Nuprl Lemma : rmin_functionality_wrt_bdd-diff

[x1,x2,y1,y2:ℕ+ ⟶ ℤ].  (bdd-diff(y1;y2)  bdd-diff(x1;x2)  bdd-diff(rmin(x1;y1);rmin(x2;y2)))


Proof




Definitions occuring in Statement :  rmin: rmin(x;y) bdd-diff: bdd-diff(f;g) nat_plus: + uall: [x:A]. B[x] implies:  Q function: x:A ⟶ B[x] int:
Definitions unfolded in proof :  uall: [x:A]. B[x] implies:  Q bdd-diff: bdd-diff(f;g) exists: x:A. B[x] rmin: rmin(x;y) member: t ∈ T nat: all: x:A. B[x] guard: {T} ge: i ≥  decidable: Dec(P) or: P ∨ Q uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) false: False not: ¬A top: Top and: P ∧ Q prop: le: A ≤ B subtype_rel: A ⊆B so_lambda: λ2x.t[x] so_apply: x[s] uiff: uiff(P;Q) cand: c∧ B iff: ⇐⇒ Q rev_implies:  Q rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :  imax_wf imax_nat nat_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_wf equal_wf le_wf less_than'_wf absval_wf subtract_wf imin_wf nat_plus_wf all_wf bdd-diff_wf imax_lb imax_ub le_functionality absval-imin-difference le_weakening
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin dependent_pairFormation dependent_set_memberEquality cut introduction extract_by_obid isectElimination setElimination rename hypothesisEquality hypothesis equalityTransitivity equalitySymmetry applyLambdaEquality dependent_functionElimination natural_numberEquality unionElimination independent_isectElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination independent_pairEquality because_Cache applyEquality functionExtensionality axiomEquality functionEquality inrFormation inlFormation

Latex:
\mforall{}[x1,x2,y1,y2:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}].    (bdd-diff(y1;y2)  {}\mRightarrow{}  bdd-diff(x1;x2)  {}\mRightarrow{}  bdd-diff(rmin(x1;y1);rmin(x2;y2)))



Date html generated: 2017_10_03-AM-08_22_20
Last ObjectModification: 2017_07_28-AM-07_22_16

Theory : reals


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