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Research-Stack/lean_binned/BinnedFormalizations.lean
allaun 475f6319ea chore(repo): push local 768-commit branch state onto clean remote baseline 
This squashes all local history (768 commits) onto the scrubbed PR #90
baseline. Individual commits were lost during filter-repo corruption;
the working tree content is preserved intact.

Build: N/A (working tree state only)
2026-06-15 22:46:50 -05:00

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import Mathlib
/- Original: z = 1/a -/
theorem eq_dc1663f465de629e (a : ℕ) (z : ℕ) : z = 1/a := by
omega
/- Original: N = {1, 2 -/
theorem eq_4a18ceaf3888bba3 (N : ℕ) : N = {1 ∧ 2 := by
omega
/- Original: k ≥ 2 where 1 x = (1 x , -/
theorem eq_0085761c3512ef7e (k : ℕ) (where : ℕ) (x : ℕ) : k ≥ 2 where 1 x = (1 x , := by
omega
/- Original: T = 2 (3 -/
theorem eq_5681ee9bcf0fa212 (T : ℕ) : T = 2 (3 := by
omega
/- Original: b = 1−a -/
theorem eq_3ed31f1ae076490c (a : ℕ) (b : ℕ) : b = 1-a := by
omega
/- Original: is = 12 − s -/
theorem eq_db937c0f244fb14f (is : ℕ) (s : ℕ) : is = 12 - s := by
omega
/- Original: ZkRT =⇒ ZUCS(1),k -/
theorem eq_840155af33c6593a (ZUCS : ℕ) (ZkRT : ℕ) (k : ℕ) : ZkRT =⇒ ZUCS(1) ∧ k := by
omega
/- Original: a ≤ b, we set [a, b] = {k ∈ ZP | a ≤ k ≤ b} -/
theorem eq_45aae6731480405b (ZP : ℕ) (a : ℕ) (b : ℕ) (k : ℕ) (set : ℕ) (we : ℕ) : a ≤ b ∧ we set [a ∧ b] = {k ∈ ZP | a ≤ k ≤ b} := by
omega
/- Original: Htx = y)g(y) (1 -/
theorem eq_0d0a8db0702e6227 (Htx : ℕ) (g : ℕ) (y : ℕ) : Htx = y)g(y) (1 := by
omega
/- Original: b = (4 -/
theorem eq_165389a1b0e1fa3a (b : ℕ) : b = (4 := by
omega
/- Original: ess ≤ λα · eP (ϕ) < eP (ϕ) , and the SRB measure µ+ = µϕ(u) has absolutely continuous conditional measures along unstable manifolds -/
theorem eq_40ce41ef24bab64a (SRB : ℕ) (absolutely : ℕ) (along : ℕ) (conditional : ℕ) (continuous : ℕ) (eP : ℕ) (ess : ℕ) (has : ℕ) (manifolds : ℕ) (measure : ℕ) (measures : ℕ) (the : ℕ) (u : ℕ) (unstable : ℕ) (µ : ℕ) (µϕ : ℕ) (λα : ℕ) (ϕ : ℕ) : ess ≤ λα * eP (ϕ) < eP (ϕ) , ∧ the SRB measure µ+ = µϕ(u) has absolutely continuous conditional measures along unstable manifolds := by
omega
/- Original: l ≥ 0, with C = (M (2 + 2CvM ))2M -/
theorem eq_41846a6477bb5031 (C : ℕ) (CvM : ℕ) (M : ℕ) (l : ℕ) : l ≥ 0 ∧ with C = (M (2 + 2CvM ))2M := by
omega
/- Original: Ep = q -/
theorem eq_a3304dc6002ba5ff (Ep : ℕ) (q : ℕ) : Ep = q := by
