Nuprl Lemma : rel-star-rel-plus2

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


Proof




Definitions occuring in Statement :  rel_plus: R+ rel_star: R^* uall: [x:A]. B[x] prop: infix_ap: y all: x:A. B[x] implies:  Q function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  rel_plus: R+ rel_star: R^* infix_ap: y uall: [x:A]. B[x] all: x:A. B[x] implies:  Q exists: x:A. B[x] member: t ∈ T prop: so_lambda: λ2x.t[x] so_apply: x[s] nat_plus: + nat: le: A ≤ B and: P ∧ Q decidable: Dec(P) or: P ∨ Q iff: ⇐⇒ Q not: ¬A rev_implies:  Q false: False uiff: uiff(P;Q) uimplies: supposing a subtract: m subtype_rel: A ⊆B top: Top less_than': less_than'(a;b) true: True ge: i ≥  satisfiable_int_formula: satisfiable_int_formula(fmla) cand: 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


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