`∀[i:ℕ]. ∀[n:ℕ+].  ((i + 1 rem n) = if (i rem n =z n - 1) then 0 else (i rem n) + 1 fi  ∈ ℤ)`

Proof

Definitions occuring in Statement :  nat_plus: `ℕ+` nat: `ℕ` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` uall: `∀[x:A]. B[x]` remainder: `n rem m` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat_plus: `ℕ+` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` top: `Top` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` ge: `i ≥ j ` nequal: `a ≠ b ∈ T ` uimplies: `b supposing a` prop: `ℙ` implies: `P `` Q` not: `¬A` false: `False` less_than': `less_than'(a;b)` and: `P ∧ Q` le: `A ≤ B` nat: `ℕ` assert: `↑b` bnot: `¬bb` guard: `{T}` sq_type: `SQType(T)` bfalse: `ff` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` true: `True` squash: `↓T` less_than: `a < b` subtract: `n - m`
Lemmas referenced :  decidable__lt nat_plus_wf istype-nat int_subtype_base equal-wf-base int_formula_prop_wf int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_and_lemma intformless_wf itermConstant_wf itermVar_wf intformeq_wf intformand_wf satisfiable-full-omega-tt nat_properties nat_plus_properties equal_wf one-rem le_wf false_wf rem_addition not_wf bnot_wf assert_wf equal-wf-T-base neg_assert_of_eq_int assert-bnot bool_subtype_base subtype_base_sq bool_cases_sqequal eqff_to_assert assert_of_eq_int eqtt_to_assert bool_wf subtract_wf eq_int_wf equal-wf-base-T uiff_transitivity iff_transitivity iff_weakening_uiff assert_of_bnot decidable__equal_int int_term_value_subtract_lemma itermSubtract_wf rem_base_case iff_weakening_equal int_formula_prop_le_lemma int_formula_prop_not_lemma intformle_wf intformnot_wf decidable__le subtract-add-cancel rem_rec_case true_wf squash_wf int_term_value_add_lemma itermAdd_wf rem_bounds_1 less_than_wf rem-1 ifthenelse_wf add_functionality_wrt_eq
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution dependent_functionElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis unionElimination universeIsType sqequalRule isect_memberEquality_alt isectElimination axiomEquality isectIsTypeImplies inhabitedIsType equalityTransitivity baseClosed applyEquality computeAll voidEquality voidElimination isect_memberEquality int_eqEquality lambdaEquality dependent_pairFormation because_Cache addEquality remainderEquality intEquality applyLambdaEquality hyp_replacement equalitySymmetry independent_isectElimination lambdaFormation independent_pairFormation dependent_set_memberEquality cumulativity independent_functionElimination instantiate promote_hyp productElimination equalityElimination impliesFunctionality imageMemberEquality universeEquality imageElimination lambdaFormation_alt addLevel

Latex:
\mforall{}[i:\mBbbN{}].  \mforall{}[n:\mBbbN{}\msupplus{}].    ((i  +  1  rem  n)  =  if  (i  rem  n  =\msubz{}  n  -  1)  then  0  else  (i  rem  n)  +  1  fi  )

Date html generated: 2020_05_19-PM-09_41_27
Last ObjectModification: 2019_12_31-PM-00_59_49

Theory : int_2

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