omega
/- Original: N = ±1 -/
theorem eq_4873edfda3319bb7 (N : ℕ) : N = ±1 := by
omega
/- Original: M = M P -/
theorem eq_5c60d8b2b41be060 (M : ℕ) (P : ℕ) : M = M P := by
omega
/- Original: M = 2 -/
theorem eq_5897ae98de4b014b (M : ℕ) : M = 2 := by
omega
/- Original: SR = 1 + 0 -/
theorem eq_f3229308596f7e0f (SR : ℕ) : SR = 1 + 0 := by
omega
/- Original: S = nK -/
theorem eq_ee1806e38c2e138f (S : ℕ) (nK : ℕ) : S = nK := by
omega
/- Original: j = ∅ -/
theorem eq_c1a65b020b4cbca9 (j : ℕ) : j = ∅ := by
omega
/- Original: r>0  r∈Z+ 12 e0= (∂ ψe ψ) (B -/
theorem eq_3cbd19f4716d6d25 (B : ℕ) (Z : ℕ) (e0 : ℕ) (r : ℕ) (ψ : ℕ) (ψe : ℕ) : r>0  r∈Z+ 12 e0= (∂ ψe ψ) (B := by
omega
/- Original: m ≥ 0, we multiply the equation for m by m2− , with m− = max(0, −m) and integrate in space and time -/
theorem eq_250c08e8451f986b (by : ℕ) (equation : ℕ) (for : ℕ) (integrate : ℕ) (m : ℕ) (m2 : ℕ) (multiply : ℕ) (space : ℕ) (the : ℕ) (time : ℕ) (we : ℕ) : m ≥ 0, we multiply the equation for m by m2- , with m- = max(0, -m) ∧ integrate in space and time := by
omega
/- Original: qi = T pi -/
theorem eq_16394787daa75309 (T : ℕ) (pi : ℕ) (qi : ℕ) : qi = T pi := by
omega
/- Original: g = −1 -/
theorem eq_ff52deca30009dcd (g : ℕ) : g = -1 := by
omega
/- Original: q = 0, Eq -/
theorem eq_32ab2ff729514bc2 (Eq : ℕ) (q : ℕ) : q = 0 ∧ Eq := by
omega
/- Original: P = ci P -/
theorem eq_86c1193b23361384 (P : ℕ) (ci : ℕ) : P = ci P := by
omega
/- Original: C = √12 -/
theorem eq_b0211e5bfd10d843 (C : ℕ) : C = √12 := by
omega
/- Original: TV = 0 -/
theorem eq_45c6d0f7052c61f2 (TV : ℕ) : TV = 0 := by
omega
/- Original: u = F u† -/
theorem eq_2e12c41db0bb746d (F : ℕ) (u : ℕ) : u = F u† := by
omega
/- Original: q > 1 and t := b/q < 1 -/
theorem eq_acdbf6dbfe3926e8 (b : ℕ) (q : ℕ) (t : ℕ) : q > 1 and t := b/q < 1 := by
omega
/- Original: cTglob = (10 -/
theorem eq_2577f85be9648be4 (cTglob : ℕ) : cTglob = (10 := by
omega
/- Original: ruf = r⋄ and Ω2uf = 4D(r⋄ ; M, ϱ) on R(1, uf , 1, ∞), where r⋄ is as in Lemma 4 -/
theorem eq_0a5b6d95dadfc3e2 (D : ℕ) (Lemma : ℕ) (M : ℕ) (R : ℕ) (as : ℕ) (is : ℕ) (on : ℕ) (r : ℕ) (ruf : ℕ) (uf : ℕ) (where : ℕ) (ϱ : ℕ) (Ω2uf : ℕ) : ruf = r⋄ ∧ Ω2uf = 4D(r⋄ ; M, ϱ) on R(1, uf , 1, ∞), where r⋄ is as in Lemma 4 := by
omega
/- Original: V = V1 + V2 , where ( V1 = χ(|x| < r)V (x), |V1 (x)| ⩽ Cr2d ⟨x⟩−D , (3 -/
