Nuprl Lemma : append_split
∀[T:Type]
∀L:T List
∀[P:T ⟶ ℙ]
((∀x:ℕ||L||. Dec(P L[x]))
⇒ (∀i,j:ℕ||L||. ((P L[i]) ⇒ P L[j] supposing i < j))
⇒ (∃L1,L2:T List
(((L = (L1 @ L2) ∈ (T List)) ∧ (∀i:ℕ||L1||. (¬(P L1[i]))) ∧ (∀i:ℕ||L2||. (P L2[i])))
∧ (∀x∈L.(P x) ⇒ (x ∈ L2)))))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
append: as @ bs,
list: T List,
int_seg: {i..j-},
less_than: a < b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
subtype_rel: A ⊆r B,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B,
ge: i ≥ j ,
le: A ≤ B,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
cons: [a / b],
iff: P ⇐⇒ Q,
sq_type: SQType(T),
rev_implies: P ⇐ Q,
l_all: (∀x∈L.P[x])
Lemmas referenced :
list_induction,
uall_wf,
all_wf,
int_seg_wf,
length_wf,
decidable_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
less_than_wf,
exists_wf,
list_wf,
equal_wf,
append_wf,
length-append,
not_wf,
l_all_wf,
l_member_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
length_of_cons_lemma,
nil_wf,
list_ind_nil_lemma,
l_all_nil,
equal-wf-base-T,
intformeq_wf,
int_formula_prop_eq_lemma,
l_all_wf_nil,
cons_wf,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
select_cons_tl_sq,
member-less_than,
select-cons-tl,
add-subtract-cancel,
false_wf,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
squash_wf,
true_wf,
select-cons-hd,
select_append_front,
iff_weakening_equal,
length_zero,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
select_cons_tl,
l_all_cons,
cons_member,
list_ind_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
because_Cache,
universeEquality,
natural_numberEquality,
hypothesis,
applyEquality,
functionExtensionality,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
isectEquality,
productEquality,
applyLambdaEquality,
setEquality,
independent_functionElimination,
baseClosed,
equalityTransitivity,
equalitySymmetry,
addEquality,
dependent_set_memberEquality,
imageMemberEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
hyp_replacement,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[P:T {}\mrightarrow{} \mBbbP{}]
((\mforall{}x:\mBbbN{}||L||. Dec(P L[x]))
{}\mRightarrow{} (\mforall{}i,j:\mBbbN{}||L||. ((P L[i]) {}\mRightarrow{} P L[j] supposing i < j))
{}\mRightarrow{} (\mexists{}L1,L2:T List
(((L = (L1 @ L2)) \mwedge{} (\mforall{}i:\mBbbN{}||L1||. (\mneg{}(P L1[i]))) \mwedge{} (\mforall{}i:\mBbbN{}||L2||. (P L2[i])))
\mwedge{} (\mforall{}x\mmember{}L.(P x) {}\mRightarrow{} (x \mmember{} L2)))))
Date html generated:
2017_10_01-AM-08_34_35
Last ObjectModification:
2017_07_26-PM-04_25_29
Theory : list!
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