diff --git "a/manifests/batches/mathlib/001663.json" "b/manifests/batches/mathlib/001663.json"
new file mode 100644--- /dev/null
+++ "b/manifests/batches/mathlib/001663.json"
@@ -0,0 +1,2126 @@
+{
+ "format": "leangpt-parquet-v2",
+ "batch": "mathlib/001663",
+ "batch_fingerprint": "151044448945a9a11ea7523fb87575691a9d317d1174a782c84e726e462508f1",
+ "evidence": {
+ "batch_fingerprint": "151044448945a9a11ea7523fb87575691a9d317d1174a782c84e726e462508f1",
+ "contamination_matches": [],
+ "failures": [
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "derivationToSquareZeroOfLift_apply",
+ "id": "derivationToSquareZeroOfLift_apply",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 77,
+ 6
+ ],
+ "file_path": "Mathlib/RingTheory/Derivation/ToSquareZero.lean",
+ "full_name": "derivationToSquareZeroOfLift_apply",
+ "start": [
+ 74,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/Derivation/ToSquareZero.lean"
+ },
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "diffToIdealOfQuotientCompEq_apply",
+ "id": "diffToIdealOfQuotientCompEq_apply",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 42,
+ 6
+ ],
+ "file_path": "Mathlib/RingTheory/Derivation/ToSquareZero.lean",
+ "full_name": "diffToIdealOfQuotientCompEq_apply",
+ "start": [
+ 39,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/Derivation/ToSquareZero.lean"
+ },
+ {
+ "error": "no internal trace for tactic 'rw [ha]'",
+ "full_name": "DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid",
+ "id": "DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 224,
+ 46
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid",
+ "start": [
+ 198,
+ 1
+ ],
+ "traced_tactics": [
+ {
+ "annotated_tactic": [
+ "\"use Multiset.replicate<\\/a> (Classical.choose<\\/a> (spec.2 hx)) p\"",
+ "[{\"full_name\":\"Multiset.replicate\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[929,5],\"def_end_pos\":[929,14]},{\"full_name\":\"Classical.choose\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[25,19],\"def_end_pos\":[25,25]}]"
+ ],
+ "state_after": "case h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 (\u2200 b \u2208 Multiset.replicate (Classical.choose \u22ef) p, Prime b) \u2227\n Associated (Multiset.replicate (Classical.choose \u22ef) p).prod x",
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 \u2203 f, (\u2200 b \u2208 f, Prime b) \u2227 Associated f.prod x",
+ "tactic": "use Multiset.replicate (Classical.choose (spec.2 hx)) p"
+ },
+ {
+ "annotated_tactic": [
+ "\"constructor\"",
+ "[]"
+ ],
+ "state_after": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 \u2200 b \u2208 Multiset.replicate (Classical.choose \u22ef) p, Prime b\n\ncase h.right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 Associated (Multiset.replicate (Classical.choose \u22ef) p).prod x",
+ "state_before": "case h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 (\u2200 b \u2208 Multiset.replicate (Classical.choose \u22ef) p, Prime b) \u2227\n Associated (Multiset.replicate (Classical.choose \u22ef) p).prod x",
+ "tactic": "constructor"
+ },
+ {
+ "annotated_tactic": [
+ "\"intro q hq\"",
+ "[]"
+ ],
+ "state_after": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\n\u22a2 Prime q",
+ "state_before": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 \u2200 b \u2208 Multiset.replicate (Classical.choose \u22ef) p, Prime b",
+ "tactic": "intro q hq"
+ },
+ {
+ "annotated_tactic": [
+ "\"have hpq := Multiset.eq_of_mem_replicate<\\/a> hq\"",
+ "[{\"full_name\":\"Multiset.eq_of_mem_replicate\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[966,9],\"def_end_pos\":[966,28]}]"
+ ],
+ "state_after": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\n\u22a2 Prime q",
+ "state_before": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\n\u22a2 Prime q",
+ "tactic": "have hpq := Multiset.eq_of_mem_replicate hq"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [hpq]\"",
+ "[]"
+ ],
+ "state_after": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\n\u22a2 Prime p",
+ "state_before": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\n\u22a2 Prime q",
+ "tactic": "rw [hpq]"
+ },
+ {
+ "annotated_tactic": [
+ "\"refine \\u27e8spec.1.ne_zero<\\/a>, spec.1.not_unit<\\/a>, ?_\\u27e9\"",
+ "[{\"full_name\":\"Irreducible.ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[231,9],\"def_end_pos\":[231,28]},{\"full_name\":\"Irreducible.not_unit\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[193,3],\"def_end_pos\":[193,11]}]"
+ ],
+ "state_after": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\n\u22a2 \u2200 (a b : R), p \u2223 a * b \u2192 p \u2223 a \u2228 p \u2223 b",
+ "state_before": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\n\u22a2 Prime p",
+ "tactic": "refine \u27e8spec.1.ne_zero, spec.1.not_unit, ?_\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"intro a b h\"",
+ "[]"
+ ],
+ "state_after": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\n\u22a2 p \u2223 a \u2228 p \u2223 b",
+ "state_before": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\n\u22a2 \u2200 (a b : R), p \u2223 a * b \u2192 p \u2223 a \u2228 p \u2223 b",
+ "tactic": "intro a b h"
+ },
+ {
+ "annotated_tactic": [
+ "\"by_cases ha : a = 0\"",
+ "[]"
+ ],
+ "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : a = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b\n\ncase neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : \u00aca = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b",
+ "state_before": "case h.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\n\u22a2 p \u2223 a \u2228 p \u2223 b",
+ "tactic": "by_cases ha : a = 0"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8m, u, rfl\\u27e9 := spec.2 ha\"",
+ "[]"
+ ],
+ "state_after": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * \u2191u * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 Classical.choose hR ^ m * \u2191u \u2228 p \u2223 b",
+ "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : \u00aca = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b",
+ "tactic": "obtain \u27e8m, u, rfl\u27e9 := spec.2 ha"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [mul_assoc<\\/a>, mul_left_comm<\\/a>, Units.dvd_mul_left<\\/a>] at h\"",
+ "[{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_left_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,22]},{\"full_name\":\"Units.dvd_mul_left\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Units.lean\",\"def_pos\":[57,9],\"def_end_pos\":[57,21]}]"
+ ],
+ "state_after": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 Classical.choose hR ^ m * \u2191u \u2228 p \u2223 b",
+ "state_before": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * \u2191u * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 Classical.choose hR ^ m * \u2191u \u2228 p \u2223 b",
+ "tactic": "rw [mul_assoc, mul_left_comm, Units.dvd_mul_left] at h"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [Units.dvd_mul_right<\\/a>]\"",
+ "[{\"full_name\":\"Units.dvd_mul_right\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Units.lean\",\"def_pos\":[38,9],\"def_end_pos\":[38,22]}]"
+ ],
+ "state_after": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 Classical.choose hR ^ m \u2228 p \u2223 b",
+ "state_before": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 Classical.choose hR ^ m * \u2191u \u2228 p \u2223 b",
+ "tactic": "rw [Units.dvd_mul_right]"
+ },
+ {
+ "annotated_tactic": [
+ "\"by_cases hm : m = 0\"",
+ "[]"
+ ],
+ "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\nhm : m = 0\n\u22a2 p \u2223 Classical.choose hR ^ m \u2228 p \u2223 b\n\ncase neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\nhm : \u00acm = 0\n\u22a2 p \u2223 Classical.choose hR ^ m \u2228 p \u2223 b",
+ "state_before": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 Classical.choose hR ^ m \u2228 p \u2223 b",
+ "tactic": "by_cases hm : m = 0"
+ },
+ {
+ "annotated_tactic": [
+ "\"left\"",
+ "[]"
+ ],
+ "state_after": "case neg.h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\nhm : \u00acm = 0\n\u22a2 p \u2223 Classical.choose hR ^ m",
+ "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\nhm : \u00acm = 0\n\u22a2 p \u2223 Classical.choose hR ^ m \u2228 p \u2223 b",
+ "tactic": "left"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8m, rfl\\u27e9 := Nat.exists_eq_succ_of_ne_zero<\\/a> hm\"",
+ "[{\"full_name\":\"Nat.exists_eq_succ_of_ne_zero\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[691,9],\"def_end_pos\":[691,34]}]"
+ ],
+ "state_after": "case neg.h.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nu : R\u02e3\nm : \u2115\nh : p \u2223 Classical.choose hR ^ m.succ * b\nha : \u00acClassical.choose hR ^ m.succ * \u2191u = 0\nhm : \u00acm.succ = 0\n\u22a2 p \u2223 Classical.choose hR ^ m.succ",
+ "state_before": "case neg.h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\nhm : \u00acm = 0\n\u22a2 p \u2223 Classical.choose hR ^ m",
+ "tactic": "obtain \u27e8m, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero hm"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [pow_succ'<\\/a>]\"",
+ "[{\"full_name\":\"pow_succ'\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[667,34],\"def_end_pos\":[667,43]}]"
+ ],
+ "state_after": "case neg.h.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nu : R\u02e3\nm : \u2115\nh : p \u2223 Classical.choose hR ^ m.succ * b\nha : \u00acClassical.choose hR ^ m.succ * \u2191u = 0\nhm : \u00acm.succ = 0\n\u22a2 p \u2223 Classical.choose hR * Classical.choose hR ^ m",
+ "state_before": "case neg.h.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nu : R\u02e3\nm : \u2115\nh : p \u2223 Classical.choose hR ^ m.succ * b\nha : \u00acClassical.choose hR ^ m.succ * \u2191u = 0\nhm : \u00acm.succ = 0\n\u22a2 p \u2223 Classical.choose hR ^ m.succ",
+ "tactic": "rw [pow_succ']"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply dvd_mul_of_dvd_left<\\/a> dvd_rfl<\\/a> _\"",
+ "[{\"full_name\":\"dvd_mul_of_dvd_left\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Basic.lean\",\"def_pos\":[87,9],\"def_end_pos\":[87,28]},{\"full_name\":\"dvd_rfl\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Basic.lean\",\"def_pos\":[141,9],\"def_end_pos\":[141,16]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case neg.h.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nu : R\u02e3\nm : \u2115\nh : p \u2223 Classical.choose hR ^ m.succ * b\nha : \u00acClassical.choose hR ^ m.succ * \u2191u = 0\nhm : \u00acm.succ = 0\n\u22a2 p \u2223 Classical.choose hR * Classical.choose hR ^ m",
+ "tactic": "apply dvd_mul_of_dvd_left dvd_rfl _"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [ha]\"",
+ "[]"
+ ],
+ "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : a = 0\n\u22a2 p \u2223 0 \u2228 p \u2223 b",
+ "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : a = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b",
+ "tactic": "rw [ha]"
+ },
+ {
+ "annotated_tactic": [
+ "\"simp only [true_or_iff<\\/a>, dvd_zero<\\/a>]\"",
+ "[{\"full_name\":\"true_or_iff\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[150,9],\"def_end_pos\":[150,20]},{\"full_name\":\"dvd_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Divisibility.lean\",\"def_pos\":[41,9],\"def_end_pos\":[41,17]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : a = 0\n\u22a2 p \u2223 0 \u2228 p \u2223 b",
+ "tactic": "simp only [true_or_iff, dvd_zero]"
+ },
+ {
+ "annotated_tactic": [
+ "\"simp only [hm, one_mul<\\/a>, pow_zero<\\/a>] at h \\u22a2\"",
+ "[{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"pow_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[651,9],\"def_end_pos\":[651,17]}]"
+ ],
+ "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\nhm : m = 0\nh : p \u2223 b\n\u22a2 p \u2223 1 \u2228 p \u2223 b",
+ "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 Classical.choose hR ^ m * b\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\nhm : m = 0\n\u22a2 p \u2223 Classical.choose hR ^ m \u2228 p \u2223 b",
+ "tactic": "simp only [hm, one_mul, pow_zero] at h \u22a2"
+ },
+ {
+ "annotated_tactic": [
+ "\"right\"",
+ "[]"
+ ],
+ "state_after": "case pos.h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\nhm : m = 0\nh : p \u2223 b\n\u22a2 p \u2223 b",
+ "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\nhm : m = 0\nh : p \u2223 b\n\u22a2 p \u2223 1 \u2228 p \u2223 b",
+ "tactic": "right"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact h\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case pos.h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (Classical.choose \u22ef) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nha : \u00acClassical.choose hR ^ m * \u2191u = 0\nhm : m = 0\nh : p \u2223 b\n\u22a2 p \u2223 b",
+ "tactic": "exact h"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [Multiset.prod_replicate<\\/a>]\"",
+ "[{\"full_name\":\"Multiset.prod_replicate\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Multiset.lean\",\"def_pos\":[130,9],\"def_end_pos\":[130,23]}]"
+ ],
+ "state_after": "case h.right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 Associated (p ^ Classical.choose \u22ef) x",
+ "state_before": "case h.right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 Associated (Multiset.replicate (Classical.choose \u22ef) p).prod x",
+ "tactic": "rw [Multiset.prod_replicate]"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact Classical.choose_spec<\\/a> (spec.2 hx)\"",
+ "[{\"full_name\":\"Classical.choose_spec\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[28,9],\"def_end_pos\":[28,20]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case h.right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := Classical.choose hR\nspec : Irreducible (Classical.choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (Classical.choose hR ^ n) x :=\n Classical.choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 Associated (p ^ Classical.choose \u22ef) x",
