Nuprl Lemma : nat-to-incomparable-property
∀[n,m:ℕ]. ¬nat-to-incomparable(n) ≤ nat-to-incomparable(m) supposing ¬(n = m ∈ ℤ)
Proof
Definitions occuring in Statement :
nat-to-incomparable: nat-to-incomparable(n),
iseg: l1 ≤ l2,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
int: ℤ,
atom: Atom,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat-to-incomparable: nat-to-incomparable(n),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
name: Name,
nat: ℕ,
rev_implies: P ⇐ Q,
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
deq-member: x ∈b L,
reduce: reduce(f;k;as),
list_ind: list_ind,
cons: [a / b],
bor: p ∨bq,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
nil: [],
it: ⋅,
exists: ∃x:A. B[x],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
top: Top,
sq_type: SQType(T),
btrue: tt,
true: True
Lemmas referenced :
list_wf,
true_wf,
squash_wf,
str-to-nat_wf,
int_subtype_base,
str-to-nat-to-str,
append-cancellation-right,
length_wf,
null_cons_lemma,
null_nil_lemma,
iseg_nil,
atom_subtype_base,
subtype_base_sq,
cons_iseg,
length_of_cons_lemma,
product_subtype_list,
length_of_nil_lemma,
list-cases,
atom-deq_wf,
assert-deq-member,
member-nat-to-str,
iseg_member,
l_member_wf,
cons_member,
member_append,
nat_wf,
equal_wf,
not_wf,
name_wf,
nat-to-incomparable_wf,
iseg_wf,
nil_wf,
cons_wf,
nat-to-str_wf,
append_wf,
iseg_append_iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
lemma_by_obid,
isectElimination,
atomEquality,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
tokenEquality,
productElimination,
independent_functionElimination,
unionElimination,
because_Cache,
voidElimination,
applyEquality,
lambdaEquality,
sqequalRule,
intEquality,
setElimination,
rename,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
inrFormation,
inlFormation,
imageElimination,
promote_hyp,
hypothesis_subsumption,
voidEquality,
instantiate,
cumulativity,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[n,m:\mBbbN{}]. \mneg{}nat-to-incomparable(n) \mleq{} nat-to-incomparable(m) supposing \mneg{}(n = m)
Date html generated:
2016_05_14-PM-03_36_17
Last ObjectModification:
2016_01_14-PM-11_19_20
Theory : decidable!equality
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