SL000J · theorem body

reflected_double_above_odd_half

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

An exact reflected doubled value necessarily lies above the odd half.

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

∀ p. ∀ h. ∀ x. ∀ m. p = 2 · h + 1 → Le(m,h) → 2 · x + m = p → Lt(h,2 · x)

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

PD0001 Le PD0002 Lt

In local proof propositions

none
Exact expanded first-order statement
forall p h x m. p = 2 * h + 1 -> (exists qst_le_magnitude. qst_le_magnitude + (m) = (h)) -> 2 * x + m = p -> (exists qst_lt_doubled. qst_lt_doubled + S (h) = (2 * x))

Proof neighborhood

Direct theorem prerequisites

add_right_cancel · Stable closed mul_comm · Stable closed zero_add · Stable closed add_assoc · Stable closed add_comm · Stable closed add_succ_left · Stable closed

Direct theorem dependents

SL000K doubling_gauss_initial_segment_complement

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

46 script commands · 19 reading checkpoints · 1 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro p
  2. L2
    intro h
  3. L3
    intro x
  4. L4
    intro m
  5. L5
    intro hpodd
  6. L6
    intro hmbelow
  7. L7
    intro hreflection
02Separate the logical casesL8–8

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

  1. L8
    cases hmbelow
03Establish hdoubleL9–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul comm.

  1. L9
    have hdouble : h + h = 2 * h
  2. L10
    trans h * 2
  3. L11
    simp [zero_add]
  4. L12
    specialize mul_comm h
  5. L13
    specialize mul_comm 2
  6. L14
    apply mul_comm
04Construct an explicit witnessL15–15

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

  1. L15
    exists x1
05Use earlier factsL16–19

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

  1. L16
    specialize add_right_cancel (x1 + S h)
  2. L17
    specialize add_right_cancel (2 * x)
  3. L18
    specialize add_right_cancel m
  4. L19
    apply add_right_cancel
06Calculate and transport equalitiesL20–22

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

  1. L20
    trans S (h + (x1 + m))
  2. L21
    trans S (x1 + h) + m
  3. L22
    congr
07Use earlier factsL23–23

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

  1. L23
    apply PA4
08Calculate and transport equalitiesL24–25

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

  1. L24
    refl
  2. L25
    trans S ((x1 + h) + m)
09Use earlier factsL26–26

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

  1. L26
    apply add_succ_left
10Calculate and transport equalitiesL27–28

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

  1. L27
    congr
  2. L28
    trans x1 + (h + m)
11Use earlier factsL29–29

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

  1. L29
    apply add_assoc
12Calculate and transport equalitiesL30–32

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

  1. L30
    trans x1 + (m + h)
  2. L31
    congr
  3. L32
    refl
13Use earlier factsL33–33

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

  1. L33
    apply add_comm
14Calculate and transport equalitiesL34–35

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

  1. L34
    trans (x1 + m) + h
  2. L35
    symm
15Use earlier factsL36–37

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

  1. L36
    apply add_assoc
  2. L37
    apply add_comm
16Calculate and transport equalitiesL38–43

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

  1. L38
    rewrite hmbelow_witness
  2. L39
    rewrite hdouble
  3. L40
    trans 2 * h + 1
  4. L41
    simp
  5. L42
    trans p
  6. L43
    symm
17Use earlier factsL44–44

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

  1. L44
    exact hpodd
18Calculate and transport equalitiesL45–45

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

  1. L45
    symm
19Use earlier factsL46–46

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

  1. L46
    exact hreflection

Library-wide reading audit

Original defined command ledger · 46 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro x
  4. 0004intro m
  5. 0005intro hpodd
  6. 0006intro hmbelow
  7. 0007intro hreflection
  8. 0008cases hmbelow
  9. 0009have hdouble : h + h = 2 * h
  10. 0010trans h * 2
  11. 0011simp [zero_add]
  12. 0012specialize mul_comm h
  13. 0013specialize mul_comm 2
  14. 0014apply mul_comm
  15. 0015exists x1
  16. 0016specialize add_right_cancel (x1 + S h)
  17. 0017specialize add_right_cancel (2 * x)
  18. 0018specialize add_right_cancel m
  19. 0019apply add_right_cancel
  20. 0020trans S (h + (x1 + m))
  21. 0021trans S (x1 + h) + m
  22. 0022congr
  23. 0023apply PA4
  24. 0024refl
  25. 0025trans S ((x1 + h) + m)
  26. 0026apply add_succ_left
  27. 0027congr
  28. 0028trans x1 + (h + m)
  29. 0029apply add_assoc
  30. 0030trans x1 + (m + h)
  31. 0031congr
  32. 0032refl
  33. 0033apply add_comm
  34. 0034trans (x1 + m) + h
  35. 0035symm
  36. 0036apply add_assoc
  37. 0037apply add_comm
  38. 0038rewrite hmbelow_witness
  39. 0039rewrite hdouble
  40. 0040trans 2 * h + 1
  41. 0041simp
  42. 0042trans p
  43. 0043symm
  44. 0044exact hpodd
  45. 0045symm
  46. 0046exact hreflection