Nuprl Lemma : div_preserves_le
∀[a,b:ℤ]. ∀[n:ℕ+]. ((a ≤ b) ⇒ ((a ÷ n) ≤ (b ÷ n)))
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
le: A ≤ B,
implies: P ⇒ Q,
divide: n ÷ m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
not: ¬A,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
uiff: uiff(P;Q),
nat: ℕ,
less_than: a < b,
squash: ↓T,
int_lower: {...i},
gt: i > j,
ge: i ≥ j
Lemmas referenced :
int_term_value_minus_lemma,
itermMinus_wf,
div_bounds_2,
div_bounds_1,
rem_bounds_2,
rem_bounds_1,
false_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract-is-int-iff,
add-is-int-iff,
div_rem_sum2,
add-commutes,
one-mul,
mul-commutes,
mul-distributes,
nequal_wf,
less_than_wf,
subtype_rel_sets,
equal_wf,
nat_plus_subtype_nat,
mul_preserves_le,
nat_plus_wf,
less_than'_wf,
le_wf,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermAdd_wf,
intformle_wf,
intformnot_wf,
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_plus_properties,
decidable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
divideEquality,
because_Cache,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
unionElimination,
addEquality,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
multiplyEquality,
setEquality,
independent_functionElimination,
remainderEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
dependent_set_memberEquality,
imageElimination,
minusEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((a \mleq{} b) {}\mRightarrow{} ((a \mdiv{} n) \mleq{} (b \mdiv{} n)))
Date html generated:
2016_05_14-AM-07_24_32
Last ObjectModification:
2016_01_14-PM-10_03_07
Theory : int_2
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