### Nuprl Lemma : rel_star_iff

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

Proof

Definitions occuring in Statement :  rel_star: `R^*` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` or: `P ∨ Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  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` cand: `A c∧ B` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` guard: `{T}` le: `A ≤ B` less_than': `less_than'(a;b)` rel_exp: `R^n` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` btrue: `tt`
Lemmas referenced :  false_wf infix_ap_wf less_than_wf add-subtract-cancel decidable__lt int_term_value_add_lemma itermAdd_wf le_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermSubtract_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties subtract_wf rel_exp_iff equal_wf and_wf or_wf rel_exp_wf nat_wf exists_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 functionEquality cumulativity universeEquality dependent_functionElimination independent_functionElimination inlFormation dependent_pairFormation dependent_set_memberEquality setElimination rename natural_numberEquality independent_isectElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll inrFormation addEquality because_Cache productEquality instantiate

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
\mforall{}x,y:T.    (x  (R\^{}*)  y  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}z:T.  ((x  (R\^{}*)  z)  \mwedge{}  (z  R  y)))  \mvee{}  (x  =  y))

Date html generated: 2016_05_14-PM-03_52_39
Last ObjectModification: 2016_01_14-PM-11_11_01

Theory : relations2

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