| {"id":"def:forget","label":"forget","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For an index \\(i : [d]\\), we define \\begin{align} \\operatorname{forget}_i : G^d & \\to G^{\\{ j : [d] \\mid j \\ne i\\} } \\\\ x & \\mapsto j \\mapsto x_j \\end{align}","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.forget","latex_link":"sect0001.html#def:forget","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.forget","lean_decl_name":"NOF.forget","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/MultidimCorners.lean","lean4_code":"def forget (i : ι) (x : ι → G) (j : {j : ι // j ≠ i}) : G := x j\n","line":6,"end_line":6,"proof_status":"formalized"} |
| {"id":"def:broadcast","label":"broadcast","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:forget","def:protocol"],"nlp_text":"Given a NOF protocol \\(P\\), the NOF broadcast on input \\(x : G^d\\) is inductively defined by \\begin{align} \\operatorname{broad}(x) : \\mathbb {N}& \\to \\operatorname{List}\\operatorname{Bool}\\\\ 0 & \\mapsto [] \\\\ t + 1 & \\mapsto \\operatorname{strat}_{t \\% d}(\\operatorname{forget}_{t \\% d}(x), \\operatorname{broad}(x, t)) :: \\operatorname{broad}(x, t) \\end{align}","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.broadcast","latex_link":"sect0002.html#def:broadcast","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.broadcast","lean_decl_name":"NOF.Protocol.broadcast","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/NOFModel.lean","lean4_code":"def broadcast (P : Protocol G d) (x : ZMod d → G) : ℕ → List Bool\n | 0 => []\n | t + 1 => P.nextBit t (forget (t : ZMod d) x) (broadcast P x t) :: broadcast P x t\n","line":19,"end_line":21,"proof_status":"formalized"} |
| {"id":"lem:forbidden-pattern-project-multicorner","label":"forbidden-pattern-project-multicorner","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:forget","def:multicorner","def:eval"],"nlp_text":"If \\((a_1, \\dots , a_d)\\) is a forbidden pattern such that \\(\\operatorname{eval}(a_i) = 1\\) for all \\(i\\), then \\[ (\\operatorname{forget}_i(a_1), \\dots , \\operatorname{forget}_i(a_d)) \\]","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.isMultidimCorner_forget_of_isForbiddenPattern","latex_link":"sect0003.html#lem:forbidden-pattern-project-multicorner","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.isMultidimCorner_forget_of_isForbiddenPattern","lean_decl_name":"NOF.isMultidimCorner_forget_of_isForbiddenPattern","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/MultidimCorners.lean","lean4_code":"lemma isMultidimCorner_forget_of_isForbiddenPattern (a : ι → ι → G) (h : IsForbiddenPattern a)\n (hS : ∀ i, ∑ j, a i j = 0) (i : ι) :\n IsMultidimCorner (fun j ↦ forget i (a j)) (forget i (a i)) := by\n rw [IsForbiddenPattern] at h\n obtain ⟨v, hv⟩ := h\n refine ⟨fun k l ↦ ?_, fun k l hneq ↦ ?_⟩\n · rw [← sub_eq_zero]\n calc\n ∑ j : {j // j ≠ i}, a k j - ∑ j : { j // j ≠ i }, a l j\n = ∑ j : {j // j ≠ i}, (a k j - a l j) := by rw [Finset.sum_sub_distrib]\n _ = ∑ j ∈ {i}ᶜ, (a k j - a l j) := (Finset.sum_subtype _ (by simp) (a k - a l)).symm\n _ = ∑ j ∈ {i}ᶜ, (a k j - a l j) + (a k i - a l i) := by simp [hv k.2, hv l.2]\n _ = ∑ j : ι, (a k j - a l j) := by rw [← Fintype.sum_eq_sum_compl_add]\n _ = 0 := by rw [Finset.sum_sub_distrib, sub_eq_zero, hS k, hS l]\n · rw [forget, forget, hv, hv l.2.symm]\n aesop\n","line":27,"end_line":42,"proof_status":"formalized"} |
| {"id":"lem:length-broadcast","label":"length-broadcast","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:broadcast"],"nlp_text":"For every NOF protocol \\(P\\), every input \\(x : G^d\\) and every time \\(t\\), \\(\\operatorname{broad}(x, t)\\) has length \\(t\\).","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.length_broadcast","latex_link":"sect0002.html#lem:length-broadcast","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.length_broadcast","lean_decl_name":"NOF.Protocol.length_broadcast","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/NOFModel.lean","lean4_code":"@[simp]\nlemma length_broadcast (P : Protocol G d) (x : ZMod d → G) : ∀ t, (broadcast P x t).length = t\n | 0 => rfl\n | t + 1 => by simpa [broadcast_succ] using length_broadcast _ _ _\n","line":32,"end_line":35,"proof_status":"formalized"} |
