Nuprl Lemma : disjoint_sublists_witness
∀[T:Type]
∀L1,L2,L:T List.
(disjoint_sublists(T;L1;L2;L)
⇒ (∃f:ℕ||L1|| + ||L2|| ⟶ ℕ||L||
(Inj(ℕ||L1|| + ||L2||;ℕ||L||;f)
∧ (∀i:ℕ||L1|| + ||L2||
(L1[i] = L[f i] ∈ T supposing i < ||L1|| ∧ L2[i - ||L1||] = L[f i] ∈ T supposing ||L1|| ≤ i)))))
Proof
Definitions occuring in Statement :
disjoint_sublists: disjoint_sublists(T;L1;L2;L),
select: L[n],
length: ||as||,
list: T List,
inject: Inj(A;B;f),
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
disjoint_sublists: disjoint_sublists(T;L1;L2;L),
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
decidable: Dec(P),
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
cand: A c∧ B,
ge: i ≥ j ,
nat: ℕ,
inject: Inj(A;B;f)
Lemmas referenced :
disjoint_sublists_wf,
list_wf,
lt_int_wf,
length_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
int_seg_wf,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
subtract_wf,
int_seg_properties,
decidable__le,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
false_wf,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
all_wf,
equal-wf-T-base,
assert_wf,
le_int_wf,
le_wf,
bnot_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
inject_wf,
select_wf,
non_neg_length,
length_wf_nat,
nat_properties,
int_subtype_base,
increasing_inj
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
universeEquality,
dependent_pairFormation,
lambdaEquality,
setElimination,
rename,
because_Cache,
unionElimination,
equalityElimination,
sqequalRule,
independent_isectElimination,
applyEquality,
functionExtensionality,
natural_numberEquality,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
addEquality,
pointwiseFunctionality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
independent_pairEquality,
axiomEquality,
productEquality,
isectEquality,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]
\mforall{}L1,L2,L:T List.
(disjoint\_sublists(T;L1;L2;L)
{}\mRightarrow{} (\mexists{}f:\mBbbN{}||L1|| + ||L2|| {}\mrightarrow{} \mBbbN{}||L||
(Inj(\mBbbN{}||L1|| + ||L2||;\mBbbN{}||L||;f)
\mwedge{} (\mforall{}i:\mBbbN{}||L1|| + ||L2||
(L1[i] = L[f i] supposing i < ||L1||
\mwedge{} L2[i - ||L1||] = L[f i] supposing ||L1|| \mleq{} i)))))
Date html generated:
2017_10_01-AM-08_35_44
Last ObjectModification:
2017_07_26-PM-04_25_53
Theory : list!
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