ND0320

SignedSupportReindex(A,B,r,s,L,M)

Actual tables and a native beta map preserve nonzero source values at bounded target indices, are injective only on nonzero source support, and cover every nonzero target value. Unequal or empty windows and inactive collisions are allowed. This is not a whole-window permutation and contains no sum equality.

Conservative notation; not a theorem, primitive, or axiom.

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

Definition in prerequisite notation

ArithTable(0,A) ∧ (ArithTable(0,B) ∧ ((∀ x. ∀ y. Lt(x,L)ArithAt(A,x,y) → ¬y = 0 → ∃ z. BetaAt(r,s,x,z) ∧ (Lt(z,M)ArithAt(B,z,y))) ∧ ((∀ x. ∀ y. ∀ z. ∀ n. ∀ m. Lt(x,L)Lt(y,L)ArithAt(A,x,n) → ¬n = 0 → ArithAt(A,y,m) → ¬m = 0 → BetaAt(r,s,x,z)BetaAt(r,s,y,z) → x = y) ∧ (∀ x. ∀ y. Lt(x,M)ArithAt(B,x,y) → ¬y = 0 → ∃ z. Lt(z,L) ∧ (BetaAt(r,s,z,x)ArithAt(A,z,y))))))

Only definitions earlier in this acyclic notation graph are used here.

Hygienic expanded first-order definition
((exists dst_positive_code_g009_definitionsource_table dst_positive_scale_g009_definitionsource_table dst_negative_code_g009_definitionsource_table dst_negative_scale_g009_definitionsource_table. ((((A)) = (((((dst_positive_code_g009_definitionsource_table) + (dst_positive_scale_g009_definitionsource_table)) * S ((dst_positive_code_g009_definitionsource_table) + (dst_positive_scale_g009_definitionsource_table)) + ((dst_positive_scale_g009_definitionsource_table) + (dst_positive_scale_g009_definitionsource_table))) + (((dst_negative_code_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table)) * S ((dst_negative_code_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table)) + ((dst_negative_scale_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table)))) * S ((((dst_positive_code_g009_definitionsource_table) + (dst_positive_scale_g009_definitionsource_table)) * S ((dst_positive_code_g009_definitionsource_table) + (dst_positive_scale_g009_definitionsource_table)) + ((dst_positive_scale_g009_definitionsource_table) + (dst_positive_scale_g009_definitionsource_table))) + (((dst_negative_code_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table)) * S ((dst_negative_code_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table)) + ((dst_negative_scale_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table)))) + ((((dst_negative_code_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table)) * S ((dst_negative_code_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table)) + ((dst_negative_scale_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table))) + (((dst_negative_code_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table)) * S ((dst_negative_code_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table)) + ((dst_negative_scale_g009_definitionsource_table) + (dst_negative_scale_g009_definitionsource_table)))))) /\ (forall dst_index_g009_definitionsource_table. (exists pvs_le_gap_g009_definitionsource_tabledomain. pvs_le_gap_g009_definitionsource_tabledomain + (dst_index_g009_definitionsource_table) = (0)) -> exists dst_positive_g009_definitionsource_table dst_negative_g009_definitionsource_table dst_value_g009_definitionsource_table. ((((exists ff_h_pvs_g009_definitionsource_tableentrypositive. ff_h_pvs_g009_definitionsource_tableentrypositive + S (dst_positive_g009_definitionsource_table) = S ((S (dst_index_g009_definitionsource_table)) * dst_positive_scale_g009_definitionsource_table)) /\ exists ff_q_pvs_g009_definitionsource_tableentrypositive. dst_positive_code_g009_definitionsource_table = ff_q_pvs_g009_definitionsource_tableentrypositive * S ((S (dst_index_g009_definitionsource_table)) * dst_positive_scale_g009_definitionsource_table) + (dst_positive_g009_definitionsource_table))) /\ (((((exists ff_h_pvs_g009_definitionsource_tableentrynegative. ff_h_pvs_g009_definitionsource_tableentrynegative + S (dst_negative_g009_definitionsource_table) = S ((S (dst_index_g009_definitionsource_table)) * dst_negative_scale_g009_definitionsource_table)) /\ exists ff_q_pvs_g009_definitionsource_tableentrynegative. dst_negative_code_g009_definitionsource_table = ff_q_pvs_g009_definitionsource_tableentrynegative * S ((S (dst_index_g009_definitionsource_table)) * dst_negative_scale_g009_definitionsource_table) + (dst_negative_g009_definitionsource_table))) /\ (exists ge_balance_positive_g009_definitionsource_tableentryvalue ge_balance_negative_g009_definitionsource_tableentryvalue. (((((dst_value_g009_definitionsource_table) = 2 * (ge_balance_positive_g009_definitionsource_tableentryvalue) /\ (ge_balance_negative_g009_definitionsource_tableentryvalue) = 0) \/ exists ge_signed_half_g009_definitionsource_tableentryvaluedecode. (((dst_value_g009_definitionsource_table) = 2 * ge_signed_half_g009_definitionsource_tableentryvaluedecode + 1 /\ (ge_balance_positive_g009_definitionsource_tableentryvalue) = 0) /\ (ge_balance_negative_g009_definitionsource_tableentryvalue) = S ge_signed_half_g009_definitionsource_tableentryvaluedecode))) /\ ((dst_positive_g009_definitionsource_table) + ge_balance_negative_g009_definitionsource_tableentryvalue = (dst_negative_g009_definitionsource_table) + ge_balance_positive_g009_definitionsource_tableentryvalue))))))))) /\ (((exists dst_positive_code_g009_definitiontarget_table dst_positive_scale_g009_definitiontarget_table dst_negative_code_g009_definitiontarget_table dst_negative_scale_g009_definitiontarget_table. ((((B)) = (((((dst_positive_code_g009_definitiontarget_table) + (dst_positive_scale_g009_definitiontarget_table)) * S ((dst_positive_code_g009_definitiontarget_table) + (dst_positive_scale_g009_definitiontarget_table)) + ((dst_positive_scale_g009_definitiontarget_table) + (dst_positive_scale_g009_definitiontarget_table))) + (((dst_negative_code_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table)) * S ((dst_negative_code_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table)) + ((dst_negative_scale_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table)))) * S ((((dst_positive_code_g009_definitiontarget_table) + (dst_positive_scale_g009_definitiontarget_table)) * S ((dst_positive_code_g009_definitiontarget_table) + (dst_positive_scale_g009_definitiontarget_table)) + ((dst_positive_scale_g009_definitiontarget_table) + (dst_positive_scale_g009_definitiontarget_table))) + (((dst_negative_code_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table)) * S ((dst_negative_code_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table)) + ((dst_negative_scale_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table)))) + ((((dst_negative_code_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table)) * S ((dst_negative_code_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table)) + ((dst_negative_scale_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table))) + (((dst_negative_code_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table)) * S ((dst_negative_code_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table)) + ((dst_negative_scale_g009_definitiontarget_table) + (dst_negative_scale_g009_definitiontarget_table)))))) /\ (forall dst_index_g009_definitiontarget_table. (exists pvs_le_gap_g009_definitiontarget_tabledomain. pvs_le_gap_g009_definitiontarget_tabledomain + (dst_index_g009_definitiontarget_table) = (0)) -> exists dst_positive_g009_definitiontarget_table dst_negative_g009_definitiontarget_table dst_value_g009_definitiontarget_table. ((((exists ff_h_pvs_g009_definitiontarget_tableentrypositive. ff_h_pvs_g009_definitiontarget_tableentrypositive + S (dst_positive_g009_definitiontarget_table) = S ((S (dst_index_g009_definitiontarget_table)) * dst_positive_scale_g009_definitiontarget_table)) /\ exists ff_q_pvs_g009_definitiontarget_tableentrypositive. dst_positive_code_g009_definitiontarget_table = ff_q_pvs_g009_definitiontarget_tableentrypositive * S ((S (dst_index_g009_definitiontarget_table)) * dst_positive_scale_g009_definitiontarget_table) + (dst_positive_g009_definitiontarget_table))) /\ (((((exists ff_h_pvs_g009_definitiontarget_tableentrynegative. ff_h_pvs_g009_definitiontarget_tableentrynegative + S (dst_negative_g009_definitiontarget_table) = S ((S (dst_index_g009_definitiontarget_table)) * dst_negative_scale_g009_definitiontarget_table)) /\ exists ff_q_pvs_g009_definitiontarget_tableentrynegative. dst_negative_code_g009_definitiontarget_table = ff_q_pvs_g009_definitiontarget_tableentrynegative * S ((S (dst_index_g009_definitiontarget_table)) * dst_negative_scale_g009_definitiontarget_table) + (dst_negative_g009_definitiontarget_table))) /\ (exists ge_balance_positive_g009_definitiontarget_tableentryvalue ge_balance_negative_g009_definitiontarget_tableentryvalue. (((((dst_value_g009_definitiontarget_table) = 2 * (ge_balance_positive_g009_definitiontarget_tableentryvalue) /\ (ge_balance_negative_g009_definitiontarget_tableentryvalue) = 0) \/ exists ge_signed_half_g009_definitiontarget_tableentryvaluedecode. (((dst_value_g009_definitiontarget_table) = 2 * ge_signed_half_g009_definitiontarget_tableentryvaluedecode + 1 /\ (ge_balance_positive_g009_definitiontarget_tableentryvalue) = 0) /\ (ge_balance_negative_g009_definitiontarget_tableentryvalue) = S ge_signed_half_g009_definitiontarget_tableentryvaluedecode))) /\ ((dst_positive_g009_definitiontarget_table) + ge_balance_negative_g009_definitiontarget_tableentryvalue = (dst_negative_g009_definitiontarget_table) + ge_balance_positive_g009_definitiontarget_tableentryvalue))))))))) /\ (((forall ssr_source_g009_definitionpreserve