Nuprl Lemma : last_index_append
∀[T:Type]. ∀[as,bs:T List]. ∀[P:T ⟶ 𝔹].
(last_index(as @ bs;x.P[x])
= if 0 <z last_index(bs;x.P[x]) then ||as|| + last_index(bs;x.P[x]) else last_index(as;x.P[x]) fi
∈ ℤ)
Proof
Definitions occuring in Statement :
last_index: last_index(L;x.P[x]),
length: ||as||,
append: as @ bs,
list: T List,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: 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,
last_index: last_index(L;x.P[x]),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
prop: ℙ,
nat: ℕ,
ge: i ≥ j ,
and: P ∧ Q,
pi1: fst(t),
cons: [a / b],
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
pi2: snd(t),
lt_int: i <z j,
le: A ≤ B,
true: True,
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bool_wf,
list_wf,
list_accum_append,
subtype_rel_list,
top_wf,
pair-eta,
list_accum_wf,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
length_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
nat_properties,
intformand_wf,
intformle_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
list_accum_nil_lemma,
length_of_nil_lemma,
add-zero,
product_subtype_list,
spread_cons_lemma,
decidable__le,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_subtype_base,
list_accum_cons_lemma,
length_of_cons_lemma,
ifthenelse_wf,
length_wf_nat,
last_index_wf,
int_seg_wf,
list_induction,
all_wf,
eqtt_to_assert,
pi2_wf,
lt_int_wf,
assert_of_lt_int,
false_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
squash_wf,
true_wf,
add_functionality_wrt_eq,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
functionEquality,
cumulativity,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
because_Cache,
sqequalHypSubstitution,
isectElimination,
thin,
universeEquality,
sqequalRule,
applyEquality,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
independent_pairEquality,
spreadEquality,
productElimination,
instantiate,
intEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
lambdaFormation,
setElimination,
rename,
intWeakElimination,
independent_pairFormation,
sqequalAxiom,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
imageElimination,
functionExtensionality,
equalityElimination,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[as,bs:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}].
...
Date html generated:
2018_05_21-PM-07_00_20
Last ObjectModification:
2017_07_26-PM-05_02_50
Theory : general
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