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Add Hermite-Biehler production tactic frontends #311

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Merged
PerAlexandersson merged 1 commit into from
Aug 3, 2026
Merged

Add Hermite-Biehler production tactic frontends #311

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44 changes: 44 additions & 0 deletions RealRooted/Tactic/Examples/HermiteBiehler.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -35,6 +35,26 @@ example {f g : ℝ[X]}
imag_pos_lc := hg,
stable := hstable

example {f g : ℝ[X]}
(hf : HasPosLeadingCoeff f) (hg : HasPosLeadingCoeff g)
(hstable : IsUpperHalfPlaneStable (hermiteBiehlerPolynomial f g)) :
f.Splits ∧ g.Splits := by
rr_hermite_biehler_splits using
real_pos_lc := hf,
imag_pos_lc := hg,
stable := hstable

example {f g : ℝ[X]}
(hf : HasPosLeadingCoeff f) (hg : HasPosLeadingCoeff g)
(hstable : IsUpperHalfPlaneStable (hermiteBiehlerPolynomial f g))
(hdegree : 1 ≤ f.natDegree) :
Prec g f := by
rr_hermite_biehler_prec using
real_pos_lc := hf,
imag_pos_lc := hg,
stable := hstable,
real_degree_pos := hdegree

example {p q : ℝ[X]}
(hp : HasNonnegCoeffs p) (hq : HasNonnegCoeffs q)
(hstable : IsUpperHalfPlaneStable (hermiteBiehlerPolynomial q p)) :
Expand Down Expand Up @@ -74,6 +94,30 @@ example {F G : Nat → ℝ[X]}
imag_pos_lc := hG,
stable := hstable

example {F G : Nat → ℝ[X]}
(hF : ∀ n : Nat, HasPosLeadingCoeff (F n))
(hG : ∀ n : Nat, HasPosLeadingCoeff (G n))
(hstable :
∀ n : Nat, IsUpperHalfPlaneStable (hermiteBiehlerPolynomial (F n) (G n))) :
∀ n : Nat, (F n).Splits ∧ (G n).Splits := by
rr_hermite_biehler_splits_sequence using
real_pos_lc := hF,
imag_pos_lc := hG,
stable := hstable

example {F G : Nat → ℝ[X]}
(hF : ∀ n : Nat, HasPosLeadingCoeff (F n))
(hG : ∀ n : Nat, HasPosLeadingCoeff (G n))
(hstable :
∀ n : Nat, IsUpperHalfPlaneStable (hermiteBiehlerPolynomial (F n) (G n)))
(hdegree : ∀ n : Nat, 1 ≤ (F n).natDegree) :
∀ n : Nat, Prec (G n) (F n) := by
rr_hermite_biehler_prec_sequence using
real_pos_lc := hF,
imag_pos_lc := hG,
stable := hstable,
real_degree_pos := hdegree

example {P Q : Nat → ℝ[X]}
(hP : ∀ n : Nat, HasNonnegCoeffs (P n))
(hQ : ∀ n : Nat, HasNonnegCoeffs (Q n))
Expand Down
79 changes: 79 additions & 0 deletions RealRooted/Tactic/HermiteBiehler.lean
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Expand Up @@ -34,6 +34,23 @@ theorem hermiteBiehlerConverse_sequence {F G : Nat → ℝ[X]}
∀ n : Nat, Prec (G n) (F n) ∨ Prec (F n) (G n) :=
fun n => RealRooted.hermiteBiehlerConverse (hF n) (hG n) (hstable n)

theorem hermiteBiehlerSplits_sequence {F G : Nat → ℝ[X]}
(hF : ∀ n : Nat, HasPosLeadingCoeff (F n))
(hG : ∀ n : Nat, HasPosLeadingCoeff (G n))
(hstable :
∀ n : Nat, IsUpperHalfPlaneStable (hermiteBiehlerPolynomial (F n) (G n))) :
∀ n : Nat, (F n).Splits ∧ (G n).Splits :=
fun n => RealRooted.splits_of_stable (hF n) (hG n) (hstable n)