ssr_value_g009_definitionpreserve. (exists pvs_gap_g009_definitionpreservesource_bound. pvs_gap_g009_definitionpreservesource_bound + S (ssr_source_g009_definitionpreserve) = ((L))) -> (exists dst_positive_code_g009_definitionpreservesource_value dst_positive_scale_g009_definitionpreservesource_value dst_negative_code_g009_definitionpreservesource_value dst_negative_scale_g009_definitionpreservesource_value dst_positive_g009_definitionpreservesource_value dst_negative_g009_definitionpreservesource_value. ((((A)) = (((((dst_positive_code_g009_definitionpreservesource_value) + (dst_positive_scale_g009_definitionpreservesource_value)) * S ((dst_positive_code_g009_definitionpreservesource_value) + (dst_positive_scale_g009_definitionpreservesource_value)) + ((dst_positive_scale_g009_definitionpreservesource_value) + (dst_positive_scale_g009_definitionpreservesource_value))) + (((dst_negative_code_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value)) * S ((dst_negative_code_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value)) + ((dst_negative_scale_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value)))) * S ((((dst_positive_code_g009_definitionpreservesource_value) + (dst_positive_scale_g009_definitionpreservesource_value)) * S ((dst_positive_code_g009_definitionpreservesource_value) + (dst_positive_scale_g009_definitionpreservesource_value)) + ((dst_positive_scale_g009_definitionpreservesource_value) + (dst_positive_scale_g009_definitionpreservesource_value))) + (((dst_negative_code_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value)) * S ((dst_negative_code_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value)) + ((dst_negative_scale_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value)))) + ((((dst_negative_code_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value)) * S ((dst_negative_code_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value)) + ((dst_negative_scale_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value))) + (((dst_negative_code_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value)) * S ((dst_negative_code_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value)) + ((dst_negative_scale_g009_definitionpreservesource_value) + (dst_negative_scale_g009_definitionpreservesource_value)))))) /\ (((((exists ff_h_pvs_g009_definitionpreservesource_valuepositive. ff_h_pvs_g009_definitionpreservesource_valuepositive + S (dst_positive_g009_definitionpreservesource_value) = S ((S (ssr_source_g009_definitionpreserve)) * dst_positive_scale_g009_definitionpreservesource_value)) /\ exists ff_q_pvs_g009_definitionpreservesource_valuepositive. dst_positive_code_g009_definitionpreservesource_value = ff_q_pvs_g009_definitionpreservesource_valuepositive * S ((S (ssr_source_g009_definitionpreserve)) * dst_positive_scale_g009_definitionpreservesource_value) + (dst_positive_g009_definitionpreservesource_value))) /\ (((((exists ff_h_pvs_g009_definitionpreservesource_valuenegative. ff_h_pvs_g009_definitionpreservesource_valuenegative + S (dst_negative_g009_definitionpreservesource_value) = S ((S (ssr_source_g009_definitionpreserve)) * dst_negative_scale_g009_definitionpreservesource_value)) /\ exists ff_q_pvs_g009_definitionpreservesource_valuenegative. dst_negative_code_g009_definitionpreservesource_value = ff_q_pvs_g009_definitionpreservesource_valuenegative * S ((S (ssr_source_g009_definitionpreserve)) * dst_negative_scale_g009_definitionpreservesource_value) + (dst_negative_g009_definitionpreservesource_value))) /\ (exists ge_balance_positive_g009_definitionpreservesource_valuevalue ge_balance_negative_g009_definitionpreservesource_valuevalue. (((((ssr_value_g009_definitionpreserve) = 2 * (ge_balance_positive_g009_definitionpreservesource_valuevalue) /\ (ge_balance_negative_g009_definitionpreservesource_valuevalue) = 0) \/ exists ge_signed_half_g009_definitionpreservesource_valuevaluedecode. (((ssr_value_g009_definitionpreserve) = 2 * ge_signed_half_g009_definitionpreservesource_valuevaluedecode + 1 /\ (ge_balance_positive_g009_definitionpreservesource_valuevalue) = 0) /\ (ge_balance_negative_g009_definitionpreservesource_valuevalue) = S ge_signed_half_g009_definitionpreservesource_valuevaluedecode))) /\ ((dst_positive_g009_definitionpreservesource_value) + ge_balance_negative_g009_definitionpreservesource_valuevalue = (dst_negative_g009_definitionpreservesource_value) + ge_balance_positive_g009_definitionpreservesource_valuevalue))))))))) -> ~(ssr_value_g009_definitionpreserve=0) -> exists ssr_target_g009_definitionpreserve. ((((exists ff_h_pvs_g009_definitionpreservemap. ff_h_pvs_g009_definitionpreservemap + S (ssr_target_g009_definitionpreserve) = S ((S (ssr_source_g009_definitionpreserve)) * (s))) /\ exists