+ "tactic": "exact Classical.choose_spec (spec.2 hx)"
+ }
+ ],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "no internal trace for tactic 'apply Associated.trans (this hp) (this hq).symm'",
+ "full_name": "DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.unique_irreducible",
+ "id": "DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.unique_irreducible",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 190,
+ 62
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.unique_irreducible",
+ "start": [
+ 169,
+ 1
+ ],
+ "traced_tactics": [
+ {
+ "annotated_tactic": [
+ "\"rcases hR with \\u27e8\\u03d6, h\\u03d6, hR\\u27e9\"",
+ "[]"
+ ],
+ "state_after": "case intro.intro\nR : Type u_1\ninst\u271d : CommRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\n\u22a2 Associated p q",
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\np q : R\nhp : Irreducible p\nhq : Irreducible q\n\u22a2 Associated p q",
+ "tactic": "rcases hR with \u27e8\u03d6, h\u03d6, hR\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"suffices \\u2200 {p : R} (_ : Irreducible<\\/a> p), Associated<\\/a> p \\u03d6 by\\n apply Associated.trans<\\/a> (this hp) (this hq).symm<\\/a>\"",
+ "[{\"full_name\":\"Irreducible\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[191,11],\"def_end_pos\":[191,22]},{\"full_name\":\"Associated\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[397,5],\"def_end_pos\":[397,15]},{\"full_name\":\"Associated.trans\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[431,19],\"def_end_pos\":[431,24]},{\"full_name\":\"Associated.symm\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[419,19],\"def_end_pos\":[419,23]}]"
+ ],
+ "state_after": "case intro.intro\nR : Type u_1\ninst\u271d : CommRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\n\u22a2 \u2200 {p : R}, Irreducible p \u2192 Associated p \u03d6",
+ "state_before": "case intro.intro\nR : Type u_1\ninst\u271d : CommRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\n\u22a2 Associated p q",
+ "tactic": "suffices \u2200 {p : R} (_ : Irreducible p), Associated p \u03d6 by\n apply Associated.trans (this hp) (this hq).symm"
+ },
+ {
+ "annotated_tactic": [
+ "\"clear hp hq p q\"",
+ "[]"
+ ],
+ "state_after": "case intro.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\n\u22a2 \u2200 {p : R}, Irreducible p \u2192 Associated p \u03d6",
+ "state_before": "case intro.intro\nR : Type u_1\ninst\u271d : CommRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\n\u22a2 \u2200 {p : R}, Irreducible p \u2192 Associated p \u03d6",
+ "tactic": "clear hp hq p q"
+ },
+ {
+ "annotated_tactic": [
+ "\"intro p hp\"",
+ "[]"
+ ],
+ "state_after": "case intro.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\n\u22a2 Associated p \u03d6",
+ "state_before": "case intro.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\n\u22a2 \u2200 {p : R}, Irreducible p \u2192 Associated p \u03d6",
+ "tactic": "intro p hp"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8n, hn\\u27e9 := hR hp.ne_zero\"",
+ "[]"
+ ],
+ "state_after": "case intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ n) p\n\u22a2 Associated p \u03d6",
+ "state_before": "case intro.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\n\u22a2 Associated p \u03d6",
+ "tactic": "obtain \u27e8n, hn\u27e9 := hR hp.ne_zero"
+ },
+ {
+ "annotated_tactic": [
+ "\"have : Irreducible<\\/a> (\\u03d6 ^ n) := hn.symm.irreducible hp\"",
+ "[{\"full_name\":\"Irreducible\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[191,11],\"def_end_pos\":[191,22]}]"
+ ],
+ "state_after": "case intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ n) p\nthis : Irreducible (\u03d6 ^ n)\n\u22a2 Associated p \u03d6",
+ "state_before": "case intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ n) p\n\u22a2 Associated p \u03d6",
+ "tactic": "have : Irreducible (\u03d6 ^ n) := hn.symm.irreducible hp"
+ },
+ {
+ "annotated_tactic": [
+ "\"rcases lt_trichotomy<\\/a> n 1 with (H | rfl | H)\"",
+ "[{\"full_name\":\"lt_trichotomy\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,22]}]"
+ ],
+ "state_after": "case intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ n) p\nthis : Irreducible (\u03d6 ^ n)\nH : n < 1\n\u22a2 Associated p \u03d6\n\ncase intro.intro.intro.inr.inl\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nhn : Associated (\u03d6 ^ 1) p\nthis : Irreducible (\u03d6 ^ 1)\n\u22a2 Associated p \u03d6\n\ncase intro.intro.intro.inr.inr\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ n) p\nthis : Irreducible (\u03d6 ^ n)\nH : 1 < n\n\u22a2 Associated p \u03d6",
+ "state_before": "case intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ n) p\nthis : Irreducible (\u03d6 ^ n)\n\u22a2 Associated p \u03d6",
+ "tactic": "rcases lt_trichotomy n 1 with (H | rfl | H)"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply Associated.trans<\\/a> (this hp) (this hq).symm<\\/a>\"",
+ "[{\"full_name\":\"Associated.trans\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[431,19],\"def_end_pos\":[431,24]},{\"full_name\":\"Associated.symm\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[419,19],\"def_end_pos\":[419,23]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\nthis : \u2200 {p : R}, Irreducible p \u2192 Associated p \u03d6\n\u22a2 Associated p q",
+ "tactic": "apply Associated.trans (this hp) (this hq).symm"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain rfl : n = 0 := by\\n clear hn this\\n revert H n\\n decide\"",
+ "[]"
+ ],
+ "state_after": "case intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nhn : Associated (\u03d6 ^ 0) p\nthis : Irreducible (\u03d6 ^ 0)\nH : 0 < 1\n\u22a2 Associated p \u03d6",
+ "state_before": "case intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ n) p\nthis : Irreducible (\u03d6 ^ n)\nH : n < 1\n\u22a2 Associated p \u03d6",
+ "tactic": "obtain rfl : n = 0 := by\n clear hn this\n revert H n\n decide"
+ },
+ {
+ "annotated_tactic": [
+ "\"simp [not_irreducible_one<\\/a>, pow_zero<\\/a>] at this\"",
+ "[{\"full_name\":\"not_irreducible_one\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[217,9],\"def_end_pos\":[217,28]},{\"full_name\":\"pow_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[651,9],\"def_end_pos\":[651,17]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nhn : Associated (\u03d6 ^ 0) p\nthis : Irreducible (\u03d6 ^ 0)\nH : 0 < 1\n\u22a2 Associated p \u03d6",
+ "tactic": "simp [not_irreducible_one, pow_zero] at this"
+ },
+ {
+ "annotated_tactic": [
+ "\"clear hn this\"",
+ "[]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nH : n < 1\n\u22a2 n = 0",
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ n) p\nthis : Irreducible (\u03d6 ^ n)\nH : n < 1\n\u22a2 n = 0",
+ "tactic": "clear hn this"
+ },
+ {
+ "annotated_tactic": [
+ "\"revert H n\"",
+ "[]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\n\u22a2 \u2200 n < 1, n = 0",
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nH : n < 1\n\u22a2 n = 0",
+ "tactic": "revert H n"
+ },
+ {
+ "annotated_tactic": [
+ "\"decide\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\n\u22a2 \u2200 n < 1, n = 0",
+ "tactic": "decide"
+ },
+ {
+ "annotated_tactic": [
+ "\"simpa only [pow_one<\\/a>] using hn.symm\"",
+ "[{\"full_name\":\"pow_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[663,7],\"def_end_pos\":[663,14]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case intro.intro.intro.inr.inl\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nhn : Associated (\u03d6 ^ 1) p\nthis : Irreducible (\u03d6 ^ 1)\n\u22a2 Associated p \u03d6",
+ "tactic": "simpa only [pow_one] using hn.symm"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8n, rfl\\u27e9 : \\u2203 k, n = 1 + k + 1 := Nat.exists_eq_add_of_lt<\\/a> H\"",
+ "[{\"full_name\":\"Nat.exists_eq_add_of_lt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Lemmas.lean\",\"def_pos\":[198,19],\"def_end_pos\":[198,38]}]"
+ ],
+ "state_after": "case intro.intro.intro.inr.inr.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ (1 + n + 1)) p\nthis : Irreducible (\u03d6 ^ (1 + n + 1))\nH : 1 < 1 + n + 1\n\u22a2 Associated p \u03d6",
+ "state_before": "case intro.intro.intro.inr.inr\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ n) p\nthis : Irreducible (\u03d6 ^ n)\nH : 1 < n\n\u22a2 Associated p \u03d6",
+ "tactic": "obtain \u27e8n, rfl\u27e9 : \u2203 k, n = 1 + k + 1 := Nat.exists_eq_add_of_lt H"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [pow_succ'<\\/a>] at this\"",
+ "[{\"full_name\":\"pow_succ'\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[667,34],\"def_end_pos\":[667,43]}]"
+ ],
+ "state_after": "case intro.intro.intro.inr.inr.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ (1 + n + 1)) p\nthis : Irreducible (\u03d6 * \u03d6 ^ (1 + n))\nH : 1 < 1 + n + 1\n\u22a2 Associated p \u03d6",
+ "state_before": "case intro.intro.intro.inr.inr.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ (1 + n + 1)) p\nthis : Irreducible (\u03d6 ^ (1 + n + 1))\nH : 1 < 1 + n + 1\n\u22a2 Associated p \u03d6",
+ "tactic": "rw [pow_succ'] at this"
+ },
+ {
+ "annotated_tactic": [
+ "\"rcases this.isUnit_or_isUnit rfl<\\/a> with (H0 | H0)\"",
+ "[{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
+ ],
+ "state_after": "case intro.intro.intro.inr.inr.intro.inl\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ (1 + n + 1)) p\nthis : Irreducible (\u03d6 * \u03d6 ^ (1 + n))\nH : 1 < 1 + n + 1\nH0 : IsUnit \u03d6\n\u22a2 Associated p \u03d6\n\ncase intro.intro.intro.inr.inr.intro.inr\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ (1 + n + 1)) p\nthis : Irreducible (\u03d6 * \u03d6 ^ (1 + n))\nH : 1 < 1 + n + 1\nH0 : IsUnit (\u03d6 ^ (1 + n))\n\u22a2 Associated p \u03d6",
+ "state_before": "case intro.intro.intro.inr.inr.intro\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ (1 + n + 1)) p\nthis : Irreducible (\u03d6 * \u03d6 ^ (1 + n))\nH : 1 < 1 + n + 1\n\u22a2 Associated p \u03d6",
+ "tactic": "rcases this.isUnit_or_isUnit rfl with (H0 | H0)"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact (h\\u03d6.not_unit H0).elim<\\/a>\"",
+ "[{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case intro.intro.intro.inr.inr.intro.inl\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ (1 + n + 1)) p\nthis : Irreducible (\u03d6 * \u03d6 ^ (1 + n))\nH : 1 < 1 + n + 1\nH0 : IsUnit \u03d6\n\u22a2 Associated p \u03d6",
+ "tactic": "exact (h\u03d6.not_unit H0).elim"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [add_comm<\\/a>, pow_succ'<\\/a>] at H0\"",
+ "[{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"pow_succ'\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[667,34],\"def_end_pos\":[667,43]}]"
+ ],
+ "state_after": "case intro.intro.intro.inr.inr.intro.inr\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ (1 + n + 1)) p\nthis : Irreducible (\u03d6 * \u03d6 ^ (1 + n))\nH : 1 < 1 + n + 1\nH0 : IsUnit (\u03d6 * \u03d6 ^ n)\n\u22a2 Associated p \u03d6",
+ "state_before": "case intro.intro.intro.inr.inr.intro.inr\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ (1 + n + 1)) p\nthis : Irreducible (\u03d6 * \u03d6 ^ (1 + n))\nH : 1 < 1 + n + 1\nH0 : IsUnit (\u03d6 ^ (1 + n))\n\u22a2 Associated p \u03d6",
+ "tactic": "rw [add_comm, pow_succ'] at H0"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact (h\\u03d6.not_unit (isUnit_of_mul_isUnit_left<\\/a> H0)).elim<\\/a>\"",
+ "[{\"full_name\":\"isUnit_of_mul_isUnit_left\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[785,9],\"def_end_pos\":[785,34]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case intro.intro.intro.inr.inr.intro.inr\nR : Type u_1\ninst\u271d : CommRing R\n\u03d6 : R\nh\u03d6 : Irreducible \u03d6\nhR : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (\u03d6 ^ n) x\np : R\nhp : Irreducible p\nn : \u2115\nhn : Associated (\u03d6 ^ (1 + n + 1)) p\nthis : Irreducible (\u03d6 * \u03d6 ^ (1 + n))\nH : 1 < 1 + n + 1\nH0 : IsUnit (\u03d6 * \u03d6 ^ n)\n\u22a2 Associated p \u03d6",
+ "tactic": "exact (h\u03d6.not_unit (isUnit_of_mul_isUnit_left H0)).elim"
+ }
+ ],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "DiscreteValuationRing.addVal_add",
+ "id": "DiscreteValuationRing.addVal_add",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 474,
+ 25
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.addVal_add",
+ "start": [
+ 473,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "DiscreteValuationRing.addVal_def'",
+ "id": "DiscreteValuationRing.addVal_def'",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 412,
+ 26
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.addVal_def'",
+ "start": [
+ 410,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "no internal trace for tactic 'rw [ha0, addVal_zero, top_le_iff, addVal_eq_top_iff] at h'",
+ "full_name": "DiscreteValuationRing.addVal_le_iff_dvd",
+ "id": "DiscreteValuationRing.addVal_le_iff_dvd",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 470,
+ 54
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.addVal_le_iff_dvd",
+ "start": [
+ 459,
+ 1
+ ],
+ "traced_tactics": [
+ {
+ "annotated_tactic": [
+ "\"have hp := Classical.choose_spec<\\/a> (exists_prime<\\/a> R)\"",
+ "[{\"full_name\":\"Classical.choose_spec\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[28,9],\"def_end_pos\":[28,20]},{\"full_name\":\"DiscreteValuationRing.exists_prime\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[113,9],\"def_end_pos\":[113,21]}]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\n\u22a2 (addVal R) a \u2264 (addVal R) b \u2194 a \u2223 b",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\n\u22a2 (addVal R) a \u2264 (addVal R) b \u2194 a \u2223 b",
+ "tactic": "have hp := Classical.choose_spec (exists_prime R)"
+ },
+ {
+ "annotated_tactic": [
+ "\"constructor <;> intro h\"",
+ "[]"
+ ],
