`∀[a:ℕ]. ∀[b:ℕ+]. ∀[r:ℕb].  ((((a * b) + r) ÷ b) = a ∈ ℤ)`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat_plus: `ℕ+` nat: `ℕ` uall: `∀[x:A]. B[x]` divide: `n ÷ m` multiply: `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` nat: `ℕ` nat_plus: `ℕ+` int_seg: `{i..j-}` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` all: `∀x:A. B[x]` guard: `{T}` ge: `i ≥ j ` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` uiff: `uiff(P;Q)` div_nrel: `Div(a;n;q)`
Lemmas referenced :  div_unique2 add_nat_wf multiply_nat_wf nat_plus_subtype_nat int_seg_subtype_nat false_wf nat_wf nat_properties int_seg_properties nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf itermAdd_wf itermMultiply_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_mul_lemma int_formula_prop_eq_lemma int_formula_prop_wf equal_wf le_wf mul_bounds_1a decidable__lt intformless_wf int_formula_prop_less_lemma int_seg_wf nat_plus_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin dependent_set_memberEquality addEquality multiplyEquality setElimination rename hypothesisEquality hypothesis because_Cache applyEquality sqequalRule natural_numberEquality independent_isectElimination independent_pairFormation lambdaFormation equalityTransitivity equalitySymmetry applyLambdaEquality productElimination dependent_functionElimination unionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination axiomEquality

Latex:
\mforall{}[a:\mBbbN{}].  \mforall{}[b:\mBbbN{}\msupplus{}].  \mforall{}[r:\mBbbN{}b].    ((((a  *  b)  +  r)  \mdiv{}  b)  =  a)

Date html generated: 2017_04_14-AM-09_15_45
Last ObjectModification: 2017_02_27-PM-03_53_08

Theory : int_2

Home Index