ND0120

IntegerVectorEqual(ab,ac,db,dc,eb,ec,fb,fc,l)

Coordinatewise equality of signed differences by balanced equality; different positive/negative representations may denote the same integer vector.

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. Exact original first-admission records.

Definition in prerequisite notation

∀ ics_index_secondwave. ∀ ics_value0_secondwave. ∀ ics_value1_secondwave. ∀ ics_value2_secondwave. ∀ ics_value3_secondwave. Lt(ics_index_secondwave,l) → BetaAt(ab,ac,ics_index_secondwave,ics_value0_secondwave) → BetaAt(db,dc,ics_index_secondwave,ics_value1_secondwave) → BetaAt(eb,ec,ics_index_secondwave,ics_value2_secondwave) → BetaAt(fb,fc,ics_index_secondwave,ics_value3_secondwave) → ics_value0_secondwave + ics_value3_secondwave = ics_value2_secondwave + ics_value1_secondwave

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

Hygienic expanded first-order definition 
forall ics_index_secondwave ics_value0_secondwave ics_value1_secondwave ics_value2_secondwave ics_value3_secondwave. (exists ics_gap_secondwave_bound. ics_gap_secondwave_bound + S (ics_index_secondwave) = (l)) -> (((exists fs_h_ics_secondwave_at0. fs_h_ics_secondwave_at0 + S (ics_value0_secondwave) = S ((S (ics_index_secondwave)) * ac)) /\ exists fs_q_ics_secondwave_at0. ab = fs_q_ics_secondwave_at0 * S ((S (ics_index_secondwave)) * ac) + (ics_value0_secondwave))) -> (((exists fs_h_ics_secondwave_at1. fs_h_ics_secondwave_at1 + S (ics_value1_secondwave) = S ((S (ics_index_secondwave)) * dc)) /\ exists fs_q_ics_secondwave_at1. db = fs_q_ics_secondwave_at1 * S ((S (ics_index_secondwave)) * dc) + (ics_value1_secondwave))) -> (((exists fs_h_ics_secondwave_at2. fs_h_ics_secondwave_at2 + S (ics_value2_secondwave) = S ((S (ics_index_secondwave)) * ec)) /\ exists fs_q_ics_secondwave_at2. eb = fs_q_ics_secondwave_at2 * S ((S (ics_index_secondwave)) * ec) + (ics_value2_secondwave))) -> (((exists fs_h_ics_secondwave_at3. fs_h_ics_secondwave_at3 + S (ics_value3_secondwave) = S ((S (ics_index_secondwave)) * fc)) /\ exists fs_q_ics_secondwave_at3. fb = fs_q_ics_secondwave_at3 * S ((S (ics_index_secondwave)) * fc) + (ics_value3_secondwave))) -> ics_value0_secondwave + ics_value3_secondwave = ics_value2_secondwave + ics_value1_secondwave

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

Direct definition dependencies

BetaAt(b,c,i,x)Lt(a,b)

Definitions depending on this notation

IntegerVectorNegate(ab,ac,db,dc,pb,pc,nb,nc,l)IntegerMatrixVectorProduct(ab,ac,db,dc,eb,ec,fb,fc,w,r,pb,pc,nb,nc)IntegerMatrixEntrywiseEqual(ab,ac,bb,bc,eb,ec,fb,fc,r,w)

Checked theorems using this definition

DL006E · integer_vector_equal_reflexiveDL006F · integer_vector_equal_transitiveDL0070 · integer_vector_equal_negatedDL0072 · integer_vector_equal_zero_from_same_componentsDL0075 · integer_vector_add_transport_inputsDL0076 · integer_vector_add_functionalDL0078 · integer_matrix_vector_product_transportDL007D · integer_column_span_transportDL0081 · integer_column_span_negate_closedDL0084 · integer_column_span_negate_existsDL0085 · matrix_integer_vector_equality_restrictDL0086 · matrix_integer_vector_equality_symmetricDL008A · matrix_integer_signed_sum_balanceDL008B · matrix_integer_alternating_prefix_balanceDL008C · matrix_integer_cofactor_fold_balanceDL0092 · matrix_integer_cofactor_streams_from_recursionDL0093 · matrix_integer_first_row_equalityDL0094 · signed_recursive_determinant_integer_invariant