theorem eq_2ddaee8030fa39eb (Cr2d : ℕ) (D : ℕ) (V : ℕ) (V1 : ℕ) (V2 : ℕ) (r : ℕ) (where : ℕ) (x : ℕ) (χ : ℕ) : V = V1 + V2 ∧ where ( V1 = χ(|x| < r)V (x) ∧ |V1 (x)| ⩽ Cr2d ⟨x⟩-D ∧ (3 := by
omega
/- Original: p =I ⊗ Φp + ϵI ⊗ Wp,Φ − ϵ2 Source0,p + O(ϵ3 ) (3 -/
theorem eq_d2028815ba9b7371 (I : ℕ) (O : ℕ) (Source0 : ℕ) (Wp : ℕ) (p : ℕ) (Φ : ℕ) (Φp : ℕ) (ϵ2 : ℕ) (ϵ3 : ℕ) (ϵI : ℕ) : p =I ⊗ Φp + ϵI ⊗ Wp ∧ Φ - ϵ2 Source0 ∧ p + O(ϵ3 ) (3 := by
omega
/- Original: x = 21 , and the previous identities imply  y (k) 12 = 0, k = 0, 1, 2, 3 -/
theorem eq_410dfb7a851615f8 (identities : ℕ) (imply : ℕ) (k : ℕ) (previous : ℕ) (the : ℕ) (x : ℕ) (y : ℕ) : x = 21 , ∧ the previous identities imply  y (k) 12 = 0, k = 0, 1, 2, 3 := by
omega
/- Original: t ≥ 0 into X tn X X tn X etΓ β(x) = [Γn β](x) β(y) γ (n) (y, x) =: β(y)γt (y, x) -/
theorem eq_069f772cfae3e2f6 (X : ℕ) (etΓ : ℕ) (into : ℕ) (n : ℕ) (t : ℕ) (tn : ℕ) (x : ℕ) (y : ℕ) (Γn : ℕ) (β : ℕ) (γ : ℕ) (γt : ℕ) : t ≥ 0 into X tn X X tn X etΓ β(x) = [Γn β](x) β(y) γ (n) (y ∧ x) =: β(y)γt (y ∧ x) := by
omega
/- Original: j=a−1 h i a,b=1, -/
theorem eq_3994fe06c226dbed (a : ℕ) (b : ℕ) (h : ℕ) (i : ℕ) (j : ℕ) : j=a-1 h i a ∧ b=1 := by
omega
/- Original: ux = − ux − P ∗ u + 2 2     1 2 1 2 1 2 2 2 = − ux − P+ ∗ u + ux + h(u) − P− ∗ u + ux + h(u) + u2 2 2 2 + h(u) − λ(t)ux -/
theorem eq_7ba3cc2c96fdf78f (P : ℕ) (h : ℕ) (t : ℕ) (u : ℕ) (u2 : ℕ) (ux : ℕ) (λ : ℕ) : ux = - ux - P * u + 2 2     1 2 1 2 1 2 2 2 = - ux - P+ * u + ux + h(u) - P- * u + ux + h(u) + u2 2 2 2 + h(u) - λ(t)ux := by
omega
/- Original: k = 4πkG σ0 C0 and χ is the Euler characteristic, i -/
theorem eq_8fa7c2d3a5dadce7 (C0 : ℕ) (Euler : ℕ) (characteristic : ℕ) (i : ℕ) (is : ℕ) (k : ℕ) (the : ℕ) (πkG : ℕ) (σ0 : ℕ) (χ : ℕ) : k = 4πkG σ0 C0 ∧ χ is the Euler characteristic, i := by
omega
/- Original: k=1 ≤ ∞ X e−C3 N δ -/
theorem eq_089337bb135cc24e (C3 : ℕ) (N : ℕ) (X : ℕ) (e : ℕ) (k : ℕ) (δ : ℕ) : k=1 ≤ ∞ X e-C3 N δ := by
omega
/- Original: m = O((n2 + n log δ −1 )/ε2 ) copies of ρunsqueezed to get outcomes v1 , · · · , v2m ∈ R2n -/
theorem eq_b13ac6b3013ec125 (O : ℕ) (R2n : ℕ) (copies : ℕ) (get : ℕ) (m : ℕ) (n : ℕ) (n2 : ℕ) (of : ℕ) (outcomes : ℕ) (to : ℕ) (v1 : ℕ) (v2m : ℕ) (δ : ℕ) (ε2 : ℕ) (ρunsqueezed : ℕ) : m = O((n2 + n log δ -1 )/ε2 ) copies of ρunsqueezed to get outcomes v1 ∧ * * * ∧ v2m ∈ R2n := by
omega
/- Original: DE = −16i + 16λ4 , then E cannot be the zero polynomial, so we must have A = 0 -/
