### Nuprl Lemma : mu-bound-unique

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

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]` implies: `P `` Q` 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` exists: `∃x:A. B[x]` prop: `ℙ` nat: `ℕ` guard: `{T}` so_lambda: `λ2x.t[x]` implies: `P `` Q` int_seg: `{i..j-}` so_apply: `x[s]` all: `∀x:A. B[x]`
Lemmas referenced :  assert_wf int_seg_wf mu-bound-property all_wf equal_wf bool_wf nat_wf mu-bound
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin dependent_pairFormation hypothesisEquality hypothesis extract_by_obid isectElimination applyEquality functionExtensionality natural_numberEquality setElimination rename because_Cache independent_isectElimination productEquality sqequalRule lambdaEquality functionEquality intEquality isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry lambdaFormation dependent_functionElimination independent_functionElimination

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

Date html generated: 2017_04_14-AM-09_19_01
Last ObjectModification: 2017_02_27-PM-03_55_46

Theory : int_2

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