### 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`
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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