`∀a,b:ℕ+. ∀c,d:ℤ.  ((((a - 1) * (b - 1)) ≤ ((a * d) - b * c)) `` (∃x:ℤ. ((c ≤ (a * x)) ∧ ((b * x) ≤ d))))`

Proof

Definitions occuring in Statement :  nat_plus: `ℕ+` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` multiply: `n * m` subtract: `n - m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` nat_plus: `ℕ+` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` subtype_rel: `A ⊆r B` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nequal: `a ≠ b ∈ T ` guard: `{T}` le: `A ≤ B` cand: `A c∧ B` uiff: `uiff(P;Q)` less_than: `a < b` squash: `↓T` int_lower: `{...i}` gt: `i > j` ge: `i ≥ j `
Lemmas referenced :  le_wf subtract_wf nat_plus_wf mul_bounds_1a nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf itermMultiply_wf int_term_value_mul_lemma mul_nat_plus equal_wf div_rem_sum subtype_rel_sets less_than_wf nequal_wf intformeq_wf int_formula_prop_eq_lemma equal-wf-base int_subtype_base rem_bounds_1 decidable__equal_int decidable__lt add-is-int-iff multiply-is-int-iff itermAdd_wf int_term_value_add_lemma false_wf rem_bounds_2 itermMinus_wf int_term_value_minus_lemma not_wf mul_preserves_le nat_plus_subtype_nat mul_cancel_in_le
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin multiplyEquality setElimination rename hypothesisEquality hypothesis natural_numberEquality because_Cache intEquality dependent_set_memberEquality dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll equalityTransitivity equalitySymmetry independent_functionElimination applyEquality setEquality applyLambdaEquality baseClosed productElimination divideEquality pointwiseFunctionality promote_hyp baseApply closedConclusion addEquality productEquality imageElimination

Latex:
\mforall{}a,b:\mBbbN{}\msupplus{}.  \mforall{}c,d:\mBbbZ{}.
((((a  -  1)  *  (b  -  1))  \mleq{}  ((a  *  d)  -  b  *  c))  {}\mRightarrow{}  (\mexists{}x:\mBbbZ{}.  ((c  \mleq{}  (a  *  x))  \mwedge{}  ((b  *  x)  \mleq{}  d))))

Date html generated: 2017_04_14-AM-09_15_25
Last ObjectModification: 2017_02_27-PM-03_53_46

Theory : int_2

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