ff_q_pvs_g009_definitionpreservemap. (r) = ff_q_pvs_g009_definitionpreservemap * S ((S (ssr_source_g009_definitionpreserve)) * (s)) + (ssr_target_g009_definitionpreserve))) /\ (((exists pvs_gap_g009_definitionpreservetarget_bound. pvs_gap_g009_definitionpreservetarget_bound + S (ssr_target_g009_definitionpreserve) = ((M))) /\ (exists dst_positive_code_g009_definitionpreservetarget_value dst_positive_scale_g009_definitionpreservetarget_value dst_negative_code_g009_definitionpreservetarget_value dst_negative_scale_g009_definitionpreservetarget_value dst_positive_g009_definitionpreservetarget_value dst_negative_g009_definitionpreservetarget_value. ((((B)) = (((((dst_positive_code_g009_definitionpreservetarget_value) + (dst_positive_scale_g009_definitionpreservetarget_value)) * S ((dst_positive_code_g009_definitionpreservetarget_value) + (dst_positive_scale_g009_definitionpreservetarget_value)) + ((dst_positive_scale_g009_definitionpreservetarget_value) + (dst_positive_scale_g009_definitionpreservetarget_value))) + (((dst_negative_code_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value)) * S ((dst_negative_code_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value)) + ((dst_negative_scale_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value)))) * S ((((dst_positive_code_g009_definitionpreservetarget_value) + (dst_positive_scale_g009_definitionpreservetarget_value)) * S ((dst_positive_code_g009_definitionpreservetarget_value) + (dst_positive_scale_g009_definitionpreservetarget_value)) + ((dst_positive_scale_g009_definitionpreservetarget_value) + (dst_positive_scale_g009_definitionpreservetarget_value))) + (((dst_negative_code_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value)) * S ((dst_negative_code_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value)) + ((dst_negative_scale_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value)))) + ((((dst_negative_code_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value)) * S ((dst_negative_code_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value)) + ((dst_negative_scale_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value))) + (((dst_negative_code_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value)) * S ((dst_negative_code_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value)) + ((dst_negative_scale_g009_definitionpreservetarget_value) + (dst_negative_scale_g009_definitionpreservetarget_value)))))) /\ (((((exists ff_h_pvs_g009_definitionpreservetarget_valuepositive. ff_h_pvs_g009_definitionpreservetarget_valuepositive + S (dst_positive_g009_definitionpreservetarget_value) = S ((S (ssr_target_g009_definitionpreserve)) * dst_positive_scale_g009_definitionpreservetarget_value)) /\ exists ff_q_pvs_g009_definitionpreservetarget_valuepositive. dst_positive_code_g009_definitionpreservetarget_value = ff_q_pvs_g009_definitionpreservetarget_valuepositive * S ((S (ssr_target_g009_definitionpreserve)) * dst_positive_scale_g009_definitionpreservetarget_value) + (dst_positive_g009_definitionpreservetarget_value))) /\ (((((exists ff_h_pvs_g009_definitionpreservetarget_valuenegative. ff_h_pvs_g009_definitionpreservetarget_valuenegative + S (dst_negative_g009_definitionpreservetarget_value) = S ((S (ssr_target_g009_definitionpreserve)) * dst_negative_scale_g009_definitionpreservetarget_value)) /\ exists ff_q_pvs_g009_definitionpreservetarget_valuenegative. dst_negative_code_g009_definitionpreservetarget_value = ff_q_pvs_g009_definitionpreservetarget_valuenegative * S ((S (ssr_target_g009_definitionpreserve)) * dst_negative_scale_g009_definitionpreservetarget_value) + (dst_negative_g009_definitionpreservetarget_value))) /\ (exists ge_balance_positive_g009_definitionpreservetarget_valuevalue ge_balance_negative_g009_definitionpreservetarget_valuevalue. (((((ssr_value_g009_definitionpreserve) = 2 * (ge_balance_positive_g009_definitionpreservetarget_valuevalue) /\ (ge_balance_negative_g009_definitionpreservetarget_valuevalue) = 0) \/ exists ge_signed_half_g009_definitionpreservetarget_valuevaluedecode. (((ssr_value_g009_definitionpreserve) = 2 * ge_signed_half_g009_definitionpreservetarget_valuevaluedecode + 1 /\ (ge_balance_positive_g009_definitionpreservetarget_valuevalue) = 0) /\ (ge_balance_negative_g009_definitionpreservetarget_valuevalue) = S ge_signed_half_g009_definitionpreservetarget_valuevaluedecode))) /\ ((dst_positive_g009_definitionpreservetarget_value) + ge_balance_negative_g009_definitionpreservetarget_valuevalue = (dst_negative_g009_definitionpreservetarget_value) + ge_balance_positive_g009_definitionpreservetarget_valuevalue))))))))))))) /\ (((forall ssr_first_g009_definitioninjective ssr_second_g009_definitioninjective ssr_image_g009_definitioninjective ssr_a_g009_definitioninjective