### Nuprl Lemma : absval-diff-symmetry

`∀[x,y:ℤ].  (|x - y| ~ |y - x|)`

Proof

Definitions occuring in Statement :  absval: `|i|` uall: `∀[x:A]. B[x]` subtract: `n - m` int: `ℤ` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` 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` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b`
Lemmas referenced :  subtype_base_sq nat_wf set_subtype_base le_wf int_subtype_base absval_unfold subtract_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf decidable__equal_int satisfiable-full-omega-tt intformand_wf intformnot_wf intformeq_wf itermSubtract_wf itermVar_wf intformless_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_formula_prop_wf decidable__le intformle_wf int_formula_prop_le_lemma eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot itermMinus_wf int_term_value_minus_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin instantiate extract_by_obid sqequalHypSubstitution isectElimination cumulativity hypothesis independent_isectElimination sqequalRule intEquality lambdaEquality natural_numberEquality hypothesisEquality minusEquality lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination because_Cache lessCases sqequalAxiom isect_memberEquality independent_pairFormation voidElimination voidEquality imageMemberEquality baseClosed imageElimination independent_functionElimination dependent_functionElimination dependent_pairFormation int_eqEquality computeAll dependent_set_memberEquality promote_hyp

Latex:
\mforall{}[x,y:\mBbbZ{}].    (|x  -  y|  \msim{}  |y  -  x|)

Date html generated: 2017_04_14-AM-09_13_48
Last ObjectModification: 2017_02_27-PM-03_51_25

Theory : int_2

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