Nuprl Lemma : pigeon-hole-implies2
∀n:ℕ
∀[m:ℕ]
∀f:ℕn ⟶ ℕm. ∀g:ℕn ⟶ ℕm. ∃i:ℕn. (∃j:ℕn [((f i) = (g j) ∈ ℤ)]) supposing Inj(ℕn;ℕm;g) supposing Inj(ℕn;ℕm;f)
supposing m < 2 * n
Proof
Definitions occuring in Statement :
inject: Inj(A;B;f),
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],
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
nat: ℕ,
inject: Inj(A;B;f),
implies: P ⇒ Q,
prop: ℙ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
int_seg: {i..j-},
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
lelt: i ≤ j < k,
guard: {T},
sq_exists: ∃x:A [B[x]],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
member-less_than,
equal_wf,
int_seg_wf,
pigeon-hole-implies-ext,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
less_than_wf,
lelt_wf,
subtract_wf,
int_seg_properties,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
sq_stable__equal,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
inject_wf,
nat_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
sq_exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
multiplyEquality,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
because_Cache,
applyEquality,
dependent_set_memberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
lessCases,
baseClosed,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
axiomSqEquality,
imageElimination,
productElimination,
functionExtensionality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
functionEquality,
applyLambdaEquality,
dependent_set_memberFormation
Latex:
\mforall{}n:\mBbbN{}
\mforall{}[m:\mBbbN{}]
\mforall{}f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m
\mforall{}g:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m. \mexists{}i:\mBbbN{}n. (\mexists{}j:\mBbbN{}n [((f i) = (g j))]) supposing Inj(\mBbbN{}n;\mBbbN{}m;g) supposing Inj(\mBbbN{}n;\mBbbN{}m;f)
supposing m < 2 * n
Date html generated:
2019_06_20-PM-01_32_21
Last ObjectModification:
2018_08_20-PM-09_32_22
Theory : list_1
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