Actual signed units and affine equations — exact evidence

Current Alpha v34 checked use; first admitted v31; not Stable. Original HA and the independently compiled Lean checker have freshly verified the complete bundle before this page was generated.

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

71 checked bundle nodes · 146 proof edges · 4704 body proof nodes.

Literal self-contained proof bundle · SHA-256 5045f1feb2f21a79ecb3cb03f95aaefeb8f01e616a4aa8640cbada3da62ae47b

dirichlet_signed_unit_candidate.py

Unchanged historical local checkpoint record · Historical snapshot manifest. Those records retain their original non-admitting flags; current authority comes from the separate freshly verified v31 release.

Exact theorem nodes and inherited prerequisites

ZU0001 dirichlet_signed_unit_self_product · actual bundle node 61
ZU0002 dirichlet_signed_unit_product_classification · actual bundle node 62
ZU0003 dirichlet_signed_unit_inverse_iff · actual bundle node 63
ZU0004 dirichlet_signed_add_cancel_left · actual bundle node 64
ZU0005 dirichlet_signed_add_solve · actual bundle node 65
ZU0006 dirichlet_signed_unit_multiply_involution · actual bundle node 66
ZU0007 dirichlet_signed_unit_multiply_cancel_right · actual bundle node 67
ZU0008 dirichlet_signed_unit_affine_solve · actual bundle node 68
ZU0009 dirichlet_signed_unit_affine_unique · actual bundle node 69
mul_eq_one_components · checked inherited prerequisite
Exact statement in the checked dependency cone
```
forall a b. a * b = 1 -> a = 1 /\ b = 1
```
mul_one · checked inherited prerequisite
Exact statement in the checked dependency cone
```
forall n. n * 1 = n
```
mul_zero_left · checked inherited prerequisite
Exact statement in the checked dependency cone
```
forall n. 0 * n = 0
```

