Nuprl Lemma : Dickson's lemma
∀p:ℕ. ∀A:ℕp ⟶ ℕ ⟶ ℕ. ∃j:ℕ. ∃i:ℕj. ∀k:ℕp. (A[k;i] ≤ A[k;j])
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s1;s2],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
int_seg: {i..j-},
exists: ∃x:A. B[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
top: Top,
subtract: n - m,
sq_stable: SqStable(P),
cand: A c∧ B,
istype: istype(T),
ge: i ≥ j ,
pi1: fst(t),
compose: f o g,
nat_plus: ℕ+
Lemmas referenced :
int_seg_wf,
le_wf,
int_seg_subtype_nat,
false_wf,
nat_wf,
istype-nat,
natrec_wf,
all_wf,
exists_wf,
subtype_rel_function,
subtype_rel_self,
decidable__int_equal,
subtype_base_sq,
int_subtype_base,
lelt_wf,
less_than_transitivity1,
less_than_irreflexivity,
less_than_wf,
subtract_wf,
decidable__lt,
istype-false,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
istype-void,
istype-int,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
primrec-wf2,
or_wf,
decidable__le,
not-le-2,
sq_stable__le,
minus-one-mul,
minus-one-mul-top,
add-swap,
add-mul-special,
zero-mul,
set_subtype_base,
not-equal-implies-less,
nat_properties,
fun_exp_wf,
fun_exp1_lemma,
fun_exp-increasing,
less-iff-le,
le-add-cancel2,
fun_exp_add1,
minus-minus,
le-add-cancel-alt,
add-member-int_seg2,
less_than_transitivity2,
le_weakening2,
less_than'_wf,
int_seg_properties,
subtract-add-cancel,
nat_plus_properties,
primrec-wf-nat-plus,
nat_plus_wf,
le_reflexive,
one-mul,
two-mul,
mul-distributes-right,
omega-shadow,
mul-distributes,
mul-associates,
mul-commutes,
mul-swap,
add_nat_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :functionIsType,
Error :universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
Error :inhabitedIsType,
sqequalRule,
Error :productIsType,
because_Cache,
applyEquality,
independent_isectElimination,
independent_pairFormation,
Error :lambdaEquality_alt,
functionEquality,
functionExtensionality,
dependent_functionElimination,
unionElimination,
instantiate,
cumulativity,
intEquality,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
productElimination,
voidElimination,
Error :unionIsType,
addEquality,
Error :isect_memberEquality_alt,
minusEquality,
Error :setIsType,
productEquality,
imageElimination,
multiplyEquality,
Error :inrFormation_alt,
sqequalIntensionalEquality,
Error :equalityIsType1,
promote_hyp,
Error :inlFormation_alt,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}p:\mBbbN{}. \mforall{}A:\mBbbN{}p {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbN{}. \mexists{}j:\mBbbN{}. \mexists{}i:\mBbbN{}j. \mforall{}k:\mBbbN{}p. (A[k;i] \mleq{} A[k;j])
Date html generated:
2019_06_20-PM-00_27_32
Last ObjectModification:
2018_09_29-PM-09_51_22
Theory : fun_1
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