+ "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : (addVal R) a \u2264 (addVal R) b\n\u22a2 a \u2223 b\n\ncase mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : a \u2223 b\n\u22a2 (addVal R) a \u2264 (addVal R) b",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\n\u22a2 (addVal R) a \u2264 (addVal R) b \u2194 a \u2223 b",
+ "tactic": "constructor <;> intro h"
+ },
+ {
+ "annotated_tactic": [
+ "\"by_cases ha0 : a = 0\"",
+ "[]"
+ ],
+ "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : (addVal R) a \u2264 (addVal R) b\nha0 : a = 0\n\u22a2 a \u2223 b\n\ncase neg\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : (addVal R) a \u2264 (addVal R) b\nha0 : \u00aca = 0\n\u22a2 a \u2223 b",
+ "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : (addVal R) a \u2264 (addVal R) b\n\u22a2 a \u2223 b",
+ "tactic": "by_cases ha0 : a = 0"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8n, ha\\u27e9 := associated_pow_irreducible<\\/a> ha0 hp.irreducible\"",
+ "[{\"full_name\":\"DiscreteValuationRing.associated_pow_irreducible\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,35]}]"
+ ],
+ "state_after": "case neg.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : (addVal R) a \u2264 (addVal R) b\nha0 : \u00aca = 0\nn : \u2115\nha : Associated a (Classical.choose \u22ef ^ n)\n\u22a2 a \u2223 b",
+ "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : (addVal R) a \u2264 (addVal R) b\nha0 : \u00aca = 0\n\u22a2 a \u2223 b",
+ "tactic": "obtain \u27e8n, ha\u27e9 := associated_pow_irreducible ha0 hp.irreducible"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [addVal<\\/a>, addValuation_apply<\\/a>, addValuation_apply<\\/a>, multiplicity_le_multiplicity_iff<\\/a>] at h\"",
+ "[{\"full_name\":\"DiscreteValuationRing.addVal\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[397,19],\"def_end_pos\":[397,25]},{\"full_name\":\"multiplicity.addValuation_apply\",\"def_path\":\"Mathlib\\/RingTheory\\/Valuation\\/PrimeMultiplicity.lean\",\"def_pos\":[22,9],\"def_end_pos\":[22,40]},{\"full_name\":\"multiplicity.addValuation_apply\",\"def_path\":\"Mathlib\\/RingTheory\\/Valuation\\/PrimeMultiplicity.lean\",\"def_pos\":[22,9],\"def_end_pos\":[22,40]},{\"full_name\":\"multiplicity.multiplicity_le_multiplicity_iff\",\"def_path\":\"Mathlib\\/RingTheory\\/Multiplicity.lean\",\"def_pos\":[233,9],\"def_end_pos\":[233,41]}]"
+ ],
+ "state_after": "case neg.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : \u2200 (n : \u2115), Classical.choose \u22ef ^ n \u2223 a \u2192 Classical.choose \u22ef ^ n \u2223 b\nha0 : \u00aca = 0\nn : \u2115\nha : Associated a (Classical.choose \u22ef ^ n)\n\u22a2 a \u2223 b",
+ "state_before": "case neg.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : (addVal R) a \u2264 (addVal R) b\nha0 : \u00aca = 0\nn : \u2115\nha : Associated a (Classical.choose \u22ef ^ n)\n\u22a2 a \u2223 b",
+ "tactic": "rw [addVal, addValuation_apply, addValuation_apply, multiplicity_le_multiplicity_iff] at h"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact ha.dvd.trans (h n ha.symm.dvd)\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case neg.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : \u2200 (n : \u2115), Classical.choose \u22ef ^ n \u2223 a \u2192 Classical.choose \u22ef ^ n \u2223 b\nha0 : \u00aca = 0\nn : \u2115\nha : Associated a (Classical.choose \u22ef ^ n)\n\u22a2 a \u2223 b",
+ "tactic": "exact ha.dvd.trans (h n ha.symm.dvd)"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [ha0, addVal_zero<\\/a>, top_le_iff<\\/a>, addVal_eq_top_iff<\\/a>] at h\"",
+ "[{\"full_name\":\"DiscreteValuationRing.addVal_zero\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[416,9],\"def_end_pos\":[416,20]},{\"full_name\":\"top_le_iff\",\"def_path\":\"Mathlib\\/Order\\/BoundedOrder.lean\",\"def_pos\":[125,9],\"def_end_pos\":[125,19]},{\"full_name\":\"DiscreteValuationRing.addVal_eq_top_iff\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[446,9],\"def_end_pos\":[446,26]}]"
+ ],
+ "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : b = 0\nha0 : a = 0\n\u22a2 a \u2223 b",
+ "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : (addVal R) a \u2264 (addVal R) b\nha0 : a = 0\n\u22a2 a \u2223 b",
+ "tactic": "rw [ha0, addVal_zero, top_le_iff, addVal_eq_top_iff] at h"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [h]\"",
+ "[]"
+ ],
+ "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : b = 0\nha0 : a = 0\n\u22a2 a \u2223 0",
+ "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : b = 0\nha0 : a = 0\n\u22a2 a \u2223 b",
+ "tactic": "rw [h]"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply dvd_zero<\\/a>\"",
+ "[{\"full_name\":\"dvd_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Divisibility.lean\",\"def_pos\":[41,9],\"def_end_pos\":[41,17]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : b = 0\nha0 : a = 0\n\u22a2 a \u2223 0",
+ "tactic": "apply dvd_zero"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [addVal<\\/a>, addValuation_apply<\\/a>, addValuation_apply<\\/a>]\"",
+ "[{\"full_name\":\"DiscreteValuationRing.addVal\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[397,19],\"def_end_pos\":[397,25]},{\"full_name\":\"multiplicity.addValuation_apply\",\"def_path\":\"Mathlib\\/RingTheory\\/Valuation\\/PrimeMultiplicity.lean\",\"def_pos\":[22,9],\"def_end_pos\":[22,40]},{\"full_name\":\"multiplicity.addValuation_apply\",\"def_path\":\"Mathlib\\/RingTheory\\/Valuation\\/PrimeMultiplicity.lean\",\"def_pos\":[22,9],\"def_end_pos\":[22,40]}]"
+ ],
+ "state_after": "case mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : a \u2223 b\n\u22a2 multiplicity (Classical.choose \u22ef) a \u2264 multiplicity (Classical.choose \u22ef) b",
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : a \u2223 b\n\u22a2 (addVal R) a \u2264 (addVal R) b",
+ "tactic": "rw [addVal, addValuation_apply, addValuation_apply]"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact multiplicity_le_multiplicity_of_dvd_right<\\/a> h\"",
+ "[{\"full_name\":\"multiplicity.multiplicity_le_multiplicity_of_dvd_right\",\"def_path\":\"Mathlib\\/RingTheory\\/Multiplicity.lean\",\"def_pos\":[262,9],\"def_end_pos\":[262,50]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na b : R\nhp : Prime (Classical.choose \u22ef)\nh : a \u2223 b\n\u22a2 multiplicity (Classical.choose \u22ef) a \u2264 multiplicity (Classical.choose \u22ef) b",
+ "tactic": "exact multiplicity_le_multiplicity_of_dvd_right h"
+ }
+ ],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "DiscreteValuationRing.addVal_mul",
+ "id": "DiscreteValuationRing.addVal_mul",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 434,
+ 25
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.addVal_mul",
+ "start": [
+ 432,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "DiscreteValuationRing.addVal_one",
+ "id": "DiscreteValuationRing.addVal_one",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 422,
+ 21
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.addVal_one",
+ "start": [
+ 421,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "DiscreteValuationRing.addVal_pow",
+ "id": "DiscreteValuationRing.addVal_pow",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 438,
+ 25
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.addVal_pow",
+ "start": [
+ 437,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "DiscreteValuationRing.addVal_zero",
+ "id": "DiscreteValuationRing.addVal_zero",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 417,
+ 22
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.addVal_zero",
+ "start": [
+ 416,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "no internal trace for tactic 'intro I'",
+ "full_name": "DiscreteValuationRing.aux_pid_of_ufd_of_unique_irreducible",
+ "id": "DiscreteValuationRing.aux_pid_of_ufd_of_unique_irreducible",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 279,
+ 27
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.aux_pid_of_ufd_of_unique_irreducible",
+ "start": [
+ 250,
+ 1
+ ],
+ "traced_tactics": [
+ {
+ "annotated_tactic": [
+ "\"constructor\"",
+ "[]"
+ ],
+ "state_after": "case principal\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\n\u22a2 \u2200 (S : Ideal R), Submodule.IsPrincipal S",
+ "state_before": "R\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\n\u22a2 IsPrincipalIdealRing R",
+ "tactic": "constructor"
+ },
+ {
+ "annotated_tactic": [
+ "\"intro I\"",
+ "[]"
+ ],
+ "state_after": "case principal\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\n\u22a2 Submodule.IsPrincipal I",
+ "state_before": "case principal\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\n\u22a2 \u2200 (S : Ideal R), Submodule.IsPrincipal S",
+ "tactic": "intro I"
+ },
+ {
+ "annotated_tactic": [
+ "\"by_cases I0 : I = \\u22a5\"",
+ "[]"
+ ],
+ "state_after": "case pos\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : I = \u22a5\n\u22a2 Submodule.IsPrincipal I\n\ncase neg\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\n\u22a2 Submodule.IsPrincipal I",
+ "state_before": "case principal\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\n\u22a2 Submodule.IsPrincipal I",
+ "tactic": "by_cases I0 : I = \u22a5"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8x, hxI, hx0\\u27e9 : \\u2203 x \\u2208 I, x \\u2260 (0 : R) := I.ne_bot_iff.mp I0\"",
+ "[]"
+ ],
+ "state_after": "case neg.intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\n\u22a2 Submodule.IsPrincipal I",
+ "state_before": "case neg\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\n\u22a2 Submodule.IsPrincipal I",
+ "tactic": "obtain \u27e8x, hxI, hx0\u27e9 : \u2203 x \u2208 I, x \u2260 (0 : R) := I.ne_bot_iff.mp I0"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8p, _, H\\u27e9 := HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible<\\/a> h\\u2081 h\\u2082\"",
+ "[{\"full_name\":\"DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[227,9],\"def_end_pos\":[227,37]}]"
+ ],
+ "state_after": "case neg.intro.intro.intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\n\u22a2 Submodule.IsPrincipal I",
+ "state_before": "case neg.intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\n\u22a2 Submodule.IsPrincipal I",
+ "tactic": "obtain \u27e8p, _, H\u27e9 := HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible h\u2081 h\u2082"
+ },
+ {
+ "annotated_tactic": [
+ "\"have ex : \\u2203 n : \\u2115, p ^ n \\u2208 I := by\\n obtain \\u27e8n, u, rfl\\u27e9 := H hx0\\n refine \\u27e8n, ?_\\u27e9\\n simpa only [Units.mul_inv_cancel_right<\\/a>] using I.mul_mem_right (\\u2191u\\u207b\\u00b9) hxI\"",
+ "[{\"full_name\":\"Units.mul_inv_cancel_right\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[298,9],\"def_end_pos\":[298,29]}]"
+ ],
+ "state_after": "case neg.intro.intro.intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 Submodule.IsPrincipal I",
+ "state_before": "case neg.intro.intro.intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\n\u22a2 Submodule.IsPrincipal I",
+ "tactic": "have ex : \u2203 n : \u2115, p ^ n \u2208 I := by\n obtain \u27e8n, u, rfl\u27e9 := H hx0\n refine \u27e8n, ?_\u27e9\n simpa only [Units.mul_inv_cancel_right] using I.mul_mem_right (\u2191u\u207b\u00b9) hxI"
+ },
+ {
+ "annotated_tactic": [
+ "\"constructor\"",
+ "[]"
+ ],
+ "state_after": "case neg.intro.intro.intro.intro.principal'\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 \u2203 a, I = Submodule.span R {a}",
+ "state_before": "case neg.intro.intro.intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 Submodule.IsPrincipal I",
+ "tactic": "constructor"
+ },
+ {
+ "annotated_tactic": [
+ "\"use p ^ Nat.find<\\/a> ex\"",
+ "[{\"full_name\":\"Nat.find\",\"def_path\":\"Mathlib\\/Init\\/Data\\/Nat\\/Lemmas.lean\",\"def_pos\":[703,15],\"def_end_pos\":[703,19]}]"
+ ],
+ "state_after": "case h\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 I = Submodule.span R {p ^ Nat.find ex}",
+ "state_before": "case neg.intro.intro.intro.intro.principal'\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 \u2203 a, I = Submodule.span R {a}",
+ "tactic": "use p ^ Nat.find ex"
+ },
+ {
+ "annotated_tactic": [
+ "\"show I = Ideal.span<\\/a> _\"",
+ "[{\"full_name\":\"Ideal.span\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[115,5],\"def_end_pos\":[115,9]}]"
+ ],
+ "state_after": "case h\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 I = span {p ^ Nat.find ex}",
+ "state_before": "case h\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 I = Submodule.span R {p ^ Nat.find ex}",
+ "tactic": "show I = Ideal.span _"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply le_antisymm<\\/a>\"",
+ "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]}]"
+ ],
+ "state_after": "case h.a\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 I \u2264 span {p ^ Nat.find ex}\n\ncase h.a\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 span {p ^ Nat.find ex} \u2264 I",
+ "state_before": "case h\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 I = span {p ^ Nat.find ex}",
+ "tactic": "apply le_antisymm"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [I0]\"",
+ "[]"
+ ],
+ "state_after": "case pos\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : I = \u22a5\n\u22a2 Submodule.IsPrincipal \u22a5",
+ "state_before": "case pos\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : I = \u22a5\n\u22a2 Submodule.IsPrincipal I",
+ "tactic": "rw [I0]"
+ },
+ {
+ "annotated_tactic": [
+ "\"use 0\"",
+ "[]"
+ ],
+ "state_after": "case h\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : I = \u22a5\n\u22a2 \u22a5 = Submodule.span R {0}",
+ "state_before": "case pos\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : I = \u22a5\n\u22a2 Submodule.IsPrincipal \u22a5",
+ "tactic": "use 0"
+ },
+ {
+ "annotated_tactic": [
+ "\"simp only [Set.singleton_zero<\\/a>, Submodule.span_zero<\\/a>]\"",
