### Nuprl Lemma : rel_plus_implies

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀x,y:T.  ((x R+ y) `` ((x R y) ∨ (∃z:T. ((x R+ z) ∧ (z R y)))))`

Proof

Definitions occuring in Statement :  rel_plus: `R+` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` rel_plus: `R+` infix_ap: `x f y` exists: `∃x:A. B[x]` member: `t ∈ T` prop: `ℙ` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` nat_plus: `ℕ+` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` rel_exp: `R^n` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` bfalse: `ff` btrue: `tt` guard: `{T}` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` cand: `A c∧ B` less_than: `a < b` squash: `↓T` true: `True`
Lemmas referenced :  infix_ap_wf rel_exp_wf false_wf le_wf nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf all_wf or_wf exists_wf rel_plus_wf nat_plus_wf primrec-wf-nat-plus nat_plus_subtype_nat eq_int_wf bool_wf equal-wf-base int_subtype_base assert_wf intformeq_wf int_formula_prop_eq_lemma equal_wf bnot_wf not_wf subtract_wf add-subtract-cancel uiff_transitivity eqtt_to_assert assert_of_eq_int iff_transitivity iff_weakening_uiff eqff_to_assert assert_of_bnot less_than_wf decidable__lt not-lt-2 less-iff-le condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates add-zero le-add-cancel equal-wf-T-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution sqequalRule productElimination thin cut instantiate introduction extract_by_obid isectElimination cumulativity hypothesisEquality because_Cache universeEquality dependent_set_memberEquality natural_numberEquality independent_pairFormation hypothesis functionExtensionality applyEquality rename setElimination dependent_functionElimination addEquality unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll functionEquality productEquality independent_functionElimination inlFormation equalitySymmetry hyp_replacement applyLambdaEquality inrFormation baseApply closedConclusion baseClosed equalityTransitivity equalityElimination impliesFunctionality imageMemberEquality minusEquality

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

Date html generated: 2017_04_17-AM-09_26_06
Last ObjectModification: 2017_02_27-PM-05_27_01

Theory : relations2

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