### Nuprl Lemma : absval-imax-difference

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

Proof

Definitions occuring in Statement :  imax: `imax(a;b)` absval: `|i|` uall: `∀[x:A]. B[x]` le: `A ≤ B` subtract: `n - m` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` le: `A ≤ B` and: `P ∧ Q` not: `¬A` implies: `P `` Q` false: `False` subtype_rel: `A ⊆r B` prop: `ℙ` true: `True` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` squash: `↓T` nat: `ℕ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b` less_than': `less_than'(a;b)`
Lemmas referenced :  less_than'_wf imax_wf absval_wf subtract_wf le_int_wf bool_wf eqtt_to_assert assert_of_le_int 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 eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot le_wf intformand_wf int_formula_prop_and_lemma squash_wf true_wf imax_unfold nat_wf 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 approximateComputation independent_functionElimination dependent_pairFormation int_eqEquality voidEquality promote_hyp instantiate independent_pairFormation cumulativity imageElimination imageMemberEquality baseClosed setElimination rename universeEquality minusEquality lessCases axiomSqEquality

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

Date html generated: 2019_06_20-PM-01_13_39
Last ObjectModification: 2018_08_20-PM-09_31_26

Theory : int_2

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