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 F n d q z. ~(d=0) -> n=d*q -> (exists dst_positive_code_keep_input dst_positive_scale_keep_input dst_negative_code_keep_input dst_negative_scale_keep_input dst_positive_keep_input dst_negative_keep_input. (((F) = (((((dst_positive_code_keep_input) + (dst_positive_scale_keep_input)) * S ((dst_positive_code_keep_input) + (dst_positive_scale_keep_input)) + ((dst_positive_scale_keep_input) + (dst_positive_scale_keep_input))) + (((dst_negative_code_keep_input) + (dst_negative_scale_keep_input)) * S ((dst_negative_code_keep_input) + (dst_negative_scale_keep_input)) + ((dst_negative_scale_keep_input) + (dst_negative_scale_keep_input)))) * S ((((dst_positive_code_keep_input) + (dst_positive_scale_keep_input)) * S ((dst_positive_code_keep_input) + (dst_positive_scale_keep_input)) + ((dst_positive_scale_keep_input) + (dst_positive_scale_keep_input))) + (((dst_negative_code_keep_input) + (dst_negative_scale_keep_input)) * S ((dst_negative_code_keep_input) + (dst_negative_scale_keep_input)) + ((dst_negative_scale_keep_input) + (dst_negative_scale_keep_input)))) + ((((dst_negative_code_keep_input) + (dst_negative_scale_keep_input)) * S ((dst_negative_code_keep_input) + (dst_negative_scale_keep_input)) + ((dst_negative_scale_keep_input) + (dst_negative_scale_keep_input))) + (((dst_negative_code_keep_input) + (dst_negative_scale_keep_input)) * S ((dst_negative_code_keep_input) + (dst_negative_scale_keep_input)) + ((dst_negative_scale_keep_input) + (dst_negative_scale_keep_input)))))) /\ (((((exists ff_h_pvs_keep_inputpositive. ff_h_pvs_keep_inputpositive + S (dst_positive_keep_input) = S ((S (d)) * dst_positive_scale_keep_input)) /\ exists ff_q_pvs_keep_inputpositive. dst_positive_code_keep_input = ff_q_pvs_keep_inputpositive * S ((S (d)) * dst_positive_scale_keep_input) + (dst_positive_keep_input))) /\ (((((exists ff_h_pvs_keep_inputnegative. ff_h_pvs_keep_inputnegative + S (dst_negative_keep_input) = S ((S (d)) * dst_negative_scale_keep_input)) /\ exists ff_q_pvs_keep_inputnegative. dst_negative_code_keep_input = ff_q_pvs_keep_inputnegative * S ((S (d)) * dst_negative_scale_keep_input) + (dst_negative_keep_input))) /\ (exists ge_balance_positive_keep_inputvalue ge_balance_negative_keep_inputvalue. (((((z) = 2 * (ge_balance_positive_keep_inputvalue) /\ (ge_balance_negative_keep_inputvalue) = 0) \/ exists ge_signed_half_keep_inputvaluedecode. (((z) = 2 * ge_signed_half_keep_inputvaluedecode + 1 /\ (ge_balance_positive_keep_inputvalue) = 0) /\ (ge_balance_negative_keep_inputvalue) = S ge_signed_half_keep_inputvaluedecode))) /\ ((dst_positive_keep_input) + ge_balance_negative_keep_inputvalue = (dst_negative_keep_input) + ge_balance_positive_keep_inputvalue))))))))) -> ((((~((d)=0)) /\ (exists dm_quotient_keep_result. (((n)=(d)*dm_quotient_keep_result) /\ (exists dst_positive_code_keep_resultinput dst_positive_scale_keep_resultinput dst_negative_code_keep_resultinput dst_negative_scale_keep_resultinput dst_positive_keep_resultinput dst_negative_keep_resultinput. (((F) = (((((dst_positive_code_keep_resultinput) + (dst_positive_scale_keep_resultinput)) * S ((dst_positive_code_keep_resultinput) + (dst_positive_scale_keep_resultinput)) + ((dst_positive_scale_keep_resultinput) + (dst_positive_scale_keep_resultinput))) + (((dst_negative_code_keep_resultinput) + (dst_negative_scale_keep_resultinput)) * S ((dst_negative_code_keep_resultinput) + (dst_negative_scale_keep_resultinput)) + ((dst_negative_scale_keep_resultinput) + (dst_negative_scale_keep_resultinput)))) * S ((((dst_positive_code_keep_resultinput) + (dst_positive_scale_keep_resultinput)) * S ((dst_positive_code_keep_resultinput) + (dst_positive_scale_keep_resultinput)) + ((dst_positive_scale_keep_resultinput) + (dst_positive_scale_keep_resultinput))) + (((dst_negative_code_keep_resultinput) + (dst_negative_scale_keep_resultinput)) * S ((dst_negative_code_keep_resultinput) + (dst_negative_scale_keep_resultinput)) + ((dst_negative_scale_keep_resultinput) + (dst_negative_scale_keep_resultinput)))) + ((((dst_negative_code_keep_resultinput) + (dst_negative_scale_keep_resultinput)) * S ((dst_negative_code_keep_resultinput) + (dst_negative_scale_keep_resultinput)) + ((dst_negative_scale_keep_resultinput) + (dst_negative_scale_keep_resultinput))) + (((dst_negative_code_keep_resultinput) + (dst_negative_scale_keep_resultinput)) * S ((dst_negative_code_keep_resultinput) + (dst_negative_scale_keep_resultinput)) + ((dst_negative_scale_keep_resultinput) + (dst_negative_scale_keep_resultinput)))))) /\ (((((exists ff_h_pvs_keep_resultinputpositive. ff_h_pvs_keep_resultinputpositive + S (dst_positive_keep_resultinput) = S ((S (d)) * dst_positive_scale_keep_resultinput)) /\ exists ff_q_pvs_keep_resultinputpositive. dst_positive_code_keep_resultinput = ff_q_pvs_keep_resultinputpositive * S ((S (d)) * dst_positive_scale_keep_resultinput) + (dst_positive_keep_resultinput))) /\ (((((exists ff_h_pvs_keep_resultinputnegative. ff_h_pvs_keep_resultinputnegative + S (dst_negative_keep_resultinput) = S ((S (d)) * dst_negative_scale_keep_resultinput)) /\ exists ff_q_pvs_keep_resultinputnegative. dst_negative_code_keep_resultinput = ff_q_pvs_keep_resultinputnegative * S ((S (d)) * dst_negative_scale_keep_resultinput) + (dst_negative_keep_resultinput))) /\ (exists ge_balance_positive_keep_resultinputvalue ge_balance_negative_keep_resultinputvalue. (((((z) = 2 * (ge_balance_positive_keep_resultinputvalue) /\ (ge_balance_negative_keep_resultinputvalue) = 0) \/ exists ge_signed_half_keep_resultinputvaluedecode. (((z) = 2 * ge_signed_half_keep_resultinputvaluedecode + 1 /\ (ge_balance_positive_keep_resultinputvalue) = 0) /\ (ge_balance_negative_keep_resultinputvalue) = S ge_signed_half_keep_resultinputvaluedecode))) /\ ((dst_positive_keep_resultinput) + ge_balance_negative_keep_resultinputvalue = (dst_negative_keep_resultinput) + ge_balance_positive_keep_resultinputvalue))))))))))))) \/ ((((d)=0 \/ ~(exists pvs_factor_keep_resultnondivisor. (n) = (d) * pvs_factor_keep_resultnondivisor)) /\ ((z)=0))))Constructive proof overview
Generated structural guide
A real quotient and a positive divisor justify retaining its actual signed input entry.
The unchanged tactic script uses 0 declared prerequisites and contains 15 exact native proof lines.
Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
none
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.
01Fix variables and assumptionsL1–8
02Separate the logical casesL9–10
03Use earlier factsL11–11
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L11
exact hd
04Construct an explicit witnessL12–12
Supply the displayed value, then prove that it has the required property.
- L12
exists q
05Separate the logical casesL13–13
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L13
split