signed_add_associative · checked inherited prerequisite

Exact statement in the checked dependency cone

forall a b c ab bc abc. (exists sa_lp_assoc_ab sa_ln_assoc_ab sa_rp_assoc_ab sa_rn_assoc_ab sa_op_assoc_ab sa_on_assoc_ab. (((a = 2 * sa_lp_assoc_ab /\ sa_ln_assoc_ab = 0) \/ exists sd_half_assoc_ab_left. ((a = 2 * sd_half_assoc_ab_left + 1 /\ sa_lp_assoc_ab = 0) /\ sa_ln_assoc_ab = S sd_half_assoc_ab_left)) /\ (((b = 2 * sa_rp_assoc_ab /\ sa_rn_assoc_ab = 0) \/ exists sd_half_assoc_ab_right. ((b = 2 * sd_half_assoc_ab_right + 1 /\ sa_rp_assoc_ab = 0) /\ sa_rn_assoc_ab = S sd_half_assoc_ab_right)) /\ (((ab = 2 * sa_op_assoc_ab /\ sa_on_assoc_ab = 0) \/ exists sd_half_assoc_ab_output. ((ab = 2 * sd_half_assoc_ab_output + 1 /\ sa_op_assoc_ab = 0) /\ sa_on_assoc_ab = S sd_half_assoc_ab_output)) /\ (sa_lp_assoc_ab + sa_rp_assoc_ab) + sa_on_assoc_ab = (sa_ln_assoc_ab + sa_rn_assoc_ab) + sa_op_assoc_ab)))) -> (exists sa_lp_assoc_abc sa_ln_assoc_abc sa_rp_assoc_abc sa_rn_assoc_abc sa_op_assoc_abc sa_on_assoc_abc. (((ab = 2 * sa_lp_assoc_abc /\ sa_ln_assoc_abc = 0) \/ exists sd_half_assoc_abc_left. ((ab = 2 * sd_half_assoc_abc_left + 1 /\ sa_lp_assoc_abc = 0) /\ sa_ln_assoc_abc = S sd_half_assoc_abc_left)) /\ (((c = 2 * sa_rp_assoc_abc /\ sa_rn_assoc_abc = 0) \/ exists sd_half_assoc_abc_right. ((c = 2 * sd_half_assoc_abc_right + 1 /\ sa_rp_assoc_abc = 0) /\ sa_rn_assoc_abc = S sd_half_assoc_abc_right)) /\ (((abc = 2 * sa_op_assoc_abc /\ sa_on_assoc_abc = 0) \/ exists sd_half_assoc_abc_output. ((abc = 2 * sd_half_assoc_abc_output + 1 /\ sa_op_assoc_abc = 0) /\ sa_on_assoc_abc = S sd_half_assoc_abc_output)) /\ (sa_lp_assoc_abc + sa_rp_assoc_abc) + sa_on_assoc_abc = (sa_ln_assoc_abc + sa_rn_assoc_abc) + sa_op_assoc_abc)))) -> (exists sa_lp_assoc_bc sa_ln_assoc_bc sa_rp_assoc_bc sa_rn_assoc_bc sa_op_assoc_bc sa_on_assoc_bc. (((b = 2 * sa_lp_assoc_bc /\ sa_ln_assoc_bc = 0) \/ exists sd_half_assoc_bc_left. ((b = 2 * sd_half_assoc_bc_left + 1 /\ sa_lp_assoc_bc = 0) /\ sa_ln_assoc_bc = S sd_half_assoc_bc_left)) /\ (((c = 2 * sa_rp_assoc_bc /\ sa_rn_assoc_bc = 0) \/ exists sd_half_assoc_bc_right. ((c = 2 * sd_half_assoc_bc_right + 1 /\ sa_rp_assoc_bc = 0) /\ sa_rn_assoc_bc = S sd_half_assoc_bc_right)) /\ (((bc = 2 * sa_op_assoc_bc /\ sa_on_assoc_bc = 0) \/ exists sd_half_assoc_bc_output. ((bc = 2 * sd_half_assoc_bc_output + 1 /\ sa_op_assoc_bc = 0) /\ sa_on_assoc_bc = S sd_half_assoc_bc_output)) /\ (sa_lp_assoc_bc + sa_rp_assoc_bc) + sa_on_assoc_bc = (sa_ln_assoc_bc + sa_rn_assoc_bc) + sa_op_assoc_bc)))) -> (exists sa_lp_assoc_target sa_ln_assoc_target sa_rp_assoc_target sa_rn_assoc_target sa_op_assoc_target sa_on_assoc_target. (((a = 2 * sa_lp_assoc_target /\ sa_ln_assoc_target = 0) \/ exists sd_half_assoc_target_left. ((a = 2 * sd_half_assoc_target_left + 1 /\ sa_lp_assoc_target = 0) /\ sa_ln_assoc_target = S sd_half_assoc_target_left)) /\ (((bc = 2 * sa_rp_assoc_target /\ sa_rn_assoc_target = 0) \/ exists sd_half_assoc_target_right. ((bc = 2 * sd_half_assoc_target_right + 1 /\ sa_rp_assoc_target = 0) /\ sa_rn_assoc_target = S sd_half_assoc_target_right)) /\ (((abc = 2 * sa_op_assoc_target /\ sa_on_assoc_target = 0) \/ exists sd_half_assoc_target_output. ((abc = 2 * sd_half_assoc_target_output + 1 /\ sa_op_assoc_target = 0) /\ sa_on_assoc_target = S sd_half_assoc_target_output)) /\ (sa_lp_assoc_target + sa_rp_assoc_target) + sa_on_assoc_target = (sa_ln_assoc_target + sa_rn_assoc_target) + sa_op_assoc_target))))

