Nuprl Lemma : rem_eq_args_z

[a:ℤ]. ∀[b:ℤ-o].  (a rem b) 0 ∈ ℤ supposing |a| |b| ∈ ℤ


Proof




Definitions occuring in Statement :  absval: |i| int_nzero: -o uimplies: supposing a uall: [x:A]. B[x] remainder: rem m natural_number: $n int: equal: t ∈ T
Definitions unfolded in proof :  all: x:A. B[x] implies:  Q member: t ∈ T prop: uall: [x:A]. B[x] subtype_rel: A ⊆B nat_plus: + uimplies: supposing a int_nzero: -o nat: 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 nequal: a ≠ b ∈  satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] guard: {T} iff: ⇐⇒ Q rev_implies:  Q bfalse: ff or: P ∨ Q sq_type: SQType(T) bnot: ¬bb ifthenelse: if then else fi  assert: b decidable: Dec(P) int_lower: {...i}
Lemmas referenced :  equal-wf-base-T int_subtype_base nat_plus_wf absval_wf nat_wf int_nzero_wf absval_unfold lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf equal_wf squash_wf true_wf nat_plus_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf equal-wf-base iff_weakening_equal rem_eq_args less_than_transitivity1 le_weakening eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot minus_mono_wrt_eq itermMinus_wf int_term_value_minus_lemma rem_antisym nequal_wf minus-zero decidable__lt absval_pos int_nzero_properties decidable__le intformnot_wf intformle_wf int_formula_prop_not_lemma int_formula_prop_le_lemma le_wf rem_sym absval_neg
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis introduction extract_by_obid sqequalHypSubstitution isectElimination thin intEquality sqequalRule baseApply closedConclusion baseClosed hypothesisEquality applyEquality setElimination rename isect_memberFormation because_Cache lambdaEquality isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry minusEquality natural_numberEquality unionElimination equalityElimination productElimination independent_isectElimination lessCases sqequalAxiom independent_pairFormation voidElimination voidEquality imageMemberEquality imageElimination independent_functionElimination universeEquality equalityUniverse levelHypothesis remainderEquality dependent_pairFormation int_eqEquality dependent_functionElimination computeAll dependent_set_memberEquality promote_hyp instantiate cumulativity

Latex:
\mforall{}[a:\mBbbZ{}].  \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}].    (a  rem  b)  =  0  supposing  |a|  =  |b|



Date html generated: 2017_04_14-AM-09_16_42
Last ObjectModification: 2017_02_27-PM-03_53_50

Theory : int_2


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