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 original first-admission records.
Statement with defined notation
∀ k. ∀ x. ∀ y. (∀ z. ∀ n. ∃ m. Pow(z,n,m)) → Pow(2,2 · k,x) → Pow(4,k,y) → x = yEvery purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.
Definitions used by this theorem
In the theorem statement
3 occurrences
In local proof propositions
2 occurrences
Exact expanded native-PA statement
forall k x y. (forall bpt_a_hj32_two_double bpt_e_hj32_two_double. exists bpt_x_hj32_two_double. (exists ff_b_bpt_value_hj32_two_double ff_c_bpt_value_hj32_two_double. ((forall ff_i_bpt_value_hj32_two_double_repeat. (exists ff_lt_bpt_value_hj32_two_double_repeat_bound. ff_lt_bpt_value_hj32_two_double_repeat_bound + S ff_i_bpt_value_hj32_two_double_repeat = bpt_e_hj32_two_double) -> (((exists ff_h_bpt_value_hj32_two_double_repeat_decoded. ff_h_bpt_value_hj32_two_double_repeat_decoded + S (bpt_a_hj32_two_double) = S ((S (ff_i_bpt_value_hj32_two_double_repeat)) * ff_c_bpt_value_hj32_two_double)) /\ exists ff_q_bpt_value_hj32_two_double_repeat_decoded. ff_b_bpt_value_hj32_two_double = ff_q_bpt_value_hj32_two_double_repeat_decoded * S ((S (ff_i_bpt_value_hj32_two_double_repeat)) * ff_c_bpt_value_hj32_two_double) + (bpt_a_hj32_two_double)))) /\ (exists ff_u_bpt_value_hj32_two_double_product ff_v_bpt_value_hj32_two_double_product. ((((exists ff_h_bpt_value_hj32_two_double_product_start. ff_h_bpt_value_hj32_two_double_product_start + S (1) = S ((S (0)) * ff_v_bpt_value_hj32_two_double_product)) /\ exists ff_q_bpt_value_hj32_two_double_product_start. ff_u_bpt_value_hj32_two_double_product = ff_q_bpt_value_hj32_two_double_product_start * S ((S (0)) * ff_v_bpt_value_hj32_two_double_product) + (1))) /\ ((((exists ff_h_bpt_value_hj32_two_double_product_terminal. ff_h_bpt_value_hj32_two_double_product_terminal + S (bpt_x_hj32_two_double) = S ((S (bpt_e_hj32_two_double)) * ff_v_bpt_value_hj32_two_double_product)) /\ exists ff_q_bpt_value_hj32_two_double_product_terminal. ff_u_bpt_value_hj32_two_double_product = ff_q_bpt_value_hj32_two_double_product_terminal * S ((S (bpt_e_hj32_two_double)) * ff_v_bpt_value_hj32_two_double_product) + (bpt_x_hj32_two_double))) /\ forall ff_i_bpt_value_hj32_two_double_product. (exists ff_lt_bpt_value_hj32_two_double_product_bound. ff_lt_bpt_value_hj32_two_double_product_bound + S ff_i_bpt_value_hj32_two_double_product = bpt_e_hj32_two_double) -> exists ff_p_bpt_value_hj32_two_double_product ff_r_bpt_value_hj32_two_double_product ff_s_bpt_value_hj32_two_double_product. ((((exists ff_h_bpt_value_hj32_two_double_product_factor. ff_h_bpt_value_hj32_two_double_product_factor + S (ff_p_bpt_value_hj32_two_double_product) = S ((S (ff_i_bpt_value_hj32_two_double_product)) * ff_c_bpt_value_hj32_two_double)) /\ exists ff_q_bpt_value_hj32_two_double_product_factor. ff_b_bpt_value_hj32_two_double = ff_q_bpt_value_hj32_two_double_product_factor * S ((S (ff_i_bpt_value_hj32_two_double_product)) * ff_c_bpt_value_hj32_two_double) + (ff_p_bpt_value_hj32_two_double_product))) /\ ((((exists ff_h_bpt_value_hj32_two_double_product_partial. ff_h_bpt_value_hj32_two_double_product_partial + S (ff_r_bpt_value_hj32_two_double_product) = S ((S (ff_i_bpt_value_hj32_two_double_product)) * ff_v_bpt_value_hj32_two_double_product)) /\ exists ff_q_bpt_value_hj32_two_double_product_partial. ff_u_bpt_value_hj32_two_double_product = ff_q_bpt_value_hj32_two_double_product_partial * S ((S (ff_i_bpt_value_hj32_two_double_product)) * ff_v_bpt_value_hj32_two_double_product) + (ff_r_bpt_value_hj32_two_double_product))) /\ ((((exists ff_h_bpt_value_hj32_two_double_product_successor. ff_h_bpt_value_hj32_two_double_product_successor + S (ff_s_bpt_value_hj32_two_double_product) = S ((S (S ff_i_bpt_value_hj32_two_double_product)) * ff_v_bpt_value_hj32_two_double_product)) /\ exists ff_q_bpt_value_hj32_two_double_product_successor. ff_u_bpt_value_hj32_two_double_product = ff_q_bpt_value_hj32_two_double_product_successor * S ((S (S ff_i_bpt_value_hj32_two_double_product)) * ff_v_bpt_value_hj32_two_double_product) + (ff_s_bpt_value_hj32_two_double_product))) /\ ff_s_bpt_value_hj32_two_double_product = ff_r_bpt_value_hj32_two_double_product * ff_p_bpt_value_hj32_two_double_product))))))))) -> (exists pa_b_hj32_two_double_left pa_c_hj32_two_double_left. ((forall pa_i_hj32_two_double_left_repeat. (exists pa_lt_hj32_two_double_left_repeat_bound. pa_lt_hj32_two_double_left_repeat_bound + S pa_i_hj32_two_double_left_repeat = 2 * k) -> (((exists pa_h_hj32_two_double_left_repeat_decoded. pa_h_hj32_two_double_left_repeat_decoded + S (2) = S ((S (pa_i_hj32_two_double_left_repeat)) * pa_c_hj32_two_double_left)) /\ exists pa_q_hj32_two_double_left_repeat_decoded. pa_b_hj32_two_double_left = pa_q_hj32_two_double_left_repeat_decoded * S ((S (pa_i_hj32_two_double_left_repeat)) * pa_c_hj32_two_double_left) + (2)))) /\ (exists pa_u_hj32_two_double_left_product pa_v_hj32_two_double_left_product. ((((exists pa_h_hj32_two_double_left_product_start. pa_h_hj32_two_double_left_product_start + S (1) = S ((S (0)) * pa_v_hj32_two_double_left_product)) /\ exists pa_q_hj32_two_double_left_product_start. pa_u_hj32_two_double_left_product = pa_q_hj32_two_double_left_product_start * S ((S (0)) * pa_v_hj32_two_double_left_product) + (1))) /\ ((((exists pa_h_hj32_two_double_left_product_terminal. pa_h_hj32_two_double_left_product_terminal + S (x) = S ((S (2 * k)) * pa_v_hj32_two_double_left_product)) /\ exists pa_q_hj32_two_double_left_product_terminal. pa_u_hj32_two_double_left_product = pa_q_hj32_two_double_left_product_terminal * S ((S (2 * k)) * pa_v_hj32_two_double_left_product) + (x))) /\ forall pa_i_hj32_two_double_left_product. (exists pa_lt_hj32_two_double_left_product_bound. pa_lt_hj32_two_double_left_product_bound + S pa_i_hj32_two_double_left_product = 2 * k) -> exists pa_p_hj32_two_double_left_product pa_r_hj32_two_double_left_product pa_s_hj32_two_double_left_product. ((((exists pa_h_hj32_two_double_left_product_factor. pa_h_hj32_two_double_left_product_factor + S (pa_p_hj32_two_double_left_product) = S ((S (pa_i_hj32_two_double_left_product)) * pa_c_hj32_two_double_left)) /\ exists pa_q_hj32_two_double_left_product_factor. pa_b_hj32_two_double_left = pa_q_hj32_two_double_left_product_factor * S ((S (pa_i_hj32_two_double_left_product)) * pa_c_hj32_two_double_left) + (pa_p_hj32_two_double_left_product))) /\ ((((exists pa_h_hj32_two_double_left_product_partial. pa_h_hj32_two_double_left_product_partial + S (pa_r_hj32_two_double_left_product) = S ((S (pa_i_hj32_two_double_left_product)) * pa_v_hj32_two_double_left_product)) /\ exists pa_q_hj32_two_double_left_product_partial. pa_u_hj32_two_double_left_product = pa_q_hj32_two_double_left_product_partial * S ((S (pa_i_hj32_two_double_left_product)) * pa_v_hj32_two_double_left_product) + (pa_r_hj32_two_double_left_product))) /\ ((((exists pa_h_hj32_two_double_left_product_successor. pa_h_hj32_two_double_left_product_successor + S (pa_s_hj32_two_double_left_product) = S ((S (S pa_i_hj32_two_double_left_product)) * pa_v_hj32_two_double_left_product)) /\ exists pa_q_hj32_two_double_left_product_successor. pa_u_hj32_two_double_left_product = pa_q_hj32_two_double_left_product_successor * S ((S (S pa_i_hj32_two_double_left_product)) * pa_v_hj32_two_double_left_product) + (pa_s_hj32_two_double_left_product))) /\ pa_s_hj32_two_double_left_product = pa_r_hj32_two_double_left_product * pa_p_hj32_two_double_left_product)))))))) -> (exists pa_b_hj32_two_double_right pa_c_hj32_two_double_right. ((forall pa_i_hj32_two_double_right_repeat. (exists pa_lt_hj32_two_double_right_repeat_bound. pa_lt_hj32_two_double_right_repeat_bound + S pa_i_hj32_two_double_right_repeat = k) -> (((exists pa_h_hj32_two_double_right_repeat_decoded. pa_h_hj32_two_double_right_repeat_decoded + S (4) = S ((S (pa_i_hj32_two_double_right_repeat)) * pa_c_hj32_two_double_right)) /\ exists pa_q_hj32_two_double_right_repeat_decoded. pa_b_hj32_two_double_right = pa_q_hj32_two_double_right_repeat_decoded * S ((S (pa_i_hj32_two_double_right_repeat)) * pa_c_hj32_two_double_right) + (4)))) /\ (exists pa_u_hj32_two_double_right_product pa_v_hj32_two_double_right_product. ((((exists pa_h_hj32_two_double_right_product_start. pa_h_hj32_two_double_right_product_start + S (1) = S ((S (0)) * pa_v_hj32_two_double_right_product)) /\ exists pa_q_hj32_two_double_right_product_start. pa_u_hj32_two_double_right_product = pa_q_hj32_two_double_right_product_start * S ((S (0)) * pa_v_hj32_two_double_right_product) + (1))) /\ ((((exists pa_h_hj32_two_double_right_product_terminal. pa_h_hj32_two_double_right_product_terminal + S (y) = S ((S (k)) * pa_v_hj32_two_double_right_product)) /\ exists pa_q_hj32_two_double_right_product_terminal. pa_u_hj32_two_double_right_product = pa_q_hj32_two_double_right_product_terminal * S ((S (k)) * pa_v_hj32_two_double_right_product) + (y))) /\ forall pa_i_hj32_two_double_right_product. (exists pa_lt_hj32_two_double_right_product_bound. pa_lt_hj32_two_double_right_product_bound + S pa_i_hj32_two_double_right_product = k) -> exists pa_p_hj32_two_double_right_product pa_r_hj32_two_double_right_product pa_s_hj32_two_double_right_product. ((((exists pa_h_hj32_two_double_right_product_factor. pa_h_hj32_two_double_right_product_factor + S (pa_p_hj32_two_double_right_product) = S ((S (pa_i_hj32_two_double_right_product)) * pa_c_hj32_two_double_right)) /\ exists pa_q_hj32_two_double_right_product_factor. pa_b_hj32_two_double_right = pa_q_hj32_two_double_right_product_factor * S ((S (pa_i_hj32_two_double_right_product)) * pa_c_hj32_two_double_right) + (pa_p_hj32_two_double_right_product))) /\ ((((exists pa_h_hj32_two_double_right_product_partial. pa_h_hj32_two_double_right_product_partial + S (pa_r_hj32_two_double_right_product) = S ((S (pa_i_hj32_two_double_right_product)) * pa_v_hj32_two_double_right_product)) /\ exists pa_q_hj32_two_double_right_product_partial. pa_u_hj32_two_double_right_product = pa_q_hj32_two_double_right_product_partial * S ((S (pa_i_hj32_two_double_right_product)) * pa_v_hj32_two_double_right_product) + (pa_r_hj32_two_double_right_product))) /\ ((((exists pa_h_hj32_two_double_right_product_successor. pa_h_hj32_two_double_right_product_successor + S (pa_s_hj32_two_double_right_product) = S ((S (S pa_i_hj32_two_double_right_product)) * pa_v_hj32_two_double_right_product)) /\ exists pa_q_hj32_two_double_right_product_successor. pa_u_hj32_two_double_right_product = pa_q_hj32_two_double_right_product_successor * S ((S (S pa_i_hj32_two_double_right_product)) * pa_v_hj32_two_double_right_product) + (pa_s_hj32_two_double_right_product))) /\ pa_s_hj32_two_double_right_product = pa_r_hj32_two_double_right_product * pa_p_hj32_two_double_right_product)))))))) -> x = yProof neighborhood
Direct theorem prerequisites
Direct theorem dependents
Definition-aware tactic body
Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.
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 assumptionsL1–6
02Establish td_seedsL7–9
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow two seed bundle from total.
- L7
have td_seeds : Pow(2,2,4) ∧ Pow(2,7,128)Definitions: Pow(2,2,4)Pow(2,7,128)Original native command in the exact edition - L8
apply pow_two_seed_bundle_from_total - L9
exact htotal
03Separate the logical casesL10–10
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L10
cases td_seeds
04Establish td_bridgeL11–20
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow mul exp from total.
- L11
have td_bridge : y = x - L12
specialize pow_mul_exp_from_total 2 - L13
specialize pow_mul_exp_from_total 2 - L14
specialize pow_mul_exp_from_total k - L15
specialize pow_mul_exp_from_total (2 * k) - L16
specialize pow_mul_exp_from_total 4 - L17
specialize pow_mul_exp_from_total y - L18
specialize pow_mul_exp_from_total x - L19
apply pow_mul_exp_from_total - L20
exact htotal
05Calculate and transport equalitiesL21–21
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L21
refl
06Use earlier factsL22–24
07Calculate and transport equalitiesL25–25
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L25
symm
08Use earlier factsL26–26
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L26
exact td_bridge
Original defined command ledger · 26 lines
- 0001
intro k - 0002
intro x - 0003
intro y - 0004
intro htotal - 0005
intro hx - 0006
intro hy - 0007
have td_seeds : Pow(2,2,4) ∧ Pow(2,7,128)Exact native replay line