signed_add_functional · checked inherited prerequisite

Exact statement in the checked dependency cone

forall left right output1 output2. (exists sa_lp_functional_left sa_ln_functional_left sa_rp_functional_left sa_rn_functional_left sa_op_functional_left sa_on_functional_left. (((left = 2 * sa_lp_functional_left /\ sa_ln_functional_left = 0) \/ exists sd_half_functional_left_left. ((left = 2 * sd_half_functional_left_left + 1 /\ sa_lp_functional_left = 0) /\ sa_ln_functional_left = S sd_half_functional_left_left)) /\ (((right = 2 * sa_rp_functional_left /\ sa_rn_functional_left = 0) \/ exists sd_half_functional_left_right. ((right = 2 * sd_half_functional_left_right + 1 /\ sa_rp_functional_left = 0) /\ sa_rn_functional_left = S sd_half_functional_left_right)) /\ (((output1 = 2 * sa_op_functional_left /\ sa_on_functional_left = 0) \/ exists sd_half_functional_left_output. ((output1 = 2 * sd_half_functional_left_output + 1 /\ sa_op_functional_left = 0) /\ sa_on_functional_left = S sd_half_functional_left_output)) /\ (sa_lp_functional_left + sa_rp_functional_left) + sa_on_functional_left = (sa_ln_functional_left + sa_rn_functional_left) + sa_op_functional_left)))) -> (exists sa_lp_functional_right sa_ln_functional_right sa_rp_functional_right sa_rn_functional_right sa_op_functional_right sa_on_functional_right. (((left = 2 * sa_lp_functional_right /\ sa_ln_functional_right = 0) \/ exists sd_half_functional_right_left. ((left = 2 * sd_half_functional_right_left + 1 /\ sa_lp_functional_right = 0) /\ sa_ln_functional_right = S sd_half_functional_right_left)) /\ (((right = 2 * sa_rp_functional_right /\ sa_rn_functional_right = 0) \/ exists sd_half_functional_right_right. ((right = 2 * sd_half_functional_right_right + 1 /\ sa_rp_functional_right = 0) /\ sa_rn_functional_right = S sd_half_functional_right_right)) /\ (((output2 = 2 * sa_op_functional_right /\ sa_on_functional_right = 0) \/ exists sd_half_functional_right_output. ((output2 = 2 * sd_half_functional_right_output + 1 /\ sa_op_functional_right = 0) /\ sa_on_functional_right = S sd_half_functional_right_output)) /\ (sa_lp_functional_right + sa_rp_functional_right) + sa_on_functional_right = (sa_ln_functional_right + sa_rn_functional_right) + sa_op_functional_right)))) -> output1 = output2

signed_add_negate_left_zero · checked inherited prerequisite

Exact statement in the checked dependency cone

forall input negated. (exists sn_pos_add_inverse_left sn_neg_add_inverse_left. (((input = 2 * sn_pos_add_inverse_left /\ sn_neg_add_inverse_left = 0) \/ exists sd_half_add_inverse_left_input. ((input = 2 * sd_half_add_inverse_left_input + 1 /\ sn_pos_add_inverse_left = 0) /\ sn_neg_add_inverse_left = S sd_half_add_inverse_left_input)) /\ ((negated = 2 * sn_neg_add_inverse_left /\ sn_pos_add_inverse_left = 0) \/ exists sd_half_add_inverse_left_output. ((negated = 2 * sd_half_add_inverse_left_output + 1 /\ sn_neg_add_inverse_left = 0) /\ sn_pos_add_inverse_left = S sd_half_add_inverse_left_output)))) -> (exists sa_lp_negate_left_zero sa_ln_negate_left_zero sa_rp_negate_left_zero sa_rn_negate_left_zero sa_op_negate_left_zero sa_on_negate_left_zero. (((negated = 2 * sa_lp_negate_left_zero /\ sa_ln_negate_left_zero = 0) \/ exists sd_half_negate_left_zero_left. ((negated = 2 * sd_half_negate_left_zero_left + 1 /\ sa_lp_negate_left_zero = 0) /\ sa_ln_negate_left_zero = S sd_half_negate_left_zero_left)) /\ (((input = 2 * sa_rp_negate_left_zero /\ sa_rn_negate_left_zero = 0) \/ exists sd_half_negate_left_zero_right. ((input = 2 * sd_half_negate_left_zero_right + 1 /\ sa_rp_negate_left_zero = 0) /\ sa_rn_negate_left_zero = S sd_half_negate_left_zero_right)) /\ (((0 = 2 * sa_op_negate_left_zero /\ sa_on_negate_left_zero = 0) \/ exists sd_half_negate_left_zero_output. ((0 = 2 * sd_half_negate_left_zero_output + 1 /\ sa_op_negate_left_zero = 0) /\ sa_on_negate_left_zero = S sd_half_negate_left_zero_output)) /\ (sa_lp_negate_left_zero + sa_rp_negate_left_zero) + sa_on_negate_left_zero = (sa_ln_negate_left_zero + sa_rn_negate_left_zero) + sa_op_negate_left_zero))))

