Nuprl Lemma : sub_functionality_wrt_le
∀[i1,i2,j1,j2:ℤ]. ((i1 - i2) ≤ (j1 - j2)) supposing ((i2 ≥ j2 ) and (i1 ≤ j1))
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
le: A ≤ B,
subtract: 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,
subtract: n - m,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
ge: i ≥ j ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
top: Top,
uiff: uiff(P;Q),
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-alt,
mul-commutes,
mul-swap,
mul-distributes,
less_than_wf,
omega-shadow,
add-zero,
mul-associates,
minus-add,
not-le-2,
zero-add,
zero-mul,
mul-distributes-right,
two-mul,
add-mul-special,
add-associates,
add-commutes,
add-swap,
one-mul,
minus-one-mul-top,
le_reflexive,
add_functionality_wrt_le,
le_wf,
ge_wf,
subtract_wf,
less_than'_wf,
minus-one-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
sqequalRule,
lemma_by_obid,
isectElimination,
hypothesisEquality,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
intEquality,
voidElimination,
multiplyEquality,
natural_numberEquality,
independent_isectElimination,
applyEquality,
voidEquality,
addEquality,
minusEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_functionElimination,
unionElimination
Latex:
\mforall{}[i1,i2,j1,j2:\mBbbZ{}]. ((i1 - i2) \mleq{} (j1 - j2)) supposing ((i2 \mgeq{} j2 ) and (i1 \mleq{} j1))
Date html generated:
2016_05_13-PM-03_40_37
Last ObjectModification:
2016_01_14-PM-06_39_05
Theory : arithmetic
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