Nuprl Lemma : disjoint_increasing_onto
∀[m,n,k:ℕ]. ∀[f:ℕn ⟶ ℕm]. ∀[g:ℕk ⟶ ℕm].
(m = (n + k) ∈ ℕ) supposing
((∀j1:ℕn. ∀j2:ℕk. (¬((f j1) = (g j2) ∈ ℤ))) and
(∀i:ℕm. ((∃j:ℕn. (i = (f j) ∈ ℤ)) ∨ (∃j:ℕk. (i = (g j) ∈ ℤ)))) and
increasing(g;k) and
increasing(f;n))
Proof
Definitions occuring in Statement :
increasing: increasing(f;k),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
or: P ∨ Q,
pi1: fst(t),
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
ge: i ≥ j ,
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
not: ¬A,
false: False,
decidable: Dec(P),
inject: Inj(A;B;f),
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
all_wf,
int_seg_wf,
not_wf,
equal_wf,
or_wf,
exists_wf,
increasing_wf,
nat_wf,
injection_le,
add_nat_wf,
sq_stable__le,
le_wf,
lelt_wf,
set_subtype_base,
int_subtype_base,
add-member-int_seg1,
add-associates,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
subtract_wf,
inject_wf,
add-commutes,
add_functionality_wrt_le,
le_reflexive,
zero-add,
one-mul,
two-mul,
mul-distributes-right,
less-iff-le,
not-lt-2,
omega-shadow,
less_than_wf,
mul-distributes,
minus-add,
mul-associates,
mul-swap,
not-le-2,
mul-commutes,
int_seg_properties,
nat_properties,
decidable__lt,
decidable__le,
subtype_base_sq,
le_weakening2,
add-is-int-iff,
le-add-cancel,
le_transitivity,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
minus-zero,
increasing_inj,
equal-wf-T-base,
assert_wf,
le_int_wf,
bnot_wf,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
le_antisymmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality,
intEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
dependent_set_memberEquality,
addEquality,
lambdaFormation,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
independent_isectElimination,
dependent_pairFormation,
unionEquality,
productEquality,
unionElimination,
productElimination,
independent_pairFormation,
sqequalIntensionalEquality,
promote_hyp,
voidElimination,
voidEquality,
minusEquality,
multiplyEquality,
addLevel,
levelHypothesis,
applyLambdaEquality,
instantiate,
cumulativity,
baseApply,
closedConclusion,
equalityElimination
Latex:
\mforall{}[m,n,k:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m]. \mforall{}[g:\mBbbN{}k {}\mrightarrow{} \mBbbN{}m].
(m = (n + k)) supposing
((\mforall{}j1:\mBbbN{}n. \mforall{}j2:\mBbbN{}k. (\mneg{}((f j1) = (g j2)))) and
(\mforall{}i:\mBbbN{}m. ((\mexists{}j:\mBbbN{}n. (i = (f j))) \mvee{} (\mexists{}j:\mBbbN{}k. (i = (g j))))) and
increasing(g;k) and
increasing(f;n))
Date html generated:
2017_04_14-AM-07_34_06
Last ObjectModification:
2017_02_27-PM-03_12_59
Theory : fun_1
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