signed_add_negate_right_zero · checked inherited prerequisite

Exact statement in the checked dependency cone

forall input negated. (exists sn_pos_add_inverse sn_neg_add_inverse. (((input = 2 * sn_pos_add_inverse /\ sn_neg_add_inverse = 0) \/ exists sd_half_add_inverse_input. ((input = 2 * sd_half_add_inverse_input + 1 /\ sn_pos_add_inverse = 0) /\ sn_neg_add_inverse = S sd_half_add_inverse_input)) /\ ((negated = 2 * sn_neg_add_inverse /\ sn_pos_add_inverse = 0) \/ exists sd_half_add_inverse_output. ((negated = 2 * sd_half_add_inverse_output + 1 /\ sn_neg_add_inverse = 0) /\ sn_pos_add_inverse = S sd_half_add_inverse_output)))) -> (exists sa_lp_negate_right_zero sa_ln_negate_right_zero sa_rp_negate_right_zero sa_rn_negate_right_zero sa_op_negate_right_zero sa_on_negate_right_zero. (((input = 2 * sa_lp_negate_right_zero /\ sa_ln_negate_right_zero = 0) \/ exists sd_half_negate_right_zero_left. ((input = 2 * sd_half_negate_right_zero_left + 1 /\ sa_lp_negate_right_zero = 0) /\ sa_ln_negate_right_zero = S sd_half_negate_right_zero_left)) /\ (((negated = 2 * sa_rp_negate_right_zero /\ sa_rn_negate_right_zero = 0) \/ exists sd_half_negate_right_zero_right. ((negated = 2 * sd_half_negate_right_zero_right + 1 /\ sa_rp_negate_right_zero = 0) /\ sa_rn_negate_right_zero = S sd_half_negate_right_zero_right)) /\ (((0 = 2 * sa_op_negate_right_zero /\ sa_on_negate_right_zero = 0) \/ exists sd_half_negate_right_zero_output. ((0 = 2 * sd_half_negate_right_zero_output + 1 /\ sa_op_negate_right_zero = 0) /\ sa_on_negate_right_zero = S sd_half_negate_right_zero_output)) /\ (sa_lp_negate_right_zero + sa_rp_negate_right_zero) + sa_on_negate_right_zero = (sa_ln_negate_right_zero + sa_rn_negate_right_zero) + sa_op_negate_right_zero))))

signed_add_total · checked inherited prerequisite

Exact statement in the checked dependency cone

forall left right. exists output. (exists sa_lp_total sa_ln_total sa_rp_total sa_rn_total sa_op_total sa_on_total. (((left = 2 * sa_lp_total /\ sa_ln_total = 0) \/ exists sd_half_total_left. ((left = 2 * sd_half_total_left + 1 /\ sa_lp_total = 0) /\ sa_ln_total = S sd_half_total_left)) /\ (((right = 2 * sa_rp_total /\ sa_rn_total = 0) \/ exists sd_half_total_right. ((right = 2 * sd_half_total_right + 1 /\ sa_rp_total = 0) /\ sa_rn_total = S sd_half_total_right)) /\ (((output = 2 * sa_op_total /\ sa_on_total = 0) \/ exists sd_half_total_output. ((output = 2 * sd_half_total_output + 1 /\ sa_op_total = 0) /\ sa_on_total = S sd_half_total_output)) /\ (sa_lp_total + sa_rp_total) + sa_on_total = (sa_ln_total + sa_rn_total) + sa_op_total))))

signed_add_zero_left · checked inherited prerequisite

Exact statement in the checked dependency cone

forall input. (exists sa_lp_zero_left sa_ln_zero_left sa_rp_zero_left sa_rn_zero_left sa_op_zero_left sa_on_zero_left. (((0 = 2 * sa_lp_zero_left /\ sa_ln_zero_left = 0) \/ exists sd_half_zero_left_left. ((0 = 2 * sd_half_zero_left_left + 1 /\ sa_lp_zero_left = 0) /\ sa_ln_zero_left = S sd_half_zero_left_left)) /\ (((input = 2 * sa_rp_zero_left /\ sa_rn_zero_left = 0) \/ exists sd_half_zero_left_right. ((input = 2 * sd_half_zero_left_right + 1 /\ sa_rp_zero_left = 0) /\ sa_rn_zero_left = S sd_half_zero_left_right)) /\ (((input = 2 * sa_op_zero_left /\ sa_on_zero_left = 0) \/ exists sd_half_zero_left_output. ((input = 2 * sd_half_zero_left_output + 1 /\ sa_op_zero_left = 0) /\ sa_on_zero_left = S sd_half_zero_left_output)) /\ (sa_lp_zero_left + sa_rp_zero_left) + sa_on_zero_left = (sa_ln_zero_left + sa_rn_zero_left) + sa_op_zero_left))))

signed_decode_normal · checked inherited prerequisite

Exact statement in the checked dependency cone

forall code pos neg. ((code = 2 * pos /\ neg = 0) \/ exists sd_half_normal. ((code = 2 * sd_half_normal + 1 /\ pos = 0) /\ neg = S sd_half_normal)) -> pos = 0 \/ neg = 0