+ "[{\"full_name\":\"Set.singleton_zero\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Pointwise\\/Basic.lean\",\"def_pos\":[93,3],\"def_end_pos\":[93,14]},{\"full_name\":\"Submodule.span_zero\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[620,9],\"def_end_pos\":[620,18]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case h\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : I = \u22a5\n\u22a2 \u22a5 = Submodule.span R {0}",
+ "tactic": "simp only [Set.singleton_zero, Submodule.span_zero]"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8n, u, rfl\\u27e9 := H hx0\"",
+ "[]"
+ ],
+ "state_after": "case intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nn : \u2115\nu : R\u02e3\nhxI : p ^ n * \u2191u \u2208 I\nhx0 : p ^ n * \u2191u \u2260 0\n\u22a2 \u2203 n, p ^ n \u2208 I",
+ "state_before": "R\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\n\u22a2 \u2203 n, p ^ n \u2208 I",
+ "tactic": "obtain \u27e8n, u, rfl\u27e9 := H hx0"
+ },
+ {
+ "annotated_tactic": [
+ "\"refine \\u27e8n, ?_\\u27e9\"",
+ "[]"
+ ],
+ "state_after": "case intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nn : \u2115\nu : R\u02e3\nhxI : p ^ n * \u2191u \u2208 I\nhx0 : p ^ n * \u2191u \u2260 0\n\u22a2 p ^ n \u2208 I",
+ "state_before": "case intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nn : \u2115\nu : R\u02e3\nhxI : p ^ n * \u2191u \u2208 I\nhx0 : p ^ n * \u2191u \u2260 0\n\u22a2 \u2203 n, p ^ n \u2208 I",
+ "tactic": "refine \u27e8n, ?_\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"simpa only [Units.mul_inv_cancel_right<\\/a>] using I.mul_mem_right (\\u2191u\\u207b\\u00b9) hxI\"",
+ "[{\"full_name\":\"Units.mul_inv_cancel_right\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[298,9],\"def_end_pos\":[298,29]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nn : \u2115\nu : R\u02e3\nhxI : p ^ n * \u2191u \u2208 I\nhx0 : p ^ n * \u2191u \u2260 0\n\u22a2 p ^ n \u2208 I",
+ "tactic": "simpa only [Units.mul_inv_cancel_right] using I.mul_mem_right (\u2191u\u207b\u00b9) hxI"
+ },
+ {
+ "annotated_tactic": [
+ "\"intro r hr\"",
+ "[]"
+ ],
+ "state_after": "case h.a\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nr : R\nhr : r \u2208 I\n\u22a2 r \u2208 span {p ^ Nat.find ex}",
+ "state_before": "case h.a\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 I \u2264 span {p ^ Nat.find ex}",
+ "tactic": "intro r hr"
+ },
+ {
+ "annotated_tactic": [
+ "\"by_cases hr0 : r = 0\"",
+ "[]"
+ ],
+ "state_after": "case pos\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nr : R\nhr : r \u2208 I\nhr0 : r = 0\n\u22a2 r \u2208 span {p ^ Nat.find ex}\n\ncase neg\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nr : R\nhr : r \u2208 I\nhr0 : \u00acr = 0\n\u22a2 r \u2208 span {p ^ Nat.find ex}",
+ "state_before": "case h.a\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nr : R\nhr : r \u2208 I\n\u22a2 r \u2208 span {p ^ Nat.find ex}",
+ "tactic": "by_cases hr0 : r = 0"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8n, u, rfl\\u27e9 := H hr0\"",
+ "[]"
+ ],
+ "state_after": "case neg.intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhr : p ^ n * \u2191u \u2208 I\nhr0 : \u00acp ^ n * \u2191u = 0\n\u22a2 p ^ n * \u2191u \u2208 span {p ^ Nat.find ex}",
+ "state_before": "case neg\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nr : R\nhr : r \u2208 I\nhr0 : \u00acr = 0\n\u22a2 r \u2208 span {p ^ Nat.find ex}",
+ "tactic": "obtain \u27e8n, u, rfl\u27e9 := H hr0"
+ },
+ {
+ "annotated_tactic": [
+ "\"simp only [mem_span_singleton<\\/a>, Units.isUnit<\\/a>, IsUnit.dvd_mul_right<\\/a>]\"",
+ "[{\"full_name\":\"Ideal.mem_span_singleton\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[503,9],\"def_end_pos\":[503,27]},{\"full_name\":\"Units.isUnit\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[689,19],\"def_end_pos\":[689,31]},{\"full_name\":\"IsUnit.dvd_mul_right\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Units.lean\",\"def_pos\":[87,9],\"def_end_pos\":[87,22]}]"
+ ],
+ "state_after": "case neg.intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhr : p ^ n * \u2191u \u2208 I\nhr0 : \u00acp ^ n * \u2191u = 0\n\u22a2 p ^ Nat.find ex \u2223 p ^ n",
+ "state_before": "case neg.intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhr : p ^ n * \u2191u \u2208 I\nhr0 : \u00acp ^ n * \u2191u = 0\n\u22a2 p ^ n * \u2191u \u2208 span {p ^ Nat.find ex}",
+ "tactic": "simp only [mem_span_singleton, Units.isUnit, IsUnit.dvd_mul_right]"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply pow_dvd_pow<\\/a>\"",
+ "[{\"full_name\":\"pow_dvd_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Basic.lean\",\"def_pos\":[157,7],\"def_end_pos\":[157,18]}]"
+ ],
+ "state_after": "case neg.intro.intro.h\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhr : p ^ n * \u2191u \u2208 I\nhr0 : \u00acp ^ n * \u2191u = 0\n\u22a2 Nat.find ex \u2264 n",
+ "state_before": "case neg.intro.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhr : p ^ n * \u2191u \u2208 I\nhr0 : \u00acp ^ n * \u2191u = 0\n\u22a2 p ^ Nat.find ex \u2223 p ^ n",
+ "tactic": "apply pow_dvd_pow"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply Nat.find_min'<\\/a>\"",
+ "[{\"full_name\":\"Nat.find_min'\",\"def_path\":\"Mathlib\\/Init\\/Data\\/Nat\\/Lemmas.lean\",\"def_pos\":[715,19],\"def_end_pos\":[715,28]}]"
+ ],
+ "state_after": "case neg.intro.intro.h.h\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhr : p ^ n * \u2191u \u2208 I\nhr0 : \u00acp ^ n * \u2191u = 0\n\u22a2 p ^ n \u2208 I",
+ "state_before": "case neg.intro.intro.h\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhr : p ^ n * \u2191u \u2208 I\nhr0 : \u00acp ^ n * \u2191u = 0\n\u22a2 Nat.find ex \u2264 n",
+ "tactic": "apply Nat.find_min'"
+ },
+ {
+ "annotated_tactic": [
+ "\"simpa only [Units.mul_inv_cancel_right<\\/a>] using I.mul_mem_right (\\u2191u\\u207b\\u00b9) hr\"",
+ "[{\"full_name\":\"Units.mul_inv_cancel_right\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[298,9],\"def_end_pos\":[298,29]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case neg.intro.intro.h.h\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhr : p ^ n * \u2191u \u2208 I\nhr0 : \u00acp ^ n * \u2191u = 0\n\u22a2 p ^ n \u2208 I",
+ "tactic": "simpa only [Units.mul_inv_cancel_right] using I.mul_mem_right (\u2191u\u207b\u00b9) hr"
+ },
+ {
+ "annotated_tactic": [
+ "\"simp only [hr0, Submodule.zero_mem<\\/a>]\"",
+ "[{\"full_name\":\"Submodule.zero_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[216,19],\"def_end_pos\":[216,27]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case pos\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\nr : R\nhr : r \u2208 I\nhr0 : r = 0\n\u22a2 r \u2208 span {p ^ Nat.find ex}",
+ "tactic": "simp only [hr0, Submodule.zero_mem]"
+ },
+ {
+ "annotated_tactic": [
+ "\"erw [Submodule.span_singleton_le_iff_mem<\\/a>]\"",
+ "[{\"full_name\":\"Submodule.span_singleton_le_iff_mem\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[624,9],\"def_end_pos\":[624,34]}]"
+ ],
+ "state_after": "case h.a\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 p ^ Nat.find ex \u2208 I",
+ "state_before": "case h.a\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 span {p ^ Nat.find ex} \u2264 I",
+ "tactic": "erw [Submodule.span_singleton_le_iff_mem]"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact Nat.find_spec<\\/a> ex\"",
+ "[{\"full_name\":\"Nat.find_spec\",\"def_path\":\"Mathlib\\/Init\\/Data\\/Nat\\/Lemmas.lean\",\"def_pos\":[707,19],\"def_end_pos\":[707,28]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case h.a\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nI : Ideal R\nI0 : \u00acI = \u22a5\nx : R\nhxI : x \u2208 I\nhx0 : x \u2260 0\np : R\nleft\u271d : Irreducible p\nH : \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (p ^ n) x\nex : \u2203 n, p ^ n \u2208 I\n\u22a2 p ^ Nat.find ex \u2208 I",
+ "tactic": "exact Nat.find_spec ex"
+ }
+ ],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "DiscreteValuationRing.exists_prime",
+ "id": "DiscreteValuationRing.exists_prime",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 114,
+ 62
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.exists_prime",
+ "start": [
+ 113,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "no internal trace for tactic 'rw [Ne, \u2190 eq_bot_iff_generator_eq_zero]'",
+ "full_name": "DiscreteValuationRing.ideal_eq_span_pow_irreducible",
+ "id": "DiscreteValuationRing.ideal_eq_span_pow_irreducible",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 361,
+ 8
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.ideal_eq_span_pow_irreducible",
+ "start": [
+ 352,
+ 1
+ ],
+ "traced_tactics": [
+ {
+ "annotated_tactic": [
+ "\"have gen_ne_zero : generator<\\/a> s \\u2260 0 := by\\n rw [Ne<\\/a>, \\u2190 eq_bot_iff_generator_eq_zero<\\/a>]\\n assumption\"",
+ "[{\"full_name\":\"Submodule.IsPrincipal.generator\",\"def_path\":\"Mathlib\\/RingTheory\\/PrincipalIdealDomain.lean\",\"def_pos\":[89,19],\"def_end_pos\":[89,28]},{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"Submodule.IsPrincipal.eq_bot_iff_generator_eq_zero\",\"def_path\":\"Mathlib\\/RingTheory\\/PrincipalIdealDomain.lean\",\"def_pos\":[114,9],\"def_end_pos\":[114,37]}]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\ngen_ne_zero : generator s \u2260 0\n\u22a2 \u2203 n, s = span {\u03d6 ^ n}",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\n\u22a2 \u2203 n, s = span {\u03d6 ^ n}",
+ "tactic": "have gen_ne_zero : generator s \u2260 0 := by\n rw [Ne, \u2190 eq_bot_iff_generator_eq_zero]\n assumption"
+ },
+ {
+ "annotated_tactic": [
+ "\"rcases associated_pow_irreducible<\\/a> gen_ne_zero hirr with \\u27e8n, u, hnu\\u27e9\"",
+ "[{\"full_name\":\"DiscreteValuationRing.associated_pow_irreducible\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,35]}]"
+ ],
+ "state_after": "case intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\ngen_ne_zero : generator s \u2260 0\nn : \u2115\nu : R\u02e3\nhnu : generator s * \u2191u = \u03d6 ^ n\n\u22a2 \u2203 n, s = span {\u03d6 ^ n}",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\ngen_ne_zero : generator s \u2260 0\n\u22a2 \u2203 n, s = span {\u03d6 ^ n}",
+ "tactic": "rcases associated_pow_irreducible gen_ne_zero hirr with \u27e8n, u, hnu\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"use n\"",
+ "[]"
+ ],
+ "state_after": "case h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\ngen_ne_zero : generator s \u2260 0\nn : \u2115\nu : R\u02e3\nhnu : generator s * \u2191u = \u03d6 ^ n\n\u22a2 s = span {\u03d6 ^ n}",
+ "state_before": "case intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\ngen_ne_zero : generator s \u2260 0\nn : \u2115\nu : R\u02e3\nhnu : generator s * \u2191u = \u03d6 ^ n\n\u22a2 \u2203 n, s = span {\u03d6 ^ n}",
+ "tactic": "use n"
+ },
+ {
+ "annotated_tactic": [
+ "\"have : span<\\/a> _ = _ := Ideal.span_singleton_generator<\\/a> s\"",
+ "[{\"full_name\":\"Ideal.span\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[115,5],\"def_end_pos\":[115,9]},{\"full_name\":\"Ideal.span_singleton_generator\",\"def_path\":\"Mathlib\\/RingTheory\\/PrincipalIdealDomain.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,46]}]"
+ ],
+ "state_after": "case h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\ngen_ne_zero : generator s \u2260 0\nn : \u2115\nu : R\u02e3\nhnu : generator s * \u2191u = \u03d6 ^ n\nthis : span {generator s} = s\n\u22a2 s = span {\u03d6 ^ n}",
+ "state_before": "case h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\ngen_ne_zero : generator s \u2260 0\nn : \u2115\nu : R\u02e3\nhnu : generator s * \u2191u = \u03d6 ^ n\n\u22a2 s = span {\u03d6 ^ n}",
+ "tactic": "have : span _ = _ := Ideal.span_singleton_generator s"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [\\u2190 this, \\u2190 hnu, span_singleton_eq_span_singleton<\\/a>]\"",
+ "[{\"full_name\":\"Ideal.span_singleton_eq_span_singleton\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[516,9],\"def_end_pos\":[516,41]}]"
+ ],
+ "state_after": "case h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\ngen_ne_zero : generator s \u2260 0\nn : \u2115\nu : R\u02e3\nhnu : generator s * \u2191u = \u03d6 ^ n\nthis : span {generator s} = s\n\u22a2 Associated (generator s) (generator s * \u2191u)",
+ "state_before": "case h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\ngen_ne_zero : generator s \u2260 0\nn : \u2115\nu : R\u02e3\nhnu : generator s * \u2191u = \u03d6 ^ n\nthis : span {generator s} = s\n\u22a2 s = span {\u03d6 ^ n}",
+ "tactic": "rw [\u2190 this, \u2190 hnu, span_singleton_eq_span_singleton]"
+ },
+ {
+ "annotated_tactic": [
+ "\"use u\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\ngen_ne_zero : generator s \u2260 0\nn : \u2115\nu : R\u02e3\nhnu : generator s * \u2191u = \u03d6 ^ n\nthis : span {generator s} = s\n\u22a2 Associated (generator s) (generator s * \u2191u)",
+ "tactic": "use u"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [Ne<\\/a>, \\u2190 eq_bot_iff_generator_eq_zero<\\/a>]\"",
+ "[{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"Submodule.IsPrincipal.eq_bot_iff_generator_eq_zero\",\"def_path\":\"Mathlib\\/RingTheory\\/PrincipalIdealDomain.lean\",\"def_pos\":[114,9],\"def_end_pos\":[114,37]}]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\n\u22a2 \u00acs = \u22a5",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\n\u22a2 generator s \u2260 0",
+ "tactic": "rw [Ne, \u2190 eq_bot_iff_generator_eq_zero]"
+ },
+ {
+ "annotated_tactic": [
+ "\"assumption\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\ns : Ideal R\nhs : s \u2260 \u22a5\n\u03d6 : R\nhirr : Irreducible \u03d6\n\u22a2 \u00acs = \u22a5",
+ "tactic": "assumption"
+ }
+ ],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "no internal trace for tactic 'intro RDVR'",
+ "full_name": "DiscreteValuationRing.iff_pid_with_one_nonzero_prime",
+ "id": "DiscreteValuationRing.iff_pid_with_one_nonzero_prime",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 145,
+ 18
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.iff_pid_with_one_nonzero_prime",
+ "start": [
+ 118,
+ 1
+ ],
+ "traced_tactics": [
