Nuprl Lemma : AF-uniform-induction3

[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
  (∀[Q:T ⟶ ℙ]. uniform-TI(T;x,y.R[x;y];t.Q[t])) supposing 
     ((∃R':T ⟶ T ⟶ ℙ(AFx,y:T.R'[x;y] ∧ (∀x,y:T.  (R[x;y]  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] uniform-TI: uniform-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] exists: x:A. B[x] not: ¬A implies:  Q and: P ∧ Q function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] uimplies: supposing a member: t ∈ T prop: so_lambda: λ2x.t[x] and: P ∧ Q so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] subtype_rel: A ⊆B implies:  Q so_apply: x[s] exists: x:A. B[x] almost-full: AFx,y:T.R[x; y] all: x:A. B[x] squash: T cand: c∧ B not: ¬A nat: guard: {T}
Lemmas referenced :  nat_wf less_than_wf not_wf all_wf almost-full_wf exists_wf AF-uniform-induction2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination hypothesis functionEquality cumulativity universeEquality instantiate because_Cache sqequalRule lambdaEquality productEquality applyEquality productElimination lambdaFormation dependent_functionElimination imageElimination introduction dependent_pairFormation independent_pairFormation setElimination rename imageMemberEquality baseClosed independent_functionElimination

\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    (\mforall{}[Q:T  {}\mrightarrow{}  \mBbbP{}].  uniform-TI(T;x,y.R[x;y];t.Q[t]))  supposing 
          ((\mexists{}R':T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}.  (AFx,y:T.R'[x;y]  \mwedge{}  (\mforall{}x,y:T.    (R[x;y]  {}\mRightarrow{}  (\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_31
Last ObjectModification: 2016_01_14-PM-06_59_47

Theory : bar-induction

Home Index