### Nuprl Lemma : modulus_base

`∀[m:ℕ+]. ∀[a:ℕm].  (a mod m ~ a)`

Proof

Definitions occuring in Statement :  modulus: `a mod n` int_seg: `{i..j-}` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` natural_number: `\$n` 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)↓` nat_plus: `ℕ+` int_seg: `{i..j-}` nequal: `a ≠ b ∈ T ` not: `¬A` implies: `P `` Q` false: `False` lelt: `i ≤ j < k` and: `P ∧ Q` guard: `{T}` all: `∀x:A. B[x]` prop: `ℙ` le: `A ≤ B` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` int_nzero: `ℤ-o` decidable: `Dec(P)` subtract: `n - m`
Lemmas referenced :  subtype_base_sq nat_wf set_subtype_base le_wf int_subtype_base value-type-has-value int-value-type less_than_transitivity1 le_weakening less_than_irreflexivity equal_wf rem_bounds_1 lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf eqff_to_assert bool_cases_sqequal bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot int_seg_wf nat_plus_wf div_bounds_1 int_seg_subtype_nat false_wf div_rem_sum subtype_rel_sets nequal_wf equal-wf-base decidable__int_equal zero-mul zero-add decidable__le not-le-2 not-equal-2 add_functionality_wrt_le add-associates add-zero le-add-cancel condition-implies-le add-commutes minus-add minus-zero minus-one-mul add-swap minus-one-mul-top le-add-cancel2 mul_preserves_le nat_plus_subtype_nat le_reflexive multiply-is-int-iff add-is-int-iff mul-commutes one-mul
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 because_Cache setElimination rename lambdaFormation productElimination dependent_functionElimination independent_functionElimination voidElimination unionElimination equalityElimination equalityTransitivity equalitySymmetry lessCases sqequalAxiom isect_memberEquality independent_pairFormation voidEquality imageMemberEquality baseClosed imageElimination dependent_pairFormation promote_hyp impliesFunctionality applyEquality setEquality dependent_set_memberEquality addEquality minusEquality multiplyEquality baseApply closedConclusion

Latex:
\mforall{}[m:\mBbbN{}\msupplus{}].  \mforall{}[a:\mBbbN{}m].    (a  mod  m  \msim{}  a)

Date html generated: 2017_04_14-AM-07_19_10
Last ObjectModification: 2017_02_27-PM-02_53_28

Theory : arithmetic

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