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

`∀[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` exists: `∃x:A. B[x]` 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` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` rel_exp: `R^n` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` btrue: `tt` decidable: `Dec(P)` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` nat_plus: `ℕ+` uiff: `uiff(P;Q)` top: `Top` true: `True` subtract: `n - m`
Lemmas referenced :  exists_wf nat_wf rel_exp_wf false_wf le_wf or_wf nat_plus_wf nat_plus_subtype_nat equal_wf decidable__equal_int subtype_base_sq int_subtype_base decidable__lt not-lt-2 not-equal-2 add_functionality_wrt_le add-associates add-zero zero-add le-add-cancel condition-implies-le add-commutes minus-add minus-zero less_than_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin cut lemma_by_obid isectElimination hypothesis lambdaEquality applyEquality hypothesisEquality unionElimination dependent_pairFormation because_Cache dependent_set_memberEquality natural_numberEquality functionEquality cumulativity universeEquality dependent_functionElimination setElimination rename instantiate intEquality independent_isectElimination independent_functionElimination inlFormation voidElimination addEquality isect_memberEquality voidEquality minusEquality equalityTransitivity equalitySymmetry inrFormation

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_53_44
Last ObjectModification: 2015_12_26-PM-06_56_56

Theory : relations2

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