Nuprl Lemma : omega-dark-shadow
∀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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