Nuprl Lemma : last_index_cons
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. ∀[a:T].
(last_index([a / L];x.P[x])
= if 0 <z last_index(L;x.P[x]) then 1 + last_index(L;x.P[x])
if P[a] then 1
else 0
fi
∈ ℤ)
Proof
Definitions occuring in Statement :
last_index: last_index(L;x.P[x]),
cons: [a / b],
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,
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
prop: ℙ,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
last_index: last_index(L;x.P[x]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
pi2: snd(t),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
true: True
Lemmas referenced :
last_index_append,
cons_wf,
nil_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
equal_wf,
squash_wf,
true_wf,
ifthenelse_wf,
lt_int_wf,
last_index_wf,
int_seg_wf,
length_wf,
list_accum_cons_lemma,
list_accum_nil_lemma,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hyp_replacement,
equalitySymmetry,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
intEquality,
natural_numberEquality,
functionExtensionality,
setElimination,
rename,
addEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination,
imageMemberEquality,
baseClosed,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. \mforall{}[a:T].
(last\_index([a / L];x.P[x])
= if 0 <z last\_index(L;x.P[x]) then 1 + last\_index(L;x.P[x])
if P[a] then 1
else 0
fi )
Date html generated:
2018_05_21-PM-07_00_28
Last ObjectModification:
2017_07_26-PM-05_02_56
Theory : general
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