### Nuprl Lemma : locally-ranked-induction

`∀[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)) `` (∀[Q:T ⟶ ℙ]. TI(T;x,y.R[y;x];x.Q[x])))))`

Proof

Definitions occuring in Statement :  trans: `Trans(T;x,y.E[x; y])` TI: `TI(T;x,y.R[x; y];t.Q[t])` int_seg: `{i..j-}` nat: `ℕ` less_than: `a < b` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` so_apply: `x[s]` 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]` member: `t ∈ T` implies: `P `` Q` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` prop: `ℙ` subtype_rel: `A ⊆r B` so_apply: `x[s1;s2]` nat: `ℕ` so_apply: `x[s]` so_lambda: `λ2x y.t[x; y]` uimplies: `b supposing a`
Lemmas referenced :  locally-ranked-is-well-founded all_wf equal_wf less_than_wf nat_wf int_seg_wf trans_wf tcWO-induction
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaFormation independent_functionElimination dependent_functionElimination cumulativity sqequalRule lambdaEquality because_Cache functionEquality intEquality applyEquality functionExtensionality universeEquality setElimination rename natural_numberEquality independent_isectElimination

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{}  (\mforall{}[Q:T  {}\mrightarrow{}  \mBbbP{}].  TI(T;x,y.R[y;x];x.Q[x])))))

Date html generated: 2017_04_14-AM-07_38_02
Last ObjectModification: 2017_02_27-PM-03_09_50

Theory : rel_1

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