Nuprl Lemma : omega-shadow-exact2
∀a:ℕ+. ∀c,d:ℤ. (∃x:ℤ. ((c ≤ (a * x)) ∧ (x ≤ d)) ⇐⇒ c ≤ (a * d))
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
nat_plus_wf,
le_weakening,
le_wf,
and_wf,
exists_wf,
less_than_wf,
omega-shadow,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
dependent_functionElimination,
multiplyEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
dependent_set_memberEquality,
introduction,
imageMemberEquality,
baseClosed,
independent_functionElimination,
because_Cache
Latex:
\mforall{}a:\mBbbN{}\msupplus{}. \mforall{}c,d:\mBbbZ{}. (\mexists{}x:\mBbbZ{}. ((c \mleq{} (a * x)) \mwedge{} (x \mleq{} d)) \mLeftarrow{}{}\mRightarrow{} c \mleq{} (a * d))
Date html generated:
2016_05_14-AM-07_23_25
Last ObjectModification:
2016_01_14-PM-10_02_33
Theory : int_2
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