| {"id":"def:valid-protocol","label":"valid-protocol","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:broadcast"],"nlp_text":"Given a function \\(F : G^d \\to \\operatorname{Bool}\\), the NOF protocol \\(P\\) is valid in \\(F\\) at time \\(t\\) on input \\(x\\) if all participants correctly guess \\(F(x)\\), namely if \\[ \\operatorname{guess}_i(\\operatorname{forget}_i(x), \\operatorname{broad}(x, t)) = F(x) \\]","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.IsValid","latex_link":"sect0002.html#def:valid-protocol","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.IsValid","lean_decl_name":"NOF.Protocol.IsValid","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/NOFModel.lean","lean4_code":"def IsValid (P : Protocol G d) (F : (ZMod d → G) → Bool) (t : ℕ) : Prop :=\n ∀ x i, P.guess i (forget i x) (broadcast P x t) = F x\n","line":29,"end_line":30,"proof_status":"formalized"} |
| {"id":"thm:det-fun-complex-eval-ge-corner-color-num","label":"det-fun-complex-eval-ge-corner-color-num","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:forbidden-pattern-project-multicorner","lem:mono-forbidden-pattern-trivial","def:det-fun-complex","def:corner-color-num"],"nlp_text":"\\[ D(\\operatorname{eval}) \\ge \\lceil \\log _2\\chi _d(G)\\rceil \\]","doc_url":null,"latex_link":"sect0003.html#thm:det-fun-complex-eval-ge-corner-color-num","lean_link":null,"lean_decl_name":"det-fun-complex-eval-ge-corner-color-num","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"thm:det-fun-complex-eval-le-corner-color-num","label":"det-fun-complex-eval-le-corner-color-num","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:forbidden-pattern-project-multicorner","lem:non-monochromatic-protocol-valid","def:det-fun-complex","def:corner-color-num"],"nlp_text":"\\[ D(\\operatorname{eval}) \\le \\lceil \\log _2\\chi _d(G)\\rceil + d \\]","doc_url":null,"latex_link":"sect0004.html#thm:det-fun-complex-eval-le-corner-color-num","lean_link":null,"lean_decl_name":"det-fun-complex-eval-le-corner-color-num","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:mono-forbidden-pattern-tip","label":"mono-forbidden-pattern-tip","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:length-broadcast","def:valid-protocol","def:forbidden-pattern"],"nlp_text":"Given \\(P\\) a NOF protocol and a time \\(t\\), if \\((a_1, \\dots , a_d)\\) is a forbidden pattern with tip \\(v\\) such that \\(\\operatorname{broad}(a_i, t)\\) equals some fixed broadcast history \\(b\\) for all \\(i\\), then \\(\\operatorname{broad}(v, t) = b\\) as well.","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsForbiddenPatternWithTip.broadcast_eq","latex_link":"sect0002.html#lem:mono-forbidden-pattern-tip","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsForbiddenPatternWithTip.broadcast_eq","lean_decl_name":"NOF.IsForbiddenPatternWithTip.broadcast_eq","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/YaelDillies/ChandraFurstLipton/368d5e4e6edadf50a32df879276214e6fba80af2/ChandraFurstLipton/NOFModel.lean","lean4_code":"lemma IsForbiddenPatternWithTip.broadcast_eq (hF : IsForbiddenPatternWithTip a v)\n (hB : ∀ i, P.broadcast (a i) t = B) : P.broadcast v t = B := by\n induction t generalizing B with\n | zero => simpa using hB 0\n | succ t ih =>\n simp_rw [Protocol.broadcast_succ] at *\n obtain _ | ⟨b, B⟩ := B\n · cases hB 0\n simp only [List.cons.injEq, forall_and] at hB\n specialize ih hB.2\n subst ih\n have h₁ : forget (t : ZMod d) v = forget (t : ZMod d) (a t) := hF.forget t\n have h₂ : P.broadcast v t = P.broadcast (a t) t := (hB.2 t).symm\n rw [h₁, h₂, hB.1]","line":51,"end_line":64,"proof_status":"formalized"} |
