### Nuprl Lemma : Dickson's lemma

`∀p:ℕ. ∀A:ℕp ⟶ ℕ ⟶ ℕ.  ∃j:ℕ. ∃i:ℕj. ∀k:ℕp. (A[k;i] ≤ A[k;j])`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` so_apply: `x[s1;s2]` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` int_seg: `{i..j-}` exists: `∃x:A. B[x]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` so_lambda: `λ2x y.t[x; y]` decidable: `Dec(P)` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` lelt: `i ≤ j < k` less_than: `a < b` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` top: `Top` subtract: `n - m` sq_stable: `SqStable(P)` cand: `A c∧ B` istype: `istype(T)` ge: `i ≥ j ` pi1: `fst(t)` compose: `f o g` nat_plus: `ℕ+`
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  Error :functionIsType,  Error :universeIsType,  introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis Error :inhabitedIsType,  sqequalRule Error :productIsType,  because_Cache applyEquality independent_isectElimination independent_pairFormation Error :lambdaEquality_alt,  functionEquality functionExtensionality dependent_functionElimination unionElimination instantiate cumulativity intEquality independent_functionElimination Error :dependent_pairFormation_alt,  Error :dependent_set_memberEquality_alt,  equalityTransitivity equalitySymmetry imageMemberEquality baseClosed productElimination voidElimination Error :unionIsType,  addEquality Error :isect_memberEquality_alt,  minusEquality Error :setIsType,  productEquality imageElimination multiplyEquality Error :inrFormation_alt,  sqequalIntensionalEquality Error :equalityIsType1,  promote_hyp Error :inlFormation_alt,  independent_pairEquality axiomEquality

Latex:
\mforall{}p:\mBbbN{}.  \mforall{}A:\mBbbN{}p  {}\mrightarrow{}  \mBbbN{}  {}\mrightarrow{}  \mBbbN{}.    \mexists{}j:\mBbbN{}.  \mexists{}i:\mBbbN{}j.  \mforall{}k:\mBbbN{}p.  (A[k;i]  \mleq{}  A[k;j])

Date html generated: 2019_06_20-PM-00_27_32
Last ObjectModification: 2018_09_29-PM-09_51_22

Theory : fun_1

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