Nuprl Lemma : disjoint_increasing_onto

[m,n,k:ℕ]. ∀[f:ℕn ⟶ ℕm]. ∀[g:ℕk ⟶ ℕm].
  (m (n k) ∈ ℕsupposing 
     ((∀j1:ℕn. ∀j2:ℕk.  ((f j1) (g j2) ∈ ℤ))) and 
     (∀i:ℕm. ((∃j:ℕn. (i (f j) ∈ ℤ)) ∨ (∃j:ℕk. (i (g j) ∈ ℤ)))) and 
     increasing(g;k) and 


Definitions occuring in Statement :  increasing: increasing(f;k) int_seg: {i..j-} nat: uimplies: supposing a uall: [x:A]. B[x] all: x:A. B[x] exists: x:A. B[x] not: ¬A or: P ∨ Q apply: a function: x:A ⟶ B[x] add: m natural_number: $n int: equal: t ∈ T
Definitions unfolded in proof :  sq_stable: SqStable(P) bool: 𝔹 unit: Unit it: btrue: tt ifthenelse: if then else fi  bfalse: ff bnot: ¬bb assert: b rev_implies:  Q iff: ⇐⇒ Q le: A ≤ B subtract: m sq_type: SQType(T) nat_plus: + less_than: a < b squash: T less_than': less_than'(a;b) true: True guard: {T} inject: Inj(A;B;f) pi1: fst(t) uiff: uiff(P;Q) lelt: i ≤ j < k ge: i ≥  decidable: Dec(P) satisfiable_int_formula: satisfiable_int_formula(fmla) top: Top and: P ∧ Q uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] nat: not: ¬A implies:  Q subtype_rel: A ⊆B int_seg: {i..j-} so_lambda: λ2x.t[x] so_apply: x[s] false: False or: P ∨ Q exists: x:A. B[x] prop:
Lemmas referenced :  le_antisymmetry subtract_nat_wf not-lt-2 mul-swap int_seg_subtype istype-false equal_functionality_wrt_subtype_rel2 not-le-2 sq_stable__le increasing_inj equal-wf-base le_int_wf bnot_wf uiff_transitivity assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int lt_int_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf less_than_wf add-is-int-iff minus-one-mul-top mul-associates mul-distributes le-add-cancel le_weakening2 istype-sqequal less_than_transitivity1 less_than_irreflexivity add-commutes add-associates minus-add minus-one-mul add-swap add-mul-special two-mul mul-distributes-right zero-mul zero-add add-zero one-mul subtype_base_sq add_functionality_wrt_le le_reflexive less-iff-le omega-shadow decidable__equal_int intformeq_wf int_formula_prop_eq_lemma le_wf add-member-int_seg1 int_seg_properties subtract_wf itermSubtract_wf int_term_value_subtract_lemma decidable__lt intformless_wf int_formula_prop_less_lemma istype-less_than inject_wf injection_le nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_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_add_lemma int_term_value_var_lemma int_formula_prop_wf istype-le int_seg_wf istype-int set_subtype_base lelt_wf int_subtype_base istype-void increasing_wf istype-nat
Rules used in proof :  dependent_set_memberEquality imageElimination equalityElimination promote_hyp multiplyEquality instantiate cumulativity minusEquality imageMemberEquality applyLambdaEquality baseApply baseClosed Error :lambdaFormation_alt,  productElimination Error :dependent_set_memberEquality_alt,  addEquality dependent_functionElimination unionElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  int_eqEquality voidElimination independent_pairFormation sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut hypothesis sqequalRule Error :functionIsType,  Error :universeIsType,  extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality Error :equalityIstype,  applyEquality intEquality Error :lambdaEquality_alt,  independent_isectElimination sqequalBase equalitySymmetry Error :isect_memberEquality_alt,  axiomEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :unionIsType,  Error :productIsType,  because_Cache functionExtensionality equalityTransitivity closedConclusion

\mforall{}[m,n,k:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}m].  \mforall{}[g:\mBbbN{}k  {}\mrightarrow{}  \mBbbN{}m].
    (m  =  (n  +  k))  supposing 
          ((\mforall{}j1:\mBbbN{}n.  \mforall{}j2:\mBbbN{}k.    (\mneg{}((f  j1)  =  (g  j2))))  and 
          (\mforall{}i:\mBbbN{}m.  ((\mexists{}j:\mBbbN{}n.  (i  =  (f  j)))  \mvee{}  (\mexists{}j:\mBbbN{}k.  (i  =  (g  j)))))  and 
          increasing(g;k)  and 

Date html generated: 2019_06_20-PM-02_12_27
Last ObjectModification: 2019_06_20-PM-02_09_00

Theory : int_2

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