signed_decode_total · checked inherited prerequisite

Exact statement in the checked dependency cone

forall code. exists pos neg. ((code = 2 * pos /\ neg = 0) \/ exists sd_half_total. ((code = 2 * sd_half_total + 1 /\ pos = 0) /\ neg = S sd_half_total))

signed_decoded_balance_implies_code_eq · checked inherited prerequisite

Exact statement in the checked dependency cone

forall code1 pos1 neg1 code2 pos2 neg2. ((code1 = 2 * pos1 /\ neg1 = 0) \/ exists sd_half_extensional_left. ((code1 = 2 * sd_half_extensional_left + 1 /\ pos1 = 0) /\ neg1 = S sd_half_extensional_left)) -> ((code2 = 2 * pos2 /\ neg2 = 0) \/ exists sd_half_extensional_right. ((code2 = 2 * sd_half_extensional_right + 1 /\ pos2 = 0) /\ neg2 = S sd_half_extensional_right)) -> pos1 + neg2 = neg1 + pos2 -> code1 = code2

signed_mul_associative · checked inherited prerequisite

Exact statement in the checked dependency cone

forall a b c ab bc abc. (exists sm_lp_assoc_ab sm_ln_assoc_ab sm_rp_assoc_ab sm_rn_assoc_ab sm_op_assoc_ab sm_on_assoc_ab. (((a = 2 * sm_lp_assoc_ab /\ sm_ln_assoc_ab = 0) \/ exists sd_half_assoc_ab_left. ((a = 2 * sd_half_assoc_ab_left + 1 /\ sm_lp_assoc_ab = 0) /\ sm_ln_assoc_ab = S sd_half_assoc_ab_left)) /\ (((b = 2 * sm_rp_assoc_ab /\ sm_rn_assoc_ab = 0) \/ exists sd_half_assoc_ab_right. ((b = 2 * sd_half_assoc_ab_right + 1 /\ sm_rp_assoc_ab = 0) /\ sm_rn_assoc_ab = S sd_half_assoc_ab_right)) /\ (((ab = 2 * sm_op_assoc_ab /\ sm_on_assoc_ab = 0) \/ exists sd_half_assoc_ab_output. ((ab = 2 * sd_half_assoc_ab_output + 1 /\ sm_op_assoc_ab = 0) /\ sm_on_assoc_ab = S sd_half_assoc_ab_output)) /\ (sm_lp_assoc_ab * sm_rp_assoc_ab + sm_ln_assoc_ab * sm_rn_assoc_ab) + sm_on_assoc_ab = (sm_lp_assoc_ab * sm_rn_assoc_ab + sm_ln_assoc_ab * sm_rp_assoc_ab) + sm_op_assoc_ab)))) -> (exists sm_lp_assoc_abc sm_ln_assoc_abc sm_rp_assoc_abc sm_rn_assoc_abc sm_op_assoc_abc sm_on_assoc_abc. (((ab = 2 * sm_lp_assoc_abc /\ sm_ln_assoc_abc = 0) \/ exists sd_half_assoc_abc_left. ((ab = 2 * sd_half_assoc_abc_left + 1 /\ sm_lp_assoc_abc = 0) /\ sm_ln_assoc_abc = S sd_half_assoc_abc_left)) /\ (((c = 2 * sm_rp_assoc_abc /\ sm_rn_assoc_abc = 0) \/ exists sd_half_assoc_abc_right. ((c = 2 * sd_half_assoc_abc_right + 1 /\ sm_rp_assoc_abc = 0) /\ sm_rn_assoc_abc = S sd_half_assoc_abc_right)) /\ (((abc = 2 * sm_op_assoc_abc /\ sm_on_assoc_abc = 0) \/ exists sd_half_assoc_abc_output. ((abc = 2 * sd_half_assoc_abc_output + 1 /\ sm_op_assoc_abc = 0) /\ sm_on_assoc_abc = S sd_half_assoc_abc_output)) /\ (sm_lp_assoc_abc * sm_rp_assoc_abc + sm_ln_assoc_abc * sm_rn_assoc_abc) + sm_on_assoc_abc = (sm_lp_assoc_abc * sm_rn_assoc_abc + sm_ln_assoc_abc * sm_rp_assoc_abc) + sm_op_assoc_abc)))) -> (exists sm_lp_assoc_bc sm_ln_assoc_bc sm_rp_assoc_bc sm_rn_assoc_bc sm_op_assoc_bc sm_on_assoc_bc. (((b = 2 * sm_lp_assoc_bc /\ sm_ln_assoc_bc = 0) \/ exists sd_half_assoc_bc_left. ((b = 2 * sd_half_assoc_bc_left + 1 /\ sm_lp_assoc_bc = 0) /\ sm_ln_assoc_bc = S sd_half_assoc_bc_left)) /\ (((c = 2 * sm_rp_assoc_bc /\ sm_rn_assoc_bc = 0) \/ exists sd_half_assoc_bc_right. ((c = 2 * sd_half_assoc_bc_right + 1 /\ sm_rp_assoc_bc = 0) /\ sm_rn_assoc_bc = S sd_half_assoc_bc_right)) /\ (((bc = 2 * sm_op_assoc_bc /\ sm_on_assoc_bc = 0) \/ exists sd_half_assoc_bc_output. ((bc = 2 * sd_half_assoc_bc_output + 1 /\ sm_op_assoc_bc = 0) /\ sm_on_assoc_bc = S sd_half_assoc_bc_output)) /\ (sm_lp_assoc_bc * sm_rp_assoc_bc + sm_ln_assoc_bc * sm_rn_assoc_bc) + sm_on_assoc_bc = (sm_lp_assoc_bc * sm_rn_assoc_bc + sm_ln_assoc_bc * sm_rp_assoc_bc) + sm_op_assoc_bc)))) -> (exists sm_lp_assoc_target sm_ln_assoc_target sm_rp_assoc_target sm_rn_assoc_target sm_op_assoc_target sm_on_assoc_target. (((a = 2 * sm_lp_assoc_target /\ sm_ln_assoc_target = 0) \/ exists sd_half_assoc_target_left. ((a = 2 * sd_half_assoc_target_left + 1 /\ sm_lp_assoc_target = 0) /\ sm_ln_assoc_target = S sd_half_assoc_target_left)) /\ (((bc = 2 * sm_rp_assoc_target /\ sm_rn_assoc_target = 0) \/ exists sd_half_assoc_target_right. ((bc = 2 * sd_half_assoc_target_right + 1 /\ sm_rp_assoc_target = 0) /\ sm_rn_assoc_target = S sd_half_assoc_target_right)) /\ (((abc = 2 * sm_op_assoc_target /\ sm_on_assoc_target = 0) \/ exists sd_half_assoc_target_output. ((abc = 2 * sd_half_assoc_target_output + 1 /\ sm_op_assoc_target = 0) /\ sm_on_assoc_target = S sd_half_assoc_target_output)) /\ (sm_lp_assoc_target * sm_rp_assoc_target + sm_ln_assoc_target * sm_rn_assoc_target) + sm_on_assoc_target = (sm_lp_assoc_target * sm_rn_assoc_target + sm_ln_assoc_target * sm_rp_assoc_target) + sm_op_assoc_target))))

