### 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: `ℤ-o` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` remainder: `n rem m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` nat: `ℕ` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` true: `True` squash: `↓T` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nequal: `a ≠ b ∈ T ` not: `¬A` false: `False` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` subtract: `n - 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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