have td_seeds : (exists pa_b_hj32_seed_two_two pa_c_hj32_seed_two_two. ((forall pa_i_hj32_seed_two_two_repeat. (exists pa_lt_hj32_seed_two_two_repeat_bound. pa_lt_hj32_seed_two_two_repeat_bound + S pa_i_hj32_seed_two_two_repeat = 2) -> (((exists pa_h_hj32_seed_two_two_repeat_decoded. pa_h_hj32_seed_two_two_repeat_decoded + S (2) = S ((S (pa_i_hj32_seed_two_two_repeat)) * pa_c_hj32_seed_two_two)) /\ exists pa_q_hj32_seed_two_two_repeat_decoded. pa_b_hj32_seed_two_two = pa_q_hj32_seed_two_two_repeat_decoded * S ((S (pa_i_hj32_seed_two_two_repeat)) * pa_c_hj32_seed_two_two) + (2)))) /\ (exists pa_u_hj32_seed_two_two_product pa_v_hj32_seed_two_two_product. ((((exists pa_h_hj32_seed_two_two_product_start. pa_h_hj32_seed_two_two_product_start + S (1) = S ((S (0)) * pa_v_hj32_seed_two_two_product)) /\ exists pa_q_hj32_seed_two_two_product_start. pa_u_hj32_seed_two_two_product = pa_q_hj32_seed_two_two_product_start * S ((S (0)) * pa_v_hj32_seed_two_two_product) + (1))) /\ ((((exists pa_h_hj32_seed_two_two_product_terminal. pa_h_hj32_seed_two_two_product_terminal + S (4) = S ((S (2)) * pa_v_hj32_seed_two_two_product)) /\ exists pa_q_hj32_seed_two_two_product_terminal. pa_u_hj32_seed_two_two_product = pa_q_hj32_seed_two_two_product_terminal * S ((S (2)) * pa_v_hj32_seed_two_two_product) + (4))) /\ forall pa_i_hj32_seed_two_two_product. (exists pa_lt_hj32_seed_two_two_product_bound. pa_lt_hj32_seed_two_two_product_bound + S pa_i_hj32_seed_two_two_product = 2) -> exists pa_p_hj32_seed_two_two_product pa_r_hj32_seed_two_two_product pa_s_hj32_seed_two_two_product. ((((exists pa_h_hj32_seed_two_two_product_factor. pa_h_hj32_seed_two_two_product_factor + S (pa_p_hj32_seed_two_two_product) = S ((S (pa_i_hj32_seed_two_two_product)) * pa_c_hj32_seed_two_two)) /\ exists pa_q_hj32_seed_two_two_product_factor. pa_b_hj32_seed_two_two = pa_q_hj32_seed_two_two_product_factor * S ((S (pa_i_hj32_seed_two_two_product)) * pa_c_hj32_seed_two_two) + (pa_p_hj32_seed_two_two_product))) /\ ((((exists pa_h_hj32_seed_two_two_product_partial. pa_h_hj32_seed_two_two_product_partial + S (pa_r_hj32_seed_two_two_product) = S ((S (pa_i_hj32_seed_two_two_product)) * pa_v_hj32_seed_two_two_product)) /\ exists pa_q_hj32_seed_two_two_product_partial. pa_u_hj32_seed_two_two_product = pa_q_hj32_seed_two_two_product_partial * S ((S (pa_i_hj32_seed_two_two_product)) * pa_v_hj32_seed_two_two_product) + (pa_r_hj32_seed_two_two_product))) /\ ((((exists pa_h_hj32_seed_two_two_product_successor. pa_h_hj32_seed_two_two_product_successor + S (pa_s_hj32_seed_two_two_product) = S ((S (S pa_i_hj32_seed_two_two_product)) * pa_v_hj32_seed_two_two_product)) /\ exists pa_q_hj32_seed_two_two_product_successor. pa_u_hj32_seed_two_two_product = pa_q_hj32_seed_two_two_product_successor * S ((S (S pa_i_hj32_seed_two_two_product)) * pa_v_hj32_seed_two_two_product) + (pa_s_hj32_seed_two_two_product))) /\ pa_s_hj32_seed_two_two_product = pa_r_hj32_seed_two_two_product * pa_p_hj32_seed_two_two_product)))))))) /\ (exists pa_b_hj32_seed_two_seven pa_c_hj32_seed_two_seven. ((forall pa_i_hj32_seed_two_seven_repeat. (exists pa_lt_hj32_seed_two_seven_repeat_bound. pa_lt_hj32_seed_two_seven_repeat_bound + S pa_i_hj32_seed_two_seven_repeat = 7) -> (((exists pa_h_hj32_seed_two_seven_repeat_decoded. pa_h_hj32_seed_two_seven_repeat_decoded + S (2) = S ((S (pa_i_hj32_seed_two_seven_repeat)) * pa_c_hj32_seed_two_seven)) /\ exists pa_q_hj32_seed_two_seven_repeat_decoded. pa_b_hj32_seed_two_seven = pa_q_hj32_seed_two_seven_repeat_decoded * S ((S (pa_i_hj32_seed_two_seven_repeat)) * pa_c_hj32_seed_two_seven) + (2)))) /\ (exists pa_u_hj32_seed_two_seven_product pa_v_hj32_seed_two_seven_product. ((((exists pa_h_hj32_seed_two_seven_product_start. pa_h_hj32_seed_two_seven_product_start + S (1) = S ((S (0)) * pa_v_hj32_seed_two_seven_product)) /\ exists pa_q_hj32_seed_two_seven_product_start. pa_u_hj32_seed_two_seven_product = pa_q_hj32_seed_two_seven_product_start * S ((S (0)) * pa_v_hj32_seed_two_seven_product) + (1))) /\ ((((exists pa_h_hj32_seed_two_seven_product_terminal. pa_h_hj32_seed_two_seven_product_terminal + S (128) = S ((S (7)) * pa_v_hj32_seed_two_seven_product)) /\ exists pa_q_hj32_seed_two_seven_product_terminal. pa_u_hj32_seed_two_seven_product = pa_q_hj32_seed_two_seven_product_terminal * S ((S (7)) * pa_v_hj32_seed_two_seven_product) + (128))) /\ forall pa_i_hj32_seed_two_seven_product. (exists pa_lt_hj32_seed_two_seven_product_bound. pa_lt_hj32_seed_two_seven_product_bound + S pa_i_hj32_seed_two_seven_product = 7) -> exists pa_p_hj32_seed_two_seven_product pa_r_hj32_seed_two_seven_product pa_s_hj32_seed_two_seven_product. ((((exists pa_h_hj32_seed_two_seven_product_factor. pa_h_hj32_seed_two_seven_product_factor + S (pa_p_hj32_seed_two_seven_product) = S ((S (pa_i_hj32_seed_two_seven_product)) * pa_c_hj32_seed_two_seven)) /\ exists pa_q_hj32_seed_two_seven_product_factor. pa_b_hj32_seed_two_seven = pa_q_hj32_seed_two_seven_product_factor * S ((S (pa_i_hj32_seed_two_seven_product)) * pa_c_hj32_seed_two_seven) + (pa_p_hj32_seed_two_seven_product))) /\ ((((exists pa_h_hj32_seed_two_seven_product_partial. pa_h_hj32_seed_two_seven_product_partial + S (pa_r_hj32_seed_two_seven_product) = S ((S (pa_i_hj32_seed_two_seven_product)) * pa_v_hj32_seed_two_seven_product)) /\ exists pa_q_hj32_seed_two_seven_product_partial. pa_u_hj32_seed_two_seven_product = pa_q_hj32_seed_two_seven_product_partial * S ((S (pa_i_hj32_seed_two_seven_product)) * pa_v_hj32_seed_two_seven_product) + (pa_r_hj32_seed_two_seven_product))) /\ ((((exists pa_h_hj32_seed_two_seven_product_successor. pa_h_hj32_seed_two_seven_product_successor + S (pa_s_hj32_seed_two_seven_product) = S ((S (S pa_i_hj32_seed_two_seven_product)) * pa_v_hj32_seed_two_seven_product)) /\ exists pa_q_hj32_seed_two_seven_product_successor. pa_u_hj32_seed_two_seven_product = pa_q_hj32_seed_two_seven_product_successor * S ((S (S pa_i_hj32_seed_two_seven_product)) * pa_v_hj32_seed_two_seven_product) + (pa_s_hj32_seed_two_seven_product))) /\ pa_s_hj32_seed_two_seven_product = pa_r_hj32_seed_two_seven_product * pa_p_hj32_seed_two_seven_product)))))))) - 0008
apply pow_two_seed_bundle_from_total - 0009
exact htotal - 0010
cases td_seeds - 0011
have td_bridge : y = x - 0012
specialize pow_mul_exp_from_total 2 - 0013
specialize pow_mul_exp_from_total 2 - 0014
specialize pow_mul_exp_from_total k - 0015
specialize pow_mul_exp_from_total (2 * k) - 0016
specialize pow_mul_exp_from_total 4 - 0017
specialize pow_mul_exp_from_total y - 0018
specialize pow_mul_exp_from_total x - 0019
apply pow_mul_exp_from_total - 0020
exact htotal - 0021
refl - 0022
exact td_seeds_left - 0023
exact hy - 0024
exact hx - 0025
symm - 0026
exact td_bridge