signed_mul_functional · checked inherited prerequisite

Exact statement in the checked dependency cone

forall left right output1 output2. (exists sm_lp_functional_left sm_ln_functional_left sm_rp_functional_left sm_rn_functional_left sm_op_functional_left sm_on_functional_left. (((left = 2 * sm_lp_functional_left /\ sm_ln_functional_left = 0) \/ exists sd_half_functional_left_left. ((left = 2 * sd_half_functional_left_left + 1 /\ sm_lp_functional_left = 0) /\ sm_ln_functional_left = S sd_half_functional_left_left)) /\ (((right = 2 * sm_rp_functional_left /\ sm_rn_functional_left = 0) \/ exists sd_half_functional_left_right. ((right = 2 * sd_half_functional_left_right + 1 /\ sm_rp_functional_left = 0) /\ sm_rn_functional_left = S sd_half_functional_left_right)) /\ (((output1 = 2 * sm_op_functional_left /\ sm_on_functional_left = 0) \/ exists sd_half_functional_left_output. ((output1 = 2 * sd_half_functional_left_output + 1 /\ sm_op_functional_left = 0) /\ sm_on_functional_left = S sd_half_functional_left_output)) /\ (sm_lp_functional_left * sm_rp_functional_left + sm_ln_functional_left * sm_rn_functional_left) + sm_on_functional_left = (sm_lp_functional_left * sm_rn_functional_left + sm_ln_functional_left * sm_rp_functional_left) + sm_op_functional_left)))) -> (exists sm_lp_functional_right sm_ln_functional_right sm_rp_functional_right sm_rn_functional_right sm_op_functional_right sm_on_functional_right. (((left = 2 * sm_lp_functional_right /\ sm_ln_functional_right = 0) \/ exists sd_half_functional_right_left. ((left = 2 * sd_half_functional_right_left + 1 /\ sm_lp_functional_right = 0) /\ sm_ln_functional_right = S sd_half_functional_right_left)) /\ (((right = 2 * sm_rp_functional_right /\ sm_rn_functional_right = 0) \/ exists sd_half_functional_right_right. ((right = 2 * sd_half_functional_right_right + 1 /\ sm_rp_functional_right = 0) /\ sm_rn_functional_right = S sd_half_functional_right_right)) /\ (((output2 = 2 * sm_op_functional_right /\ sm_on_functional_right = 0) \/ exists sd_half_functional_right_output. ((output2 = 2 * sd_half_functional_right_output + 1 /\ sm_op_functional_right = 0) /\ sm_on_functional_right = S sd_half_functional_right_output)) /\ (sm_lp_functional_right * sm_rp_functional_right + sm_ln_functional_right * sm_rn_functional_right) + sm_on_functional_right = (sm_lp_functional_right * sm_rn_functional_right + sm_ln_functional_right * sm_rp_functional_right) + sm_op_functional_right)))) -> output1 = output2

