Actual signed units and affine equations — Exact Proof Explorer

Classify every actual signed product equal to one and solve the affine equation needed for finite inversion.

9 theorem bodies · 36 proof edges · 401 tactic lines · 4 layers

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

9 theorems
0123
ZU0001 · dirichlet_signed_unit_self_product

Each of the two canonical signed units has an actual signed square equal to positive one.

layer 0 · 47 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
ZU0002 · dirichlet_signed_unit_product_classification

An actual signed product is positive one only for the two equal canonical units; mixed signs are constructively impossible.

layer 0 · 151 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
ZU0003 · dirichlet_signed_unit_inverse_iff

The finite signed-unit predicate is equivalent to existence of an actual signed multiplicative inverse, not defined by an inverse oracle.

layer 1 · 21 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
ZU0004 · dirichlet_signed_add_cancel_left

Cancellation of a common canonical signed summand follows by constructing its actual additive inverse.

layer 0 · 43 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
ZU0005 · dirichlet_signed_add_solve

Construct the signed addend taking any canonical signed r to any canonical signed e, including zero and negative values.

layer 0 · 26 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
ZU0007 · dirichlet_signed_unit_multiply_cancel_right

An actual signed unit can be cancelled from a common right factor without cancelling arbitrary zero or nonunit factors.

layer 2 · 24 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
ZU0008 · dirichlet_signed_unit_affine_solve

Given either actual signed unit, construct both x and its actual product y with r+y=e; no difference, product or solution witness is supplied.

layer 2 · 24 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
ZU0009 · dirichlet_signed_unit_affine_unique

Any two witnessed solutions of the same unit-affine equation have equal canonical inputs and equal actual product outputs.

layer 3 · 32 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Exactly 9 displayed theorems have independently verified Alpha checked-use authority; none is admitted to Stable. Body-only enrollment never grants checked theorem use.