ND0143

SignedBalance(z,p,n)

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

The original canonical code z represents the integer difference p−n; these supplied components need not be normalized.

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

Definition in prerequisite notation

∃ sb_pos_lowerlayer. ∃ sb_neg_lowerlayer. SignedDecode(z,sb_pos_lowerlayer,sb_neg_lowerlayer) ∧ p + sb_neg_lowerlayer = n + sb_pos_lowerlayer

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

Hygienic expanded first-order definition 
exists sb_pos_lowerlayer sb_neg_lowerlayer. (((z = 2 * sb_pos_lowerlayer /\ sb_neg_lowerlayer = 0) \/ exists sd_half_lowerlayer. ((z = 2 * sd_half_lowerlayer + 1 /\ sb_pos_lowerlayer = 0) /\ sb_neg_lowerlayer = S sd_half_lowerlayer)) /\ p + sb_neg_lowerlayer = n + sb_pos_lowerlayer)

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

Direct definition dependencies

SignedDecode(z,p,n)

Definitions depending on this notation

ArithTable(N,F)ArithAt(F,i,z)SignedPrefixSum(F,l,z)

Checked theorems using this definition

SS0001 · divisor_signed_table_at_from_componentsSS0002 · divisor_signed_table_at_to_componentsSS0003 · divisor_signed_table_from_componentsSS0006 · divisor_signed_table_lookupSS0007 · divisor_signed_table_at_functionalSS0009 · divisor_signed_sum_from_componentsSS000A · divisor_signed_sum_to_componentsSS000B · divisor_signed_sum_exists_from_componentsSS000C · divisor_signed_sum_functionalSS000F · divisor_signed_balance_negateSS0010 · divisor_signed_balance_negate_introSS0011 · divisor_signed_negate_fixed_zeroSS0013 · divisor_signed_table_equality_component_balanceSS0015 · divisor_signed_sum_negation_transportSS0016 · divisor_signed_sum_successor_introSS0017 · divisor_signed_sum_successor_decomposeSS001A · divisor_signed_table_reindex_from_componentsSS001D · divisor_signed_sum_component_reindex