FS003W · theorem body

four_square_parity_even_coordinate_pair_selection

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

Every even sum of four naturals constructively selects one of the three partitions into two equal-parity pairs.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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.

Statement with defined notation

∀ a. ∀ b. ∀ c. ∀ d. Even(a + b + c + d) → (Even(a) ∧ Even(b) ∨ Odd(a) ∧ Odd(b)) ∧ (Even(c) ∧ Even(d) ∨ Odd(c) ∧ Odd(d)) ∨ ((Even(a) ∧ Even(c) ∨ Odd(a) ∧ Odd(c)) ∧ (Even(b) ∧ Even(d) ∨ Odd(b) ∧ Odd(d)) ∨ (Even(a) ∧ Even(d) ∨ Odd(a) ∧ Odd(d)) ∧ (Even(b) ∧ Even(c) ∨ Odd(b) ∧ Odd(c)))

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0009 Even PD0010 Odd

In local proof propositions

PD0009 Even PD0010 Odd
Exact expanded first-order statement
forall a b c d. (exists fsps_even_selection_sum. (a + b + c + d) = 2 * fsps_even_selection_sum) -> (((((((exists fsd_even_first_fsps_coordinate_selection_ab. a = 2 * fsd_even_first_fsps_coordinate_selection_ab) /\ (exists fsd_even_second_fsps_coordinate_selection_ab. b = 2 * fsd_even_second_fsps_coordinate_selection_ab)) \/ ((exists fsd_odd_first_fsps_coordinate_selection_ab. a = 2 * fsd_odd_first_fsps_coordinate_selection_ab + 1) /\ (exists fsd_odd_second_fsps_coordinate_selection_ab. b = 2 * fsd_odd_second_fsps_coordinate_selection_ab + 1)))) /\ ((((exists fsd_even_first_fsps_coordinate_selection_cd. c = 2 * fsd_even_first_fsps_coordinate_selection_cd) /\ (exists fsd_even_second_fsps_coordinate_selection_cd. d = 2 * fsd_even_second_fsps_coordinate_selection_cd)) \/ ((exists fsd_odd_first_fsps_coordinate_selection_cd. c = 2 * fsd_odd_first_fsps_coordinate_selection_cd + 1) /\ (exists fsd_odd_second_fsps_coordinate_selection_cd. d = 2 * fsd_odd_second_fsps_coordinate_selection_cd + 1))))) \/ (((((((exists fsd_even_first_fsps_coordinate_selection_crossed_ac. a = 2 * fsd_even_first_fsps_coordinate_selection_crossed_ac) /\ (exists fsd_even_second_fsps_coordinate_selection_crossed_ac. c = 2 * fsd_even_second_fsps_coordinate_selection_crossed_ac)) \/ ((exists fsd_odd_first_fsps_coordinate_selection_crossed_ac. a = 2 * fsd_odd_first_fsps_coordinate_selection_crossed_ac + 1) /\ (exists fsd_odd_second_fsps_coordinate_selection_crossed_ac. c = 2 * fsd_odd_second_fsps_coordinate_selection_crossed_ac + 1)))) /\ ((((exists fsd_even_first_fsps_coordinate_selection_crossed_bd. b = 2 * fsd_even_first_fsps_coordinate_selection_crossed_bd) /\ (exists fsd_even_second_fsps_coordinate_selection_crossed_bd. d = 2 * fsd_even_second_fsps_coordinate_selection_crossed_bd)) \/ ((exists fsd_odd_first_fsps_coordinate_selection_crossed_bd. b = 2 * fsd_odd_first_fsps_coordinate_selection_crossed_bd + 1) /\ (exists fsd_odd_second_fsps_coordinate_selection_crossed_bd. d = 2 * fsd_odd_second_fsps_coordinate_selection_crossed_bd + 1))))) \/ (((((exists fsd_even_first_fsps_coordinate_selection_crossed_ad. a = 2 * fsd_even_first_fsps_coordinate_selection_crossed_ad) /\ (exists fsd_even_second_fsps_coordinate_selection_crossed_ad. d = 2 * fsd_even_second_fsps_coordinate_selection_crossed_ad)) \/ ((exists fsd_odd_first_fsps_coordinate_selection_crossed_ad. a = 2 * fsd_odd_first_fsps_coordinate_selection_crossed_ad + 1) /\ (exists fsd_odd_second_fsps_coordinate_selection_crossed_ad. d = 2 * fsd_odd_second_fsps_coordinate_selection_crossed_ad + 1)))) /\ ((((exists fsd_even_first_fsps_coordinate_selection_crossed_bc. b = 2 * fsd_even_first_fsps_coordinate_selection_crossed_bc) /\ (exists fsd_even_second_fsps_coordinate_selection_crossed_bc. c = 2 * fsd_even_second_fsps_coordinate_selection_crossed_bc)) \/ ((exists fsd_odd_first_fsps_coordinate_selection_crossed_bc. b = 2 * fsd_odd_first_fsps_coordinate_selection_crossed_bc + 1) /\ (exists fsd_odd_second_fsps_coordinate_selection_crossed_bc. c = 2 * fsd_odd_second_fsps_coordinate_selection_crossed_bc + 1)))))))))