+ {
+ "annotated_tactic": [
+ "\"constructor\"",
+ "[]"
+ ],
+ "state_after": "case mp\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 DiscreteValuationRing R \u2192 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime\n\ncase mpr\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 (IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime) \u2192 DiscreteValuationRing R",
+ "state_before": "R\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 DiscreteValuationRing R \u2194 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "tactic": "constructor"
+ },
+ {
+ "annotated_tactic": [
+ "\"intro RDVR\"",
+ "[]"
+ ],
+ "state_after": "case mp\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\n\u22a2 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "state_before": "case mp\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 DiscreteValuationRing R \u2192 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "tactic": "intro RDVR"
+ },
+ {
+ "annotated_tactic": [
+ "\"rcases id<\\/a> RDVR with \\u27e8Rlocal\\u27e9\"",
+ "[{\"full_name\":\"id\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[33,15],\"def_end_pos\":[33,17]}]"
+ ],
+ "state_after": "case mp.mk\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "state_before": "case mp\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\n\u22a2 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "tactic": "rcases id RDVR with \u27e8Rlocal\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"constructor\"",
+ "[]"
+ ],
+ "state_after": "case mp.mk.left\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 IsPrincipalIdealRing R\n\ncase mp.mk.right\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "state_before": "case mp.mk\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "tactic": "constructor"
+ },
+ {
+ "annotated_tactic": [
+ "\"use LocalRing.maximalIdeal<\\/a> R\"",
+ "[{\"full_name\":\"LocalRing.maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[106,5],\"def_end_pos\":[106,17]}]"
+ ],
+ "state_after": "case h\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) (maximalIdeal R) \u2227 \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = maximalIdeal R",
+ "state_before": "case mp.mk.right\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "tactic": "use LocalRing.maximalIdeal R"
+ },
+ {
+ "annotated_tactic": [
+ "\"constructor\"",
+ "[]"
+ ],
+ "state_after": "case h.left\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) (maximalIdeal R)\n\ncase h.right\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = maximalIdeal R",
+ "state_before": "case h\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) (maximalIdeal R) \u2227 \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = maximalIdeal R",
+ "tactic": "constructor"
+ },
+ {
+ "annotated_tactic": [
+ "\"assumption\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case mp.mk.left\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 IsPrincipalIdealRing R",
+ "tactic": "assumption"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact \\u27e8Rlocal, inferInstance<\\/a>\\u27e9\"",
+ "[{\"full_name\":\"inferInstance\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[99,8],\"def_end_pos\":[99,21]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case h.left\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) (maximalIdeal R)",
+ "tactic": "exact \u27e8Rlocal, inferInstance\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"rintro Q \\u27e8hQ1, hQ2\\u27e9\"",
+ "[]"
+ ],
+ "state_after": "case h.right.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nQ : Ideal R\nhQ1 : Q \u2260 \u22a5\nhQ2 : Q.IsPrime\n\u22a2 Q = maximalIdeal R",
+ "state_before": "case h.right\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\n\u22a2 \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = maximalIdeal R",
+ "tactic": "rintro Q \u27e8hQ1, hQ2\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8q, rfl\\u27e9 := (IsPrincipalIdealRing.principal<\\/a> Q).1\"",
+ "[{\"full_name\":\"IsPrincipalIdealRing.principal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[48,3],\"def_end_pos\":[48,12]}]"
+ ],
+ "state_after": "case h.right.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nq : R\nhQ1 : Submodule.span R {q} \u2260 \u22a5\nhQ2 : IsPrime (Submodule.span R {q})\n\u22a2 Submodule.span R {q} = maximalIdeal R",
+ "state_before": "case h.right.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nQ : Ideal R\nhQ1 : Q \u2260 \u22a5\nhQ2 : Q.IsPrime\n\u22a2 Q = maximalIdeal R",
+ "tactic": "obtain \u27e8q, rfl\u27e9 := (IsPrincipalIdealRing.principal Q).1"
+ },
+ {
+ "annotated_tactic": [
+ "\"have hq : q \\u2260 0 := by\\n rintro rfl\\n apply hQ1\\n simp\"",
+ "[]"
+ ],
+ "state_after": "case h.right.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nq : R\nhQ1 : Submodule.span R {q} \u2260 \u22a5\nhQ2 : IsPrime (Submodule.span R {q})\nhq : q \u2260 0\n\u22a2 Submodule.span R {q} = maximalIdeal R",
+ "state_before": "case h.right.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nq : R\nhQ1 : Submodule.span R {q} \u2260 \u22a5\nhQ2 : IsPrime (Submodule.span R {q})\n\u22a2 Submodule.span R {q} = maximalIdeal R",
+ "tactic": "have hq : q \u2260 0 := by\n rintro rfl\n apply hQ1\n simp"
+ },
+ {
+ "annotated_tactic": [
+ "\"erw [span_singleton_prime<\\/a> hq] at hQ2\"",
+ "[{\"full_name\":\"Ideal.span_singleton_prime\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[531,9],\"def_end_pos\":[531,29]}]"
+ ],
+ "state_after": "case h.right.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nq : R\nhQ1 : Submodule.span R {q} \u2260 \u22a5\nhQ2 : Prime q\nhq : q \u2260 0\n\u22a2 Submodule.span R {q} = maximalIdeal R",
+ "state_before": "case h.right.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nq : R\nhQ1 : Submodule.span R {q} \u2260 \u22a5\nhQ2 : IsPrime (Submodule.span R {q})\nhq : q \u2260 0\n\u22a2 Submodule.span R {q} = maximalIdeal R",
+ "tactic": "erw [span_singleton_prime hq] at hQ2"
+ },
+ {
+ "annotated_tactic": [
+ "\"replace hQ2 := hQ2.irreducible\"",
+ "[]"
+ ],
+ "state_after": "case h.right.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nq : R\nhQ1 : Submodule.span R {q} \u2260 \u22a5\nhq : q \u2260 0\nhQ2 : Irreducible q\n\u22a2 Submodule.span R {q} = maximalIdeal R",
+ "state_before": "case h.right.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nq : R\nhQ1 : Submodule.span R {q} \u2260 \u22a5\nhQ2 : Prime q\nhq : q \u2260 0\n\u22a2 Submodule.span R {q} = maximalIdeal R",
+ "tactic": "replace hQ2 := hQ2.irreducible"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [irreducible_iff_uniformizer<\\/a>] at hQ2\"",
+ "[{\"full_name\":\"DiscreteValuationRing.irreducible_iff_uniformizer\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,36]}]"
+ ],
+ "state_after": "case h.right.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nq : R\nhQ1 : Submodule.span R {q} \u2260 \u22a5\nhq : q \u2260 0\nhQ2 : maximalIdeal R = span {q}\n\u22a2 Submodule.span R {q} = maximalIdeal R",
+ "state_before": "case h.right.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nq : R\nhQ1 : Submodule.span R {q} \u2260 \u22a5\nhq : q \u2260 0\nhQ2 : Irreducible q\n\u22a2 Submodule.span R {q} = maximalIdeal R",
+ "tactic": "rw [irreducible_iff_uniformizer] at hQ2"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact hQ2.symm\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case h.right.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nq : R\nhQ1 : Submodule.span R {q} \u2260 \u22a5\nhq : q \u2260 0\nhQ2 : maximalIdeal R = span {q}\n\u22a2 Submodule.span R {q} = maximalIdeal R",
+ "tactic": "exact hQ2.symm"
+ },
+ {
+ "annotated_tactic": [
+ "\"rintro rfl\"",
+ "[]"
+ ],
+ "state_after": "R\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nhQ1 : Submodule.span R {0} \u2260 \u22a5\nhQ2 : IsPrime (Submodule.span R {0})\n\u22a2 False",
+ "state_before": "R\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nq : R\nhQ1 : Submodule.span R {q} \u2260 \u22a5\nhQ2 : IsPrime (Submodule.span R {q})\n\u22a2 q \u2260 0",
+ "tactic": "rintro rfl"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply hQ1\"",
+ "[]"
+ ],
+ "state_after": "R\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nhQ1 : Submodule.span R {0} \u2260 \u22a5\nhQ2 : IsPrime (Submodule.span R {0})\n\u22a2 Submodule.span R {0} = \u22a5",
+ "state_before": "R\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nhQ1 : Submodule.span R {0} \u2260 \u22a5\nhQ2 : IsPrime (Submodule.span R {0})\n\u22a2 False",
+ "tactic": "apply hQ1"
+ },
+ {
+ "annotated_tactic": [
+ "\"simp\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "R\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing\u271d : IsPrincipalIdealRing R\ntoLocalRing\u271d : LocalRing R\nRlocal : maximalIdeal R \u2260 \u22a5\nhQ1 : Submodule.span R {0} \u2260 \u22a5\nhQ2 : IsPrime (Submodule.span R {0})\n\u22a2 Submodule.span R {0} = \u22a5",
+ "tactic": "simp"
+ },
+ {
+ "annotated_tactic": [
+ "\"rintro \\u27e8RPID, Punique\\u27e9\"",
+ "[]"
+ ],
+ "state_after": "case mpr.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime\n\u22a2 DiscreteValuationRing R",
+ "state_before": "case mpr\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 (IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime) \u2192 DiscreteValuationRing R",
+ "tactic": "rintro \u27e8RPID, Punique\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"haveI : LocalRing<\\/a> R := LocalRing.of_unique_nonzero_prime<\\/a> Punique\"",
+ "[{\"full_name\":\"LocalRing\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[41,7],\"def_end_pos\":[41,16]},{\"full_name\":\"LocalRing.of_unique_nonzero_prime\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[80,9],\"def_end_pos\":[80,32]}]"
+ ],
+ "state_after": "case mpr.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime\nthis : LocalRing R\n\u22a2 DiscreteValuationRing R",
+ "state_before": "case mpr.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime\n\u22a2 DiscreteValuationRing R",
+ "tactic": "haveI : LocalRing R := LocalRing.of_unique_nonzero_prime Punique"
+ },
+ {
+ "annotated_tactic": [
+ "\"refine { not_a_field' := ?_ }\"",
+ "[]"
+ ],
+ "state_after": "case mpr.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime\nthis : LocalRing R\n\u22a2 maximalIdeal R \u2260 \u22a5",
+ "state_before": "case mpr.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime\nthis : LocalRing R\n\u22a2 DiscreteValuationRing R",
+ "tactic": "refine { not_a_field' := ?_ }"
+ },
+ {
+ "annotated_tactic": [
+ "\"rcases Punique with \\u27e8P, \\u27e8hP1, hP2\\u27e9, _\\u27e9\"",
+ "[]"
+ ],
+ "state_after": "case mpr.intro.intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright\u271d : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = P\nhP1 : P \u2260 \u22a5\nhP2 : P.IsPrime\n\u22a2 maximalIdeal R \u2260 \u22a5",
+ "state_before": "case mpr.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime\nthis : LocalRing R\n\u22a2 maximalIdeal R \u2260 \u22a5",
+ "tactic": "rcases Punique with \u27e8P, \u27e8hP1, hP2\u27e9, _\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"have hPM : P \\u2264 maximalIdeal<\\/a> R := le_maximalIdeal<\\/a> hP2.1\"",
+ "[{\"full_name\":\"LocalRing.maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[106,5],\"def_end_pos\":[106,17]},{\"full_name\":\"LocalRing.le_maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[136,9],\"def_end_pos\":[136,24]}]"
+ ],
+ "state_after": "case mpr.intro.intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright\u271d : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = P\nhP1 : P \u2260 \u22a5\nhP2 : P.IsPrime\nhPM : P \u2264 maximalIdeal R\n\u22a2 maximalIdeal R \u2260 \u22a5",
+ "state_before": "case mpr.intro.intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright\u271d : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = P\nhP1 : P \u2260 \u22a5\nhP2 : P.IsPrime\n\u22a2 maximalIdeal R \u2260 \u22a5",
+ "tactic": "have hPM : P \u2264 maximalIdeal R := le_maximalIdeal hP2.1"
+ },
+ {
+ "annotated_tactic": [
+ "\"intro h\"",
+ "[]"
+ ],
+ "state_after": "case mpr.intro.intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright\u271d : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = P\nhP1 : P \u2260 \u22a5\nhP2 : P.IsPrime\nhPM : P \u2264 maximalIdeal R\nh : maximalIdeal R = \u22a5\n\u22a2 False",
+ "state_before": "case mpr.intro.intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright\u271d : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = P\nhP1 : P \u2260 \u22a5\nhP2 : P.IsPrime\nhPM : P \u2264 maximalIdeal R\n\u22a2 maximalIdeal R \u2260 \u22a5",
+ "tactic": "intro h"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [h, le_bot_iff<\\/a>] at hPM\"",
+ "[{\"full_name\":\"le_bot_iff\",\"def_path\":\"Mathlib\\/Order\\/BoundedOrder.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,19]}]"
+ ],
+ "state_after": "case mpr.intro.intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright\u271d : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = P\nhP1 : P \u2260 \u22a5\nhP2 : P.IsPrime\nhPM : P = \u22a5\nh : maximalIdeal R = \u22a5\n\u22a2 False",
+ "state_before": "case mpr.intro.intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright\u271d : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = P\nhP1 : P \u2260 \u22a5\nhP2 : P.IsPrime\nhPM : P \u2264 maximalIdeal R\nh : maximalIdeal R = \u22a5\n\u22a2 False",
+ "tactic": "rw [h, le_bot_iff] at hPM"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact hP1 hPM\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case mpr.intro.intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2074 : CommRing R\u271d\ninst\u271d\u00b3 : IsDomain R\u271d\ninst\u271d\u00b2 : DiscreteValuationRing R\u271d\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright\u271d : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = P\nhP1 : P \u2260 \u22a5\nhP2 : P.IsPrime\nhPM : P = \u22a5\nh : maximalIdeal R = \u22a5\n\u22a2 False",
