Nuprl Lemma : AF-induction3

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`  (∀[Q:T ⟶ ℙ]. 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])))`

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]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` and: `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` prop: `ℙ` so_lambda: `λ2x.t[x]` and: `P ∧ Q` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` implies: `P `` 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: `A c∧ B` not: `¬A` nat: `ℕ` guard: `{T}`
Lemmas referenced :  nat_wf less_than_wf not_wf all_wf almost-full_wf exists_wf AF-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

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
((\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_29
Last ObjectModification: 2016_01_14-PM-06_59_46

Theory : bar-induction

Home Index