hcju/isabelleps huggingface dataset:
subspace_kernel: shows "local.subspace {x. f x = (0::'c)}"
case_prodI2: fixes p :: "'f \<times> 'g"   and c :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
prems_6: shows "trms\<^sub>s\<^sub>t (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S') \<subseteq> SMP M"
dg_Dom_vrange: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
unit_imp_no_irreducible_divisors: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and p :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times x one"   and "comm_semiring_1.irreducible times one zero p" shows "\<not> dvd.dvd times p x"
R_def: shows "R = cbox (case (real_of_int l - \<delta>, snd p) of (x, y) \<Rightarrow> (x, real_of_int y)) (real_of_int l + \<delta>, real_of_int (snd p) + \<delta>)"
ceiling_add_one: fixes x :: "'b" shows "\<lceil>x + (1::'b)\<rceil> = \<lceil>x\<rceil> + 1"
abscissa: fixes x :: "real"   and y :: "real" shows "abscissa (Point x y) = x"
stepsize_inverse: fixes L :: "real"   and h :: "real"   and B :: "real"   and r :: "real"   and p :: "nat"   and T2 :: "real"   and T1 :: "real" assumes "0 \<le> L"   and "0 \<le> h"   and "0 \<le> B"   and "0 \<le> r"   and "0 < p"   and "T2 \<le> T1"   and "0 \<le> T2"   and "h \<le> root p (r * L / B / (exp (L * T1 + 1) - 1))" shows "B / L * (exp (L * T2 + 1) - 1) * h ^ p \<le> r"
fls_pow_conv_fps_pow: fixes f :: "'a fls"   and n :: "nat" assumes "0 \<le> fls_subdegree f" shows "f ^ n = fps_to_fls (fls_regpart f ^ n)"
conc_def: fixes w :: "'e list"   and x :: "nat \<Rightarrow> 'e" shows "w \<frown> x \<equiv> \<lambda>n. if n < length w then w ! n else x (n - length w)"
g_rel: shows "(g, ga) \<in> br g.\<alpha> g.invar"
inf_top_left: fixes x :: "'a" shows "top \<sqinter> x = x"
sel: fixes x3 :: "nat \<times> nat" shows "un_SAT_PUSH' (SAT_PUSH' x3) = x3"
Init_stp_act_rev: fixes P :: "state \<Rightarrow> bool" shows "Init P \<equiv> Init $P"
RPDi_MSkip: fixes i :: "nat"   and n :: "nat" shows "RPDi i (MSkip n) \<subseteq> MSkip ` {i. i \<le> n}"
IH: fixes r :: "'trm" assumes "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm" shows "FvarsT (rawpsubstT r txs) \<subseteq> FvarsT r - snd ` set txs \<union> \<Union> {uu_. \<exists>t x. uu_ = FvarsT t \<and> (t, x) \<in> set txs}"
minus_commute: fixes a :: "complex"   and b :: "complex" shows "- a - b = - b - a"
b: shows "\<forall>a\<in>A \<union> B. b \<sqsubseteq> a"
comp_R_2: shows "\<lbrakk>\<lbrakk>u\<^sub>1\<rbrakk>\<rbrakk> \<odot> \<lbrakk>\<lbrakk>\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk> = Maps.MkArr (src r\<^sub>0s\<^sub>1.p\<^sub>0) (trg u) (Maps.Comp \<lbrakk>u\<^sub>1\<rbrakk> \<lbrakk>\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk>)"
bell00_def: shows "|\<beta>\<^sub>0\<^sub>0\<rangle> \<equiv> complex_of_real (1 / sqrt 2) \<cdot>\<^sub>m |Matrix.vec 4 (\<lambda>i. if i = 0 \<or> i = 3 then 1 else 0)\<rangle>"
mat_of_cols_carrier_3: fixes n :: "nat"   and vs :: "'a Matrix.vec list" shows "dim_col (mat_of_cols n vs) = length vs"
measure_subadditive_finite: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "finite I"   and "A ` I \<subseteq> sets M"   and "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>" shows "Sigma_Algebra.measure M (\<Union> (A ` I)) \<le> (\<Sum>i\<in>I. Sigma_Algebra.measure M (A i))"
error_def: fixes exception :: "'f" shows "error exception = Prog (\<lambda>h. Inl exception)"
