Nuprl Lemma : rem_gen_base_case

[a:ℤ]. ∀[n:ℤ-o].  (a rem n) a ∈ ℤ supposing |a| < |n|


Proof




Definitions occuring in Statement :  absval: |i| int_nzero: -o less_than: a < b uimplies: supposing a uall: [x:A]. B[x] remainder: rem m int: equal: t ∈ T
Definitions unfolded in proof :  all: x:A. B[x] implies:  Q member: t ∈ T prop: uall: [x:A]. B[x] nat: subtype_rel: A ⊆B nat_plus: + true: True squash: T uimplies: supposing a guard: {T} iff: ⇐⇒ Q and: P ∧ Q rev_implies:  Q decidable: Dec(P) or: P ∨ Q int_nzero: -o so_lambda: λ2x.t[x] so_apply: x[s] nequal: a ≠ b ∈  not: ¬A false: False satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] top: Top subtract: m less_than': less_than'(a;b) uiff: uiff(P;Q) le: A ≤ B
Lemmas referenced :  less_than_wf absval_wf nat_wf nat_plus_wf equal_wf rem_base_case iff_weakening_equal squash_wf true_wf absval_pos nat_plus_subtype_nat subtype_rel_self decidable__le le_wf rem_sym_1a subtype_rel_sets nequal_wf full-omega-unsat 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 int_subtype_base nat_plus_properties minus_minus_cancel intformnot_wf intformle_wf itermMinus_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_minus_lemma absval_sym int_nzero_wf minus-zero minus-add add-commutes condition-implies-le le-add-cancel add-zero zero-add add_functionality_wrt_le not-equal-2 not-lt-2 false_wf decidable__lt satisfiable-full-omega-tt int_nzero_properties rem_sym_2
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality applyEquality lambdaEquality sqequalRule intEquality because_Cache natural_numberEquality imageElimination independent_isectElimination imageMemberEquality baseClosed equalityTransitivity equalitySymmetry productElimination independent_functionElimination instantiate universeEquality cumulativity dependent_functionElimination unionElimination dependent_set_memberEquality setEquality approximateComputation dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation minusEquality remainderEquality axiomEquality addEquality isect_memberFormation computeAll

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



Date html generated: 2019_06_20-PM-01_14_54
Last ObjectModification: 2018_09_17-PM-05_56_08

Theory : int_2


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