### 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: `ℕ+` uall: `∀[x:A]. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` bdd-diff: `bdd-diff(f;g)` exists: `∃x:A. B[x]` rmin: `rmin(x;y)` member: `t ∈ T` nat: `ℕ` all: `∀x:A. B[x]` guard: `{T}` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` le: `A ≤ B` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)` cand: `A c∧ B` iff: `P `⇐⇒` Q` rev_implies: `P `` 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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