Nuprl Lemma : int-list-index-append
∀[x:ℤ]. ∀[L1,L2:ℤ List].
(int-list-index(L1 @ L2;x) ~ if int-list-member(x;L1)
then int-list-index(L1;x)
else ||L1|| + int-list-index(L2;x)
fi )
Proof
Definitions occuring in Statement :
int-list-index: int-list-index(L;x),
int-list-member: int-list-member(i;xs),
length: ||as||,
append: as @ bs,
list: T List,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
add: n + m,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
int-list-index: int-list-index(L;x),
lelt: i ≤ j < k,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
subtract: n - m
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
ge_wf,
istype-less_than,
list-cases,
list_ind_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-void,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
list_ind_cons_lemma,
length_of_cons_lemma,
istype-nat,
list_wf,
int-list-member_wf,
nil_wf,
eqtt_to_assert,
assert-int-list-member,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
l_member_wf,
zero-add,
int-list-index_wf,
int_seg_wf,
length_wf,
cons_wf,
append_wf,
lelt_wf,
eq_int_wf,
assert_of_eq_int,
istype-false,
add_nat_plus,
add_nat_wf,
length_wf_nat,
length-append,
add-is-int-iff,
false_wf,
decidable__lt,
nat_plus_properties,
neg_assert_of_eq_int,
cons_member,
add-member-int_seg2,
int_seg_subtype,
non_neg_length,
length_append,
subtype_rel_list,
top_wf,
int_seg_properties,
int_seg_subtype_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
isect_memberEquality_alt,
axiomSqEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
intEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalBase,
equalityElimination,
cumulativity,
addEquality,
pointwiseFunctionality,
productIsType,
minusEquality,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}[x:\mBbbZ{}]. \mforall{}[L1,L2:\mBbbZ{} List].
(int-list-index(L1 @ L2;x) \msim{} if int-list-member(x;L1)
then int-list-index(L1;x)
else ||L1|| + int-list-index(L2;x)
fi )
Date html generated:
2020_05_20-AM-08_08_37
Last ObjectModification:
2020_01_31-AM-09_41_04
Theory : general
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