Nuprl Lemma : absval-imin-difference

[a,b,c,d:ℤ].  (|imin(a;b) imin(c;d)| ≤ imax(|a c|;|b d|))


Proof




Definitions occuring in Statement :  imin: imin(a;b) imax: imax(a;b) absval: |i| uall: [x:A]. B[x] le: A ≤ B subtract: m int:
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T le: A ≤ B and: P ∧ Q not: ¬A implies:  Q false: False subtype_rel: A ⊆B prop: true: True all: x:A. B[x] bool: 𝔹 unit: Unit it: btrue: tt uiff: uiff(P;Q) uimplies: supposing a ifthenelse: if then else fi  bfalse: ff exists: x:A. B[x] or: P ∨ Q sq_type: SQType(T) guard: {T} bnot: ¬bb assert: b decidable: Dec(P) satisfiable_int_formula: satisfiable_int_formula(fmla) top: Top nat: squash: T iff: ⇐⇒ Q rev_implies:  Q less_than: a < b less_than': less_than'(a;b)
Lemmas referenced :  less_than'_wf imax_wf absval_wf subtract_wf imin_wf le_int_wf bool_wf eqtt_to_assert assert_of_le_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot le_wf decidable__le full-omega-unsat intformnot_wf intformle_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf nat_wf intformand_wf int_formula_prop_and_lemma squash_wf true_wf imin_unfold imax_unfold subtype_rel_self iff_weakening_equal lt_int_wf assert_of_lt_int top_wf less_than_wf itermSubtract_wf int_term_value_subtract_lemma intformless_wf itermConstant_wf int_formula_prop_less_lemma int_term_value_constant_lemma itermMinus_wf int_term_value_minus_lemma absval_unfold not_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality lambdaEquality dependent_functionElimination hypothesisEquality because_Cache extract_by_obid isectElimination hypothesis applyEquality axiomEquality equalityTransitivity equalitySymmetry intEquality isect_memberEquality voidElimination natural_numberEquality lambdaFormation unionElimination equalityElimination independent_isectElimination dependent_pairFormation promote_hyp instantiate independent_functionElimination approximateComputation int_eqEquality voidEquality cumulativity setElimination rename independent_pairFormation imageElimination imageMemberEquality baseClosed universeEquality minusEquality lessCases axiomSqEquality

Latex:
\mforall{}[a,b,c,d:\mBbbZ{}].    (|imin(a;b)  -  imin(c;d)|  \mleq{}  imax(|a  -  c|;|b  -  d|))



Date html generated: 2019_06_20-PM-01_13_46
Last ObjectModification: 2018_08_20-PM-09_31_31

Theory : int_2


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