### Nuprl Lemma : rel-star-iff-rel-plus-or

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀x,y:T.  (x (R^*) y `⇐⇒` (x R+ y) ∨ (x = y ∈ T))`

Proof

Definitions occuring in Statement :  rel_plus: `R+` rel_star: `R^*` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` or: `P ∨ Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  rel_plus: `R+` rel_star: `R^*` infix_ap: `x f y` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` or: `P ∨ Q` subtype_rel: `A ⊆r B` exists: `∃x:A. B[x]` nat: `ℕ` decidable: `Dec(P)` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` nat_plus: `ℕ+` le: `A ≤ B` not: `¬A` false: `False` uiff: `uiff(P;Q)` top: `Top` less_than': `less_than'(a;b)` true: `True` subtract: `n - m` rel_exp: `R^n` eq_int: `(i =z j)` ifthenelse: `if b then t else f fi ` btrue: `tt`
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation independent_pairFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesis lambdaEquality applyEquality hypothesisEquality unionElimination functionEquality cumulativity universeEquality productElimination dependent_functionElimination setElimination rename natural_numberEquality instantiate intEquality independent_isectElimination because_Cache independent_functionElimination inrFormation inlFormation dependent_pairFormation dependent_set_memberEquality voidElimination addEquality isect_memberEquality voidEquality minusEquality equalityTransitivity equalitySymmetry

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    \mforall{}x,y:T.    (x  rel\_star(T;  R)  y  \mLeftarrow{}{}\mRightarrow{}  (x  R\msupplus{}  y)  \mvee{}  (x  =  y))

Date html generated: 2016_05_14-PM-03_52_46
Last ObjectModification: 2015_12_26-PM-06_57_02

Theory : relations2

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