Original question and givens
Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-1,0), R=(2,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
G001 — Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-1,0), R=(2,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).
Step-by-step derivation
S001 · implies
Before
source roles = [qf-density12-length-triangle-defect, density12_length_triangle_defect, integer_euclidean_axis]
After
derived = [[path_via_middle, direct_endpoint_length, path_minus_direct, equality]]
Conditions: none
Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources.
Step identifiers and dependencies
Rule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001
S002 · calculation
Before
Axis ORIGINAL y minus x := sub(-1, 0)
After
Axis ORIGINAL y minus x = -1
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
Axis ORIGINAL z minus y := sub(2, -1)
After
Axis ORIGINAL z minus y = 3
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
Axis ORIGINAL z minus x := sub(2, 0)
After
Axis ORIGINAL z minus x = 2
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
Axis PQ ORIGINAL negative branch negation := sub(0, -1)
After
Axis PQ ORIGINAL negative branch negation = 1
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
source roles = [Axis PQ, -1]
After
derived = [No, 1]
Conditions: none
Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator.
Step identifiers and dependencies
Rule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S005
S007 · implies
Before
source roles = [Axis QR, 3]
After
derived = [Yes, 3]
Conditions: none
Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator.
Step identifiers and dependencies
Rule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S006
S008 · implies
Before
source roles = [Axis PR, 2]
After
derived = [Yes, 2]
Conditions: none
Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator.
Step identifiers and dependencies
Rule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S007
S009 · calculation
Before
Axis ORIGINAL path via middle := sum(1, 3)
After
Axis ORIGINAL path via middle = 4
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
Axis ORIGINAL path minus direct := sub(4, 2)
After
Axis ORIGINAL path minus direct = 2
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · implies
Before
source roles = [0, -1, 2]
After
derived = [4, 2, 2, No]
Conditions: none
Take the absolute lengths of BOTH legs before adding;the direct endpoint length uses only abs(z-x).The gap is0 when the middle lies in the CLOSED endpoint segment,including endpoints,and positive outside.This exact axis instance is not a general infinite-metric proof.
Step identifiers and dependencies
Rule: Length_axis_PATH_not_endpoint_distance_closed_segment_equality · Uses: G001, S010