Nuprl Lemma : last_index_wf
∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹]. (last_index(L;x.P[x]) ∈ ℕ||L|| + 1)
Proof
Definitions occuring in Statement :
last_index: last_index(L;x.P[x]),
length: ||as||,
list: T List,
int_seg: {i..j-},
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
last_index: last_index(L;x.P[x]),
member: t ∈ T,
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
pi2: snd(t),
lelt: i ≤ j < k,
and: P ∧ Q,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
false: False,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
prop: ℙ,
so_apply: x[s],
nat: ℕ,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
sq_type: SQType(T),
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
true: True
Lemmas referenced :
int_seg_wf,
length_wf,
int_seg_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
false_wf,
lelt_wf,
equal_wf,
bool_wf,
list_wf,
nat_properties,
intformle_wf,
int_formula_prop_le_lemma,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
length_of_nil_lemma,
list_accum_nil_lemma,
decidable__le,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
length_of_cons_lemma,
list_accum_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
subtype_rel_product,
int_seg_subtype,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
thin,
productEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
addEquality,
cumulativity,
hypothesisEquality,
hypothesis,
setElimination,
rename,
because_Cache,
lambdaFormation,
productElimination,
sqequalRule,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
functionEquality,
universeEquality,
intWeakElimination,
axiomEquality,
applyEquality,
dependent_pairEquality,
hypothesis_subsumption,
applyLambdaEquality,
instantiate,
functionExtensionality,
equalityElimination,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. (last\_index(L;x.P[x]) \mmember{} \mBbbN{}||L|| + 1)
Date html generated:
2018_05_21-PM-07_00_11
Last ObjectModification:
2017_07_26-PM-05_02_43
Theory : general
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