ssr_b_g009_definitioninjective. (exists pvs_gap_g009_definitioninjectivefirst_bound. pvs_gap_g009_definitioninjectivefirst_bound + S (ssr_first_g009_definitioninjective) = ((L))) -> (exists pvs_gap_g009_definitioninjectivesecond_bound. pvs_gap_g009_definitioninjectivesecond_bound + S (ssr_second_g009_definitioninjective) = ((L))) -> (exists dst_positive_code_g009_definitioninjectivefirst_value dst_positive_scale_g009_definitioninjectivefirst_value dst_negative_code_g009_definitioninjectivefirst_value dst_negative_scale_g009_definitioninjectivefirst_value dst_positive_g009_definitioninjectivefirst_value dst_negative_g009_definitioninjectivefirst_value. ((((A)) = (((((dst_positive_code_g009_definitioninjectivefirst_value) + (dst_positive_scale_g009_definitioninjectivefirst_value)) * S ((dst_positive_code_g009_definitioninjectivefirst_value) + (dst_positive_scale_g009_definitioninjectivefirst_value)) + ((dst_positive_scale_g009_definitioninjectivefirst_value) + (dst_positive_scale_g009_definitioninjectivefirst_value))) + (((dst_negative_code_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value)) * S ((dst_negative_code_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value)) + ((dst_negative_scale_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value)))) * S ((((dst_positive_code_g009_definitioninjectivefirst_value) + (dst_positive_scale_g009_definitioninjectivefirst_value)) * S ((dst_positive_code_g009_definitioninjectivefirst_value) + (dst_positive_scale_g009_definitioninjectivefirst_value)) + ((dst_positive_scale_g009_definitioninjectivefirst_value) + (dst_positive_scale_g009_definitioninjectivefirst_value))) + (((dst_negative_code_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value)) * S ((dst_negative_code_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value)) + ((dst_negative_scale_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value)))) + ((((dst_negative_code_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value)) * S ((dst_negative_code_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value)) + ((dst_negative_scale_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value))) + (((dst_negative_code_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value)) * S ((dst_negative_code_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value)) + ((dst_negative_scale_g009_definitioninjectivefirst_value) + (dst_negative_scale_g009_definitioninjectivefirst_value)))))) /\ (((((exists ff_h_pvs_g009_definitioninjectivefirst_valuepositive. ff_h_pvs_g009_definitioninjectivefirst_valuepositive + S (dst_positive_g009_definitioninjectivefirst_value) = S ((S (ssr_first_g009_definitioninjective)) * dst_positive_scale_g009_definitioninjectivefirst_value)) /\ exists ff_q_pvs_g009_definitioninjectivefirst_valuepositive. dst_positive_code_g009_definitioninjectivefirst_value = ff_q_pvs_g009_definitioninjectivefirst_valuepositive * S ((S (ssr_first_g009_definitioninjective)) * dst_positive_scale_g009_definitioninjectivefirst_value) + (dst_positive_g009_definitioninjectivefirst_value))) /\ (((((exists ff_h_pvs_g009_definitioninjectivefirst_valuenegative. ff_h_pvs_g009_definitioninjectivefirst_valuenegative + S (dst_negative_g009_definitioninjectivefirst_value) = S ((S (ssr_first_g009_definitioninjective)) * dst_negative_scale_g009_definitioninjectivefirst_value)) /\ exists ff_q_pvs_g009_definitioninjectivefirst_valuenegative. dst_negative_code_g009_definitioninjectivefirst_value = ff_q_pvs_g009_definitioninjectivefirst_valuenegative * S ((S (ssr_first_g009_definitioninjective)) * dst_negative_scale_g009_definitioninjectivefirst_value) + (dst_negative_g009_definitioninjectivefirst_value))) /\ (exists ge_balance_positive_g009_definitioninjectivefirst_valuevalue ge_balance_negative_g009_definitioninjectivefirst_valuevalue. (((((ssr_a_g009_definitioninjective) = 2 * (ge_balance_positive_g009_definitioninjectivefirst_valuevalue) /\ (ge_balance_negative_g009_definitioninjectivefirst_valuevalue) = 0) \/ exists ge_signed_half_g009_definitioninjectivefirst_valuevaluedecode. (((ssr_a_g009_definitioninjective) = 2 * ge_signed_half_g009_definitioninjectivefirst_valuevaluedecode + 1 /\ (ge_balance_positive_g009_definitioninjectivefirst_valuevalue) = 0) /\ (ge_balance_negative_g009_definitioninjectivefirst_valuevalue) = S ge_signed_half_g009_definitioninjectivefirst_valuevaluedecode))) /\ ((dst_positive_g009_definitioninjectivefirst_value) + ge_balance_negative_g009_definitioninjectivefirst_valuevalue = (dst_negative_g009_definitioninjectivefirst_value) + ge_balance_positive_g009_definitioninjectivefirst_valuevalue))))))))) -> ~(ssr_a_g009_definitioninjective=0) -> (exists dst_positive_code_g009_definitioninjectivesecond_value dst_positive_scale_g009_definitioninjectivesecond_value