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: equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T imin: imin(a;b) imax: imax(a;b) uimplies: supposing a has-value: (a)↓ all: x:A. B[x] implies:  Q bool: 𝔹 unit: Unit it: btrue: tt uiff: uiff(P;Q) and: P ∧ Q ifthenelse: if then else fi  bfalse: ff exists: x:A. B[x] prop: 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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