Nuprl Lemma : least-equiv-induction2

`∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ].`
`  ∀x,y:A.  ((x least-equiv(A;R) y) `` {∀[P:A ⟶ ℙ]. ((∀x,y:A.  ((x R y) `` (P[x] `⇐⇒` P[y]))) `` P[x] `` P[y])})`

Proof

Definitions occuring in Statement :  least-equiv: `least-equiv(A;R)` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` infix_ap: `x f y` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` member: `t ∈ T` prop: `ℙ` so_apply: `x[s]` so_lambda: `λ2x.t[x]` infix_ap: `x f y`
Lemmas referenced :  least-equiv-induction all_wf iff_wf least-equiv_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality independent_functionElimination hypothesis applyEquality sqequalRule lambdaEquality functionEquality cumulativity universeEquality dependent_functionElimination

Latex:
\mforall{}[A:Type].  \mforall{}[R:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].
\mforall{}x,y:A.
((x  least-equiv(A;R)  y)
{}\mRightarrow{}  \{\mforall{}[P:A  {}\mrightarrow{}  \mBbbP{}].  ((\mforall{}x,y:A.    ((x  R  y)  {}\mRightarrow{}  (P[x]  \mLeftarrow{}{}\mRightarrow{}  P[y])))  {}\mRightarrow{}  P[x]  {}\mRightarrow{}  P[y])\})

Date html generated: 2018_05_21-PM-00_52_07
Last ObjectModification: 2018_05_04-AM-10_25_19

Theory : relations2

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