Nuprl Lemma : int-list-index-property
∀x:ℤ. ∀L:ℤ List. (((x ∈ L) ⇐⇒ int-list-index(L;x) < ||L||) ∧ ((x ∈ L) ⇒ (L[int-list-index(L;x)] = x ∈ ℤ)))
Proof
Definitions occuring in Statement :
int-list-index: int-list-index(L;x),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
so_apply: x[s],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
cand: A c∧ B,
not: ¬A,
false: False,
ge: i ≥ j ,
guard: {T},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
int-list-index: int-list-index(L;x),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
cons: [a / b],
le: A ≤ B
Lemmas referenced :
list_induction,
iff_wf,
l_member_wf,
less_than_wf,
int-list-index_wf,
int_seg_wf,
length_wf,
equal-wf-base,
int_subtype_base,
list_subtype_base,
list_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
length_nil,
non_neg_length,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
list_ind_cons_lemma,
length_of_cons_lemma,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
intformnot_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
equal_wf,
cons_wf,
cons_member,
subtype_base_sq,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
select-cons-tl,
int_seg_subtype_nat,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
intEquality,
sqequalRule,
lambdaEquality,
productEquality,
hypothesisEquality,
hypothesis,
applyEquality,
setElimination,
rename,
natural_numberEquality,
addEquality,
functionEquality,
baseApply,
closedConclusion,
baseClosed,
because_Cache,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
unionElimination,
equalityElimination,
dependent_set_memberEquality,
imageMemberEquality,
pointwiseFunctionality,
promote_hyp,
inlFormation,
instantiate,
cumulativity,
imageElimination,
inrFormation
Latex:
\mforall{}x:\mBbbZ{}. \mforall{}L:\mBbbZ{} List.
(((x \mmember{} L) \mLeftarrow{}{}\mRightarrow{} int-list-index(L;x) < ||L||) \mwedge{} ((x \mmember{} L) {}\mRightarrow{} (L[int-list-index(L;x)] = x)))
Date html generated:
2018_05_21-PM-07_32_24
Last ObjectModification:
2017_07_26-PM-05_07_31
Theory : general
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