### 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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