Nuprl Lemma : disjoint_increasing_onto

∀[m,n,k:ℕ]. ∀[f:ℕn ⟶ ℕm]. ∀[g:ℕk ⟶ ℕm].
  (m = (n + k) ∈ ℕ) supposing 
     ((∀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 
     increasing(f;n))


Proof



Definitions occuring in Statement :  increasing: increasing(f;k),  int_seg: {i..j-},  nat: ℕ,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  exists: ∃x:A. B[x],  not: ¬A,  or: P ∨ Q,  apply: f a,  function: x:A ⟶ B[x],  add: n + m,  natural_number: $n,  int: ℤ,  equal: s = t ∈ T
Definitions unfolded in proof :  sq_stable: SqStable(P),  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  ifthenelse: if b then t else f fi ,  bfalse: ff,  bnot: ¬bb,  assert: ↑b,  rev_implies: P ⇐ Q,  iff: P ⇐⇒ Q,  le: A ≤ B,  subtract: n - m,  sq_type: SQType(T),  nat_plus: ℕ+,  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 ≥ j ,  decidable: Dec(P),  satisfiable_int_formula: satisfiable_int_formula(fmla),  top: Top,  and: P ∧ Q,  uall: ∀[x:A]. B[x],  member: t ∈ T,  uimplies: b supposing a,  all: ∀x:A. B[x],  nat: ℕ,  not: ¬A,  implies: P ⇒ Q,  subtype_rel: A ⊆r 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: ℙ
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
Latex:
\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 
          increasing(f;n))


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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