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
MX0046 · signed_support_incidence_row_sum_valueMX0047 · signed_support_incidence_column_sum_valueMX0048 · signed_support_incidence_row_sums_equalMX0049 · signed_support_incidence_column_sums_equalMX004A · signed_support_reindex_sum_equalMX004B · signed_support_reindex_sum_existsMX0055 · dirichlet_coprime_grid_support_reindex