Nuprl Lemma : accum_split_wf
∀[A,T:Type]. ∀[f:(T List × A) ⟶ 𝔹]. ∀[g:(T List × A) ⟶ A]. ∀[x:A]. ∀[L:T List].
(accum_split(g;x;f;L) ∈ {p:(T List × A) List × T List × A| let LL,L2 = p in is_accum_splitting(T;A;L;LL;L2;f;g;x)} )
Proof
Definitions occuring in Statement :
accum_split: accum_split(g;x;f;L),
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
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 :
squash: ↓T,
less_than: a < b,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
prop: ℙ,
and: P ∧ Q,
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,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
bfalse: ff,
cons: [a / b],
iff: P ⇐⇒ Q,
true: True,
cand: A c∧ B,
concat: concat(ll),
it: ⋅,
nil: [],
select: L[n],
so_apply: x[s1;s2;s3],
so_lambda: so_lambda3,
append: as @ bs,
pi2: snd(t),
pi1: fst(t),
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
accum_split: accum_split(g;x;f;L),
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
less_than': less_than'(a;b),
subtract: n - m,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
int_iseg: {i...j},
unit: Unit,
bool: 𝔹,
spreadn: spread3,
listp: A List+,
nat_plus: ℕ+
Lemmas referenced :
istype-universe,
bool_wf,
list_wf,
length_wf_nat,
istype-nat,
int_term_value_add_lemma,
itermAdd_wf,
top_wf,
subtype_rel_list,
null_wf3,
decidable__assert,
length_wf,
non_neg_length,
subtype_rel_self,
istype-le,
decidable__lt,
decidable__le,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
subtract_wf,
decidable__equal_int,
subtract-1-ge-0,
int_seg_wf,
int_seg_properties,
istype-less_than,
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,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
length_of_cons_lemma,
null_cons_lemma,
product_subtype_list,
is_accum_splitting_wf,
iseg_wf,
istype-assert,
iseg_nil,
istype-void,
l_all_nil,
reduce_hd_cons_lemma,
reduce_nil_lemma,
istype-base,
stuck-spread,
list_ind_nil_lemma,
map_nil_lemma,
nil_wf,
list_accum_nil_lemma,
length_of_nil_lemma,
null_nil_lemma,
list-cases,
iff_weakening_equal,
false_wf,
subtract-is-int-iff,
le-add-cancel2,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-associates,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
istype-false,
subtract_nat_wf,
length_firstn_eq,
true_wf,
squash_wf,
less_than_wf,
firstn_wf,
last-lemma-sq,
list_accum_append,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
assert_of_null,
uiff_transitivity,
not_wf,
bnot_wf,
assert_wf,
equal-wf-T-base,
append_wf,
eqtt_to_assert,
last_wf,
cons_wf,
list_accum_cons_lemma,
list_ind_cons_lemma,
pi1_wf_top,
map_wf,
concat_wf,
append_back_nil,
iseg_single,
map_cons_lemma,
length-append,
select_wf,
equal_wf,
select_append_front,
pi2_wf,
select_append_back,
length-singleton,
add-member-int_seg2,
map_append_sq,
concat_append,
concat-single,
l_all_single,
all_wf,
l_all_append,
add-is-int-iff,
nat_plus_properties,
add_nat_plus,
hd-append-sq,
add-subtract-cancel,
length_append,
le_wf,
select-cons-hd,
length_cons,
length_nil,
append_assoc,
btrue_neq_bfalse,
not_assert_elim,
assert_elim,
iseg_append_single
Rules used in proof :
universeEquality,
functionIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
addEquality,
imageElimination,
hypothesis_subsumption,
promote_hyp,
productIsType,
dependent_set_memberEquality_alt,
applyLambdaEquality,
because_Cache,
instantiate,
applyEquality,
unionElimination,
productElimination,
inhabitedIsType,
functionIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
voidElimination,
universeIsType,
independent_pairFormation,
Error :memTop,
dependent_functionElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
sqequalRule,
rename,
setElimination,
hypothesis,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation_alt,
thin,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalityIstype,
baseClosed,
productEquality,
independent_pairEquality,
closedConclusion,
baseApply,
pointwiseFunctionality,
minusEquality,
imageMemberEquality,
equalityElimination,
intEquality,
cumulativity,
dependent_pairEquality_alt,
hyp_replacement,
setIsType,
functionEquality,
voidEquality
Latex:
\mforall{}[A,T:Type]. \mforall{}[f:(T List \mtimes{} A) {}\mrightarrow{} \mBbbB{}]. \mforall{}[g:(T List \mtimes{} A) {}\mrightarrow{} A]. \mforall{}[x:A]. \mforall{}[L:T List].
(accum\_split(g;x;f;L) \mmember{} \{p:(T List \mtimes{} A) List \mtimes{} T List \mtimes{} A|
let LL,L2 = p
in is\_accum\_splitting(T;A;L;LL;L2;f;g;x)\} )
Date html generated:
2020_05_20-AM-08_12_08
Last ObjectModification:
2020_01_22-PM-03_22_52
Theory : general
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