signed_mul_of_decoded_equation · checked inherited prerequisite

Exact statement in the checked dependency cone

forall left right output lp ln rp rn op on. ((left = 2 * lp /\ ln = 0) \/ exists sd_half_mul_intro_left. ((left = 2 * sd_half_mul_intro_left + 1 /\ lp = 0) /\ ln = S sd_half_mul_intro_left)) -> ((right = 2 * rp /\ rn = 0) \/ exists sd_half_mul_intro_right. ((right = 2 * sd_half_mul_intro_right + 1 /\ rp = 0) /\ rn = S sd_half_mul_intro_right)) -> ((output = 2 * op /\ on = 0) \/ exists sd_half_mul_intro_output. ((output = 2 * sd_half_mul_intro_output + 1 /\ op = 0) /\ on = S sd_half_mul_intro_output)) -> (lp * rp + ln * rn) + on = (lp * rn + ln * rp) + op -> (exists sm_lp_intro sm_ln_intro sm_rp_intro sm_rn_intro sm_op_intro sm_on_intro. (((left = 2 * sm_lp_intro /\ sm_ln_intro = 0) \/ exists sd_half_intro_left. ((left = 2 * sd_half_intro_left + 1 /\ sm_lp_intro = 0) /\ sm_ln_intro = S sd_half_intro_left)) /\ (((right = 2 * sm_rp_intro /\ sm_rn_intro = 0) \/ exists sd_half_intro_right. ((right = 2 * sd_half_intro_right + 1 /\ sm_rp_intro = 0) /\ sm_rn_intro = S sd_half_intro_right)) /\ (((output = 2 * sm_op_intro /\ sm_on_intro = 0) \/ exists sd_half_intro_output. ((output = 2 * sd_half_intro_output + 1 /\ sm_op_intro = 0) /\ sm_on_intro = S sd_half_intro_output)) /\ (sm_lp_intro * sm_rp_intro + sm_ln_intro * sm_rn_intro) + sm_on_intro = (sm_lp_intro * sm_rn_intro + sm_ln_intro * sm_rp_intro) + sm_op_intro))))

signed_mul_one_right · checked inherited prerequisite

Exact statement in the checked dependency cone

forall input. (exists sm_lp_one_right sm_ln_one_right sm_rp_one_right sm_rn_one_right sm_op_one_right sm_on_one_right. (((input = 2 * sm_lp_one_right /\ sm_ln_one_right = 0) \/ exists sd_half_one_right_left. ((input = 2 * sd_half_one_right_left + 1 /\ sm_lp_one_right = 0) /\ sm_ln_one_right = S sd_half_one_right_left)) /\ (((2 = 2 * sm_rp_one_right /\ sm_rn_one_right = 0) \/ exists sd_half_one_right_right. ((2 = 2 * sd_half_one_right_right + 1 /\ sm_rp_one_right = 0) /\ sm_rn_one_right = S sd_half_one_right_right)) /\ (((input = 2 * sm_op_one_right /\ sm_on_one_right = 0) \/ exists sd_half_one_right_output. ((input = 2 * sd_half_one_right_output + 1 /\ sm_op_one_right = 0) /\ sm_on_one_right = S sd_half_one_right_output)) /\ (sm_lp_one_right * sm_rp_one_right + sm_ln_one_right * sm_rn_one_right) + sm_on_one_right = (sm_lp_one_right * sm_rn_one_right + sm_ln_one_right * sm_rp_one_right) + sm_op_one_right))))

