Nuprl Lemma : list_accum_split
∀[T:Type]. ∀[i:ℕ]. ∀[l:T List]. ∀[f:Top ⟶ T ⟶ Top]. ∀[y:Top].
accumulate (with value x and list item a):
f[x;a]
over list:
l
with starting value:
y) ~ accumulate (with value x and list item a):
f[x;a]
over list:
nth_tl(i;l)
with starting value:
accumulate (with value x and list item a):
f[x;a]
over list:
firstn(i;l)
with starting value:
y))
supposing i < ||l||
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
length: ||as||,
nth_tl: nth_tl(n;as),
list_accum: list_accum,
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
or: P ∨ Q,
decidable: Dec(P),
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
lt_int: i <z j,
le_int: i ≤z j,
nth_tl: nth_tl(n;as),
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
it: ⋅,
nil: [],
guard: {T},
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
cons: [a / b],
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
subtype_rel: A ⊆r B,
subtract: n - m,
firstn: firstn(n;as),
list_accum: list_accum,
append: as @ bs,
list_ind: list_ind
Lemmas referenced :
nat_wf,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
list_wf,
top_wf,
length_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
list_ind_cons_lemma,
decidable__equal_int,
set_subtype_base,
subtype_base_sq,
equal_wf,
le_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
spread_cons_lemma,
product_subtype_list,
list_ind_nil_lemma,
list-cases,
int_subtype_base,
colength_wf_list,
equal-wf-T-base,
decidable__lt,
nth_tl_decomp,
list_accum_nil_lemma,
firstn_wf,
select_wf,
firstn_decomp
Rules used in proof :
universeEquality,
because_Cache,
universeIsType,
unionElimination,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomSqEquality,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
cumulativity,
instantiate,
baseClosed,
addEquality,
dependent_set_memberEquality,
applyLambdaEquality,
productElimination,
hypothesis_subsumption,
promote_hyp,
applyEquality,
functionIsType,
inhabitedIsType,
callbyvalueReduce,
sqleReflexivity
Latex:
\mforall{}[T:Type]. \mforall{}[i:\mBbbN{}]. \mforall{}[l:T List]. \mforall{}[f:Top {}\mrightarrow{} T {}\mrightarrow{} Top]. \mforall{}[y:Top].
accumulate (with value x and list item a):
f[x;a]
over list:
l
with starting value:
y) \msim{} accumulate (with value x and list item a):
f[x;a]
over list:
nth\_tl(i;l)
with starting value:
accumulate (with value x and list item a):
f[x;a]
over list:
firstn(i;l)
with starting value:
y))
supposing i < ||l||
Date html generated:
2019_10_15-AM-10_53_54
Last ObjectModification:
2018_09_27-AM-10_10_58
Theory : list!
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