Nuprl Lemma : interleaving_occurence_onto
∀[A:Type]
∀L,L1,L2:A List. ∀f1:ℕ||L1|| ⟶ ℕ||L||. ∀f2:ℕ||L2|| ⟶ ℕ||L||.
∀j:ℕ||L||. ((∃k:ℕ||L1||. (j = (f1 k) ∈ ℤ)) ∨ (∃k:ℕ||L2||. (j = (f2 k) ∈ ℤ)))
supposing interleaving_occurence(A;L1;L2;L;f1;f2)
Proof
Definitions occuring in Statement :
interleaving_occurence: interleaving_occurence(T;L1;L2;L;f1;f2),
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
interleaving_occurence: interleaving_occurence(T;L1;L2;L;f1;f2),
and: P ∧ Q,
increasing: increasing(f;k),
int_seg: {i..j-},
lelt: i ≤ j < k,
nat: ℕ,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
subtype_rel: A ⊆r B,
subtract: n - m,
finite': finite'(T),
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
inject: Inj(A;B;f),
less_than': less_than'(a;b),
surject: Surj(A;B;f),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
member-less_than,
int_seg_wf,
length_wf,
nat_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
false_wf,
lelt_wf,
add-member-int_seg2,
decidable__le,
subtract_wf,
intformle_wf,
int_formula_prop_le_lemma,
equal_wf,
interleaving_occurence_wf,
list_wf,
nsub_finite',
lt_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
less_than_wf,
int_seg_subtype,
int_seg_properties,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
le_int_wf,
le_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
nat_wf,
increasing_inj,
length_wf_nat,
decidable__equal_int,
non_neg_length,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesis,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
natural_numberEquality,
cumulativity,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
functionEquality,
universeEquality,
independent_functionElimination,
addEquality,
imageElimination,
equalityElimination,
inlFormation,
inrFormation
Latex:
\mforall{}[A:Type]
\mforall{}L,L1,L2:A List. \mforall{}f1:\mBbbN{}||L1|| {}\mrightarrow{} \mBbbN{}||L||. \mforall{}f2:\mBbbN{}||L2|| {}\mrightarrow{} \mBbbN{}||L||.
\mforall{}j:\mBbbN{}||L||. ((\mexists{}k:\mBbbN{}||L1||. (j = (f1 k))) \mvee{} (\mexists{}k:\mBbbN{}||L2||. (j = (f2 k))))
supposing interleaving\_occurence(A;L1;L2;L;f1;f2)
Date html generated:
2017_10_01-AM-08_37_31
Last ObjectModification:
2017_07_26-PM-04_26_31
Theory : list!
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