### Nuprl Lemma : rel-star-rel-plus2

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀x,y,z:T.  ((x R y) `` (y (R^*) z) `` (x R+ z))`

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]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
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]` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat_plus: `ℕ+` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` false: `False` uiff: `uiff(P;Q)` uimplies: `b supposing a` subtract: `n - m` subtype_rel: `A ⊆r B` top: `Top` less_than': `less_than'(a;b)` true: `True` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` cand: `A c∧ B`
Lemmas referenced :  equal_wf and_wf int_formula_prop_less_lemma intformless_wf subtract_wf infix_ap_wf add-subtract-cancel le_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermAdd_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties rel_exp_iff2 nat_plus_subtype_nat less_than_wf le-add-cancel add-zero add-associates add_functionality_wrt_le add-commutes minus-one-mul-top zero-add minus-one-mul minus-add condition-implies-le not-lt-2 false_wf decidable__lt rel_exp_wf nat_wf exists_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin cut lemma_by_obid isectElimination hypothesis lambdaEquality applyEquality hypothesisEquality functionEquality cumulativity universeEquality dependent_pairFormation dependent_set_memberEquality addEquality setElimination rename natural_numberEquality dependent_functionElimination unionElimination independent_pairFormation voidElimination independent_functionElimination independent_isectElimination isect_memberEquality voidEquality intEquality because_Cache minusEquality int_eqEquality computeAll inlFormation productEquality instantiate

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

Date html generated: 2016_05_14-PM-03_53_37
Last ObjectModification: 2016_01_14-PM-11_10_47

Theory : relations2

Home Index