Nuprl Lemma : list_split_wf
∀[T:Type]. ∀[f:(T List) ⟶ 𝔹]. ∀[L:T List].
(list_split(f;L) ∈ {p:T List List × (T List)| let LL,L2 = p in is_list_splitting(T;L;LL;L2;f)} )
Proof
Definitions occuring in Statement :
list_split: list_split(f;L),
is_list_splitting: is_list_splitting(T;L;LL;L2;f),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
spread: spread def,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
bfalse: ff,
cons: [a / b],
true: True,
cand: A c∧ B,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
concat: concat(ll),
is_list_splitting: is_list_splitting(T;L;LL;L2;f),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
list_split: list_split(f;L),
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
uiff: uiff(P;Q),
iseg: l1 ≤ l2,
int_iseg: {i...j},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
subtract: n - m,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
non_neg_length,
length_wf,
decidable__assert,
null_wf3,
subtype_rel_list,
top_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
length_wf_nat,
list_wf,
bool_wf,
istype-universe,
length_of_cons_lemma,
null_cons_lemma,
product_subtype_list,
is_list_splitting_wf,
iseg_wf,
assert_wf,
l_all_nil,
list_ind_nil_lemma,
reduce_nil_lemma,
nil_wf,
list_accum_nil_lemma,
length_of_nil_lemma,
null_nil_lemma,
list-cases,
btrue_neq_bfalse,
append_is_nil,
not_assert_elim,
btrue_wf,
last-lemma-sq,
firstn_wf,
less_than_wf,
squash_wf,
true_wf,
length_firstn_eq,
subtract_nat_wf,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
subtract-is-int-iff,
false_wf,
iff_weakening_equal,
list_accum_append,
list_accum_cons_lemma,
cons_wf,
last_wf,
append-nil,
concat_wf,
equal_wf,
append_wf,
istype-assert,
iseg_single,
list_ind_cons_lemma,
ifthenelse_wf,
equal-wf-T-base,
bnot_wf,
not_wf,
length-append,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
concat-single,
concat_append,
bfalse_wf,
assert_elim,
l_all_cons,
all_wf,
l_all_append,
le_wf,
append_assoc,
iseg_append_single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
productElimination,
unionElimination,
applyEquality,
instantiate,
because_Cache,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
imageElimination,
addEquality,
isectIsTypeImplies,
functionIsType,
universeEquality,
promote_hyp,
baseClosed,
equalityIsType3,
independent_pairEquality,
equalityIsType1,
imageMemberEquality,
minusEquality,
equalityIstype,
pointwiseFunctionality,
baseApply,
closedConclusion,
productEquality,
equalityElimination,
functionEquality,
hyp_replacement,
intEquality,
equalityIsType4
Latex:
\mforall{}[T:Type]. \mforall{}[f:(T List) {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List].
(list\_split(f;L) \mmember{} \{p:T List List \mtimes{} (T List)| let LL,L2 = p in is\_list\_splitting(T;L;LL;L2;f)\} )
Date html generated:
2019_10_15-AM-11_15_40
Last ObjectModification:
2019_06_25-PM-02_35_09
Theory : general
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