### Nuprl Lemma : absval_ifthenelse

`∀[x:ℤ]. (|x| ~ if 0 <z x then x else -x fi )`

Proof

Definitions occuring in Statement :  absval: `|i|` ifthenelse: `if b then t else f fi ` lt_int: `i <z j` uall: `∀[x:A]. B[x]` minus: `-n` natural_number: `\$n` int: `ℤ` sqequal: `s ~ t`
Definitions unfolded in proof :  absval: `|i|` uall: `∀[x:A]. B[x]` member: `t ∈ T` has-value: `(a)↓` 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` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` not: `¬A` false: `False` prop: `ℙ` 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`
Lemmas referenced :  value-type-has-value int-value-type lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut callbyvalueReduce extract_by_obid sqequalHypSubstitution isectElimination thin intEquality independent_isectElimination hypothesis hypothesisEquality natural_numberEquality lambdaFormation unionElimination equalityElimination because_Cache productElimination lessCases sqequalAxiom isect_memberEquality independent_pairFormation voidElimination voidEquality imageMemberEquality baseClosed imageElimination independent_functionElimination dependent_pairFormation equalityTransitivity equalitySymmetry promote_hyp dependent_functionElimination instantiate cumulativity

Latex:
\mforall{}[x:\mBbbZ{}].  (|x|  \msim{}  if  0  <z  x  then  x  else  -x  fi  )

Date html generated: 2017_04_14-AM-07_33_13
Last ObjectModification: 2017_02_27-PM-03_07_12

Theory : int_1

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