+ "tactic": "exact hP1 hPM"
+ }
+ ],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "no internal trace for tactic 'ring'",
+ "full_name": "DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal",
+ "id": "DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 88,
+ 54
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal",
+ "start": [
+ 75,
+ 1
+ ],
+ "traced_tactics": [
+ {
+ "annotated_tactic": [
+ "\"have h2 : \\u00acIsUnit<\\/a> \\u03d6 := show \\u03d6 \\u2208 maximalIdeal<\\/a> R from h.symm \\u25b8 Submodule.mem_span_singleton_self<\\/a> \\u03d6\"",
+ "[{\"full_name\":\"IsUnit\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[653,5],\"def_end_pos\":[653,11]},{\"full_name\":\"LocalRing.maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[106,5],\"def_end_pos\":[106,17]},{\"full_name\":\"Submodule.mem_span_singleton_self\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[474,9],\"def_end_pos\":[474,32]}]"
+ ],
+ "state_after": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\n\u22a2 Irreducible \u03d6",
+ "state_before": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\n\u22a2 Irreducible \u03d6",
+ "tactic": "have h2 : \u00acIsUnit \u03d6 := show \u03d6 \u2208 maximalIdeal R from h.symm \u25b8 Submodule.mem_span_singleton_self \u03d6"
+ },
+ {
+ "annotated_tactic": [
+ "\"refine \\u27e8h2, ?_\\u27e9\"",
+ "[]"
+ ],
+ "state_after": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\n\u22a2 \u2200 (a b : R), \u03d6 = a * b \u2192 IsUnit a \u2228 IsUnit b",
+ "state_before": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\n\u22a2 Irreducible \u03d6",
+ "tactic": "refine \u27e8h2, ?_\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"intro a b hab\"",
+ "[]"
+ ],
+ "state_after": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = a * b\n\u22a2 IsUnit a \u2228 IsUnit b",
+ "state_before": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\n\u22a2 \u2200 (a b : R), \u03d6 = a * b \u2192 IsUnit a \u2228 IsUnit b",
+ "tactic": "intro a b hab"
+ },
+ {
+ "annotated_tactic": [
+ "\"by_contra! h\"",
+ "[]"
+ ],
+ "state_after": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh\u271d : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = a * b\nh : \u00acIsUnit a \u2227 \u00acIsUnit b\n\u22a2 False",
+ "state_before": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = a * b\n\u22a2 IsUnit a \u2228 IsUnit b",
+ "tactic": "by_contra! h"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8ha : a \\u2208 maximalIdeal<\\/a> R, hb : b \\u2208 maximalIdeal<\\/a> R\\u27e9 := h\"",
+ "[{\"full_name\":\"LocalRing.maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[106,5],\"def_end_pos\":[106,17]},{\"full_name\":\"LocalRing.maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[106,5],\"def_end_pos\":[106,17]}]"
+ ],
+ "state_after": "case intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = a * b\nha : a \u2208 maximalIdeal R\nhb : b \u2208 maximalIdeal R\n\u22a2 False",
+ "state_before": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh\u271d : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = a * b\nh : \u00acIsUnit a \u2227 \u00acIsUnit b\n\u22a2 False",
+ "tactic": "obtain \u27e8ha : a \u2208 maximalIdeal R, hb : b \u2208 maximalIdeal R\u27e9 := h"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [h, mem_span_singleton'<\\/a>] at ha hb\"",
+ "[{\"full_name\":\"Ideal.mem_span_singleton'\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[176,9],\"def_end_pos\":[176,28]}]"
+ ],
+ "state_after": "case intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = a * b\nha : \u2203 a_1, a_1 * \u03d6 = a\nhb : \u2203 a, a * \u03d6 = b\n\u22a2 False",
+ "state_before": "case intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = a * b\nha : a \u2208 maximalIdeal R\nhb : b \u2208 maximalIdeal R\n\u22a2 False",
+ "tactic": "rw [h, mem_span_singleton'] at ha hb"
+ },
+ {
+ "annotated_tactic": [
+ "\"rcases ha with \\u27e8a, rfl\\u27e9\"",
+ "[]"
+ ],
+ "state_after": "case intro.intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\nb : R\nhb : \u2203 a, a * \u03d6 = b\na : R\nhab : \u03d6 = a * \u03d6 * b\n\u22a2 False",
+ "state_before": "case intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = a * b\nha : \u2203 a_1, a_1 * \u03d6 = a\nhb : \u2203 a, a * \u03d6 = b\n\u22a2 False",
+ "tactic": "rcases ha with \u27e8a, rfl\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"rcases hb with \\u27e8b, rfl\\u27e9\"",
+ "[]"
+ ],
+ "state_after": "case intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = a * \u03d6 * (b * \u03d6)\n\u22a2 False",
+ "state_before": "case intro.intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\nb : R\nhb : \u2203 a, a * \u03d6 = b\na : R\nhab : \u03d6 = a * \u03d6 * b\n\u22a2 False",
+ "tactic": "rcases hb with \u27e8b, rfl\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [show a * \\u03d6 * (b * \\u03d6) = \\u03d6 * (\\u03d6 * (a * b)) by ring] at hab\"",
+ "[]"
+ ],
+ "state_after": "case intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = \u03d6 * (\u03d6 * (a * b))\n\u22a2 False",
+ "state_before": "case intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = a * \u03d6 * (b * \u03d6)\n\u22a2 False",
+ "tactic": "rw [show a * \u03d6 * (b * \u03d6) = \u03d6 * (\u03d6 * (a * b)) by ring] at hab"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply h\\u03d6\"",
+ "[]"
+ ],
+ "state_after": "case intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = \u03d6 * (\u03d6 * (a * b))\n\u22a2 \u03d6 = 0",
+ "state_before": "case intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = \u03d6 * (\u03d6 * (a * b))\n\u22a2 False",
+ "tactic": "apply h\u03d6"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply eq_zero_of_mul_eq_self_right<\\/a> _ hab.symm\"",
+ "[{\"full_name\":\"eq_zero_of_mul_eq_self_right\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[267,9],\"def_end_pos\":[267,37]}]"
+ ],
+ "state_after": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = \u03d6 * (\u03d6 * (a * b))\n\u22a2 \u03d6 * (a * b) \u2260 1",
+ "state_before": "case intro.intro.intro\nR\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = \u03d6 * (\u03d6 * (a * b))\n\u22a2 \u03d6 = 0",
+ "tactic": "apply eq_zero_of_mul_eq_self_right _ hab.symm"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact fun hh => h2 (isUnit_of_dvd_one<\\/a> \\u27e8_, hh.symm\\u27e9)\"",
+ "[{\"full_name\":\"isUnit_of_dvd_one\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Units.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,26]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = \u03d6 * (\u03d6 * (a * b))\n\u22a2 \u03d6 * (a * b) \u2260 1",
+ "tactic": "exact fun hh => h2 (isUnit_of_dvd_one \u27e8_, hh.symm\u27e9)"
+ },
+ {
+ "annotated_tactic": [
+ "\"ring\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "R\u271d : Type u\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\ninst\u271d\u00b3 : DiscreteValuationRing R\u271d\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u03d6 : R\nh\u03d6 : \u03d6 \u2260 0\nh : maximalIdeal R = span {\u03d6}\nh2 : \u00acIsUnit \u03d6\na b : R\nhab : \u03d6 = a * \u03d6 * (b * \u03d6)\n\u22a2 a * \u03d6 * (b * \u03d6) = \u03d6 * (\u03d6 * (a * b))",
+ "tactic": "ring"
+ }
+ ],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "DiscreteValuationRing.not_a_field",
+ "id": "DiscreteValuationRing.not_a_field",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 63,
+ 15
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.not_a_field",
+ "start": [
+ 62,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "DiscreteValuationRing.not_isField",
+ "id": "DiscreteValuationRing.not_isField",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 68,
+ 64
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.not_isField",
+ "start": [
+ 67,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "no internal trace for tactic 'rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]'",
+ "full_name": "DiscreteValuationRing.unit_mul_pow_congr_pow",
+ "id": "DiscreteValuationRing.unit_mul_pow_congr_pow",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 377,
+ 39
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.unit_mul_pow_congr_pow",
+ "start": [
+ 364,
+ 1
+ ],
+ "traced_tactics": [
+ {
+ "annotated_tactic": [
+ "\"have key : Associated<\\/a> (Multiset.replicate<\\/a> m p).prod<\\/a> (Multiset.replicate<\\/a> n q).prod<\\/a> := by\\n rw [Multiset.prod_replicate<\\/a>, Multiset.prod_replicate<\\/a>, Associated<\\/a>]\\n refine \\u27e8u * v\\u207b\\u00b9, ?_\\u27e9\\n simp only [Units.val_mul<\\/a>]\\n rw [mul_left_comm<\\/a>, \\u2190 mul_assoc<\\/a>, h, mul_right_comm<\\/a>, Units.mul_inv<\\/a>, one_mul<\\/a>]\"",
+ "[{\"full_name\":\"Associated\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[397,5],\"def_end_pos\":[397,15]},{\"full_name\":\"Multiset.replicate\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[929,5],\"def_end_pos\":[929,14]},{\"full_name\":\"Multiset.prod\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Multiset.lean\",\"def_pos\":[40,5],\"def_end_pos\":[40,9]},{\"full_name\":\"Multiset.replicate\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[929,5],\"def_end_pos\":[929,14]},{\"full_name\":\"Multiset.prod\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Multiset.lean\",\"def_pos\":[40,5],\"def_end_pos\":[40,9]},{\"full_name\":\"Multiset.prod_replicate\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Multiset.lean\",\"def_pos\":[130,9],\"def_end_pos\":[130,23]},{\"full_name\":\"Multiset.prod_replicate\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Multiset.lean\",\"def_pos\":[130,9],\"def_end_pos\":[130,23]},{\"full_name\":\"Associated\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[397,5],\"def_end_pos\":[397,15]},{\"full_name\":\"Units.val_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[226,9],\"def_end_pos\":[226,16]},{\"full_name\":\"mul_left_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,22]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_right_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[190,9],\"def_end_pos\":[190,23]},{\"full_name\":\"Units.mul_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[265,9],\"def_end_pos\":[265,16]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\n\u22a2 m = n",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\n\u22a2 m = n",
+ "tactic": "have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by\n rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]\n refine \u27e8u * v\u207b\u00b9, ?_\u27e9\n simp only [Units.val_mul]\n rw [mul_left_comm, \u2190 mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]"
+ },
+ {
+ "annotated_tactic": [
+ "\"have := by\\n refine Multiset.card_eq_card_of_rel<\\/a> (UniqueFactorizationMonoid.factors_unique<\\/a> ?_ ?_ key)\\n all_goals\\n intro x hx\\n obtain rfl := Multiset.eq_of_mem_replicate<\\/a> hx\\n assumption\"",
+ "[{\"full_name\":\"Multiset.card_eq_card_of_rel\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[2884,9],\"def_end_pos\":[2884,28]},{\"full_name\":\"UniqueFactorizationMonoid.factors_unique\",\"def_path\":\"Mathlib\\/RingTheory\\/UniqueFactorizationDomain.lean\",\"def_pos\":[278,9],\"def_end_pos\":[278,23]},{\"full_name\":\"Multiset.eq_of_mem_replicate\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[966,9],\"def_end_pos\":[966,28]}]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\nthis : Multiset.card (Multiset.replicate m p) = Multiset.card (Multiset.replicate n q)\n\u22a2 m = n",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\n\u22a2 m = n",
+ "tactic": "have := by\n refine Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique ?_ ?_ key)\n all_goals\n intro x hx\n obtain rfl := Multiset.eq_of_mem_replicate hx\n assumption"
+ },
+ {
+ "annotated_tactic": [
+ "\"simpa only [Multiset.card_replicate<\\/a>]\"",
+ "[{\"full_name\":\"Multiset.card_replicate\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[958,17],\"def_end_pos\":[958,31]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\nthis : Multiset.card (Multiset.replicate m p) = Multiset.card (Multiset.replicate n q)\n\u22a2 m = n",
+ "tactic": "simpa only [Multiset.card_replicate]"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [Multiset.prod_replicate<\\/a>, Multiset.prod_replicate<\\/a>, Associated<\\/a>]\"",
+ "[{\"full_name\":\"Multiset.prod_replicate\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Multiset.lean\",\"def_pos\":[130,9],\"def_end_pos\":[130,23]},{\"full_name\":\"Multiset.prod_replicate\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Multiset.lean\",\"def_pos\":[130,9],\"def_end_pos\":[130,23]},{\"full_name\":\"Associated\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[397,5],\"def_end_pos\":[397,15]}]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\n\u22a2 \u2203 u, p ^ m * \u2191u = q ^ n",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\n\u22a2 Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod",
+ "tactic": "rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]"
+ },
+ {
+ "annotated_tactic": [
+ "\"refine \\u27e8u * v\\u207b\\u00b9, ?_\\u27e9\"",
+ "[]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\n\u22a2 p ^ m * \u2191(u * v\u207b\u00b9) = q ^ n",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\n\u22a2 \u2203 u, p ^ m * \u2191u = q ^ n",
+ "tactic": "refine \u27e8u * v\u207b\u00b9, ?_\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"simp only [Units.val_mul<\\/a>]\"",
+ "[{\"full_name\":\"Units.val_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[226,9],\"def_end_pos\":[226,16]}]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\n\u22a2 p ^ m * (\u2191u * \u2191v\u207b\u00b9) = q ^ n",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\n\u22a2 p ^ m * \u2191(u * v\u207b\u00b9) = q ^ n",
