### Nuprl Lemma : least-equiv-induction

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

Proof

Definitions occuring in Statement :  least-equiv: `least-equiv(A;R)` uall: `∀[x:A]. B[x]` prop: `ℙ` 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]` implies: `P `` Q` all: `∀x:A. B[x]` infix_ap: `x f y` least-equiv: `least-equiv(A;R)` transitive-reflexive-closure: `R^*` or: `P ∨ Q` prop: `ℙ` and: `P ∧ Q` member: `t ∈ T` so_apply: `x[s]` so_lambda: `λ2x.t[x]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  and_wf equal_wf transitive-closure-induction or_wf infix_ap_wf least-equiv_wf all_wf iff_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution sqequalRule unionElimination thin cut hypothesis addLevel hyp_replacement equalitySymmetry dependent_set_memberEquality independent_pairFormation hypothesisEquality introduction extract_by_obid isectElimination applyLambdaEquality setElimination rename productElimination applyEquality levelHypothesis lambdaEquality independent_functionElimination instantiate cumulativity universeEquality because_Cache dependent_functionElimination functionEquality

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

Date html generated: 2018_05_21-PM-00_52_06
Last ObjectModification: 2018_05_04-AM-10_21_29

Theory : relations2

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