dst_negative_code_g009_definitioninjectivesecond_value dst_negative_scale_g009_definitioninjectivesecond_value dst_positive_g009_definitioninjectivesecond_value dst_negative_g009_definitioninjectivesecond_value. ((((A)) = (((((dst_positive_code_g009_definitioninjectivesecond_value) + (dst_positive_scale_g009_definitioninjectivesecond_value)) * S ((dst_positive_code_g009_definitioninjectivesecond_value) + (dst_positive_scale_g009_definitioninjectivesecond_value)) + ((dst_positive_scale_g009_definitioninjectivesecond_value) + (dst_positive_scale_g009_definitioninjectivesecond_value))) + (((dst_negative_code_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value)) * S ((dst_negative_code_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value)) + ((dst_negative_scale_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value)))) * S ((((dst_positive_code_g009_definitioninjectivesecond_value) + (dst_positive_scale_g009_definitioninjectivesecond_value)) * S ((dst_positive_code_g009_definitioninjectivesecond_value) + (dst_positive_scale_g009_definitioninjectivesecond_value)) + ((dst_positive_scale_g009_definitioninjectivesecond_value) + (dst_positive_scale_g009_definitioninjectivesecond_value))) + (((dst_negative_code_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value)) * S ((dst_negative_code_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value)) + ((dst_negative_scale_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value)))) + ((((dst_negative_code_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value)) * S ((dst_negative_code_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value)) + ((dst_negative_scale_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value))) + (((dst_negative_code_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value)) * S ((dst_negative_code_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value)) + ((dst_negative_scale_g009_definitioninjectivesecond_value) + (dst_negative_scale_g009_definitioninjectivesecond_value)))))) /\ (((((exists ff_h_pvs_g009_definitioninjectivesecond_valuepositive. ff_h_pvs_g009_definitioninjectivesecond_valuepositive + S (dst_positive_g009_definitioninjectivesecond_value) = S ((S (ssr_second_g009_definitioninjective)) * dst_positive_scale_g009_definitioninjectivesecond_value)) /\ exists ff_q_pvs_g009_definitioninjectivesecond_valuepositive. dst_positive_code_g009_definitioninjectivesecond_value = ff_q_pvs_g009_definitioninjectivesecond_valuepositive * S ((S (ssr_second_g009_definitioninjective)) * dst_positive_scale_g009_definitioninjectivesecond_value) + (dst_positive_g009_definitioninjectivesecond_value))) /\ (((((exists ff_h_pvs_g009_definitioninjectivesecond_valuenegative. ff_h_pvs_g009_definitioninjectivesecond_valuenegative + S (dst_negative_g009_definitioninjectivesecond_value) = S ((S (ssr_second_g009_definitioninjective)) * dst_negative_scale_g009_definitioninjectivesecond_value)) /\ exists ff_q_pvs_g009_definitioninjectivesecond_valuenegative. dst_negative_code_g009_definitioninjectivesecond_value = ff_q_pvs_g009_definitioninjectivesecond_valuenegative * S ((S (ssr_second_g009_definitioninjective)) * dst_negative_scale_g009_definitioninjectivesecond_value) + (dst_negative_g009_definitioninjectivesecond_value))) /\ (exists ge_balance_positive_g009_definitioninjectivesecond_valuevalue ge_balance_negative_g009_definitioninjectivesecond_valuevalue. (((((ssr_b_g009_definitioninjective) = 2 * (ge_balance_positive_g009_definitioninjectivesecond_valuevalue) /\ (ge_balance_negative_g009_definitioninjectivesecond_valuevalue) = 0) \/ exists ge_signed_half_g009_definitioninjectivesecond_valuevaluedecode. (((ssr_b_g009_definitioninjective) = 2 * ge_signed_half_g009_definitioninjectivesecond_valuevaluedecode + 1 /\ (ge_balance_positive_g009_definitioninjectivesecond_valuevalue) = 0) /\ (ge_balance_negative_g009_definitioninjectivesecond_valuevalue) = S ge_signed_half_g009_definitioninjectivesecond_valuevaluedecode))) /\ ((dst_positive_g009_definitioninjectivesecond_value) + ge_balance_negative_g009_definitioninjectivesecond_valuevalue = (dst_negative_g009_definitioninjectivesecond_value) + ge_balance_positive_g009_definitioninjectivesecond_valuevalue))))))))) -> ~(ssr_b_g009_definitioninjective=0) -> (((exists ff_h_pvs_g009_definitioninjectivefirst_map. ff_h_pvs_g009_definitioninjectivefirst_map + S (ssr_image_g009_definitioninjective) = S ((S (ssr_first_g009_definitioninjective)) * (s))) /\ exists ff_q_pvs_g009_definitioninjectivefirst_map. (r) = ff_q_pvs_g009_definitioninjectivefirst_map * S ((S (ssr_first_g009_definitioninjective)) * (s)) + (ssr_image_g009_definitioninjective))) -> (((exists ff_h_pvs_g009_definitioninjectivesecond_map. ff_h_pvs_g009_definitioninjectivesecond_map + S (ssr_image_g009_definitioninjective) = S ((S (ssr_second_g009_definitioninjective)) * (s))) /\ exists ff_q_pvs_g009_definitioninjectivesecond_map. (r) = ff_q_pvs_g009_definitioninjectivesecond_map * S ((S (ssr_second_g009_definitioninjective)) * (s)) + (ssr_image_g009_definitioninjective))) -> ssr_first_g009_definitioninjective=ssr_second_g009_definitioninjective) /\ (forall ssr_target_g009_definitioncover ssr_value_g009_definitioncover. (exists pvs_gap_g009_definitioncovertarget_bound. pvs_gap_g009_definitioncovertarget_bound + S (ssr_target_g009_definitioncover) = ((M))) -> (exists dst_positive_code_g009_definitioncovertarget_value dst_positive_scale_g009_definitioncovertarget_value dst_negative_code_g009_definitioncovertarget_value dst_negative_scale_g009_definitioncovertarget_value dst_positive_g009_definitioncovertarget_value dst_negative_g009_definitioncovertarget_value. ((((B)) = (((((dst_positive_code_g009_definitioncovertarget_value) + (dst_positive_scale_g009_definitioncovertarget_value)) * S ((dst_positive_code_g009_definitioncovertarget_value) + (dst_positive_scale_g009_definitioncovertarget_value)) + ((dst_positive_scale_g009_definitioncovertarget_value) + (dst_positive_scale_g009_definitioncovertarget_value))) + (((dst_negative_code_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value)) * S ((dst_negative_code_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value)) + ((dst_negative_scale_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value)))) * S ((((dst_positive_code_g009_definitioncovertarget_value) + (dst_positive_scale_g009_definitioncovertarget_value)) * S ((dst_positive_code_g009_definitioncovertarget_value) + (dst_positive_scale_g009_definitioncovertarget_value)) + ((dst_positive_scale_g009_definitioncovertarget_value) + (dst_positive_scale_g009_definitioncovertarget_value))) + (((dst_negative_code_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value)) * S ((dst_negative_code_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value)) + ((dst_negative_scale_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value)))) + ((((dst_negative_code_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value)) * S ((dst_negative_code_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value)) + ((dst_negative_scale_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value))) + (((dst_negative_code_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value)) * S ((dst_negative_code_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value)) + ((dst_negative_scale_g009_definitioncovertarget_value) + (dst_negative_scale_g009_definitioncovertarget_value)))))) /\ (((((exists ff_h_pvs_g009_definitioncovertarget_valuepositive. ff_h_pvs_g009_definitioncovertarget_valuepositive + S (dst_positive_g009_definitioncovertarget_value) = S ((S (ssr_target_g009_definitioncover)) * dst_positive_scale_g009_definitioncovertarget_value)) /\ exists ff_q_pvs_g009_definitioncovertarget_valuepositive. dst_positive_code_g009_definitioncovertarget_value = ff_q_pvs_g009_definitioncovertarget_valuepositive * S ((S (ssr_target_g009_definitioncover)) * dst_positive_scale_g009_definitioncovertarget_value) + (dst_positive_g009_definitioncovertarget_value))) /\ (((((exists ff_h_pvs_g009_definitioncovertarget_valuenegative. ff_h_pvs_g009_definitioncovertarget_valuenegative + S (dst_negative_g009_definitioncovertarget_value) = S ((S (ssr_target_g009_definitioncover)) * dst_negative_scale_g009_definitioncovertarget_value)) /\ exists ff_q_pvs_g009_definitioncovertarget_valuenegative. dst_negative_code_g009_definitioncovertarget_value = ff_q_pvs_g009_definitioncovertarget_valuenegative * S ((S (ssr_target_g009_definitioncover)) * dst_negative_scale_g009_definitioncovertarget_value) + (dst_negative_g009_definitioncovertarget_value))) /\ (exists ge_balance_positive_g009_definitioncovertarget_valuevalue ge_balance_negative_g009_definitioncovertarget_valuevalue. (((((ssr_value_g009_definitioncover) = 2 * (ge_balance_positive_g009_definitioncovertarget_valuevalue) /\ (ge_balance_negative_g009_definitioncovertarget_valuevalue) = 0) \/ exists ge_signed_half_g009_definitioncovertarget_valuevaluedecode. (((ssr_value_g009_definitioncover) = 2 * ge_signed_half_g009_definitioncovertarget_valuevaluedecode + 1 /\ (ge_balance_positive_g009_definitioncovertarget_valuevalue) = 0) /\ (ge_balance_negative_g009_definitioncovertarget_valuevalue) = S ge_signed_half_g009_definitioncovertarget_valuevaluedecode))) /\ ((dst_positive_g009_definitioncovertarget_value) + ge_balance_negative_g009_definitioncovertarget_valuevalue = (dst_negative_g009_definitioncovertarget_value) + ge_balance_positive_g009_definitioncovertarget_valuevalue))))))))) -> ~(ssr_value_g009_definitioncover=0) -> exists