### 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

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