TS0034 · theorem body

even_double_sum_reflects_even_tail

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

If an additive even block plus a tail is even, the tail has its own constructive additive half witness.

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

forall a b h. (a + a) + b = h + h -> exists k. b = k + k

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

none

In local proof propositions

PD0009 Even PD0010 Odd
Exact expanded first-order statement
forall a b h. (a + a) + b = h + h -> exists k. b = k + k

Proof neighborhood

Direct theorem prerequisites

TS0033 pairing_double_equals_two_mul even_sum_parity_cases · Alpha closed even_not_odd · Stable closed

Direct theorem dependents

TS0036 square_factor_even_valuation_reflects_cofactor

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

31 script commands · 17 reading checkpoints · 3 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)
pairing double equals two mul
01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro h
  4. L4
    intro heven
02Establish hsumL5–5

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

  1. L5
    have hsum : Even(a + a + b)Definitions: Even(a + a + b)Original native command in the exact edition
03Construct an explicit witnessL6–6

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

  1. L6
    exists h
04Calculate and transport equalitiesL7–7

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

  1. L7
    trans h + h
05Use earlier factsL8–9

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

  1. L8
    exact heven
  2. L9
    apply pairing_double_equals_two_mul
06Establish hdoubleL10–10

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

  1. L10
    have hdouble : Even(a + a)Definitions: Even(a + a)Original native command in the exact edition
07Construct an explicit witnessL11–11

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

  1. L11
    exists a
08Use earlier factsL12–12

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

  1. L12
    apply pairing_double_equals_two_mul
09Establish hcasesL13–17

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 hcases : Even(a + a) ∧ Even(b) ∨ Odd(a + a) ∧ Odd(b)Definitions: Even(a + a)Even(b)Odd(a + a)Odd(b)Original native command in the exact edition
  2. L14
    specialize even_sum_parity_cases (a + a)
  3. L15
    specialize even_sum_parity_cases b
  4. L16
    apply even_sum_parity_cases
  5. L17
    exact hsum
10Separate the logical casesL18–20

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

  1. L18
    cases hcases
  2. L19
    cases hcases_left
  3. L20
    cases hcases_left_right
11Construct an explicit witnessL21–21

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

  1. L21
    exists x
12Calculate and transport equalitiesL22–22

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

  1. L22
    trans 2 * x
13Use earlier factsL23–23

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

  1. L23
    exact hcases_left_right_witness
14Calculate and transport equalitiesL24–24

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

  1. L24
    symm
15Use earlier factsL25–25

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

  1. L25
    apply pairing_double_equals_two_mul
16Separate the logical casesL26–27

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

  1. L26
    cases hcases_right
  2. L27
    exfalso
17Use earlier factsL28–31

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

  1. L28
    specialize even_not_odd (a + a)
  2. L29
    apply even_not_odd
  3. L30
    exact hdouble
  4. L31
    exact hcases_right_left

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro heven
  5. 0005have hsum : Even(a + a + b)
    Exact native replay linehave hsum : exists k. (a + a) + b = 2 * k
  6. 0006exists h
  7. 0007trans h + h
  8. 0008exact heven
  9. 0009apply pairing_double_equals_two_mul
  10. 0010have hdouble : Even(a + a)
    Exact native replay linehave hdouble : exists k. a + a = 2 * k
  11. 0011exists a
  12. 0012apply pairing_double_equals_two_mul
  13. 0013have hcases : Even(a + a) ∧ Even(b) ∨ Odd(a + a) ∧ Odd(b)
    Exact native replay linehave hcases : (((exists k. a + a = 2 * k) /\ (exists k. b = 2 * k)) \/ ((exists k. a + a = 2 * k + 1) /\ (exists k. b = 2 * k + 1)))
  14. 0014specialize even_sum_parity_cases (a + a)
  15. 0015specialize even_sum_parity_cases b
  16. 0016apply even_sum_parity_cases
  17. 0017exact hsum
  18. 0018cases hcases
  19. 0019cases hcases_left
  20. 0020cases hcases_left_right
  21. 0021exists x
  22. 0022trans 2 * x
  23. 0023exact hcases_left_right_witness
  24. 0024symm
  25. 0025apply pairing_double_equals_two_mul
  26. 0026cases hcases_right
  27. 0027exfalso
  28. 0028specialize even_not_odd (a + a)
  29. 0029apply even_not_odd
  30. 0030exact hdouble
  31. 0031exact hcases_right_left