ssr_source_g009_definitioncover. ((exists pvs_gap_g009_definitioncoversource_bound. pvs_gap_g009_definitioncoversource_bound + S (ssr_source_g009_definitioncover) = ((L))) /\ (((((exists ff_h_pvs_g009_definitioncovermap. ff_h_pvs_g009_definitioncovermap + S (ssr_target_g009_definitioncover) = S ((S (ssr_source_g009_definitioncover)) * (s))) /\ exists ff_q_pvs_g009_definitioncovermap. (r) = ff_q_pvs_g009_definitioncovermap * S ((S (ssr_source_g009_definitioncover)) * (s)) + (ssr_target_g009_definitioncover))) /\ (exists dst_positive_code_g009_definitioncoversource_value dst_positive_scale_g009_definitioncoversource_value dst_negative_code_g009_definitioncoversource_value dst_negative_scale_g009_definitioncoversource_value dst_positive_g009_definitioncoversource_value dst_negative_g009_definitioncoversource_value. ((((A)) = (((((dst_positive_code_g009_definitioncoversource_value) + (dst_positive_scale_g009_definitioncoversource_value)) * S ((dst_positive_code_g009_definitioncoversource_value) + (dst_positive_scale_g009_definitioncoversource_value)) + ((dst_positive_scale_g009_definitioncoversource_value) + (dst_positive_scale_g009_definitioncoversource_value))) + (((dst_negative_code_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value)) * S ((dst_negative_code_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value)) + ((dst_negative_scale_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value)))) * S ((((dst_positive_code_g009_definitioncoversource_value) + (dst_positive_scale_g009_definitioncoversource_value)) * S ((dst_positive_code_g009_definitioncoversource_value) + (dst_positive_scale_g009_definitioncoversource_value)) + ((dst_positive_scale_g009_definitioncoversource_value) + (dst_positive_scale_g009_definitioncoversource_value))) + (((dst_negative_code_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value)) * S ((dst_negative_code_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value)) + ((dst_negative_scale_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value)))) + ((((dst_negative_code_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value)) * S ((dst_negative_code_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value)) + ((dst_negative_scale_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value))) + (((dst_negative_code_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value)) * S ((dst_negative_code_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value)) + ((dst_negative_scale_g009_definitioncoversource_value) + (dst_negative_scale_g009_definitioncoversource_value)))))) /\ (((((exists ff_h_pvs_g009_definitioncoversource_valuepositive. ff_h_pvs_g009_definitioncoversource_valuepositive + S (dst_positive_g009_definitioncoversource_value) = S ((S (ssr_source_g009_definitioncover)) * dst_positive_scale_g009_definitioncoversource_value)) /\ exists ff_q_pvs_g009_definitioncoversource_valuepositive. dst_positive_code_g009_definitioncoversource_value = ff_q_pvs_g009_definitioncoversource_valuepositive * S ((S (ssr_source_g009_definitioncover)) * dst_positive_scale_g009_definitioncoversource_value) + (dst_positive_g009_definitioncoversource_value))) /\ (((((exists ff_h_pvs_g009_definitioncoversource_valuenegative. ff_h_pvs_g009_definitioncoversource_valuenegative + S (dst_negative_g009_definitioncoversource_value) = S ((S (ssr_source_g009_definitioncover)) * dst_negative_scale_g009_definitioncoversource_value)) /\ exists ff_q_pvs_g009_definitioncoversource_valuenegative. dst_negative_code_g009_definitioncoversource_value = ff_q_pvs_g009_definitioncoversource_valuenegative * S ((S (ssr_source_g009_definitioncover)) * dst_negative_scale_g009_definitioncoversource_value) + (dst_negative_g009_definitioncoversource_value))) /\ (exists ge_balance_positive_g009_definitioncoversource_valuevalue ge_balance_negative_g009_definitioncoversource_valuevalue. (((((ssr_value_g009_definitioncover) = 2 * (ge_balance_positive_g009_definitioncoversource_valuevalue) /\ (ge_balance_negative_g009_definitioncoversource_valuevalue) = 0) \/ exists ge_signed_half_g009_definitioncoversource_valuevaluedecode. (((ssr_value_g009_definitioncover) = 2 * ge_signed_half_g009_definitioncoversource_valuevaluedecode + 1 /\ (ge_balance_positive_g009_definitioncoversource_valuevalue) = 0) /\ (ge_balance_negative_g009_definitioncoversource_valuevalue) = S ge_signed_half_g009_definitioncoversource_valuevaluedecode))) /\ ((dst_positive_g009_definitioncoversource_value) + ge_balance_negative_g009_definitioncoversource_valuevalue = (dst_negative_g009_definitioncoversource_value) + ge_balance_positive_g009_definitioncoversource_valuevalue))))))))))))))))))))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

Definitions depending on this notation

none

Checked theorems using this definition