signed_mul_to_decoded_equation · checked inherited prerequisite

Exact statement in the checked dependency cone

forall left right output lp ln rp rn op on. ((left = 2 * lp /\ ln = 0) \/ exists sd_half_mul_elim_left. ((left = 2 * sd_half_mul_elim_left + 1 /\ lp = 0) /\ ln = S sd_half_mul_elim_left)) -> ((right = 2 * rp /\ rn = 0) \/ exists sd_half_mul_elim_right. ((right = 2 * sd_half_mul_elim_right + 1 /\ rp = 0) /\ rn = S sd_half_mul_elim_right)) -> ((output = 2 * op /\ on = 0) \/ exists sd_half_mul_elim_output. ((output = 2 * sd_half_mul_elim_output + 1 /\ op = 0) /\ on = S sd_half_mul_elim_output)) -> (exists sm_lp_elim sm_ln_elim sm_rp_elim sm_rn_elim sm_op_elim sm_on_elim. (((left = 2 * sm_lp_elim /\ sm_ln_elim = 0) \/ exists sd_half_elim_left. ((left = 2 * sd_half_elim_left + 1 /\ sm_lp_elim = 0) /\ sm_ln_elim = S sd_half_elim_left)) /\ (((right = 2 * sm_rp_elim /\ sm_rn_elim = 0) \/ exists sd_half_elim_right. ((right = 2 * sd_half_elim_right + 1 /\ sm_rp_elim = 0) /\ sm_rn_elim = S sd_half_elim_right)) /\ (((output = 2 * sm_op_elim /\ sm_on_elim = 0) \/ exists sd_half_elim_output. ((output = 2 * sd_half_elim_output + 1 /\ sm_op_elim = 0) /\ sm_on_elim = S sd_half_elim_output)) /\ (sm_lp_elim * sm_rp_elim + sm_ln_elim * sm_rn_elim) + sm_on_elim = (sm_lp_elim * sm_rn_elim + sm_ln_elim * sm_rp_elim) + sm_op_elim)))) -> (lp * rp + ln * rn) + on = (lp * rn + ln * rp) + op

signed_mul_total · checked inherited prerequisite

Exact statement in the checked dependency cone

forall left right. exists output. (exists sm_lp_total sm_ln_total sm_rp_total sm_rn_total sm_op_total sm_on_total. (((left = 2 * sm_lp_total /\ sm_ln_total = 0) \/ exists sd_half_total_left. ((left = 2 * sd_half_total_left + 1 /\ sm_lp_total = 0) /\ sm_ln_total = S sd_half_total_left)) /\ (((right = 2 * sm_rp_total /\ sm_rn_total = 0) \/ exists sd_half_total_right. ((right = 2 * sd_half_total_right + 1 /\ sm_rp_total = 0) /\ sm_rn_total = S sd_half_total_right)) /\ (((output = 2 * sm_op_total /\ sm_on_total = 0) \/ exists sd_half_total_output. ((output = 2 * sd_half_total_output + 1 /\ sm_op_total = 0) /\ sm_on_total = S sd_half_total_output)) /\ (sm_lp_total * sm_rp_total + sm_ln_total * sm_rn_total) + sm_on_total = (sm_lp_total * sm_rn_total + sm_ln_total * sm_rp_total) + sm_op_total))))

signed_negate_total · checked inherited prerequisite

Exact statement in the checked dependency cone

forall input. exists output. (exists sn_pos_total sn_neg_total. (((input = 2 * sn_pos_total /\ sn_neg_total = 0) \/ exists sd_half_total_input. ((input = 2 * sd_half_total_input + 1 /\ sn_pos_total = 0) /\ sn_neg_total = S sd_half_total_input)) /\ ((output = 2 * sn_neg_total /\ sn_pos_total = 0) \/ exists sd_half_total_output. ((output = 2 * sd_half_total_output + 1 /\ sn_neg_total = 0) /\ sn_pos_total = S sd_half_total_output))))

zero_add · checked inherited prerequisite
Exact statement in the checked dependency cone
```
forall n. 0 + n = n
```