Nuprl Lemma : strict-fun-connected-induction

  ∀f:T ⟶ T
    ∀[R:T ⟶ T ⟶ ℙ]
      ((∀x,y,z:T.  (y is f*(z)  (R[y;z] ∨ (y z ∈ T))  R[x;z]) supposing ((¬(x y ∈ T)) and (x (f y) ∈ T)))
       {∀x,y:T.  (x f+(y)  R[x;y])})


Definitions occuring in Statement :  strict-fun-connected: f+(x) fun-connected: is f*(x) uimplies: supposing a uall: [x:A]. B[x] prop: guard: {T} so_apply: x[s1;s2] all: x:A. B[x] not: ¬A implies:  Q or: P ∨ Q apply: a function: x:A ⟶ B[x] universe: Type equal: t ∈ T
Definitions unfolded in proof :  guard: {T} uall: [x:A]. B[x] all: x:A. B[x] implies:  Q member: t ∈ T so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] or: P ∨ Q prop: uimplies: supposing a not: ¬A false: False so_lambda: λ2x.t[x] subtype_rel: A ⊆B so_apply: x[s] strict-fun-connected: f+(x) and: P ∧ Q
Lemmas referenced :  fun-connected-induction or_wf equal_wf fun-connected_wf not_wf all_wf isect_wf strict-fun-connected_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_functionElimination lambdaEquality applyEquality functionExtensionality cumulativity hypothesis independent_functionElimination inrFormation because_Cache axiomEquality rename voidElimination inlFormation functionEquality universeEquality independent_isectElimination unionElimination productElimination equalitySymmetry

    \mforall{}f:T  {}\mrightarrow{}  T
        \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}]
                    (y  is  f*(z)  {}\mRightarrow{}  (R[y;z]  \mvee{}  (y  =  z))  {}\mRightarrow{}  R[x;z])  supposing  ((\mneg{}(x  =  y))  and  (x  =  (f  y))))
            {}\mRightarrow{}  \{\mforall{}x,y:T.    (x  =  f+(y)  {}\mRightarrow{}  R[x;y])\})

Date html generated: 2018_05_21-PM-07_45_27
Last ObjectModification: 2017_07_26-PM-05_22_51

Theory : general

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