Proof neighborhood

Direct theorem prerequisites

add_assoc · Stable closed even_sum_parity_cases · Alpha closed FS003V four_square_parity_odd_blocks_crossed_selection

Direct theorem dependents

FS003X four_square_parity_even_norm_pair_selection

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

30 script commands · 11 reading checkpoints · 2 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (1)
four square parity odd blocks crossed selection
01Fix variables and assumptionsL1–5

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro heven
02Establish hgroupedL6–6

Establish this local claim before using it. It is not an additional assumption.

  1. L6
    have hgrouped : Even(a + b + (c + d))Definitions: Even(a + b + (c + d))Original native command in the exact edition
03Separate the logical casesL7–7

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L7
    cases heven
04Construct an explicit witnessL8–8

Supply the displayed value, then prove that it has the required property.

  1. L8
    exists x
05Calculate and transport equalitiesL9–10

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L9
    trans a + b + c + d
  2. L10
    symm
06Use earlier factsL11–12

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L11
    apply add_assoc
  2. L12
    exact heven_witness
07Establish hblocksL13–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply even sum parity cases.

  1. L13
    have hblocks : Even(a + b) ∧ Even(c + d) ∨ Odd(a + b) ∧ Odd(c + d)Definitions: Even(a + b)Even(c + d)Odd(a + b)Odd(c + d)Original native command in the exact edition
  2. L14
    specialize even_sum_parity_cases (a + b)
  3. L15
    specialize even_sum_parity_cases (c + d)
  4. L16
    apply even_sum_parity_cases
  5. L17
    exact hgrouped
08Separate the logical casesL18–21

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L18
    cases hblocks
  2. L19
    cases hblocks_left
  3. L20
    left
  4. L21
    split
09Use earlier factsL22–25

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L22
    apply even_sum_parity_cases
  2. L23
    exact hblocks_left_left
  3. L24
    apply even_sum_parity_cases
  4. L25
    exact hblocks_left_right
10Separate the logical casesL26–27

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L26
    cases hblocks_right
  2. L27
    right
11Use earlier factsL28–30

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L28
    apply four_square_parity_odd_blocks_crossed_selection
  2. L29
    exact hblocks_right_left
  3. L30
    exact hblocks_right_right

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro heven
  6. 0006have hgrouped : Even(a + b + (c + d))
    Exact native replay linehave hgrouped : exists fsps_even_grouped. ((a + b) + (c + d)) = 2 * fsps_even_grouped
  7. 0007cases heven
  8. 0008exists x
  9. 0009trans a + b + c + d
  10. 0010symm
  11. 0011apply add_assoc
  12. 0012exact heven_witness
  13. 0013have hblocks : Even(a + b) ∧ Even(c + d) ∨ Odd(a + b) ∧ Odd(c + d)
    Exact native replay linehave hblocks : (((exists fsps_even_block_first_even. (a + b) = 2 * fsps_even_block_first_even) /\ (exists fsps_even_block_second_even. (c + d) = 2 * fsps_even_block_second_even)) \/ ((exists fsps_odd_block_first_odd. (a + b) = 2 * fsps_odd_block_first_odd + 1) /\ (exists fsps_odd_block_second_odd. (c + d) = 2 * fsps_odd_block_second_odd + 1)))
  14. 0014specialize even_sum_parity_cases (a + b)
  15. 0015specialize even_sum_parity_cases (c + d)
  16. 0016apply even_sum_parity_cases
  17. 0017exact hgrouped
  18. 0018cases hblocks
  19. 0019cases hblocks_left
  20. 0020left
  21. 0021split
  22. 0022apply even_sum_parity_cases
  23. 0023exact hblocks_left_left
  24. 0024apply even_sum_parity_cases
  25. 0025exact hblocks_left_right
  26. 0026cases hblocks_right
  27. 0027right
  28. 0028apply four_square_parity_odd_blocks_crossed_selection
  29. 0029exact hblocks_right_left
  30. 0030exact hblocks_right_right