| {"id":"def:rand-protocol-complex","label":"rand-protocol-complex","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:valid-protocol"],"nlp_text":"The communication complexity of a randomised NOF protocol \\(P\\) for \\(F\\) with error \\(\\epsilon \\) is the smallest time \\(t\\) such that \\[ \\mathbb P(x \\mid P \\text{ is not valid at time } t) \\le \\epsilon \\]","doc_url":null,"latex_link":"sect0005.html#def:rand-protocol-complex","lean_link":null,"lean_decl_name":"rand-protocol-complex","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:trivial-protocol-valid","label":"trivial-protocol-valid","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:valid-protocol","def:trivial-protocol"],"nlp_text":"For all \\(F\\), the trivial protocol for \\(F\\) is valid in time \\(d\\left\\lceil \\log _2 n\\right\\rceil \\).","doc_url":null,"latex_link":"sect0002.html#lem:trivial-protocol-valid","lean_link":null,"lean_decl_name":"trivial-protocol-valid","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:det-protocol-complex","label":"det-protocol-complex","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:valid-protocol"],"nlp_text":"The communication complexity of a NOF protocol \\(P\\) for \\(F\\) is the smallest time \\(t\\) such that \\(P\\) is valid in \\(F\\) at time \\(t\\) on all inputs \\(x\\), or \\(\\infty \\) if no such \\(t\\) exists.","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.complexity","latex_link":"sect0002.html#def:det-protocol-complex","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.complexity","lean_decl_name":"NOF.Protocol.complexity","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/NOFModel.lean","lean4_code":"noncomputable\ndef complexity (P : Protocol G d) (F : (ZMod d → G) → Bool) : ENat :=\n ⨅ (t : ℕ) (_ : IsValid P F t), t\n","line":37,"end_line":39,"proof_status":"formalized"} |
| {"id":"lem:non-monochromatic-protocol-valid","label":"non-monochromatic-protocol-valid","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:valid-protocol","def:non-monochromatic-protocol"],"nlp_text":"If \\(c\\) is a coloring such that all monochromatic forbidden patterns are trivial, then the non-monochromatic protocol for \\(c\\) is valid in time \\(\\lceil \\log _2\\chi _d(G)\\rceil + d\\).","doc_url":null,"latex_link":"sect0004.html#lem:non-monochromatic-protocol-valid","lean_link":null,"lean_decl_name":"non-monochromatic-protocol-valid","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:rand-eq-test-protocol-eval-valid","label":"rand-eq-test-protocol-eval-valid","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:valid-protocol","def:rand-eq-test-protocol-eval"],"nlp_text":"The randomised equality testing protocol is valid for \\(\\operatorname{eval}\\) at time \\(2d\\).","doc_url":null,"latex_link":"sect0005.html#lem:rand-eq-test-protocol-eval-valid","lean_link":null,"lean_decl_name":"rand-eq-test-protocol-eval-valid","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"cor:det-fun-complex-eval-ge-corner-free-num","label":"det-fun-complex-eval-ge-corner-free-num","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["thm:det-fun-complex-eval-ge-corner-color-num","lem:corner-num-lower"],"nlp_text":"\\[ D(\\operatorname{eval}) \\ge d\\log _2\\frac N{r_d(G)} \\]","doc_url":null,"latex_link":"sect0003.html#cor:det-fun-complex-eval-ge-corner-free-num","lean_link":null,"lean_decl_name":"det-fun-complex-eval-ge-corner-free-num","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"cor:det-fun-complex-eval-le-corner-free-num","label":"det-fun-complex-eval-le-corner-free-num","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["thm:det-fun-complex-eval-le-corner-color-num","lem:corner-num-upper"],"nlp_text":"\\[ D(\\operatorname{eval}) \\le 2d\\log _2\\frac N{r_d(G)} \\]","doc_url":null,"latex_link":"sect0004.html#cor:det-fun-complex-eval-le-corner-free-num","lean_link":null,"lean_decl_name":"det-fun-complex-eval-le-corner-free-num","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:mono-forbidden-pattern-trivial","label":"mono-forbidden-pattern-trivial","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:mono-forbidden-pattern-tip","def:eval"],"nlp_text":"Given \\(P\\) a NOF protocol valid in time \\(t\\) for \\(\\operatorname{eval}\\), all monochromatic forbidden patterns are trivial.","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.trivial_of_isForbiddenPattern_of_isValid_eval","latex_link":"sect0003.html#lem:mono-forbidden-pattern-trivial","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.trivial_of_isForbiddenPattern_of_isValid_eval","lean_decl_name":"NOF.trivial_of_isForbiddenPattern_of_isValid_eval","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/LowerBoundEval.lean","lean4_code":"lemma