theorem eq_6b1aad59341606ce (A : ℕ) (DE : ℕ) (E : ℕ) (be : ℕ) (cannot : ℕ) (have : ℕ) (i : ℕ) (must : ℕ) (polynomial : ℕ) (so : ℕ) (the : ℕ) (we : ℕ) (zero : ℕ) (λ4 : ℕ) : DE = -16i + 16λ4 ∧ then E cannot be the zero polynomial ∧ so we must have A = 0 := by
omega
/- Original: H = µ10 B), while boundary conditions are naturally expressed with the inclusion map i : ∂Ω → Ω and its pullback action on forms -/
theorem eq_2a1f22b692aa87c9 (B : ℕ) (H : ℕ) (action : ℕ) (are : ℕ) (boundary : ℕ) (conditions : ℕ) (expressed : ℕ) (forms : ℕ) (i : ℕ) (inclusion : ℕ) (its : ℕ) (map : ℕ) (naturally : ℕ) (on : ℕ) (pullback : ℕ) (the : ℕ) (while : ℕ) (µ10 : ℕ) (Ω : ℕ) : H = µ10 B), while boundary conditions are naturally expressed with the inclusion map i : ∂Ω → Ω ∧ its pullback action on forms := by
omega
/- Original: C ≤ γ dte + γ dte e N −∞ = 1+ Z ∞ dteγt Pr(gk − E[gk ] ≥ t) 0 0 √ 2πγC N e γ2 C 2 2 -/
theorem eq_2ceef993fadb8617 (C : ℕ) (E : ℕ) (N : ℕ) (Pr : ℕ) (Z : ℕ) (dte : ℕ) (dteγt : ℕ) (e : ℕ) (gk : ℕ) (t : ℕ) (γ : ℕ) (γ2 : ℕ) (πγC : ℕ) : C ≤ γ dte + γ dte e N -∞ = 1+ Z ∞ dteγt Pr(gk - E[gk ] ≥ t) 0 0 √ 2πγC N e γ2 C 2 2 := by
omega
/- Original: G = (V, E) be a connected, locally finite, infinite graph -/
theorem eq_2927625930f9ceca (E : ℕ) (G : ℕ) (V : ℕ) (a : ℕ) (be : ℕ) (connected : ℕ) (finite : ℕ) (graph : ℕ) (infinite : ℕ) (locally : ℕ) : G = (V ∧ E) be a connected ∧ locally finite ∧ infinite graph := by
omega
/- Original: S = [M∗ T M] and T = [MSM∗ ], and AT = [MM∗ ] -/
theorem eq_b2589194317b5375 (AT : ℕ) (M : ℕ) (MM : ℕ) (MSM : ℕ) (S : ℕ) (T : ℕ) : S = [M* T M] ∧ T = [MSM* ], and AT = [MM* ] := by
omega
/- Original: j = OK (a−K N ), when s − j ∈ J−(N − aN ), N − aN Kd -/
theorem eq_3b5cf8d2d4956d3c (J : ℕ) (K : ℕ) (Kd : ℕ) (N : ℕ) (OK : ℕ) (a : ℕ) (aN : ℕ) (j : ℕ) (s : ℕ) (when : ℕ) : j = OK (a-K N ) ∧ when s - j ∈ J-(N - aN ) ∧ N - aN Kd := by
omega
/- Original: i=k (97) 1 lim inf − lnq pk (X0n−1 ) ≥ Hq,k -/
theorem eq_e5af0670d5ebda22 (Hq : ℕ) (X0n : ℕ) (i : ℕ) (inf : ℕ) (k : ℕ) (lim : ℕ) (lnq : ℕ) (pk : ℕ) : i=k (97) 1 lim inf - lnq pk (X0n-1 ) ≥ Hq ∧ k := by
omega
/- Original: q = N1 ∥m∥2 depends on m, we have ∂i q = 2m N -/
theorem eq_526b05d361948008 (N : ℕ) (N1 : ℕ) (depends : ℕ) (have : ℕ) (i : ℕ) (m : ℕ) (on : ℕ) (q : ℕ) (we : ℕ) : q = N1 ∥m∥2 depends on m ∧ we have ∂i q = 2m N := by
omega
