### 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])))`

Proof

Definitions occuring in Statement :  almost-full: `AFx,y:T.R[x; y]` TI: `TI(T;x,y.R[x; y];t.Q[t])` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` so_apply: `x[s]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` almost-full: `AFx,y:T.R[x; y]` all: `∀x:A. B[x]` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` implies: `P `` Q` not: `¬A` false: `False` prop: `ℙ` TI: `TI(T;x,y.R[x; y];t.Q[t])` subtype_rel: `A ⊆r 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

Latex:
\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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