Nuprl Lemma : last_index_property
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List].
((↑P[L[last_index(L;x.P[x]) - 1]]) ∧ (¬(∃x∈nth_tl(last_index(L;x.P[x]);L). ↑P[x])) supposing 0 < last_index(L;x.P[x])
∧ ¬(∃x∈L. ↑P[x]) supposing last_index(L;x.P[x]) = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
last_index: last_index(L;x.P[x]),
l_exists: (∃x∈L. P[x]),
select: L[n],
nth_tl: nth_tl(n;as),
list: T List,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ 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],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
pi2: snd(t),
select: L[n],
last_index: last_index(L;x.P[x]),
assert: ↑b,
l_exists: (∃x∈L. P[x]),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtract: n - m,
nth_tl: nth_tl(n;as),
le_int: i ≤z j,
lt_int: i <z j,
tl: tl(l),
true: True
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
assert_witness,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
product_subtype_list,
colength-cons-not-zero,
istype-nat,
colength_wf_list,
istype-false,
istype-le,
select_wf,
subtract_wf,
last_index_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
non_neg_length,
decidable__lt,
length_wf,
length_wf_nat,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
le_wf,
list_wf,
bool_wf,
istype-universe,
equal-wf-base,
assert_wf,
l_exists_wf_nil,
less_than_wf,
equal_wf,
pi2_wf,
ifthenelse_wf,
nil_wf,
list_accum_wf,
nth_tl_nil,
base_wf,
stuck-spread,
list_accum_nil_lemma,
length_of_nil_lemma,
last_index_cons,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
assert-bnot,
iff_weakening_uiff,
select-cons-tl,
add-associates,
add-commutes,
add-swap,
zero-add,
cons_wf,
length_of_cons_lemma,
length_cons,
le_int_wf,
assert_of_le_int,
reduce_tl_cons_lemma,
l_exists_cons,
l_exists_wf,
l_member_wf,
istype-assert
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,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
unionElimination,
promote_hyp,
hypothesis_subsumption,
equalityIsType1,
dependent_set_memberEquality_alt,
because_Cache,
applyEquality,
closedConclusion,
imageElimination,
addEquality,
instantiate,
equalityIsType4,
baseApply,
baseClosed,
intEquality,
functionIsType,
universeEquality,
isect_memberFormation,
spreadEquality,
lambdaEquality,
productEquality,
lambdaFormation,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
cumulativity,
equalityElimination,
equalityIsType3,
setIsType,
unionIsType
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List].
((\muparrow{}P[L[last\_index(L;x.P[x]) - 1]]) \mwedge{} (\mneg{}(\mexists{}x\mmember{}nth\_tl(last\_index(L;x.P[x]);L). \muparrow{}P[x]))
supposing 0 < last\_index(L;x.P[x])
\mwedge{} \mneg{}(\mexists{}x\mmember{}L. \muparrow{}P[x]) supposing last\_index(L;x.P[x]) = 0)
Date html generated:
2019_10_15-AM-11_10_27
Last ObjectModification:
2018_10_18-PM-11_52_11
Theory : general
Home
Index