/- Original: di=1 |i⟩⟨i|A + |d⟩⟨d|A , trHB (|V0 ⟩⟩⟨⟨V0 |) = P′ (1 − ε2 ) di=1 |i⟩⟨i|A , if d is odd , if d is even so trHB (|V0 ⟩⟩⟨⟨V0 |) ≤ IA -/
theorem eq_2fd8a5739608720e (A : ℕ) (IA : ℕ) (P : ℕ) (V0 : ℕ) (d : ℕ) (di : ℕ) (even : ℕ) (i : ℕ) (is : ℕ) (odd : ℕ) (so : ℕ) (trHB : ℕ) (ε2 : ℕ) : di=1 |i⟩⟨i|A + |d⟩⟨d|A ∧ trHB (|V0 ⟩⟩⟨⟨V0 |) = P′ (1 - ε2 ) di=1 |i⟩⟨i|A ∧ if d is odd ∧ if d is even so trHB (|V0 ⟩⟩⟨⟨V0 |) ≤ IA := by
omega
/- Original: i=0 τ (35) which satisfies T (A) ∈ gTI for all A and T (A) = A for all A ∈ gTI -/
theorem eq_d725a0ae5e609f46 (A : ℕ) (T : ℕ) (all : ℕ) (for : ℕ) (gTI : ℕ) (i : ℕ) (satisfies : ℕ) (which : ℕ) (τ : ℕ) : i=0 τ (35) which satisfies T (A) ∈ gTI for all A ∧ T (A) = A for all A ∈ gTI := by
omega
/- Original: M ≥ g independent branches: Cmulti = M · (CR + Cprep ) + O(N M 2 ) -/
theorem eq_ee0fe7334f580135 (CR : ℕ) (Cmulti : ℕ) (Cprep : ℕ) (M : ℕ) (N : ℕ) (O : ℕ) (branches : ℕ) (g : ℕ) (independent : ℕ) : M ≥ g independent branches: Cmulti = M * (CR + Cprep ) + O(N M 2 ) := by
omega
/- Original: E = π2∗ (T ∗ M ) ∗ E-mail: jorge -/
theorem eq_347dab94660d5126 (E : ℕ) (M : ℕ) (T : ℕ) (jorge : ℕ) (mail : ℕ) (π2 : ℕ) : E = π2* (T * M ) * E-mail: jorge := by
omega
/- Original: Q = L ⋉ U , where L = LI = Q ∩ Θ(Q) -/
theorem eq_772db3539479dd4b (L : ℕ) (LI : ℕ) (Q : ℕ) (U : ℕ) (where : ℕ) (Θ : ℕ) : Q = L ⋉ U ∧ where L = LI = Q ∩ Θ(Q) := by
omega
/- Original: B = B(x, 4r) of radius 4r > 0 that is contained inside of B (k) ∩ S c -/
theorem eq_7f3b54e3d118c8d5 (B : ℕ) (S : ℕ) (c : ℕ) (contained : ℕ) (inside : ℕ) (is : ℕ) (k : ℕ) (of : ℕ) (r : ℕ) (radius : ℕ) (that : ℕ) (x : ℕ) : B = B(x ∧ 4r) of radius 4r > 0 that is contained inside of B (k) ∩ S c := by
omega
/- Original: u = 1 + cn(ϕ, m) = and hence cn(ϕK , m) = 2 , 1 + x2 (1 − x2 ) -/
theorem eq_cc987483e73ebf5c (cn : ℕ) (hence : ℕ) (m : ℕ) (u : ℕ) (x2 : ℕ) (ϕ : ℕ) (ϕK : ℕ) : u = 1 + cn(ϕ, m) = ∧ hence cn(ϕK , m) = 2 , 1 + x2 (1 - x2 ) := by
omega
/- Original: n ≥ 1, we set     k X X n Gn := F ⊗ Gn , where Gn := σ X :0≤k ≤2 −1 -/
theorem eq_aa3fd0e09ced1b7f (F : ℕ) (Gn : ℕ) (X : ℕ) (k : ℕ) (n : ℕ) (set : ℕ) (we : ℕ) (where : ℕ) (σ : ℕ) : n ≥ 1, we set     k X X n Gn := F ⊗ Gn , where Gn := σ X :0≤k ≤2 -1 := by
omega
/- Original: XB > qn | E(B)) = P(XRκ−k > qn ) -/
theorem eq_710b5f57b38d8dae (B : ℕ) (E : ℕ) (P : ℕ) (XB : ℕ) (XRκ : ℕ) (k : ℕ) (qn : ℕ) : XB > qn | E(B)) = P(XRκ-k > qn ) := by
omega