o_4: shows "xdstp \<in> set (if fst (dports m) = 0 \<and> snd (dports m) = - 1 then [None] else if fst (dports m) \<le> snd (dports m) then map (Some \<circ> (\<lambda>pfx. (pfxm_prefix pfx, ~~ pfxm_mask pfx))) (wordinterval_CIDR_split_prefixmatch (WordInterval (fst (dports m)) (snd (dports m)))) else [])"
tensor_preserves_is_tangle: fixes x :: "wall"   and y :: "wall" assumes "is_tangle_diagram x"   and "is_tangle_diagram y" shows "is_tangle_diagram (x \<otimes> y)"
Aik: shows "A $ mod_type_class.from_nat i $ mod_type_class.from_nat k = matrix_to_iarray A !! i !! k"
beta_reduces_fst: fixes P :: "'a trm"   and P' :: "'a trm" assumes "P \<rightarrow>\<beta>\<^sup>* P'" shows "Fst P \<rightarrow>\<beta>\<^sup>* Fst P'"
simple_refinement_safe_RelS_Eg1Eg2: fixes Some_R :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) set" assumes "\<And>c mds mem mem'. (\<langle>c, mds, mem\<rangle>\<^sub>A, \<langle>c, mds, mem'\<rangle>\<^sub>A) \<in> Some_R \<Longrightarrow> A.low_mds_eq mds mem mem'" shows "simple_refinement_safe Some_R RelS_Eg1Eg2 abs_steps"
simps_3: fixes m1 :: "real mat convnet"   and m2 :: "real mat convnet" shows "remove_weights (Pool m1 m2) = Pool (remove_weights m1) (remove_weights m2)"
verts_in_translation_finite_2: fixes X :: "'b allegorical_term" shows "finite (edges (translation X))"
cong_iff_lin: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and uminus :: "'d \<Rightarrow> 'd"   and division_segment :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and m :: "'d" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo a b m = (\<exists>k. b = plus a (times m k))"
carrier_stalk_def: shows "stk2.carrier_stalk \<equiv> dlim \<FF>2 \<rho>2 (stk1.neighborhoods x)"
inf_Inf: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "inf x y = Inf (insert x (\<eta> y))"
enum_ex_char_def: fixes P :: "char \<Rightarrow> bool" shows "enum_class.enum_ex P = list_ex P enum_class.enum"
open_contains_ball: fixes S :: "'b set" shows "open S = (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "list.set (map f xs) = f ` list.set xs"
simps_2: fixes s :: "'a \<Rightarrow> 'b option"   and ops :: "('a, 'b) sas_plus_operator list"   and opss :: "('a, 'b) sas_plus_operator list list" shows "execute_parallel_plan_sas_plus s (ops # opss) = (if are_all_operators_applicable_in s ops \<and> are_all_operator_effects_consistent ops then execute_parallel_plan_sas_plus (execute_parallel_operator_sas_plus s ops) opss else s)"
exec_sequence_appD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and xs :: "('e, 'd, 'f) com list"   and ys :: "('e, 'd, 'f) com list"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>sequence Seq (xs @ ys),Normal s\<rangle> \<Rightarrow> t" shows "\<exists>s'. \<Gamma>\<turnstile> \<langle>sequence Seq xs,Normal s\<rangle> \<Rightarrow> s' \<and> \<Gamma>\<turnstile> \<langle>sequence Seq ys,s'\<rangle> \<Rightarrow> t"
semilattice_neutr_axioms: shows "semilattice_neutr (\<union>#) {#}"
sim21_size_def: fixes P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and xcpfrs :: "'a option \<times> 'a frame list"   and xcpfrs' :: "'a option \<times> 'a frame list" shows "sim21_size P xcpfrs xcpfrs' = ((xcpfrs, xcpfrs') \<in> inv_image (less_than <*lex*> same_fst (\<lambda>n. True) (\<lambda>n. {(pcxcp, pcxcp'). sim21_size_aux n pcxcp pcxcp'})) (\<lambda>(xcp, frs). (length frs, case frs of (stk, loc, C, M, pc) # frs \<Rightarrow> (length (instrs_of P C M), pc, xcp))))"
