### Nuprl Lemma : absval_unfold3

`∀[x:ℤ]. (|x| ~ if x <z 0 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` sq_type: `SQType(T)` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` 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` ifthenelse: `if b then t else f fi ` cand: `A c∧ B` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` bnot: `¬bb` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` prop: `ℙ` decidable: `Dec(P)` le: `A ≤ B` subtract: `n - m`
Lemmas referenced :  value-type-has-value int-value-type subtype_base_sq int_subtype_base istype-int lt_int_wf eqtt_to_assert assert_of_lt_int istype-top istype-void less_than_transitivity2 le_weakening2 less_than_irreflexivity eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf less_than_wf iff_weakening_uiff assert_of_bnot istype-less_than istype-assert decidable__int_equal istype-false not-equal-2 not-lt-2 add_functionality_wrt_le add-associates add-zero zero-add le-add-cancel condition-implies-le add-commutes minus-add minus-zero
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep Error :isect_memberFormation_alt,  introduction cut callbyvalueReduce extract_by_obid sqequalHypSubstitution isectElimination thin intEquality independent_isectElimination hypothesis hypothesisEquality instantiate cumulativity dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination axiomSqEquality natural_numberEquality Error :inhabitedIsType,  Error :lambdaFormation_alt,  unionElimination equalityElimination productElimination lessCases Error :isect_memberEquality_alt,  Error :isectIsTypeImplies,  independent_pairFormation voidElimination imageMemberEquality baseClosed imageElimination because_Cache Error :dependent_pairFormation_alt,  Error :equalityIstype,  promote_hyp Error :functionIsType,  Error :universeIsType,  minusEquality addEquality

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

Date html generated: 2019_06_20-AM-11_24_22
Last ObjectModification: 2019_01_31-AM-09_41_21

Theory : arithmetic

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