Nuprl Lemma : int_seg_subtype_special
∀[n,m:ℤ]. ({n + 1..m-} ⊆r {n..m-})
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
lelt: i ≤ j < k,
so_apply: x[s],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
subtype_rel_sets,
and_wf,
le_wf,
less_than_wf,
decidable__le,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
istype-void,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
because_Cache,
Error :lambdaEquality_alt,
addEquality,
hypothesisEquality,
natural_numberEquality,
hypothesis,
Error :inhabitedIsType,
independent_isectElimination,
setElimination,
rename,
Error :setIsType,
Error :universeIsType,
Error :lambdaFormation_alt,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
voidElimination,
independent_functionElimination,
applyEquality,
Error :isect_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
minusEquality,
axiomEquality
Latex:
\mforall{}[n,m:\mBbbZ{}]. (\{n + 1..m\msupminus{}\} \msubseteq{}r \{n..m\msupminus{}\})
Date html generated:
2019_06_20-AM-11_23_48
Last ObjectModification:
2018_09_28-AM-11_05_37
Theory : arithmetic
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