### Nuprl Lemma : rel_plus_closure

`∀[T:Type]. ∀[R,R2:T ⟶ T ⟶ ℙ].`
`  (Trans(T)(R2[_1;_2]) `` (∀x,y:T.  ((x R y) `` (x R2 y))) `` (∀x,y:T.  ((x R+ y) `` (x R2 y))))`

Proof

Definitions occuring in Statement :  rel_plus: `R+` trans: `Trans(T;x,y.E[x; y])` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` rel_plus: `R+` infix_ap: `x f y` exists: `∃x:A. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` prop: `ℙ` nat: `ℕ` le: `A ≤ B` 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]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` rel_exp: `R^n` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` rev_implies: `P `` Q` trans: `Trans(T;x,y.E[x; y])`
Lemmas referenced :  rel_exp_one 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 primrec-wf-nat-plus nat_plus_subtype_nat nat_plus_wf rel_plus_wf trans_wf 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 exists_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
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution sqequalRule productElimination thin promote_hyp cut hypothesis dependent_functionElimination hypothesisEquality independent_functionElimination introduction extract_by_obid isectElimination instantiate cumulativity because_Cache universeEquality dependent_set_memberEquality natural_numberEquality independent_pairFormation functionExtensionality applyEquality rename setElimination addEquality unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll functionEquality baseApply closedConclusion baseClosed equalityTransitivity equalitySymmetry productEquality equalityElimination impliesFunctionality

Latex:
\mforall{}[T:Type].  \mforall{}[R,R2:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
(Trans(T)(R2[\$_{1}\$;\$_{2}\$])  {}\mRightarrow{}  (\mforall{}x,y:T.    ((x  R  y)  {}\mRightarrow{}  (x  R\000C2  y)))  {}\mRightarrow{}  (\mforall{}x,y:T.    ((x  R\msupplus{}  y)  {}\mRightarrow{}  (x  R2  y))))

Date html generated: 2017_04_17-AM-09_26_53
Last ObjectModification: 2017_02_27-PM-05_27_50

Theory : relations2

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