| import Mathlib |
| import OAI.GroupTheory.SimpleOvergroups.FirstFamilies |
| import OAI.GroupTheory.SimpleOvergroups.NamedActions |
| import OAI.GroupTheory.SimpleOvergroups.ThreeCodes |
|
|
| namespace OAI |
|
|
|
|
| namespace SimpleFInftyOvergroups.ContextualEncoding |
| open FiniteAlgebraicEnvelopes Bootstrap BoundedEncoding |
| noncomputable section |
| open scoped Classical |
|
|
| structure TestCertificate {A S Q : Type*} |
| (C : DelimiterCode A S) (f : List A → Part (Option Q)) (n : ℕ) : Prop where |
| total : ∀ w, (C.encode w).length ≤ n → (f w).Dom |
| nonzero : Search.value f [] ≠ none |
| contextual : CodeTable.Contextual C (Search.value f) n |
| ideal : CodeTable.ZeroIdeal C (Search.value f) n |
|
|
| namespace TestCertificate |
| variable {A S Q : Type*} [DecidableEq A] [DecidableEq Q] |
| (C : DelimiterCode A S) (f : List A → Part (Option Q)) {n : ℕ} |
|
|
| omit [DecidableEq A] [DecidableEq Q] in |
| theorem of_value (q : List A → Option Q) |
| (hf : ∀ w, (C.encode w).length ≤ n → q w ∈ f w) |
| (h0 : q [] ≠ none) (hc : CodeTable.Contextual C q n) (hz : CodeTable.ZeroIdeal C q n) : |
| TestCertificate C f n := by |
| have he (w : List A) (hw : (C.encode w).length ≤ n) : Search.value f w = q w := |
| Part.mem_unique (Search.value_mem (hf w hw).1) (hf w hw) |
| refine ⟨fun w hw => (hf w hw).1, ?_, ?_, ?_⟩ |
| · rw [he [] (Nat.zero_le _)] |
| exact h0 |
| · intro a b u v hu hv h |
| have hlu : (C.encode u).length ≤ n := by |
| simp only [DelimiterCode.encode_append,List.length_append] at hu |
| omega |
| have hlv : (C.encode v).length ≤ n := by |
| simp only [DelimiterCode.encode_append,List.length_append] at hv |
| omega |
| rw [he u hlu,he v hlv] at h |
| rw [he _ hu,he _ hv] |
| exact hc a b u v hu hv h |
| · intro a b u hu h |
| have hlu : (C.encode u).length ≤ n := by |
| simp only [DelimiterCode.encode_append,List.length_append] at hu |
| omega |
| rw [he u hlu] at h |
| rw [he _ hu] |
| exact hz a b u hu h |
| end TestCertificate |
|
|
| namespace GroupFamily |
| variable {G : Type*} [Group G] {k : ℕ} (s : Fin k → G) |
| theorem certificate (C : DelimiterCode (Fin k × Bool) ThreeCodes.Alphabet) (n : ℕ) : |
| TestCertificate C (test s) n := by |
| apply TestCertificate.of_value C (test s) (fun w => some (WordProblem.testedClass s w)) |
| · intro w hw; exact Part.mem_some _ |
| · exact Option.some_ne_none _ |
| · intro a b u v hu hv h |
| exact congrArg some (context s (Option.some.inj h) a b) |
| · intro a b u hu h |
| cases h |
| end GroupFamily |
| |
| namespace ProductFamily |
| variable {A : Type*} [DecidableEq A] {m n : ℕ} (C : WordCalculus (FreeMonoid A) m) |
| (code : DelimiterCode (Letter A) ThreeCodes.Alphabet) |
| (cost : A → ℕ) (hc : ∀ w, grade C w = wordCost cost w) |
| (hp : ∀ a, 10 * ProductFamily.cost cost a + 2 ≤ (code.block a).length) |
| (hm : n / 10 ≤ m) {f : List A → Part (List A)} |
| (hf : ∀ u, C.grade (FreeMonoid.ofList u) ≤ n / 10 → |
| (C.normal (FreeMonoid.ofList u)).toList ∈ f u) |
|
|
| include hc hp hm hf |
| theorem certificate : TestCertificate code (test C.omega.toList f) n := by |
| have hb (w : List (Letter A)) (hw : (code.encode w).length ≤ n) : grade C w ≤ n / 10 := by |
| rw [hc] |
| exact code.cost_le_quota _ hp hw |
| apply TestCertificate.of_value code (test C.omega.toList f) (value C) |
| · intro w hw |
| exact test_full C w (hb w hw) hf |
| · exact value_nil C |
| · intro a b u v hu hv h |
| exact value_context C h a b ((hb _ hu).trans hm) ((hb _ hv).trans hm) |
| · intro a b u hu h |
| exact value_zero_context C h a b ((hb _ hu).trans hm) |
| end ProductFamily |
|
|
| namespace ActionFamily |
| open NamedAction |
| variable {G X V Y K L : Type*} [Group V] [MulAction V Y] |
| [DecidableEq G] [DecidableEq X] [DecidableEq K] [DecidableEq L] |
| {N : Operations G X} {m n : ℕ} {C : ActionCalculus V Y K L m} |
| (I : Interpretation N C) (code : DelimiterCode (Letter G X) ThreeCodes.Alphabet) |
| (hp : ∀ a, 10 * N.letterCost a + 2 ≤ (code.block a).length) (hm : 3 * (n / 10) ≤ m) |
| (f : List (Letter G X) → Part (Result K L)) |
| (hf : ∀ w, N.wordCost w ≤ n / 10 → N.totalTest I.gkey I.xkey w ∈ f w) |
|
|
| include I hp hm hf |
| omit [DecidableEq G] [DecidableEq X] [DecidableEq K] in |
| theorem certificate : TestCertificate code f n := by |
| have hb (w : List (Letter G X)) (hw : (code.encode w).length ≤ n) : N.wordCost w ≤ n / 10 := |
| code.cost_le_quota _ hp hw |
| apply TestCertificate.of_value code f (N.totalTest I.gkey I.xkey) |
| · intro w hw |
| exact hf w (hb w hw) |
| · exact I.totalTest_nil |
| · intro a b u v hu hv h |
| exact I.totalTest_context h (hb _ hu) (hb _ hv) hm |
| · intro a b u hu h |
| exact I.totalTest_zero_context h (hb _ hu) hm |
| end ActionFamily |
| end |
| end SimpleFInftyOvergroups.ContextualEncoding |
|
|
| end OAI |
|
|