trivial_of_isForbiddenPattern_of_isValid_eval (ha : IsForbiddenPattern a)\n (hP : P.IsValid eval t) (hE : ∀ i, eval (a i) = true) (hB : ∀ i, P.broadcast (a i) t = B) :\n ∀ i j, a i = a j := by\n obtain ⟨v, ha⟩ := ha\n have : P.broadcast v t = B := ha.broadcast_eq hB\n have h i : eval (a i) = eval v :=\n calc\n _ = P.guess i (forget i (a i)) (P.broadcast (a i) t) := by rw [hP]\n _ = P.guess i (forget i v) (P.broadcast v t) := by rw [hB, ha.broadcast_eq hB, ha.forget]\n _ = eval v := by rw [hP]\n suffices h : ∀ i, a i = v by simp [h]\n intro i\n ext j\n obtain hij | rfl := ne_or_eq i j\n · exact ha hij\n simp only [hE, Bool.true_eq, eval_eq_true, forall_const] at h\n rw [← sub_eq_zero]\n calc\n a i i - v i = ∑ j, (a i j - v j) := by\n rw [Fintype.sum_eq_single]; simpa [eq_comm, sub_eq_zero] using @ha i\n _ = ∑ j, a i j - ∑ j, v j := by rw [Finset.sum_sub_distrib]\n _ = 0 := by simpa [h] using hE i\n","line":18,"end_line":39,"proof_status":"formalized"} |
| {"id":"def:rand-complex","label":"rand-complex","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:rand-protocol-complex"],"nlp_text":"The randomised communication complexity of a function \\(F\\) with error \\(\\epsilon \\), denoted \\(R_\\epsilon (F)\\), is the minimum of the randomised communication complexity of \\(P\\) when \\(P\\) ranges over all randomised NOF protocols.","doc_url":null,"latex_link":"sect0005.html#def:rand-complex","lean_link":null,"lean_decl_name":"rand-complex","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:trivial-bound-det-fun-complex","label":"trivial-bound-det-fun-complex","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:trivial-protocol-valid","def:det-fun-complex"],"nlp_text":"The communication complexity of any function \\(F\\) is at most \\(d\\left\\lceil \\log _2 n\\right\\rceil \\).","doc_url":null,"latex_link":"sect0002.html#lem:trivial-bound-det-fun-complex","lean_link":null,"lean_decl_name":"trivial-bound-det-fun-complex","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:det-fun-complex","label":"det-fun-complex","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:det-protocol-complex"],"nlp_text":"The deterministic communication complexity of a function \\(F\\), denoted \\(D(F)\\), is the minimum of the communication complexity of \\(P\\) when \\(P\\) ranges over all NOF protocols.","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.funComplexity","latex_link":"sect0002.html#def:det-fun-complex","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.funComplexity","lean_decl_name":"NOF.funComplexity","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/NOFModel.lean","lean4_code":"noncomputable def funComplexity (F : (ZMod d → G) → Bool) := ⨅ P : Protocol G d, P.complexity F\n","line":46,"end_line":46,"proof_status":"formalized"} |
| {"id":"thm:eval-rand-complexity","label":"eval-rand-complexity","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["lem:rand-eq-test-protocol-eval-valid","def:rand-complex","def:eval"],"nlp_text":"\\[ R_\\epsilon (\\operatorname{eval}) \\le 2d\\lceil \\log _2\\epsilon ^{-1}\\rceil \\]","doc_url":null,"latex_link":"sect0005.html#thm:eval-rand-complexity","lean_link":null,"lean_decl_name":"eval-rand-complexity","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:corner-num-upper","label":"corner-num-upper","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:corner-free-num","def:corner-color-num"],"nlp_text":"\\[ r_d(G) \\chi _d(G) \\le 2d N^d \\log N \\]","doc_url":null,"latex_link":"sect0001.html#lem:corner-num-upper","lean_link":null,"lean_decl_name":"corner-num-upper","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:corner-num-lower","label":"corner-num-lower","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:corner-free-num","def:corner-color-num"],"nlp_text":"\\[ r_d(G) \\chi _d(G) \\ge N^d \\]","doc_url":null,"latex_link":"sect0001.html#lem:corner-num-lower","lean_link":null,"lean_decl_name":"corner-num-lower","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:corner-free-num","label":"corner-free-num","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:multicorner"],"nlp_text":"The \\(d\\)-dimensional corner-free number of \\(G\\), denoted \\(r_d(G)\\) is the size of the largest set \\(A\\) in \\(G^d\\) such that \\(A\\) doesn’t contain a non-trivial corner.","doc_url":null,"latex_link":"sect0001.html#def:corner-free-num","lean_link":null,"lean_decl_name":"corner-free-num","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:protocol","label":"protocol","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A NOF protocol \\(P\\) consists of maps \\begin{align} \\operatorname{strat}: & [d] \\to G^{d - 1} \\to \\operatorname{List}\\operatorname{Bool}\\to \\operatorname{Bool}\\\\ \\operatorname{guess}: & [d] \\to G^{d - 1} \\to \\operatorname{List}\\operatorname{Bool}\\to \\operatorname{Bool}\\end{align}","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol","latex_link":"sect0002.html#def:protocol","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol","lean_decl_name":"NOF.Protocol","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"./ChandraFurstLipton/LowerBoundEval.lean","lean4_code":"def Protocol.trivial (hd : 3 ≤ d) (F : (ZMod d → G) → Bool) : Protocol G d where\n nextBit i x B := by\n refine (Nat.bits (Fintype.equivFin G (x ⟨i + 1, ?_ ⟩))).getI (B.length / d)\n rw [Ne, add_eq_left, ← Nat.cast_one, ZMod.natCast_eq_zero_iff, Nat.dvd_one]\n omega\n guess i x B := F fun j ↦\n if h : j = i then\n (Fintype.equivFin G).symm <| (ZMod.finEquiv _).symm\n (BitVec.toNat (BitVec.ofBoolListLE (getBits B i.val d)))\n else\n x ⟨j, h⟩\n\n","line":50,"end_line":61,"proof_status":"formalized"} |
| {"id":"def:rand-eq-test-protocol-eval","label":"rand-eq-test-protocol-eval","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:protocol"],"nlp_text":"The randomised equality testing protocol for \\(\\operatorname{eval}\\) has domain \\(\\Omega := (\\operatorname{Bool}^d)^{\\lceil \\log _2\\epsilon ^{-1}\\rceil }\\) with the uniform measure and is defined by making participant \\(i\\) do “Compute \\[ a_{i, k} = \\sum _{j \\ne i} \\omega _{j, k}x_j \\]","doc_url":null,"latex_link":"sect0005.html#def:rand-eq-test-protocol-eval","lean_link":null,"lean_decl_name":"rand-eq-test-protocol-eval","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:trivial-protocol","label":"trivial-protocol","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:protocol"],"nlp_text":"For all \\(F\\), we define the trivial protocol by making participant \\(i\\) do \"Send the \\(t / d\\)-th bit of the number of participant \\(i + 1\\)\" and \"Compute \\(x_i\\) from the binary representation given by participant \\(i - 1\\), then compute \\(F(x)\\)\".","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.trivial","latex_link":"sect0002.html#def:trivial-protocol","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.Protocol.trivial","lean_decl_name":"NOF.Protocol.trivial","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/LowerBoundEval.lean","lean4_code":"noncomputable\ndef Protocol.trivial (hd : 3 ≤ d) (F : (ZMod d → G) → Bool) : Protocol G d where\n nextBit i x B := by\n refine (Nat.bits (Fintype.equivFin G (x ⟨i + 1, ?_ ⟩))).getI (B.length / d)\n rw [Ne, add_eq_left, ← Nat.cast_one, ZMod.natCast_eq_zero_iff, Nat.dvd_one]\n omega\n guess i x B := F fun j ↦\n if h : j = i then\n (Fintype.equivFin G).symm <| (ZMod.finEquiv _).symm\n (BitVec.toNat (BitVec.ofBoolListLE (getBits B i.val d)))\n else\n x ⟨j, h⟩\n","line":49,"end_line":60,"proof_status":"statement_formalized"} |
