### Nuprl Lemma : mu-unique

`∀[f:ℕ ⟶ 𝔹]. ∀[x:ℕ].  mu(f) = x ∈ ℤ supposing (↑(f x)) ∧ (∀y:ℕx. (¬↑(f y)))`

Proof

Definitions occuring in Statement :  mu: `mu(f)` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` nat: `ℕ` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` le: `A ≤ B` less_than': `less_than'(a;b)` int_seg: `{i..j-}` lelt: `i ≤ j < k` cand: `A c∧ B` less_than: `a < b`
Lemmas referenced :  int_formula_prop_eq_lemma intformeq_wf decidable__equal_int not_wf all_wf assert_wf lelt_wf int_formula_prop_less_lemma intformless_wf decidable__lt subtype_rel_self false_wf int_seg_subtype_nat int_seg_wf nat_wf bool_wf subtype_rel_dep_function le_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermAdd_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties mu-bound-unique
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin lemma_by_obid isectElimination dependent_set_memberEquality addEquality setElimination rename hypothesisEquality natural_numberEquality hypothesis dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll applyEquality because_Cache lambdaFormation productEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality independent_functionElimination

Latex:
\mforall{}[f:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[x:\mBbbN{}].    mu(f)  =  x  supposing  (\muparrow{}(f  x))  \mwedge{}  (\mforall{}y:\mBbbN{}x.  (\mneg{}\muparrow{}(f  y)))

Date html generated: 2016_05_14-AM-07_30_08
Last ObjectModification: 2016_01_14-PM-09_58_12

Theory : int_2

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