Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
Exact expanded first-order arithmetic statement
forall u. ((exists v. (exists sto_ap_inverse_forward sto_an_inverse_forward sto_bp_inverse_forward sto_bn_inverse_forward sto_cp_inverse_forward sto_cn_inverse_forward. (((((u) = 2 * (sto_ap_inverse_forward) /\ (sto_an_inverse_forward) = 0) \/ exists ge_signed_half_inverse_forwardleft. (((u) = 2 * ge_signed_half_inverse_forwardleft + 1 /\ (sto_ap_inverse_forward) = 0) /\ (sto_an_inverse_forward) = S ge_signed_half_inverse_forwardleft))) /\ ((((((v) = 2 * (sto_bp_inverse_forward) /\ (sto_bn_inverse_forward) = 0) \/ exists ge_signed_half_inverse_forwardright. (((v) = 2 * ge_signed_half_inverse_forwardright + 1 /\ (sto_bp_inverse_forward) = 0) /\ (sto_bn_inverse_forward) = S ge_signed_half_inverse_forwardright))) /\ ((((((2) = 2 * (sto_cp_inverse_forward) /\ (sto_cn_inverse_forward) = 0) \/ exists ge_signed_half_inverse_forwardoutput. (((2) = 2 * ge_signed_half_inverse_forwardoutput + 1 /\ (sto_cp_inverse_forward) = 0) /\ (sto_cn_inverse_forward) = S ge_signed_half_inverse_forwardoutput))) /\ ((sto_ap_inverse_forward * sto_bp_inverse_forward + sto_an_inverse_forward * sto_bn_inverse_forward) + sto_cn_inverse_forward = (sto_ap_inverse_forward * sto_bn_inverse_forward + sto_an_inverse_forward * sto_bp_inverse_forward) + sto_cp_inverse_forward)))))))) -> (((u) = 2 \/ (u) = 1))) /\ ((((u) = 2 \/ (u) = 1)) -> exists v. (exists sto_ap_inverse_backward sto_an_inverse_backward sto_bp_inverse_backward sto_bn_inverse_backward sto_cp_inverse_backward sto_cn_inverse_backward. (((((u) = 2 * (sto_ap_inverse_backward) /\ (sto_an_inverse_backward) = 0) \/ exists ge_signed_half_inverse_backwardleft. (((u) = 2 * ge_signed_half_inverse_backwardleft + 1 /\ (sto_ap_inverse_backward) = 0) /\ (sto_an_inverse_backward) = S ge_signed_half_inverse_backwardleft))) /\ ((((((v) = 2 * (sto_bp_inverse_backward) /\ (sto_bn_inverse_backward) = 0) \/ exists ge_signed_half_inverse_backwardright. (((v) = 2 * ge_signed_half_inverse_backwardright + 1 /\ (sto_bp_inverse_backward) = 0) /\ (sto_bn_inverse_backward) = S ge_signed_half_inverse_backwardright))) /\ ((((((2) = 2 * (sto_cp_inverse_backward) /\ (sto_cn_inverse_backward) = 0) \/ exists ge_signed_half_inverse_backwardoutput. (((2) = 2 * ge_signed_half_inverse_backwardoutput + 1 /\ (sto_cp_inverse_backward) = 0) /\ (sto_cn_inverse_backward) = S ge_signed_half_inverse_backwardoutput))) /\ ((sto_ap_inverse_backward * sto_bp_inverse_backward + sto_an_inverse_backward * sto_bn_inverse_backward) + sto_cn_inverse_backward = (sto_ap_inverse_backward * sto_bn_inverse_backward + sto_an_inverse_backward * sto_bp_inverse_backward) + sto_cp_inverse_backward))))))))Constructive proof overview
Generated structural guide
The finite signed-unit predicate is equivalent to existence of an actual signed multiplicative inverse, not defined by an inverse oracle.
The unchanged tactic script uses 2 declared prerequisites and contains 21 exact native proof lines.
Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
Direct dependents
Formal native tactic body
Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.
Read the argument
Proof checkpoints
This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.
Named ingredients (2)
01Fix variables and assumptionsL1–1
Work with arbitrary variables or the premises of the current implication.
- L1
intro u
02Separate the logical casesL2–2
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L2
split
03Fix variables and assumptionsL3–3
Work with arbitrary variables or the premises of the current implication.
- L3
intro h
04Separate the logical casesL4–4
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L4
cases h
05Establish hcL5–9
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply dirichlet signed unit product classification.
06Separate the logical casesL10–12
07Use earlier factsL13–13
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L13
exact hc_left_left
08Separate the logical casesL14–15
09Use earlier factsL16–16
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L16
exact hc_right_left
10Fix variables and assumptionsL17–17
Work with arbitrary variables or the premises of the current implication.
- L17
intro hu
11Construct an explicit witnessL18–18
Supply the displayed value, then prove that it has the required property.
- L18
exists u
Original exact command ledger · 21 lines
- 0001
intro u - 0002
split - 0003
intro h - 0004
cases h - 0005
have hc : (u=2 /\ x=2) \/ (u=1 /\ x=1) - 0006
specialize dirichlet_signed_unit_product_classification (u) - 0007
specialize dirichlet_signed_unit_product_classification (x) - 0008
apply dirichlet_signed_unit_product_classification - 0009
exact h_witness - 0010
cases hc - 0011
cases hc_left - 0012
left - 0013
exact hc_left_left - 0014
cases hc_right - 0015
right - 0016
exact hc_right_left - 0017
intro hu - 0018
exists u - 0019
specialize dirichlet_signed_unit_self_product (u) - 0020
apply dirichlet_signed_unit_self_product - 0021
exact hu