Nuprl Lemma : omega-shadow
∀a,b:ℕ+. ∀c,d,x:ℤ. ((c ≤ (a * x)) ⇒ ((b * x) ≤ d) ⇒ ((b * c) ≤ (a * d)))
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
nat_plus_wf,
le_wf,
le-add-cancel,
add-associates,
add_functionality_wrt_le,
add-commutes,
add-swap,
minus-add,
minus-one-mul-top,
mul-swap,
minus-one-mul,
condition-implies-le,
not-le-2,
false_wf,
decidable__le,
int_subtype_base,
less_than_wf,
set_subtype_base,
multiply-is-int-iff,
nat_plus_subtype_nat,
mul_preserves_le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
multiplyEquality,
setElimination,
rename,
applyEquality,
hypothesis,
sqequalRule,
independent_isectElimination,
because_Cache,
productElimination,
baseApply,
closedConclusion,
baseClosed,
intEquality,
lambdaEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
isect_memberEquality,
voidEquality,
addEquality,
minusEquality
Latex:
\mforall{}a,b:\mBbbN{}\msupplus{}. \mforall{}c,d,x:\mBbbZ{}. ((c \mleq{} (a * x)) {}\mRightarrow{} ((b * x) \mleq{} d) {}\mRightarrow{} ((b * c) \mleq{} (a * d)))
Date html generated:
2016_05_13-PM-03_32_21
Last ObjectModification:
2016_01_14-PM-06_41_13
Theory : arithmetic
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