elims: fixes x :: "('form, 'rule) graph_node"   and y :: "('form, 'var) antecedent list" assumes "inPorts' x = y" obtains r :: "'rule" where "x = Rule r__" and "y = antecedent r__"   | r :: "'form" where "x = Assumption ra__" and "y = []"   | r :: "'form" where "x = Conclusion r" and "y = [plain_ant r]"   | "x = Helper" and "y = [plain_ant anyP]"
has_parametrized_subprob_densityI: fixes M :: "'a measure"   and N :: "'a \<Rightarrow> 'b measure"   and R :: "'b measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> N x = density R (f x)"   and "\<And>x. x \<in> space M \<Longrightarrow> subprob_space (N x)"   and "(\<lambda>(x, y). f x y) \<in> borel_measurable (M \<Otimes>\<^sub>M R)" shows "has_parametrized_subprob_density M N R f"
chambers_2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "C0 \<sim> g ` C0"
fst_mismatch_def: shows "i.c\<^sub>0 = (x \<cdot> y) ! \<^bold>|i.\<alpha>\<^bold>|"
negligible_Int: fixes s :: "'a set"   and t :: "'a set" assumes "negligible s \<or> negligible t" shows "negligible (s \<inter> t)"
rot_0: fixes f :: "'b \<Rightarrow> 'b"   and f' :: "'b \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'b"   and X :: "'b set" assumes "c1_on_open_R2 f f' X" shows "c1_on_open_R2.rot (0::'b) = (0::'b)"
v_eq: fixes i :: "nat" shows "(1 - (1 / 2) ^ i) * \<alpha> = ennreal ((1 - (1 / 2) ^ i) * \<alpha>')"
append_exec_meth: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and n :: "nat"   and ins' :: "'addr instr list" assumes "exec_meth ci P ins xt t h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"   and "jump_ok ins 0 n" shows "exec_meth ci P (ins' @ ins) (shift (length ins') xt) t h (stk, loc, length ins' + pc, xcp) ta h' (stk', loc', length ins' + pc', xcp')"
mark_loop_locs_def: shows "mark_loop_locs = prefixed ''mark_loop''"
cmult: fixes c :: "'a"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "\<Theta>[F](\<lambda>x. c * f x) = \<Theta>[F](f)"
atLeastLessThan_empty_iff2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.preorder less_eq less" shows "({} = ord.atLeastLessThan less_eq less a b) = (\<not> less a b)"
abs_summable_on_altdef: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "(f abs_summable_on A) = set_integrable (count_space UNIV) A f"
HBL2: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "Fvars \<phi> = {}"   and "Fvars \<chi> = {}" shows "bprv (imp (cnj (PP \<langle>\<phi>\<rangle>) (PP \<langle>imp \<phi> \<chi>\<rangle>)) (PP \<langle>\<chi>\<rangle>))"
igSwapIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "igSwapIGOp MOD \<equiv> \<forall>zs z1 z2 delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSwap MOD zs z1 z2 (igOp MOD delta inp binp) = igOp MOD delta (igSwapInp MOD zs z1 z2 inp) (igSwapBinp MOD zs z1 z2 binp)"
one_closed: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme" assumes "subgroup H G" shows "\<one>\<^bsub>G\<^esub> \<in> H"
AccLoop: fixes i :: "AbstrLevelsID"   and C :: "CSet"   and XS :: "CSet set"   and S :: "CSet"   and ZS :: "CSet set" shows "Acc i C = XS \<union> Acc i S \<and> Acc i S = ZS \<union> Acc i C \<longrightarrow> Acc i C = XS \<union> ZS \<union> {C, S}"
perm_1: fixes pi :: "(name \<times> name) list"   and x3 :: "subject"   and x1 :: "name"   and x2 :: "pi" shows "pi \<bullet> x3\<guillemotleft>x1\<guillemotright> \<prec> x2 = pi \<bullet> x3\<guillemotleft>pi \<bullet> x1\<guillemotright> \<prec> pi \<bullet> x2"
