Nuprl Lemma : accum_split_prefix2
∀[A,T:Type]. ∀[x:A]. ∀[g:(T List × A) ⟶ A]. ∀[f:(T List × A) ⟶ 𝔹]. ∀[L:T List]. ∀[ZZ:(T List × A) List].
∀[Z,X:T List × A].
accum_split(g;x;f;concat(map(λp.(fst(p));ZZ @ [Z]))) = <ZZ, Z> ∈ ((T List × A) List × T List × A)
supposing accum_split(g;x;f;L) = <ZZ @ [Z], X> ∈ ((T List × A) List × T List × A)
Proof
Definitions occuring in Statement :
accum_split: accum_split(g;x;f;L),
concat: concat(ll),
map: map(f;as),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
pi1: fst(t),
so_apply: x[s],
all: ∀x:A. B[x],
guard: {T},
accum_split: accum_split(g;x;f;L),
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
and: P ∧ Q,
pi2: snd(t),
or: P ∨ Q,
uiff: uiff(P;Q),
not: ¬A,
false: False,
cons: [a / b],
spreadn: spread3,
decidable: Dec(P),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
cand: A c∧ B,
nat: ℕ,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
listp: A List+,
squash: ↓T,
concat: concat(ll),
sq_type: SQType(T),
bnot: ¬bb
Lemmas referenced :
last_induction,
all_wf,
list_wf,
equal_wf,
accum_split_wf,
append_wf,
cons_wf,
nil_wf,
concat_wf,
map_wf,
is_accum_splitting_wf,
bool_wf,
list_accum_nil_lemma,
list-cases,
null_nil_lemma,
btrue_wf,
null_cons_lemma,
bfalse_wf,
append_is_nil,
and_wf,
null_wf3,
btrue_neq_bfalse,
product_subtype_list,
length_of_nil_lemma,
length-append,
subtype_rel_list,
top_wf,
list_accum_append,
list_accum_cons_lemma,
set_wf,
decidable__assert,
list_ind_nil_lemma,
list_ind_cons_lemma,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_null,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
length_wf,
length_of_cons_lemma,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
hd_wf,
listp_properties,
cons_neq_nil,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
reduce_hd_cons_lemma,
tl_wf,
reduce_tl_cons_lemma,
map_cons_lemma,
map_nil_lemma,
concat-single,
accum_split_inverse,
reduce_nil_lemma,
pi1_wf_top,
subtype_rel_product,
squash_wf,
true_wf,
last_lemma,
last_wf,
iff_weakening_equal,
bool_cases,
subtype_base_sq,
bool_subtype_base,
general-append-cancellation,
ge_wf,
length_cons_ge_one,
map_append_sq,
concat_append,
bool_cases_sqequal,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
productEquality,
cumulativity,
hypothesis,
because_Cache,
functionEquality,
functionExtensionality,
applyEquality,
independent_pairEquality,
productElimination,
setElimination,
rename,
setEquality,
independent_functionElimination,
lambdaFormation,
spreadEquality,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
voidElimination,
voidEquality,
independent_pairFormation,
applyLambdaEquality,
unionElimination,
independent_isectElimination,
dependent_set_memberEquality,
promote_hyp,
hypothesis_subsumption,
equalityElimination,
baseClosed,
impliesFunctionality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
addEquality,
minusEquality,
hyp_replacement,
imageMemberEquality,
imageElimination,
equalityUniverse,
levelHypothesis,
instantiate,
inrFormation
Latex:
\mforall{}[A,T:Type]. \mforall{}[x:A]. \mforall{}[g:(T List \mtimes{} A) {}\mrightarrow{} A]. \mforall{}[f:(T List \mtimes{} A) {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List].
\mforall{}[ZZ:(T List \mtimes{} A) List]. \mforall{}[Z,X:T List \mtimes{} A].
accum\_split(g;x;f;concat(map(\mlambda{}p.(fst(p));ZZ @ [Z]))) = <ZZ, Z>
supposing accum\_split(g;x;f;L) = <ZZ @ [Z], X>
Date html generated:
2018_05_21-PM-08_07_48
Last ObjectModification:
2017_07_26-PM-05_43_30
Theory : general
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