### Nuprl Lemma : modulus_wf

`∀[a:ℤ]. ∀[n:ℤ-o].  (a mod n ∈ ℕ)`

Proof

Definitions occuring in Statement :  modulus: `a mod n` int_nzero: `ℤ-o` nat: `ℕ` uall: `∀[x:A]. B[x]` member: `t ∈ T` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` modulus: `a mod n` has-value: `(a)↓` uimplies: `b supposing a` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` not: `¬A` implies: `P `` Q` false: `False` prop: `ℙ` less_than: `a < b` and: `P ∧ Q` less_than': `less_than'(a;b)` true: `True` squash: `↓T` top: `Top` nat: `ℕ` all: `∀x:A. B[x]` uiff: `uiff(P;Q)` gt: `i > j` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` decidable: `Dec(P)` subtract: `n - m` subtype_rel: `A ⊆r B` le: `A ≤ B`
Lemmas referenced :  value-type-has-value int-value-type equal_wf less_than_wf not-gt-2 le_wf int_nzero_wf rem_bounds_z lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot decidable__lt false_wf not-lt-2 less-iff-le condition-implies-le minus-one-mul minus-minus minus-one-mul-top add_functionality_wrt_le add-associates zero-add add-swap add-commutes add-zero le-add-cancel absval_unfold2 add_functionality_wrt_lt le_reflexive less_than_transitivity2 le_weakening2 add-mul-special zero-mul subtract_wf absval_wf nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule callbyvalueReduce extract_by_obid sqequalHypSubstitution isectElimination thin intEquality independent_isectElimination hypothesis remainderEquality because_Cache setElimination rename lambdaFormation independent_functionElimination voidElimination hypothesisEquality natural_numberEquality lessCases independent_pairFormation baseClosed equalityTransitivity equalitySymmetry imageMemberEquality axiomSqEquality isect_memberEquality voidEquality imageElimination productElimination dependent_set_memberEquality dependent_functionElimination axiomEquality unionElimination equalityElimination dependent_pairFormation promote_hyp instantiate cumulativity impliesFunctionality minusEquality applyEquality lambdaEquality addEquality multiplyEquality

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

Date html generated: 2019_06_20-AM-11_25_35
Last ObjectModification: 2018_08_20-PM-09_28_29

Theory : arithmetic

Home Index