### Nuprl Lemma : imin-imax-distributive

`∀[x,y,z:ℤ].  (imin(imax(x;y);imax(x;z)) = imax(x;imin(y;z)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  imin: `imin(a;b)` imax: `imax(a;b)` uall: `∀[x:A]. B[x]` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` imin: `imin(a;b)` imax: `imax(a;b)` uimplies: `b supposing a` has-value: `(a)↓` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` decidable: `Dec(P)`
Lemmas referenced :  value-type-has-value int-value-type 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 satisfiable-full-omega-tt intformand_wf intformle_wf itermVar_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf decidable__equal_int intformeq_wf int_formula_prop_eq_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis intEquality sqequalRule sqequalHypSubstitution isect_memberEquality isectElimination thin hypothesisEquality axiomEquality because_Cache extract_by_obid independent_isectElimination callbyvalueReduce lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination dependent_pairFormation promote_hyp dependent_functionElimination instantiate cumulativity independent_functionElimination voidElimination natural_numberEquality lambdaEquality int_eqEquality voidEquality independent_pairFormation computeAll

Latex:
\mforall{}[x,y,z:\mBbbZ{}].    (imin(imax(x;y);imax(x;z))  =  imax(x;imin(y;z)))

Date html generated: 2017_04_14-AM-09_14_36
Last ObjectModification: 2017_02_27-PM-03_52_17

Theory : int_2

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