Nuprl Lemma : pigeon-hole-implies-ext
∀n:ℕ. ∀[m:ℕ]. ∀f:ℕn ⟶ ℕm. ∃i:ℕn. (∃j:ℕi [((f i) = (f j) ∈ ℤ)]) supposing m < n
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
int_seg_decide: int_seg_decide(d;i;j),
it: ⋅,
genrec-ap: genrec-ap,
pi1: fst(t),
pigeon-hole-implies,
decidable__exists_int_seg,
decidable__equal_int,
any: any x,
decidable__int_equal,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
pigeon-hole-implies,
lifting-strict-decide,
strict4-spread,
lifting-strict-callbyvalue,
strict4-decide,
lifting-strict-int_eq,
decidable__exists_int_seg,
decidable__equal_int,
decidable__int_equal
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}[m:\mBbbN{}]. \mforall{}f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m. \mexists{}i:\mBbbN{}n. (\mexists{}j:\mBbbN{}i [((f i) = (f j))]) supposing m < n
Date html generated:
2018_05_21-PM-00_38_56
Last ObjectModification:
2018_05_18-AM-08_16_16
Theory : list_1
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