/- Original: N ≥ 0 to see that fk+ (N ) ≤ C sup Assume that gm −−−−−→ 0 hence q = 0 -/
theorem eq_682a7c79a2c07abc (Assume : ℕ) (C : ℕ) (N : ℕ) (fk : ℕ) (gm : ℕ) (hence : ℕ) (q : ℕ) (see : ℕ) (sup : ℕ) (that : ℕ) (to : ℕ) : N ≥ 0 to see that fk+ (N ) ≤ C sup Assume that gm -----→ 0 hence q = 0 := by
omega
/- Original: K< is the cone given by  K< = (H(e))e∈En,d | L(e) < L(e′ ) if e, e′ satisfy condition (2)(b)(ii) of Definition 4 -/
theorem eq_bc1cdbda2c7b279f (Definition : ℕ) (En : ℕ) (H : ℕ) (K : ℕ) (L : ℕ) (b : ℕ) (by : ℕ) (condition : ℕ) (cone : ℕ) (d : ℕ) (e : ℕ) (given : ℕ) (ii : ℕ) (is : ℕ) (of : ℕ) (satisfy : ℕ) (the : ℕ) : K< is the cone given by  K< = (H(e))e∈En ∧ d | L(e) < L(e′ ) if e ∧ e′ satisfy condition (2)(b)(ii) of Definition 4 := by
omega
/- Original: dS = Γo Z ji± nΓi dS = 0 Γi hold -/
theorem eq_094035ea8dbecdbc (Z : ℕ) (dS : ℕ) (hold : ℕ) (ji : ℕ) (nΓi : ℕ) (Γi : ℕ) (Γo : ℕ) : dS = Γo Z ji± nΓi dS = 0 Γi hold := by
omega
/- Original: m =p n=1 1 1 K L   (3 -/
theorem eq_cf28a9fb490522ae (K : ℕ) (L : ℕ) (m : ℕ) (n : ℕ) (p : ℕ) : m =p n=1 1 1 K L   (3 := by
omega
/- Original: n≥1 γ1 ,··· ,γn ∈PL N A∩(γ1 ∪···∪γn )̸=∅ n Y φn (γ1 , -/
theorem eq_cdd78fef1b6ac967 (A : ℕ) (N : ℕ) (PL : ℕ) (Y : ℕ) (n : ℕ) (γ1 : ℕ) (γn : ℕ) (φn : ℕ) : n≥1 γ1 ∧ *** ∧ γn ∈PL N A∩(γ1 ∪***∪γn )̸=∅ n Y φn (γ1 := by
omega
/- Original: e = −1, ηm = 1, ηf = −1 -/
theorem eq_10fc93294da04990 (e : ℕ) (ηf : ℕ) (ηm : ℕ) : e = -1 ∧ ηm = 1 ∧ ηf = -1 := by
omega
/- Original: e = R+ (Γ)−1 Γ, Γ e2 = R+ (Γ2 )−1 Γ2 -/
theorem eq_b9b5adde4c75eb99 (R : ℕ) (e : ℕ) (e2 : ℕ) (Γ : ℕ) (Γ2 : ℕ) : e = R+ (Γ)-1 Γ ∧ Γ e2 = R+ (Γ2 )-1 Γ2 := by
omega
/- Original: R = RU(a), pushing both sides through the MPS tensors should give the same virtual operators on the boundary -/
theorem eq_3810af9531ba920b (MPS : ℕ) (R : ℕ) (RU : ℕ) (a : ℕ) (both : ℕ) (boundary : ℕ) (give : ℕ) (on : ℕ) (operators : ℕ) (pushing : ℕ) (same : ℕ) (should : ℕ) (sides : ℕ) (tensors : ℕ) (the : ℕ) (through : ℕ) (virtual : ℕ) : R = RU(a) ∧ pushing both sides through the MPS tensors should give the same virtual operators on the boundary := by
omega
/- Original: ADM = HV take values from −∞ to ∞ due to this subtraction -/
theorem eq_90ae24f93c2aba19 (ADM : ℕ) (HV : ℕ) (due : ℕ) (from : ℕ) (subtraction : ℕ) (take : ℕ) (this : ℕ) (to : ℕ) (values : ℕ) : ADM = HV take values from -∞ to ∞ due to this subtraction := by
omega