theorem hermiteBiehlerPrec_sequence {F G : Nat → ℝ[X]}
(hF : ∀ n : Nat, HasPosLeadingCoeff (F n))
(hG : ∀ n : Nat, HasPosLeadingCoeff (G n))
(hstable :
∀ n : Nat, IsUpperHalfPlaneStable (hermiteBiehlerPolynomial (F n) (G n)))
(hdegree : ∀ n : Nat, 1 ≤ (F n).natDegree) :
∀ n : Nat, Prec (G n) (F n) :=
fun n => RealRooted.prec_of_stable_general (hF n) (hG n) (hstable n) (hdegree n)

theorem hermiteBiehlerOddEven_rightHalfPlaneStable_sequence {P Q : Nat → ℝ[X]}
(hP : ∀ n : Nat, HasNonnegCoeffs (P n))
(hQ : ∀ n : Nat, HasNonnegCoeffs (Q n))
Expand Down Expand Up @@ -74,6 +91,21 @@ syntax (name := rr_hermite_biehler_converse_named)
"stable" ":=" term :
tactic

syntax (name := rr_hermite_biehler_splits_named)
"rr_hermite_biehler_splits" " using "
"real_pos_lc" ":=" term ","
"imag_pos_lc" ":=" term ","
"stable" ":=" term :
tactic

syntax (name := rr_hermite_biehler_prec_named)
"rr_hermite_biehler_prec" " using "
"real_pos_lc" ":=" term ","
"imag_pos_lc" ":=" term ","
"stable" ":=" term ","
"real_degree_pos" ":=" term :
tactic

syntax (name := rr_hermite_biehler_odd_even_hurwitz_named)
"rr_hermite_biehler_odd_even_hurwitz" " using "
"odd_nonneg" ":=" term ","
Expand Down Expand Up @@ -102,6 +134,21 @@ syntax (name := rr_hermite_biehler_converse_sequence_named)
"stable" ":=" term :
tactic

syntax (name := rr_hermite_biehler_splits_sequence_named)
"rr_hermite_biehler_splits_sequence" " using "
"real_pos_lc" ":=" term ","
"imag_pos_lc" ":=" term ","
"stable" ":=" term :
tactic

syntax (name := rr_hermite_biehler_prec_sequence_named)
"rr_hermite_biehler_prec_sequence" " using "
"real_pos_lc" ":=" term ","
"imag_pos_lc" ":=" term ","
"stable" ":=" term ","
"real_degree_pos" ":=" term :
tactic

syntax (name := rr_hermite_biehler_odd_even_hurwitz_sequence_named)
"rr_hermite_biehler_odd_even_hurwitz_sequence" " using "
"odd_nonneg" ":=" term ","
Expand Down Expand Up @@ -144,6 +191,21 @@ macro_rules
stable := $hstable:term) =>
`(tactic|
exact RealRooted.hermiteBiehlerConverse $hf $hg $hstable)
| `(tactic|
rr_hermite_biehler_splits using
real_pos_lc := $hf:term,
imag_pos_lc := $hg:term,
stable := $hstable:term) =>
`(tactic|
exact RealRooted.splits_of_stable $hf $hg $hstable)
| `(tactic|
rr_hermite_biehler_prec using
real_pos_lc := $hf:term,
imag_pos_lc := $hg:term,
stable := $hstable:term,
real_degree_pos := $hdegree:term) =>
`(tactic|
exact RealRooted.prec_of_stable_general $hf $hg $hstable $hdegree)
| `(tactic|
rr_hermite_biehler_odd_even_hurwitz using
odd_nonneg := $hp:term,
Expand Down Expand Up @@ -175,6 +237,23 @@ macro_rules
`(tactic|
exact RealRooted.Tactic.hermiteBiehlerConverse_sequence
$hf $hg $hstable)
| `(tactic|
rr_hermite_biehler_splits_sequence using
real_pos_lc := $hf:term,
imag_pos_lc := $hg:term,
stable := $hstable:term) =>
`(tactic|
exact RealRooted.Tactic.hermiteBiehlerSplits_sequence
$hf $hg $hstable)
| `(tactic|
rr_hermite_biehler_prec_sequence using
real_pos_lc := $hf:term,
imag_pos_lc := $hg:term,
stable := $hstable:term,
real_degree_pos := $hdegree:term) =>
`(tactic|
exact RealRooted.Tactic.hermiteBiehlerPrec_sequence
$hf $hg $hstable $hdegree)
| `(tactic|
rr_hermite_biehler_odd_even_hurwitz_sequence using
odd_nonneg := $hp:term,
Expand Down
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