### Nuprl Lemma : modulus-is-rem

`∀[a:ℕ]. ∀[n:ℤ-o].  (a mod n ~ a rem n)`

Proof

Definitions occuring in Statement :  modulus: `a mod n` int_nzero: `ℤ-o` nat: `ℕ` uall: `∀[x:A]. B[x]` remainder: `n rem m` 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]` modulus: `a mod n` has-value: `(a)↓` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` not: `¬A` implies: `P `` Q` false: `False` prop: `ℙ` all: `∀x:A. B[x]` 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` decidable: `Dec(P)` or: `P ∨ Q` nat_plus: `ℕ+` le: `A ≤ B` guard: `{T}` int_lower: `{...i}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` subtype_rel: `A ⊆r B` bfalse: `ff` exists: `∃x:A. B[x]` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` gt: `i > j`
Lemmas referenced :  subtype_base_sq nat_wf set_subtype_base le_wf int_subtype_base value-type-has-value int-value-type equal_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf decidable__lt rem_bounds_1 less_than_transitivity1 less_than_irreflexivity rem_bounds_4 decidable__le false_wf not-le-2 not-equal-2 condition-implies-le minus-zero add-zero minus-add minus-minus add-swap add-commutes add-associates zero-add add_functionality_wrt_le le-add-cancel not-lt-2 eqff_to_assert bool_cases_sqequal bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot not-gt-2 int_nzero_wf
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 callbyvalueReduce remainderEquality setElimination rename because_Cache lambdaFormation independent_functionElimination voidElimination unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination lessCases sqequalAxiom isect_memberEquality independent_pairFormation voidEquality imageMemberEquality baseClosed imageElimination dependent_functionElimination dependent_set_memberEquality minusEquality addEquality applyEquality dependent_pairFormation promote_hyp impliesFunctionality

Latex:
\mforall{}[a:\mBbbN{}].  \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}].    (a  mod  n  \msim{}  a  rem  n)

Date html generated: 2017_04_14-AM-07_19_04
Last ObjectModification: 2017_02_27-PM-02_53_16

Theory : arithmetic

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