BT00UX · Bertrand theorem

primorial_factor_prefix_restrict_add

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

Restrict a selector prefix of length a+l to its first a entries.

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. ∀ l. (∀ x. Lt(x,a + l) → ∃ y. BetaAt(b,c,x,y) ∧ (Prime(S x) ∧ y = S x ∨ ¬Prime(S x) ∧ y = 1)) → ∀ x. Lt(x,a) → ∃ y. BetaAt(b,c,x,y) ∧ (Prime(S x) ∧ y = S x ∨ ¬Prime(S x) ∧ y = 1)

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

PD0002 Lt PD0004 Prime PD0013 BetaAt

8 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall a b c l. (forall bpr_index_bpfpra_source. (exists bpr_gap_bpfpra_source_bound. bpr_gap_bpfpra_source_bound + S (bpr_index_bpfpra_source) = a + l) -> exists bpr_value_bpfpra_source. ((((exists bpr_height_bpfpra_source_decoded. bpr_height_bpfpra_source_decoded + S (bpr_value_bpfpra_source) = S ((S (bpr_index_bpfpra_source)) * c)) /\ exists bpr_quotient_bpfpra_source_decoded. b = bpr_quotient_bpfpra_source_decoded * S ((S (bpr_index_bpfpra_source)) * c) + (bpr_value_bpfpra_source))) /\ (((((~(S (bpr_index_bpfpra_source) = 1) /\ forall bpr_left_bpfpra_source_choice_prime bpr_right_bpfpra_source_choice_prime. S (bpr_index_bpfpra_source) = bpr_left_bpfpra_source_choice_prime * bpr_right_bpfpra_source_choice_prime -> bpr_left_bpfpra_source_choice_prime = 1 \/ bpr_right_bpfpra_source_choice_prime = 1)) /\ bpr_value_bpfpra_source = S (bpr_index_bpfpra_source)) \/ (~((~(S (bpr_index_bpfpra_source) = 1) /\ forall bpr_left_bpfpra_source_choice_prime bpr_right_bpfpra_source_choice_prime. S (bpr_index_bpfpra_source) = bpr_left_bpfpra_source_choice_prime * bpr_right_bpfpra_source_choice_prime -> bpr_left_bpfpra_source_choice_prime = 1 \/ bpr_right_bpfpra_source_choice_prime = 1)) /\ bpr_value_bpfpra_source = 1))))) -> (forall bpr_index_bpfpra_target. (exists bpr_gap_bpfpra_target_bound. bpr_gap_bpfpra_target_bound + S (bpr_index_bpfpra_target) = a) -> exists bpr_value_bpfpra_target. ((((exists bpr_height_bpfpra_target_decoded. bpr_height_bpfpra_target_decoded + S (bpr_value_bpfpra_target) = S ((S (bpr_index_bpfpra_target)) * c)) /\ exists bpr_quotient_bpfpra_target_decoded. b = bpr_quotient_bpfpra_target_decoded * S ((S (bpr_index_bpfpra_target)) * c) + (bpr_value_bpfpra_target))) /\ (((((~(S (bpr_index_bpfpra_target) = 1) /\ forall bpr_left_bpfpra_target_choice_prime bpr_right_bpfpra_target_choice_prime. S (bpr_index_bpfpra_target) = bpr_left_bpfpra_target_choice_prime * bpr_right_bpfpra_target_choice_prime -> bpr_left_bpfpra_target_choice_prime = 1 \/ bpr_right_bpfpra_target_choice_prime = 1)) /\ bpr_value_bpfpra_target = S (bpr_index_bpfpra_target)) \/ (~((~(S (bpr_index_bpfpra_target) = 1) /\ forall bpr_left_bpfpra_target_choice_prime bpr_right_bpfpra_target_choice_prime. S (bpr_index_bpfpra_target) = bpr_left_bpfpra_target_choice_prime * bpr_right_bpfpra_target_choice_prime -> bpr_left_bpfpra_target_choice_prime = 1 \/ bpr_right_bpfpra_target_choice_prime = 1)) /\ bpr_value_bpfpra_target = 1)))))

Proof neighborhood

Direct theorem prerequisites

BT0013 le_add_right BT001D lt_of_lt_of_le

Direct theorem dependents

BT00UY primorial_prefix_interval_split

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

Read the argument

Proof checkpoints

16 script commands · 2 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.

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 (2)
le add rightlt of lt of le
01Fix variables and assumptionsL1–7

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 l
  5. L5
    intro hsource
  6. L6
    intro i
  7. L7
    intro hi
02Use earlier factsL8–16

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

  1. L8
    apply hsource
  2. L9
    specialize lt_of_lt_of_le i
  3. L10
    specialize lt_of_lt_of_le a
  4. L11
    specialize lt_of_lt_of_le (a + l)
  5. L12
    apply lt_of_lt_of_le
  6. L13
    exact hi
  7. L14
    specialize le_add_right a
  8. L15
    specialize le_add_right l
  9. L16
    exact le_add_right

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro hsource
  6. 0006intro i
  7. 0007intro hi
  8. 0008apply hsource
  9. 0009specialize lt_of_lt_of_le i
  10. 0010specialize lt_of_lt_of_le a
  11. 0011specialize lt_of_lt_of_le (a + l)
  12. 0012apply lt_of_lt_of_le
  13. 0013exact hi
  14. 0014specialize le_add_right a
  15. 0015specialize le_add_right l
  16. 0016exact le_add_right