/- Original: i=1 where vi = |ei ⟩⟨ei+1 | for i = 1, -/
theorem eq_4b9a9d818949e851 (ei : ℕ) (for : ℕ) (i : ℕ) (vi : ℕ) (where : ℕ) : i=1 where vi = |ei ⟩⟨ei+1 | for i = 1, := by
omega
/- Original: x > a, ψ− (x, k) = e−ikx , x < 0 -/
theorem eq_203a31a08446bc90 (a : ℕ) (e : ℕ) (ikx : ℕ) (k : ℕ) (x : ℕ) (ψ : ℕ) : x > a ∧ ψ- (x ∧ k) = e-ikx ∧ x < 0 := by
omega
/- Original: j = N, λ j,i = −N -/
theorem eq_6dd2330a87fae20a (N : ℕ) (i : ℕ) (j : ℕ) (λ : ℕ) : j = N ∧ λ j ∧ i = -N := by
omega
/- Original: PLi = Li ⊗ t, TORAL CHERN–SIMONS TQFT 31 be the toral Maslov–Kashiwara index of Proposition 2 -/
theorem eq_5032a7d91908ad76 (CHERN : ℕ) (Kashiwara : ℕ) (Li : ℕ) (Maslov : ℕ) (PLi : ℕ) (Proposition : ℕ) (SIMONS : ℕ) (TORAL : ℕ) (TQFT : ℕ) (be : ℕ) (index : ℕ) (of : ℕ) (t : ℕ) (the : ℕ) (toral : ℕ) : PLi = Li ⊗ t ∧ TORAL CHERN–SIMONS TQFT 31 be the toral Maslov–Kashiwara index of Proposition 2 := by
omega
/- Original: m = (2p + 3) -/
theorem eq_540e17d250b0af50 (m : ℕ) (p : ℕ) : m = (2p + 3) := by
omega
/- Original: X = H(P µ×µ X) -/
theorem eq_6c3ba7cc1fb845ce (H : ℕ) (P : ℕ) (X : ℕ) (µ : ℕ) : X = H(P µ*µ X) := by
omega
/- Original: b = 0, and Q b − 1 Hilbert-Schmidt -/
theorem eq_3b6ae611d6b382ed (Hilbert : ℕ) (Q : ℕ) (Schmidt : ℕ) (b : ℕ) : b = 0, ∧ Q b - 1 Hilbert-Schmidt := by
omega
/- Original: j = 0), it is locally pure gauge -/
theorem eq_97721d62fd2a1ccb (gauge : ℕ) (is : ℕ) (it : ℕ) (j : ℕ) (locally : ℕ) (pure : ℕ) : j = 0) ∧ it is locally pure gauge := by
omega
/- Original: n = √ · 2n−2 -/
theorem eq_5a5fbddb6b6a87a3 (n : ℕ) : n = √ * 2n-2 := by
omega
/- Original: k = 2, as they need some refinement for general k-point correlation functions -/
theorem eq_4e29398f0be55f03 (as : ℕ) (correlation : ℕ) (for : ℕ) (functions : ℕ) (general : ℕ) (k : ℕ) (need : ℕ) (point : ℕ) (refinement : ℕ) (some : ℕ) (they : ℕ) : k = 2 ∧ as they need some refinement for general k-point correlation functions := by
omega
/- Original: Qinst = -/
theorem eq_747a27fec4690e42 (Qinst : ℕ) : Qinst = := by
omega
/- Original: t = 0) = 0) -/
theorem eq_584ccf70dc2448ec (t : ℕ) : t = 0) = 0) := by
omega
/- Original: KLk = KLk (g) is the full subcategory of b g-modules that satisfy the following properties -/
theorem eq_70d1ccc18e7340b6 (KLk : ℕ) (b : ℕ) (following : ℕ) (full : ℕ) (g : ℕ) (is : ℕ) (modules : ℕ) (of : ℕ) (properties : ℕ) (satisfy : ℕ) (subcategory : ℕ) (that : ℕ) (the : ℕ) : KLk = KLk (g) is the full subcategory of b g-modules that satisfy the following properties := by
