### 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: `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 ` 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: `ℕ` squash: `↓T` 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 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

Home Index