TS000G

two_strict_values_sum_below_double

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Two natural values each strictly below p have sum strictly below 2p.

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.

Exact expanded first-order arithmetic statement

forall a b p. (exists ftsp_gap_sum_first. ftsp_gap_sum_first + S (a) = p) -> (exists ftsp_gap_sum_second. ftsp_gap_sum_second + S (b) = p) -> (exists ftsp_gap_sum_result. ftsp_gap_sum_result + S (a + b) = p + p)

Constructive proof overview

Generated structural guide

Two natural values each strictly below p have sum strictly below 2p.

The unchanged tactic script uses 3 declared prerequisites and contains 23 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized add_succ_left Stable theorem; checked-use authorized

Direct dependents

TS000H prime_floor_bounded_two_square_norm_below_double

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

23 script commands · 9 reading checkpoints · 0 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.

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 p
  4. L4
    intro hfirst
  5. L5
    intro hsecond
02Separate the logical casesL6–7

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

  1. L6
    cases hfirst
  2. L7
    cases hsecond
03Construct an explicit witnessL8–8

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

  1. L8
    exists x + x1 + 1
04Calculate and transport equalitiesL9–17

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

  1. L9
    rewrite <- hfirst_witness
  2. L10
    rewrite <- hsecond_witness
  3. L11
    simp [add_assoc, add_comm, add_succ_left]
  4. L12
    congr
  5. L13
    congr
  6. L14
    congr
  7. L15
    refl
  8. L16
    trans (x + b) + a
  9. L17
    symm
05Use earlier factsL18–18

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

  1. L18
    apply add_assoc
06Calculate and transport equalitiesL19–20

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

  1. L19
    trans (b + x) + a
  2. L20
    congr
07Use earlier factsL21–21

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

  1. L21
    apply add_comm
08Calculate and transport equalitiesL22–22

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

  1. L22
    refl
09Use earlier factsL23–23

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

  1. L23
    apply add_assoc

Library-wide reading audit

Original exact command ledger · 23 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro p
  4. 0004intro hfirst
  5. 0005intro hsecond
  6. 0006cases hfirst
  7. 0007cases hsecond
  8. 0008exists x + x1 + 1
  9. 0009rewrite <- hfirst_witness
  10. 0010rewrite <- hsecond_witness
  11. 0011simp [add_assoc, add_comm, add_succ_left]
  12. 0012congr
  13. 0013congr
  14. 0014congr
  15. 0015refl
  16. 0016trans (x + b) + a
  17. 0017symm
  18. 0018apply add_assoc
  19. 0019trans (b + x) + a
  20. 0020congr
  21. 0021apply add_comm
  22. 0022refl
  23. 0023apply add_assoc