feat(RingTheory): Every connected graded bialgebra is a Hopf algebra

[hawkrobe]

---

see #PR reviews > #31898 implement HopfAlgebra for TensorAlgebra @ 💬.

• depends on: [Merged by Bors] - refactor(RingTheory/HopfAlgebra): ofConvInverse constructor #39785

I used Claude Code to plan and audit.

[github-actions]

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[github-actions]

PR summary a99a90748b

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.Coalgebra.Graded (new file)	1078
Mathlib.RingTheory.GradedAlgebra.Connected (new file)	1091
Mathlib.RingTheory.Bialgebra.Graded (new file)	1137
Mathlib.RingTheory.HopfAlgebra.Graded (new file)	1167

---

Declarations diff (regex)

+ GradedAlgebra.IsConnected
+ GradedAlgebra.IsConnected.mem_zero_iff
+ GradedCoalgebra
+ apply_mem_of_mem_bigradedPart
+ comul_sub_tmul_one_mem_lower
+ convPow_apply_eq_zero_of_lt
+ counit_eq_counit_proj_zero
+ eq_counit_smul_one
+ id_mul_takeuchiT
+ lTensor_proj_zero_comul
+ neg_takeuchiF_pow_apply_of_mem
+ ofGradedConnected
+ proj_zero_eq_algebraMap_comp_counit
+ rTensor_proj_zero_comul
+ takeuchiAntipode
+ takeuchiAntipode_apply_eq_takeuchiT_of_le
+ takeuchiAntipode_apply_of_mem
+ takeuchiAntipode_lTensor_comul_eq
+ takeuchiAntipode_mul_lTensor_comul
+ takeuchiAntipode_mul_rTensor_comul
+ takeuchiAntipode_rTensor_comul_eq
+ takeuchiF
+ takeuchiT
+ takeuchiT_mul_id
+ toConv_id_eq
+ zeroLEquiv
+ zeroLEquiv_apply
+ zeroLEquiv_symm_apply_coe

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit a99a907).

• +33 new declarations
• −0 removed declarations

+Bialgebra.comul_sub_tmul_one_mem_lower
+Bialgebra.counit_eq_counit_proj_zero
+Bialgebra.lTensor_proj_zero_comul
+Bialgebra.proj_zero_eq_algebraMap_comp_counit
+Bialgebra.rTensor_proj_zero_comul
+GradedAlgebra.IsConnected
+GradedAlgebra.IsConnected.casesOn
+GradedAlgebra.IsConnected.eq_counit_smul_one
+GradedAlgebra.IsConnected.eq_span_one
+GradedAlgebra.IsConnected.mem_zero_iff
+GradedAlgebra.IsConnected.mk
+GradedAlgebra.IsConnected.rec
+GradedAlgebra.IsConnected.recOn
+GradedAlgebra.IsConnected.zeroLEquiv
+GradedAlgebra.IsConnected.zeroLEquiv_apply
+GradedAlgebra.IsConnected.zeroLEquiv_symm_apply_coe
+GradedCoalgebra
+GradedCoalgebra.casesOn
+GradedCoalgebra.comul_mem
+GradedCoalgebra.counit_eq_zero_of_ne_zero
+GradedCoalgebra.mk
+GradedCoalgebra.rec
+GradedCoalgebra.recOn
+HopfAlgebra.id_mul_takeuchiT
+HopfAlgebra.ofGradedConnected
+HopfAlgebra.takeuchiAntipode
+HopfAlgebra.takeuchiAntipode_apply_of_mem
+HopfAlgebra.takeuchiAntipode_mul_lTensor_comul
+HopfAlgebra.takeuchiAntipode_mul_rTensor_comul
+HopfAlgebra.takeuchiF
+HopfAlgebra.takeuchiT
+HopfAlgebra.takeuchiT_mul_id
+apply_mem_of_mem_bigradedPart

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit a99a90748b
Reference commit 7ffc1c27cb

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - refactor(RingTheory/HopfAlgebra): ofConvInverse constructor #39785
  By Dependent Issues (🤖). Happy coding!

[mathlib-bors]

This pull request is now in draft mode. No active bors state needed cleanup.

While this PR remains draft, bors will ignore commands on this PR. Mark it ready for review before using commands like bors r+ or bors try.