### Nuprl Lemma : locally-ranked-is-well-founded

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`  (Trans(T;x,y.R[y;x])`
`  `` (∀k:ℕ. ∀rank:T ⟶ ℕ. ∀l:T ⟶ ℕk.`
`        ((∀x,y:T.  (((l x) = (l y) ∈ ℤ) `` R[x;y] `` rank x < rank y)) `` tcWO(T;x,y.R[y;x]))))`

Proof

Definitions occuring in Statement :  trans: `Trans(T;x,y.E[x; y])` tcWO: `tcWO(T;x,y.>[x; y])` int_seg: `{i..j-}` nat: `ℕ` less_than: `a < b` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `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]` implies: `P `` Q` all: `∀x:A. B[x]` tcWO: `tcWO(T;x,y.>[x; y])` and: `P ∧ Q` member: `t ∈ T` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` squash: `↓T` int_seg: `{i..j-}` so_lambda: `λ2x.t[x]` nat: `ℕ` so_apply: `x[s]` uimplies: `b supposing a` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` guard: `{T}` trans: `Trans(T;x,y.E[x; y])` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` sq_stable: `SqStable(P)` lelt: `i ≤ j < k` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` subtract: `n - m` top: `Top` true: `True` ge: `i ≥ j ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b` cand: `A c∧ B`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  Error :lambdaFormation_alt,  independent_pairFormation Error :universeIsType,  cut applyEquality hypothesisEquality hypothesis thin because_Cache sqequalHypSubstitution sqequalRule introduction imageElimination imageMemberEquality baseClosed Error :functionIsType,  Error :equalityIstype,  extract_by_obid isectElimination intEquality Error :lambdaEquality_alt,  natural_numberEquality setElimination rename independent_isectElimination sqequalBase equalitySymmetry instantiate universeEquality Error :inhabitedIsType,  dependent_functionElimination independent_functionElimination unionElimination equalityElimination productElimination Error :dependent_set_memberEquality_alt,  addEquality equalityTransitivity Error :dependent_pairFormation_alt,  promote_hyp cumulativity voidElimination applyLambdaEquality Error :isect_memberEquality_alt,  minusEquality baseApply closedConclusion Error :productIsType

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
(Trans(T;x,y.R[y;x])
{}\mRightarrow{}  (\mforall{}k:\mBbbN{}.  \mforall{}rank:T  {}\mrightarrow{}  \mBbbN{}.  \mforall{}l:T  {}\mrightarrow{}  \mBbbN{}k.
((\mforall{}x,y:T.    (((l  x)  =  (l  y))  {}\mRightarrow{}  R[x;y]  {}\mRightarrow{}  rank  x  <  rank  y))  {}\mRightarrow{}  tcWO(T;x,y.R[y;x]))))

Date html generated: 2019_06_20-PM-00_30_10
Last ObjectModification: 2019_01_04-AM-11_46_07

Theory : rel_1

Home Index