+ "tactic": "simp only [Units.val_mul]"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [mul_left_comm<\\/a>, \\u2190 mul_assoc<\\/a>, h, mul_right_comm<\\/a>, Units.mul_inv<\\/a>, one_mul<\\/a>]\"",
+ "[{\"full_name\":\"mul_left_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,22]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_right_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[190,9],\"def_end_pos\":[190,23]},{\"full_name\":\"Units.mul_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[265,9],\"def_end_pos\":[265,16]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\n\u22a2 p ^ m * (\u2191u * \u2191v\u207b\u00b9) = q ^ n",
+ "tactic": "rw [mul_left_comm, \u2190 mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]"
+ },
+ {
+ "annotated_tactic": [
+ "\"refine Multiset.card_eq_card_of_rel<\\/a> (UniqueFactorizationMonoid.factors_unique<\\/a> ?_ ?_ key)\"",
+ "[{\"full_name\":\"Multiset.card_eq_card_of_rel\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[2884,9],\"def_end_pos\":[2884,28]},{\"full_name\":\"UniqueFactorizationMonoid.factors_unique\",\"def_path\":\"Mathlib\\/RingTheory\\/UniqueFactorizationDomain.lean\",\"def_pos\":[278,9],\"def_end_pos\":[278,23]}]"
+ ],
+ "state_after": "case refine_1\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\n\u22a2 \u2200 x \u2208 Multiset.replicate m p, Irreducible x\n\ncase refine_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\n\u22a2 \u2200 x \u2208 Multiset.replicate n q, Irreducible x",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\n\u22a2 ?m.49829",
+ "tactic": "refine Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique ?_ ?_ key)"
+ },
+ {
+ "annotated_tactic": [
+ "\"all_goals\\n intro x hx\\n obtain rfl := Multiset.eq_of_mem_replicate<\\/a> hx\\n assumption\"",
+ "[{\"full_name\":\"Multiset.eq_of_mem_replicate\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[966,9],\"def_end_pos\":[966,28]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case refine_1\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\n\u22a2 \u2200 x \u2208 Multiset.replicate m p, Irreducible x\n\ncase refine_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\n\u22a2 \u2200 x \u2208 Multiset.replicate n q, Irreducible x",
+ "tactic": "all_goals\n intro x hx\n obtain rfl := Multiset.eq_of_mem_replicate hx\n assumption"
+ },
+ {
+ "annotated_tactic": [
+ "\"intro x hx\"",
+ "[]"
+ ],
+ "state_after": "case refine_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\nx : R\nhx : x \u2208 Multiset.replicate n q\n\u22a2 Irreducible x",
+ "state_before": "case refine_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\n\u22a2 \u2200 x \u2208 Multiset.replicate n q, Irreducible x",
+ "tactic": "intro x hx"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain rfl := Multiset.eq_of_mem_replicate<\\/a> hx\"",
+ "[{\"full_name\":\"Multiset.eq_of_mem_replicate\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[966,9],\"def_end_pos\":[966,28]}]"
+ ],
+ "state_after": "case refine_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np : R\nhp : Irreducible p\nu v : R\u02e3\nm n : \u2115\nx : R\nhq : Irreducible x\nh : \u2191u * p ^ m = \u2191v * x ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n x).prod\nhx : x \u2208 Multiset.replicate n x\n\u22a2 Irreducible x",
+ "state_before": "case refine_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * p ^ m = \u2191v * q ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod\nx : R\nhx : x \u2208 Multiset.replicate n q\n\u22a2 Irreducible x",
+ "tactic": "obtain rfl := Multiset.eq_of_mem_replicate hx"
+ },
+ {
+ "annotated_tactic": [
+ "\"assumption\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case refine_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\np : R\nhp : Irreducible p\nu v : R\u02e3\nm n : \u2115\nx : R\nhq : Irreducible x\nh : \u2191u * p ^ m = \u2191v * x ^ n\nkey : Associated (Multiset.replicate m p).prod (Multiset.replicate n x).prod\nhx : x \u2208 Multiset.replicate n x\n\u22a2 Irreducible x",
+ "tactic": "assumption"
+ }
+ ],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ },
+ {
+ "error": "theorem has no selected tactic transitions",
+ "full_name": "Irreducible.maximalIdeal_eq",
+ "id": "Irreducible.maximalIdeal_eq",
+ "reason": "import_failure",
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 101,
+ 39
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "Irreducible.maximalIdeal_eq",
+ "start": [
+ 99,
+ 1
+ ],
+ "traced_tactics": [],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ },
+ "source_file": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean"
+ }
+ ],
+ "input_proofs": 35,
+ "input_units": 2,
+ "proofs": 15,
+ "published_proofs": 12,
+ "published_transitions": 48,
+ "replay_failures": [],
+ "transitions": 79,
+ "validation_failures": [
+ {
+ "errors": [
+ "internal transition boundary mismatch",
+ "internal transition boundary mismatch"
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "proof_id": "mathlib:DiscreteValuationRing.addVal_eq_top_iff",
+ "provenance": {
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 456,
+ 22
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.addVal_eq_top_iff",
+ "start": [
+ 446,
+ 1
+ ],
+ "traced_tactics": [
+ {
+ "annotated_tactic": [
+ "\"have hi := (Classical.choose_spec<\\/a> (exists_prime<\\/a> R)).irreducible<\\/a>\"",
+ "[{\"full_name\":\"Classical.choose_spec\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[28,9],\"def_end_pos\":[28,20]},{\"full_name\":\"DiscreteValuationRing.exists_prime\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[113,9],\"def_end_pos\":[113,21]},{\"full_name\":\"Prime.irreducible\",\"def_path\":\"Mathlib\\/Algebra\\/Associated.lean\",\"def_pos\":[357,19],\"def_end_pos\":[357,36]}]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\n\u22a2 (addVal R) a = \u22a4 \u2194 a = 0",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\n\u22a2 (addVal R) a = \u22a4 \u2194 a = 0",
+ "tactic": "have hi := (Classical.choose_spec (exists_prime R)).irreducible"
+ },
+ {
+ "annotated_tactic": [
+ "\"constructor\"",
+ "[]"
+ ],
+ "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\n\u22a2 (addVal R) a = \u22a4 \u2192 a = 0\n\ncase mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\n\u22a2 a = 0 \u2192 (addVal R) a = \u22a4",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\n\u22a2 (addVal R) a = \u22a4 \u2194 a = 0",
+ "tactic": "constructor"
+ },
+ {
+ "annotated_tactic": [
+ "\"contrapose\"",
+ "[]"
+ ],
+ "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\n\u22a2 \u00aca = 0 \u2192 \u00ac(addVal R) a = \u22a4",
+ "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\n\u22a2 (addVal R) a = \u22a4 \u2192 a = 0",
+ "tactic": "contrapose"
+ },
+ {
+ "annotated_tactic": [
+ "\"intro h\"",
+ "[]"
+ ],
+ "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\nh : \u00aca = 0\n\u22a2 \u00ac(addVal R) a = \u22a4",
+ "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\n\u22a2 \u00aca = 0 \u2192 \u00ac(addVal R) a = \u22a4",
+ "tactic": "intro h"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8n, ha\\u27e9 := associated_pow_irreducible<\\/a> h hi\"",
+ "[{\"full_name\":\"DiscreteValuationRing.associated_pow_irreducible\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,35]}]"
+ ],
+ "state_after": "case mp.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\nh : \u00aca = 0\nn : \u2115\nha : Associated a (Classical.choose \u22ef ^ n)\n\u22a2 \u00ac(addVal R) a = \u22a4",
+ "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\nh : \u00aca = 0\n\u22a2 \u00ac(addVal R) a = \u22a4",
+ "tactic": "obtain \u27e8n, ha\u27e9 := associated_pow_irreducible h hi"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8u, rfl\\u27e9 := ha.symm\"",
+ "[]"
+ ],
+ "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\nhi : Irreducible (Classical.choose \u22ef)\nn : \u2115\nu : R\u02e3\nh : \u00acClassical.choose \u22ef ^ n * \u2191u = 0\nha : Associated (Classical.choose \u22ef ^ n * \u2191u) (Classical.choose \u22ef ^ n)\n\u22a2 \u00ac(addVal R) (Classical.choose \u22ef ^ n * \u2191u) = \u22a4",
+ "state_before": "case mp.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\nh : \u00aca = 0\nn : \u2115\nha : Associated a (Classical.choose \u22ef ^ n)\n\u22a2 \u00ac(addVal R) a = \u22a4",
+ "tactic": "obtain \u27e8u, rfl\u27e9 := ha.symm"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [mul_comm<\\/a>, addVal_def'<\\/a> u hi n]\"",
+ "[{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"DiscreteValuationRing.addVal_def'\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[410,9],\"def_end_pos\":[410,20]}]"
+ ],
+ "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\nhi : Irreducible (Classical.choose \u22ef)\nn : \u2115\nu : R\u02e3\nh : \u00acClassical.choose \u22ef ^ n * \u2191u = 0\nha : Associated (Classical.choose \u22ef ^ n * \u2191u) (Classical.choose \u22ef ^ n)\n\u22a2 \u00ac\u2191n = \u22a4",
+ "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\nhi : Irreducible (Classical.choose \u22ef)\nn : \u2115\nu : R\u02e3\nh : \u00acClassical.choose \u22ef ^ n * \u2191u = 0\nha : Associated (Classical.choose \u22ef ^ n * \u2191u) (Classical.choose \u22ef ^ n)\n\u22a2 \u00ac(addVal R) (Classical.choose \u22ef ^ n * \u2191u) = \u22a4",
+ "tactic": "rw [mul_comm, addVal_def' u hi n]"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact PartENat.natCast_ne_top<\\/a> _\"",
+ "[{\"full_name\":\"PartENat.natCast_ne_top\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/PartENat.lean\",\"def_pos\":[380,9],\"def_end_pos\":[380,23]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\nhi : Irreducible (Classical.choose \u22ef)\nn : \u2115\nu : R\u02e3\nh : \u00acClassical.choose \u22ef ^ n * \u2191u = 0\nha : Associated (Classical.choose \u22ef ^ n * \u2191u) (Classical.choose \u22ef ^ n)\n\u22a2 \u00ac\u2191n = \u22a4",
+ "tactic": "exact PartENat.natCast_ne_top _"
+ },
+ {
+ "annotated_tactic": [
+ "\"rintro rfl\"",
+ "[]"
+ ],
+ "state_after": "case mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\nhi : Irreducible (Classical.choose \u22ef)\n\u22a2 (addVal R) 0 = \u22a4",
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\na : R\nhi : Irreducible (Classical.choose \u22ef)\n\u22a2 a = 0 \u2192 (addVal R) a = \u22a4",
+ "tactic": "rintro rfl"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact addVal_zero<\\/a>\"",
+ "[{\"full_name\":\"DiscreteValuationRing.addVal_zero\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[416,9],\"def_end_pos\":[416,20]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\nhi : Irreducible (Classical.choose \u22ef)\n\u22a2 (addVal R) 0 = \u22a4",
+ "tactic": "exact addVal_zero"
+ }
+ ],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ }
+ },
+ "theorem_name": "DiscreteValuationRing.addVal_eq_top_iff"
+ },
+ {
+ "errors": [
+ "internal transition boundary mismatch",
+ "internal transition boundary mismatch",
+ "internal transition boundary mismatch"
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "proof_id": "mathlib:DiscreteValuationRing.of_ufd_of_unique_irreducible",
+ "provenance": {
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 303,
+ 91
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.of_ufd_of_unique_irreducible",
+ "start": [
+ 286,
+ 1
+ ],
+ "traced_tactics": [
+ {
+ "annotated_tactic": [
+ "\"rw [iff_pid_with_one_nonzero_prime<\\/a>]\"",
+ "[{\"full_name\":\"DiscreteValuationRing.iff_pid_with_one_nonzero_prime\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,39]}]"
+ ],
+ "state_after": "R\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\n\u22a2 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "state_before": "R\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\n\u22a2 DiscreteValuationRing R",
+ "tactic": "rw [iff_pid_with_one_nonzero_prime]"
+ },
+ {
+ "annotated_tactic": [
+ "\"haveI PID : IsPrincipalIdealRing<\\/a> R := aux_pid_of_ufd_of_unique_irreducible<\\/a> R h\\u2081 h\\u2082\"",
+ "[{\"full_name\":\"IsPrincipalIdealRing\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[47,7],\"def_end_pos\":[47,27]},{\"full_name\":\"DiscreteValuationRing.aux_pid_of_ufd_of_unique_irreducible\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[250,9],\"def_end_pos\":[250,45]}]"
+ ],
+ "state_after": "R\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\n\u22a2 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "state_before": "R\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\n\u22a2 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "tactic": "haveI PID : IsPrincipalIdealRing R := aux_pid_of_ufd_of_unique_irreducible R h\u2081 h\u2082"
+ },
+ {
+ "annotated_tactic": [
+ "\"obtain \\u27e8p, hp\\u27e9 := h\\u2081\"",
+ "[]"
+ ],
+ "state_after": "case intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n\u22a2 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "state_before": "R\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2081 : \u2203 p, Irreducible p\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\n\u22a2 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "tactic": "obtain \u27e8p, hp\u27e9 := h\u2081"
+ },
+ {
+ "annotated_tactic": [
+ "\"refine \\u27e8PID, \\u27e8Ideal.span<\\/a> {p}, \\u27e8?_, ?_\\u27e9, ?_\\u27e9\\u27e9\"",
+ "[{\"full_name\":\"Ideal.span\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[115,5],\"def_end_pos\":[115,9]}]"
+ ],
+ "state_after": "case intro.refine_1\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n\u22a2 span {p} \u2260 \u22a5\n\ncase intro.refine_2\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n\u22a2 (span {p}).IsPrime\n\ncase intro.refine_3\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n\u22a2 \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = span {p}",
+ "state_before": "case intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n\u22a2 IsPrincipalIdealRing R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 P.IsPrime",
