### Nuprl Lemma : imax_lb

`∀[a,b,c:ℤ].  uiff(imax(a;b) ≤ c;{(a ≤ c) ∧ (b ≤ c)})`

Proof

Definitions occuring in Statement :  imax: `imax(a;b)` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` guard: `{T}` le: `A ≤ B` and: `P ∧ Q` int: `ℤ`
Definitions unfolded in proof :  guard: `{T}` imax: `imax(a;b)` has-value: `(a)↓` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing 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 ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` le: `A ≤ B` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  value-type-has-value int-value-type le_int_wf bool_wf eqtt_to_assert assert_of_le_int decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf less_than'_wf le_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot ifthenelse_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep callbyvalueReduce cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin intEquality independent_isectElimination hypothesis hypothesisEquality because_Cache lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_pairFormation isect_memberFormation dependent_functionElimination natural_numberEquality dependent_pairFormation lambdaEquality int_eqEquality isect_memberEquality voidElimination voidEquality computeAll independent_pairEquality axiomEquality productEquality promote_hyp instantiate cumulativity independent_functionElimination

Latex:
\mforall{}[a,b,c:\mBbbZ{}].    uiff(imax(a;b)  \mleq{}  c;\{(a  \mleq{}  c)  \mwedge{}  (b  \mleq{}  c)\})

Date html generated: 2017_04_14-AM-09_14_08
Last ObjectModification: 2017_02_27-PM-03_51_31

Theory : int_2

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