DL003D — matrix_rank_selector_decidable

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.

This branch proves the finite determinant/rank/span substrate. It does not claim Smith or Hermite normal form, lattice index equals determinant, determinant multiplicativity, lattice reduction, or geometry-of-numbers theorems.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ l. ∀ B. FiniteMatrixSelector(b,c,l,B) ∨ ¬FiniteMatrixSelector(b,c,l,B)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

FiniteMatrixSelector(b,c,l,B) · 2 Lt(a,b) · 8 BetaAt(b,c,i,x) · 6

Actual proof prerequisites

matrix_rank_bounded_prefix_decidable matrix_rank_injective_prefix_decidable

Original expanded first-order statement

forall b c l B. (((forall fom_index_mrf_selector_yesbound. (exists fom_gap_mrf_selector_yesbound_index_bound. fom_gap_mrf_selector_yesbound_index_bound + S (fom_index_mrf_selector_yesbound) = l) -> exists fom_value_mrf_selector_yesbound. ((((exists fom_beta_height_mrf_selector_yesbound_entry. fom_beta_height_mrf_selector_yesbound_entry + S (fom_value_mrf_selector_yesbound) = S ((S (fom_index_mrf_selector_yesbound)) * c)) /\ exists fom_beta_quotient_mrf_selector_yesbound_entry. b = fom_beta_quotient_mrf_selector_yesbound_entry * S ((S (fom_index_mrf_selector_yesbound)) * c) + (fom_value_mrf_selector_yesbound))) /\ (exists fom_gap_mrf_selector_yesbound_value_bound. fom_gap_mrf_selector_yesbound_value_bound + S (fom_value_mrf_selector_yesbound) = B))) /\ (forall mdr_i_selector_yesdistinct mdr_j_selector_yesdistinct mdr_a_selector_yesdistinct. (exists mdr_gap_selector_yesdistincti. mdr_gap_selector_yesdistincti + S (mdr_i_selector_yesdistinct) = (l)) -> (exists mdr_gap_selector_yesdistinctj. mdr_gap_selector_yesdistinctj + S (mdr_j_selector_yesdistinct) = (l)) -> (((exists ff_h_mdr_selector_yesdistinctfirst. ff_h_mdr_selector_yesdistinctfirst + S (mdr_a_selector_yesdistinct) = S ((S (mdr_i_selector_yesdistinct)) * c)) /\ exists ff_q_mdr_selector_yesdistinctfirst. b = ff_q_mdr_selector_yesdistinctfirst * S ((S (mdr_i_selector_yesdistinct)) * c) + (mdr_a_selector_yesdistinct))) -> (((exists ff_h_mdr_selector_yesdistinctsecond. ff_h_mdr_selector_yesdistinctsecond + S (mdr_a_selector_yesdistinct) = S ((S (mdr_j_selector_yesdistinct)) * c)) /\ exists ff_q_mdr_selector_yesdistinctsecond. b = ff_q_mdr_selector_yesdistinctsecond * S ((S (mdr_j_selector_yesdistinct)) * c) + (mdr_a_selector_yesdistinct))) -> mdr_i_selector_yesdistinct = mdr_j_selector_yesdistinct))) \/ ~(((forall fom_index_mrf_selector_nobound. (exists fom_gap_mrf_selector_nobound_index_bound. fom_gap_mrf_selector_nobound_index_bound + S (fom_index_mrf_selector_nobound) = l) -> exists fom_value_mrf_selector_nobound. ((((exists fom_beta_height_mrf_selector_nobound_entry. fom_beta_height_mrf_selector_nobound_entry + S (fom_value_mrf_selector_nobound) = S ((S (fom_index_mrf_selector_nobound)) * c)) /\ exists fom_beta_quotient_mrf_selector_nobound_entry. b = fom_beta_quotient_mrf_selector_nobound_entry * S ((S (fom_index_mrf_selector_nobound)) * c) + (fom_value_mrf_selector_nobound))) /\ (exists fom_gap_mrf_selector_nobound_value_bound. fom_gap_mrf_selector_nobound_value_bound + S (fom_value_mrf_selector_nobound) = B))) /\ (forall mdr_i_selector_nodistinct mdr_j_selector_nodistinct mdr_a_selector_nodistinct. (exists mdr_gap_selector_nodistincti. mdr_gap_selector_nodistincti + S (mdr_i_selector_nodistinct) = (l)) -> (exists mdr_gap_selector_nodistinctj. mdr_gap_selector_nodistinctj + S (mdr_j_selector_nodistinct) = (l)) -> (((exists ff_h_mdr_selector_nodistinctfirst. ff_h_mdr_selector_nodistinctfirst + S (mdr_a_selector_nodistinct) = S ((S (mdr_i_selector_nodistinct)) * c)) /\ exists ff_q_mdr_selector_nodistinctfirst. b = ff_q_mdr_selector_nodistinctfirst * S ((S (mdr_i_selector_nodistinct)) * c) + (mdr_a_selector_nodistinct))) -> (((exists ff_h_mdr_selector_nodistinctsecond. ff_h_mdr_selector_nodistinctsecond + S (mdr_a_selector_nodistinct) = S ((S (mdr_j_selector_nodistinct)) * c)) /\ exists ff_q_mdr_selector_nodistinctsecond. b = ff_q_mdr_selector_nodistinctsecond * S ((S (mdr_j_selector_nodistinct)) * c) + (mdr_a_selector_nodistinct))) -> mdr_i_selector_nodistinct = mdr_j_selector_nodistinct)))

Complete tactic proof in conservative notation

All 31 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

31 script commands · 14 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 (2)

matrix rank bounded prefix decidable matrix rank injective prefix decidable

01Fix variables and assumptionsL1–4

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

L1
intro b
L2
intro c
L3
intro l
L4
intro B

02Establish hboundL5–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply matrix rank bounded prefix decidable.

L5
have hbound : (∀ x. Lt(x,l) → ∃ y. BetaAt(b,c,x,y) ∧ Lt(y,B)) ∨ ¬(∀ x. Lt(x,l) → ∃ y. BetaAt(b,c,x,y) ∧ Lt(y,B))Definitions: Lt(x,l)BetaAt(b,c,x,y)Lt(y,B)Original native command in the exact edition
L6
specialize matrix_rank_bounded_prefix_decidable (b)
L7
specialize matrix_rank_bounded_prefix_decidable (c)
L8
specialize matrix_rank_bounded_prefix_decidable (l)
L9
specialize matrix_rank_bounded_prefix_decidable (B)
L10
apply matrix_rank_bounded_prefix_decidable

03Separate the logical casesL11–11