Nuprl Lemma : pigeon-hole
∀[n,m:ℕ]. ∀[f:ℕn ⟶ ℕm]. n ≤ m supposing Inj(ℕn;ℕm;f)
Proof
Definitions occuring in Statement :
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
prop: ℙ,
gt: i > j,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
true: True,
inject: Inj(A;B;f),
subtype_rel: A ⊆r B
Lemmas referenced :
finite-partition,
less_than'_wf,
inject_wf,
int_seg_wf,
nat_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
sum_bound,
false_wf,
le_wf,
length_wf,
int_seg_properties,
decidable__lt,
lelt_wf,
intformless_wf,
int_formula_prop_less_lemma,
le_weakening2,
select_wf,
non_neg_length,
length_wf_nat,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
sqequalRule,
independent_pairEquality,
lambdaEquality,
because_Cache,
isectElimination,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
isect_memberEquality,
functionEquality,
voidElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
independent_pairFormation,
dependent_set_memberEquality,
lambdaFormation,
applyEquality,
functionExtensionality,
imageMemberEquality,
baseClosed,
applyLambdaEquality
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m]. n \mleq{} m supposing Inj(\mBbbN{}n;\mBbbN{}m;f)
Date html generated:
2018_05_21-PM-00_38_32
Last ObjectModification:
2018_05_19-AM-06_45_50
Theory : list_1
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