omega
/- Original: i ≥ 0, γ1 + γ2 + i = γ − 1, and l1 + l2 = l -/
theorem eq_aa2457246f273f8f (i : ℕ) (l : ℕ) (l1 : ℕ) (l2 : ℕ) (γ : ℕ) (γ1 : ℕ) (γ2 : ℕ) : i ≥ 0, γ1 + γ2 + i = γ - 1, ∧ l1 + l2 = l := by
omega
/- Original: f = λ0 (F ⊗ id)f , so (F ⊗ id)f =  -/
theorem eq_f08aae76038d585a (F : ℕ) (f : ℕ) (id : ℕ) (so : ℕ) (λ0 : ℕ) : f = λ0 (F ⊗ id)f ∧ so (F ⊗ id)f =  := by
omega
/- Original: k = i term is shown in red horizontal lines -/
theorem eq_cbb575aa4c4bb9c0 (horizontal : ℕ) (i : ℕ) (is : ℕ) (k : ℕ) (lines : ℕ) (red : ℕ) (shown : ℕ) (term : ℕ) : k = i term is shown in red horizontal lines := by
omega
/- Original: N =1 where ψ̃ denotes the normalized LQG coherent state -/
theorem eq_e239715ab54a14e7 (LQG : ℕ) (N : ℕ) (coherent : ℕ) (denotes : ℕ) (normalized : ℕ) (state : ℕ) (the : ℕ) (where : ℕ) (ψ : ℕ) : N =1 where ψ̃ denotes the normalized LQG coherent state := by
omega
/- Original: j=1 Therefore, we can apply Theorem 2 -/
theorem eq_43d1ba4864a0e012 (Theorem : ℕ) (Therefore : ℕ) (apply : ℕ) (can : ℕ) (j : ℕ) (we : ℕ) : j=1 Therefore ∧ we can apply Theorem 2 := by
omega
/- Original: A = *-alg(J) -/
theorem eq_8667a4abcdfb0e33 (A : ℕ) (J : ℕ) (alg : ℕ) : A = *-alg(J) := by
omega
/- Original: dKdr = f1 drtt when e2ψ = f holds in vacuum -/
theorem eq_2eee55494a48e808 (dKdr : ℕ) (drtt : ℕ) (e2ψ : ℕ) (f : ℕ) (f1 : ℕ) (holds : ℕ) (vacuum : ℕ) (when : ℕ) : dKdr = f1 drtt when e2ψ = f holds in vacuum := by
omega
/- Original: s = 12 , similar to (3 -/
theorem eq_e73d75157c00b15f (s : ℕ) (similar : ℕ) (to : ℕ) : s = 12 ∧ similar to (3 := by
omega
/- Original: I = Id is defined as in Subsection 2 -/
theorem eq_2309f4fa60e78526 (I : ℕ) (Id : ℕ) (Subsection : ℕ) (as : ℕ) (defined : ℕ) (is : ℕ) : I = Id is defined as in Subsection 2 := by
omega
/- Original: P = 1000 for 2-bead and 3-bead polymer molecules, and choose P = 1000, 2000, 5000, 10000 for 4-bead polymer molecules -/
theorem eq_e6bb6012128aeb31 (P : ℕ) (bead : ℕ) (choose : ℕ) (for : ℕ) (molecules : ℕ) (polymer : ℕ) : P = 1000 for 2-bead ∧ 3-bead polymer molecules, and choose P = 1000, 2000, 5000, 10000 for 4-bead polymer molecules := by
omega
/- Original: i = ais 1 Ks -/
theorem eq_dea3ec296a54c888 (Ks : ℕ) (ais : ℕ) (i : ℕ) : i = ais 1 Ks := by
omega
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