| {"id":"def:non-monochromatic-protocol","label":"non-monochromatic-protocol","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:protocol"],"nlp_text":"Given a coloring \\(c : \\{ x \\mid \\operatorname{eval}x = 1\\} \\to [C]\\), writing \\(a_i\\) the vector whose \\(j\\)-th coordinate is \\(x_j\\) except when \\(j = i\\) in which case it is \\(-\\sum _{j \\ne i} x_j\\), we define the non-monochromatic protocol for \\(c\\) by making participant \\(i\\) do “Send the \\(t / d\\)-th bit of \\(c(a_i)\\) until time \\(\\lceil \\log _2\\chi _d(G)\\rceil \\), then send \\(1\\) iff \\(c(a_i)\\) agrees with the broadcast from time \\(1\\) to time \\(\\lceil \\log _2\\chi _d(G)\\rceil \\) read as a color” and “Send \\(1\\) iff the broadcasts from time \\(\\lceil \\log _2\\chi _d(G)\\rceil \\) to time \\(\\lceil \\log _2\\chi _d(G)\\rceil + d\\) were all \\(1\\)”.","doc_url":null,"latex_link":"sect0004.html#def:non-monochromatic-protocol","lean_link":null,"lean_decl_name":"non-monochromatic-protocol","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:forbidden-pattern","label":"forbidden-pattern","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We say a tuple \\((a_1, \\dots , a_d) : (G^d)^d\\) is a forbidden pattern with tip \\(v : G^d\\) if \\[ a_{i, j} = v_j \\]","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsForbiddenPatternWithTip","latex_link":"sect0001.html#def:forbidden-pattern","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsForbiddenPatternWithTip","lean_decl_name":"NOF.IsForbiddenPatternWithTip","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"./ChandraFurstLipton/NOFModel.lean","lean4_code":"lemma IsForbiddenPatternWithTip.broadcast_eq (hF : IsForbiddenPatternWithTip a v)\n (hB : ∀ i, P.broadcast (a i) t = B) : P.broadcast v t = B := by\n induction t generalizing B with\n | zero => simpa using hB 0\n | succ t ih =>\n simp_rw [Protocol.broadcast_succ] at *\n obtain _ | ⟨b, B⟩ := B\n · cases hB 0\n simp only [List.cons.injEq, forall_and] at hB\n specialize ih hB.2\n subst ih\n have h₁ : forget (t : ZMod d) v = forget (t : ZMod d) (a t) := hF.forget t\n have h₂ : P.broadcast v t = P.broadcast (a t) t := (hB.2 t).symm\n rw [h₁, h₂, hB.1]\n\n","line":51,"end_line":65,"proof_status":"formalized"} |
| {"id":"def:multicorner","label":"multicorner","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:forbidden-pattern"],"nlp_text":"A multidimensional corner in \\(d\\) dimensions is a tuple of the form \\((x, x + ce_1, \\dots , x + ce_d)\\) for some \\(x : G^d\\) and \\(c : G\\), where \\(ce_i\\) is the vector of all zeroes except in position \\(i\\) where it is \\(c\\). Such a corner is said to be trivial if \\(c = 0\\).","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsMultidimCorner","latex_link":"sect0001.html#def:multicorner","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.IsMultidimCorner","lean_decl_name":"NOF.IsMultidimCorner","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/MultidimCorners.lean","lean4_code":"structure IsMultidimCorner (a : ι → ι → G) (b : ι → G) : Prop where\n sum_eq_sum : ∀ i j, ∑ k, a i k = ∑ k, a j k\n isForbiddenPatternWithTip : IsForbiddenPatternWithTip a b\n","line":21,"end_line":23,"proof_status":"formalized"} |
| {"id":"def:corner-color-num","label":"corner-color-num","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":["def:multicorner"],"nlp_text":"The \\(d\\)-dimensional corner-coloring number of \\(G\\), denoted \\(\\chi _d(G)\\), is the smallest number of colors one needs to color \\(G^d\\) such that no non-trivial \\(d\\)-dimensional corner is monochromatic.","doc_url":null,"latex_link":"sect0001.html#def:corner-color-num","lean_link":null,"lean_decl_name":"corner-color-num","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:eval","label":"eval","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://YaelDillies.github.io/ChandraFurstLipton/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The \\(\\operatorname{eval}\\) function is defined by \\begin{align} \\operatorname{eval}: G^d & \\to \\operatorname{Bool}\\\\ x & \\mapsto \\begin{cases} 1 & \\text{ if } \\sum _i x_i = 0 \\\\ 0 & \\text{ else} \\end{cases}\\end{align}","doc_url":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.eval","latex_link":"sect0003.html#def:eval","lean_link":"https://YaelDillies.github.io/ChandraFurstLipton/docs/find/#doc/NOF.eval","lean_decl_name":"NOF.eval","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ChandraFurstLipton","source_path":"ChandraFurstLipton/LowerBoundEval.lean","lean4_code":"def eval (x : ι → G) : Bool :=\n ∑ i, x i == 0\n","line":11,"end_line":12,"proof_status":"formalized"} |
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