QmpTr: fixes H :: "'a set"   and N :: "'a set" assumes "G \<triangleright> H"   and "G \<triangleright> N"   and "H \<subseteq> N" shows "Qm\<^bsub>G H,N\<^esub> \<in> gHom (G / H) (G / N)"
strategy_progression_def: fixes \<sigma> :: "'b list \<Rightarrow> 'b"   and p :: "'b list" shows "strategy_progression \<sigma> p = lappend (llist_of (prefixes p)) (ltl (iterates (augment_list \<sigma>) p))"
split: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c hp list \<Rightarrow> 'b"   and hp :: "'c hp" shows "P (case hp of Hp x xa \<Rightarrow> f x xa) = (\<forall>x1 x2. hp = Hp x1 x2 \<longrightarrow> P (f x1 x2))"
arr_char: fixes f :: "'c local.term set \<times> 'c local.term set" shows "CC.arr f = (arr (fst f) \<and> arr (snd f))"
simps_1: fixes k :: "'b" shows "nnf (Atom k) = Atom k"
prv_imp_refl: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (imp \<phi> \<phi>)"
list_digraph_simps_2: fixes G_list :: "'b list \<times> ('b \<times> 'b) list" shows "parcs (list_digraph G_list) = set (snd G_list)"
addSyncEmpty: fixes sa :: "'c event list"   and A :: "'c event set"   and u :: "'c event list"   and y1 :: "'c event" shows "sa setinterleaves (([], u), A) \<and> y1 \<in> A \<longrightarrow> sa @ [y1] setinterleaves (([y1], u @ [y1]), A)"
upper_asymptotic_density_bdd_interval_1: shows "upper_asymptotic_density {} = 0"
iT_cut_ge_1: fixes n :: "nat"   and t :: "nat" shows "[\<dots>n] \<down>\<ge> t = (if n < t then {} else [t\<dots>,n - t])"
MaxR_f_def: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and acc :: "'a option" assumes "MaxR r" shows "MaxR.MaxR_f r x acc = (if x \<in> Field r then Some (case acc of None \<Rightarrow> x | Some y \<Rightarrow> MaxR.maxR r x y) else acc)"
simps: fixes \<delta> :: "('a, 'b) ta_rule set"   and f :: "'b"   and ts :: "'b tree list" shows "faccs \<delta> (NODE f ts) = (let Qs = map (faccs \<delta>) ts in {q. Bex \<delta> (r_matchc q f Qs)})"
assoc_in_hom_2: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "\<guillemotleft>\<a>\<^sub>D[f, g, h] : (D.dom f \<star>\<^sub>D D.dom g) \<star>\<^sub>D D.dom h \<Rightarrow>\<^sub>D D.cod f \<star>\<^sub>D D.cod g \<star>\<^sub>D D.cod h\<guillemotright>"
atom_map_sup_pres: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "bounded_lattice.atom_map less_eq less bot (sup x y) = bounded_lattice.atom_map less_eq less bot x \<union> bounded_lattice.atom_map less_eq less bot y"
zero_in_Polys: fixes X :: "'b set" shows "0 \<in> P[X]"
open_euler_infinite_label: fixes G :: "('c, 'c \<times> nat \<times> 'c) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "fin_digraph G"   and "tail G = fst"   and "head G = snd \<circ> snd"   and "Digraph_Component.connected G"   and "u \<in> verts G"   and "v \<in> verts G"   and "\<And>w. \<lbrakk>w \<in> verts G; u \<noteq> w; v \<noteq> w\<rbrakk> \<Longrightarrow> in_degree G w = out_degree G w"   and "in_degree G u + 1 = out_degree G u"   and "out_degree G v + 1 = in_degree G v" shows "\<exists>p. pre_digraph.euler_trail G u p v"
thetaSeqWD_sym: shows "sym thetaSeqWD"
vreal_of_real_impl'': shows "disjnt (range vreal_of_real_impl'') (elts \<omega>)"
UINF_cong: fixes A :: "'d set"   and F :: "'d \<Rightarrow> ('e, 'f) uexpr"   and G :: "'d \<Rightarrow> ('e, 'f) uexpr" assumes "\<And>P. P \<in> A \<Longrightarrow> F P = G P" shows "(\<Sqinter> P \<in> A \<bullet> F P) = (\<Sqinter> P \<in> A \<bullet> G P)"
