Nuprl Lemma : add_cancel_in_le
∀[a,b,n:ℤ]. a ≤ b supposing (a + n) ≤ (b + n)
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
add: n + m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
top: Top,
subtract: n - m,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
decidable__le,
le-add-cancel,
mul-associates,
mul-distributes,
less_than_wf,
omega-shadow,
add-zero,
minus-add,
zero-add,
zero-mul,
mul-distributes-right,
two-mul,
add-mul-special,
add-commutes,
add-swap,
one-mul,
minus-one-mul,
add-associates,
le_reflexive,
subtract_wf,
add_functionality_wrt_le,
not-le-2,
le_wf,
less_than'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
sqequalRule,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
lemma_by_obid,
isectElimination,
axiomEquality,
addEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
voidElimination,
independent_isectElimination,
natural_numberEquality,
multiplyEquality,
voidEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_functionElimination,
unionElimination
Latex:
\mforall{}[a,b,n:\mBbbZ{}]. a \mleq{} b supposing (a + n) \mleq{} (b + n)
Date html generated:
2016_05_13-PM-03_39_47
Last ObjectModification:
2016_01_14-PM-06_38_28
Theory : arithmetic
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