Nuprl Lemma : AF-induction2

[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
  (∀[Q:T ⟶ ℙ]. TI(T;x,y.R[x;y];t.Q[t])) supposing (AFx,y:T.¬R[x;y] and (∀x,y,z:T.  (R[x;y]  R[y;z]  R[x;z])))


Definitions occuring in Statement :  almost-full: AFx,y:T.R[x; y] TI: TI(T;x,y.R[x; y];t.Q[t]) uimplies: supposing a uall: [x:A]. B[x] prop: so_apply: x[s1;s2] so_apply: x[s] all: x:A. B[x] not: ¬A implies:  Q function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] uimplies: supposing a member: t ∈ T almost-full: AFx,y:T.R[x; y] all: x:A. B[x] squash: T so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] implies:  Q not: ¬A false: False prop: TI: TI(T;x,y.R[x; y];t.Q[t]) subtype_rel: A ⊆B so_lambda: λ2x.t[x] so_apply: x[s] guard: {T}
Lemmas referenced :  almost-full_wf all_wf false_wf subtype_rel_sets not_wf AF-induction nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut introduction sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality imageElimination hypothesis imageMemberEquality baseClosed functionEquality lemma_by_obid rename isectElimination applyEquality independent_isectElimination lambdaFormation independent_functionElimination voidElimination because_Cache setElimination setEquality universeEquality cumulativity

\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    (\mforall{}[Q:T  {}\mrightarrow{}  \mBbbP{}].  TI(T;x,y.R[x;y];t.Q[t]))  supposing 
          (AFx,y:T.\mneg{}R[x;y]  and 
          (\mforall{}x,y,z:T.    (R[x;y]  {}\mRightarrow{}  R[y;z]  {}\mRightarrow{}  R[x;z])))

Date html generated: 2016_05_13-PM-03_51_25
Last ObjectModification: 2016_01_14-PM-06_59_49

Theory : bar-induction

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