+ "tactic": "refine \u27e8PID, \u27e8Ideal.span {p}, \u27e8?_, ?_\u27e9, ?_\u27e9\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [Submodule.ne_bot_iff<\\/a>]\"",
+ "[{\"full_name\":\"Submodule.ne_bot_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Lattice.lean\",\"def_pos\":[97,19],\"def_end_pos\":[97,29]}]"
+ ],
+ "state_after": "case intro.refine_1\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n\u22a2 \u2203 x \u2208 span {p}, x \u2260 0",
+ "state_before": "case intro.refine_1\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n\u22a2 span {p} \u2260 \u22a5",
+ "tactic": "rw [Submodule.ne_bot_iff]"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact \\u27e8p, Ideal.mem_span_singleton.mpr (dvd_refl<\\/a> p), hp.ne_zero\\u27e9\"",
+ "[{\"full_name\":\"dvd_refl\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Basic.lean\",\"def_pos\":[137,9],\"def_end_pos\":[137,17]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case intro.refine_1\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n\u22a2 \u2203 x \u2208 span {p}, x \u2260 0",
+ "tactic": "exact \u27e8p, Ideal.mem_span_singleton.mpr (dvd_refl p), hp.ne_zero\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"rwa [Ideal.span_singleton_prime<\\/a> hp.ne_zero, \\u2190 UniqueFactorizationMonoid.irreducible_iff_prime<\\/a>]\"",
+ "[{\"full_name\":\"Ideal.span_singleton_prime\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[531,9],\"def_end_pos\":[531,29]},{\"full_name\":\"UniqueFactorizationMonoid.irreducible_iff_prime\",\"def_path\":\"Mathlib\\/RingTheory\\/UniqueFactorizationDomain.lean\",\"def_pos\":[194,13],\"def_end_pos\":[194,34]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case intro.refine_2\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n\u22a2 (span {p}).IsPrime",
+ "tactic": "rwa [Ideal.span_singleton_prime hp.ne_zero, \u2190 UniqueFactorizationMonoid.irreducible_iff_prime]"
+ },
+ {
+ "annotated_tactic": [
+ "\"intro I\"",
+ "[]"
+ ],
+ "state_after": "case intro.refine_3\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\n\u22a2 (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) I \u2192 I = span {p}",
+ "state_before": "case intro.refine_3\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n\u22a2 \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) y \u2192 y = span {p}",
+ "tactic": "intro I"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [\\u2190 Submodule.IsPrincipal.span_singleton_generator<\\/a> I]\"",
+ "[{\"full_name\":\"Submodule.IsPrincipal.span_singleton_generator\",\"def_path\":\"Mathlib\\/RingTheory\\/PrincipalIdealDomain.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,33]}]"
+ ],
+ "state_after": "case intro.refine_3\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\n\u22a2 (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) (Submodule.span R {Submodule.IsPrincipal.generator I}) \u2192\n Submodule.span R {Submodule.IsPrincipal.generator I} = span {p}",
+ "state_before": "case intro.refine_3\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\n\u22a2 (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) I \u2192 I = span {p}",
+ "tactic": "rw [\u2190 Submodule.IsPrincipal.span_singleton_generator I]"
+ },
+ {
+ "annotated_tactic": [
+ "\"rintro \\u27e8I0, hI\\u27e9\"",
+ "[]"
+ ],
+ "state_after": "case intro.refine_3.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\nI0 : Submodule.span R {Submodule.IsPrincipal.generator I} \u2260 \u22a5\nhI : IsPrime (Submodule.span R {Submodule.IsPrincipal.generator I})\n\u22a2 Submodule.span R {Submodule.IsPrincipal.generator I} = span {p}",
+ "state_before": "case intro.refine_3\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\n\u22a2 (fun P => P \u2260 \u22a5 \u2227 P.IsPrime) (Submodule.span R {Submodule.IsPrincipal.generator I}) \u2192\n Submodule.span R {Submodule.IsPrincipal.generator I} = span {p}",
+ "tactic": "rintro \u27e8I0, hI\u27e9"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply span_singleton_eq_span_singleton.mpr\"",
+ "[]"
+ ],
+ "state_after": "case intro.refine_3.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\nI0 : Submodule.span R {Submodule.IsPrincipal.generator I} \u2260 \u22a5\nhI : IsPrime (Submodule.span R {Submodule.IsPrincipal.generator I})\n\u22a2 Associated (Submodule.IsPrincipal.generator I) p",
+ "state_before": "case intro.refine_3.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\nI0 : Submodule.span R {Submodule.IsPrincipal.generator I} \u2260 \u22a5\nhI : IsPrime (Submodule.span R {Submodule.IsPrincipal.generator I})\n\u22a2 Submodule.span R {Submodule.IsPrincipal.generator I} = span {p}",
+ "tactic": "apply span_singleton_eq_span_singleton.mpr"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply h\\u2082 _ hp\"",
+ "[]"
+ ],
+ "state_after": "R\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\nI0 : Submodule.span R {Submodule.IsPrincipal.generator I} \u2260 \u22a5\nhI : IsPrime (Submodule.span R {Submodule.IsPrincipal.generator I})\n\u22a2 Irreducible (Submodule.IsPrincipal.generator I)",
+ "state_before": "case intro.refine_3.intro\nR\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\nI0 : Submodule.span R {Submodule.IsPrincipal.generator I} \u2260 \u22a5\nhI : IsPrime (Submodule.span R {Submodule.IsPrincipal.generator I})\n\u22a2 Associated (Submodule.IsPrincipal.generator I) p",
+ "tactic": "apply h\u2082 _ hp"
+ },
+ {
+ "annotated_tactic": [
+ "\"erw [Ne<\\/a>, span_singleton_eq_bot<\\/a>] at I0\"",
+ "[{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"Ideal.span_singleton_eq_bot\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[199,9],\"def_end_pos\":[199,30]}]"
+ ],
+ "state_after": "R\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\nI0 : \u00acSubmodule.IsPrincipal.generator I = 0\nhI : IsPrime (Submodule.span R {Submodule.IsPrincipal.generator I})\n\u22a2 Irreducible (Submodule.IsPrincipal.generator I)",
+ "state_before": "R\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\nI0 : Submodule.span R {Submodule.IsPrincipal.generator I} \u2260 \u22a5\nhI : IsPrime (Submodule.span R {Submodule.IsPrincipal.generator I})\n\u22a2 Irreducible (Submodule.IsPrincipal.generator I)",
+ "tactic": "erw [Ne, span_singleton_eq_bot] at I0"
+ },
+ {
+ "annotated_tactic": [
+ "\"rwa [UniqueFactorizationMonoid.irreducible_iff_prime<\\/a>, \\u2190 Ideal.span_singleton_prime<\\/a> I0]\"",
+ "[{\"full_name\":\"UniqueFactorizationMonoid.irreducible_iff_prime\",\"def_path\":\"Mathlib\\/RingTheory\\/UniqueFactorizationDomain.lean\",\"def_pos\":[194,13],\"def_end_pos\":[194,34]},{\"full_name\":\"Ideal.span_singleton_prime\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[531,9],\"def_end_pos\":[531,29]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "R\u271d : Type u_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : UniqueFactorizationMonoid R\nh\u2082 : \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\nI0 : \u00acSubmodule.IsPrincipal.generator I = 0\nhI : IsPrime (Submodule.span R {Submodule.IsPrincipal.generator I})\n\u22a2 Irreducible (Submodule.IsPrincipal.generator I)",
+ "tactic": "rwa [UniqueFactorizationMonoid.irreducible_iff_prime, \u2190 Ideal.span_singleton_prime I0]"
+ }
+ ],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ }
+ },
+ "theorem_name": "DiscreteValuationRing.of_ufd_of_unique_irreducible"
+ },
+ {
+ "errors": [
+ "internal transition boundary mismatch",
+ "internal transition boundary mismatch"
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "proof_id": "mathlib:DiscreteValuationRing.unit_mul_pow_congr_unit",
+ "provenance": {
+ "selection": {
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+ "end": [
+ 388,
+ 46
+ ],
+ "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean",
+ "full_name": "DiscreteValuationRing.unit_mul_pow_congr_unit",
+ "start": [
+ 380,
+ 1
+ ],
+ "traced_tactics": [
+ {
+ "annotated_tactic": [
+ "\"obtain rfl : m = n := unit_mul_pow_congr_pow<\\/a> hirr hirr u v m n h\"",
+ "[{\"full_name\":\"DiscreteValuationRing.unit_mul_pow_congr_pow\",\"def_path\":\"Mathlib\\/RingTheory\\/DiscreteValuationRing\\/Basic.lean\",\"def_pos\":[364,9],\"def_end_pos\":[364,31]}]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u2191u * \u03d6 ^ m = \u2191v * \u03d6 ^ m\n\u22a2 u = v",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm n : \u2115\nh : \u2191u * \u03d6 ^ m = \u2191v * \u03d6 ^ n\n\u22a2 u = v",
+ "tactic": "obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [\\u2190 sub_eq_zero<\\/a>] at h\"",
+ "[{\"full_name\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]}]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u2191u * \u03d6 ^ m - \u2191v * \u03d6 ^ m = 0\n\u22a2 u = v",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u2191u * \u03d6 ^ m = \u2191v * \u03d6 ^ m\n\u22a2 u = v",
+ "tactic": "rw [\u2190 sub_eq_zero] at h"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [\\u2190 sub_mul<\\/a>, mul_eq_zero<\\/a>] at h\"",
+ "[{\"full_name\":\"sub_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[401,7],\"def_end_pos\":[401,14]},{\"full_name\":\"mul_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[282,9],\"def_end_pos\":[282,20]}]"
+ ],
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u2191u - \u2191v = 0 \u2228 \u03d6 ^ m = 0\n\u22a2 u = v",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u2191u * \u03d6 ^ m - \u2191v * \u03d6 ^ m = 0\n\u22a2 u = v",
+ "tactic": "rw [\u2190 sub_mul, mul_eq_zero] at h"
+ },
+ {
+ "annotated_tactic": [
+ "\"cases' h with h h\"",
+ "[]"
+ ],
+ "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u2191u - \u2191v = 0\n\u22a2 u = v\n\ncase inr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u03d6 ^ m = 0\n\u22a2 u = v",
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u2191u - \u2191v = 0 \u2228 \u03d6 ^ m = 0\n\u22a2 u = v",
+ "tactic": "cases' h with h h"
+ },
+ {
+ "annotated_tactic": [
+ "\"rw [sub_eq_zero<\\/a>] at h\"",
+ "[{\"full_name\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]}]"
+ ],
+ "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u2191u = \u2191v\n\u22a2 u = v",
+ "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u2191u - \u2191v = 0\n\u22a2 u = v",
+ "tactic": "rw [sub_eq_zero] at h"
+ },
+ {
+ "annotated_tactic": [
+ "\"exact mod_cast h\"",
+ "[]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u2191u = \u2191v\n\u22a2 u = v",
+ "tactic": "exact mod_cast h"
+ },
+ {
+ "annotated_tactic": [
+ "\"apply (hirr.ne_zero (pow_eq_zero<\\/a> h)).elim<\\/a>\"",
+ "[{\"full_name\":\"pow_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[186,7],\"def_end_pos\":[186,18]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+ ],
+ "state_after": "no goals",
+ "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DiscreteValuationRing R\n\u03d6 : R\nhirr : Irreducible \u03d6\nu v : R\u02e3\nm : \u2115\nh : \u03d6 ^ m = 0\n\u22a2 u = v",
+ "tactic": "apply (hirr.ne_zero (pow_eq_zero h)).elim"
+ }
+ ],
+ "url": "https://github.com/leanprover-community/mathlib4"
+ }
+ },
+ "theorem_name": "DiscreteValuationRing.unit_mul_pow_congr_unit"
+ }
+ ],
+ "validation_rejected_proofs": 3,
+ "zero_transition_proofs": 0
+ },
+ "shards": [
+ {
+ "path": "data/train/mathlib-001663-0000-530fafc348ba7ba4.parquet",
+ "sha256": "530fafc348ba7ba42bbd99129cdb6cffe510cb51535d2b4401988e83c4fcd7c6",
+ "bytes": 168304,
+ "rows": 46,
+ "revision": "df8c9c479578582ca5384c5c62608ea4677707ca"
+ },
+ {
+ "path": "data/internal_test/mathlib-001663-0000-7cd5a5ab464059c3.parquet",
+ "sha256": "7cd5a5ab464059c393f7fcfb1ad02d459acc64ea33589d6daa7558443c575fca",
+ "bytes": 55267,
+ "rows": 2,
+ "revision": "df8c9c479578582ca5384c5c62608ea4677707ca"
+ }
+ ],
+ "proof_ids": [
+ "mathlib:DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible",
+ "mathlib:DiscreteValuationRing.addVal_def",
+ "mathlib:DiscreteValuationRing.addVal_uniformizer",
+ "mathlib:DiscreteValuationRing.associated_of_irreducible",
+ "mathlib:DiscreteValuationRing.associated_pow_irreducible",
+ "mathlib:DiscreteValuationRing.eq_unit_mul_pow_irreducible",
+ "mathlib:DiscreteValuationRing.exists_irreducible",
+ "mathlib:DiscreteValuationRing.irreducible_iff_uniformizer",
+ "mathlib:DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization",
+ "mathlib:Irreducible.addVal_pow",
+ "mathlib:liftOfDerivationToSquareZero_mk_apply",
+ "mathlib:liftOfDerivationToSquareZero_mk_apply'"
+ ],
+ "proof_keys": [
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Derivation/ToSquareZero.lean\",null,null,\"liftOfDerivationToSquareZero_mk_apply\"]",
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Derivation/ToSquareZero.lean\",null,null,\"liftOfDerivationToSquareZero_mk_apply'\"]",
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean\",null,null,\"DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible\"]",
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean\",null,null,\"DiscreteValuationRing.addVal_def\"]",
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean\",null,null,\"DiscreteValuationRing.addVal_uniformizer\"]",
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean\",null,null,\"DiscreteValuationRing.associated_of_irreducible\"]",
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean\",null,null,\"DiscreteValuationRing.associated_pow_irreducible\"]",
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean\",null,null,\"DiscreteValuationRing.eq_unit_mul_pow_irreducible\"]",
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean\",null,null,\"DiscreteValuationRing.exists_irreducible\"]",
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean\",null,null,\"DiscreteValuationRing.irreducible_iff_uniformizer\"]",
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean\",null,null,\"DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization\"]",
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DiscreteValuationRing/Basic.lean\",null,null,\"Irreducible.addVal_pow\"]"
+ ]
+}
\ No newline at end of file