelt_wf,
non_neg_length,
cons_member,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
select-cons-tl,
add-subtract-cancel,
decidable__equal_int,
int_subtype_base,
squash_wf,
true_wf,
select_cons_tl,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
because_Cache,
hypothesis,
functionEquality,
natural_numberEquality,
productEquality,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
equalityElimination,
promote_hyp,
instantiate,
universeEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
addEquality
Latex:
\mforall{}[A:Type]
\mforall{}d:A List. \mforall{}x:A. \mforall{}eq:EqDecider(A).
((x \mmember{} d) {}\mRightarrow{} (\mexists{}i:\mBbbN{}||d||. ((\mforall{}j:\mBbbN{}i. (\mneg{}(d[j] = x))) \mwedge{} (d[i] = x))))
Date html generated:
2017_04_17-AM-09_15_43
Last ObjectModification:
2017_02_27-PM-05_21_32
Theory : decidable!equality
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