### Nuprl Lemma : strict-fun-connected-induction

`∀[T:Type]`
`  ∀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])})`

Proof

Definitions occuring in Statement :  strict-fun-connected: `y = f+(x)` fun-connected: `y is f*(x)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` or: `P ∨ Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  guard: `{T}` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` or: `P ∨ Q` prop: `ℙ` uimplies: `b supposing a` not: `¬A` false: `False` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` strict-fun-connected: `y = 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

Latex:
\mforall{}[T:Type]
\mforall{}f:T  {}\mrightarrow{}  T
\mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}]
((\mforall{}x,y,z:T.
(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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