integral_mult_right_zero: fixes M :: "'b measure"   and c :: "'a"   and f :: "'b \<Rightarrow> 'a" shows "LINT x|M. c * f x = c * integral\<^sup>L M f"
RDirProd_Quot_list_is_ring: fixes n :: "nat"   and I :: "nat \<Rightarrow> 'a set" assumes "\<And>i. i \<le> n \<Longrightarrow> ideal (I i) R" shows "ring (RDirProd_list (map (\<lambda>i. R Quot I i) [0..<Suc n]))"
le_cases: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less_eq y x"
prems: fixes j :: "nat" assumes "Suc i < length (seq j)" shows "thesis"
irreducible\<^sub>d_mE: fixes f :: "int poly" assumes "irreducible\<^sub>d_m f" obtains "0 < degree_m f" and "\<And>g h. \<lbrakk>degree_m g < degree_m f; degree_m h < degree_m f\<rbrakk> \<Longrightarrow> Mp f \<noteq> Mp (g * h)"
index_vec: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "i < n" shows "Matrix.vec n f $v i = f i"
listrelp_betas: fixes ts :: "lam list"   and ts' :: "lam list"   and t :: "lam"   and t' :: "lam" assumes "listrelp (\<rightarrow>\<^sub>\<beta>\<^sup>*) ts ts'"   and "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" shows "t \<degree>\<degree> ts \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<degree>\<degree> ts'"
proj4_eq_flag: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> kD rt" shows "\<pi>\<^sub>4 (the (rt dip)) = the (flag rt dip)"
simps_23: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "char list \<Rightarrow> 'a" shows "(case Boolean of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f2"
scale_right_distrib: fixes a :: "'d"   and x :: "'d"   and y :: "'d" shows "a * (x + y) = a * x + a * y"
obsv_trans_def: shows "obsv_trans = (\<Union>r S v d_f. \<Union> (range (obsv_round r S v d_f)))\<^sup>="
inv_in_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "C.iso f"   and "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>C.inv f : b \<rightarrow>\<^sub>C a\<guillemotright>"
lub_unique: fixes S :: "'a set"   and x :: "'a"   and L :: "'a" assumes "S \<subseteq> A"   and "isLub S cl x"   and "isLub S cl L" shows "x = L"
chain_pred_terminates: fixes F :: "'b upred \<Rightarrow> 'b upred"   and E :: "nat \<Rightarrow> 'b upred" assumes "constr F E"   and "Monotonic F" shows "(\<Or>range E \<and> \<mu> F) = (\<Or>range E \<and> \<nu> F)"
list_Previous_Cons: fixes xs :: "'d list"   and init :: "'d"   and n :: "nat" shows "xs\<^bsup>\<leftarrow>' init\<^esup> n = (init # xs) ! n"
return_spmf_None_parametric: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_spmf A (return_pmf None) (return_pmf None)"
llist_of_eq_LNil_conv: fixes xs :: "'c list" shows "(llist_of xs = LNil) = (xs = [])"
simply_connected_eq_contractible_loop_any: fixes S :: "'b set" shows "simply_connected S = (\<forall>p a. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<and> a \<in> S \<longrightarrow> homotopic_loops S p (linepath a a))"
flip_blinfun_apply: fixes f :: "'b \<Rightarrow>\<^sub>L 'c \<Rightarrow>\<^sub>L 'a"   and a :: "'c"   and b :: "'b" shows "blinfun_apply (blinfun_apply (flip_blinfun f) a) b = blinfun_apply (blinfun_apply f b) a"
b: shows "(\<Squnion>n. integral\<^sup>N M (induced_function A (F n))) = \<integral>\<^sup>+ x. (\<Squnion>n. induced_function A (F n) x) \<partial>M"
compose_append: fixes ss :: "('c \<Rightarrow> ('d, 'c) Term.term) list"   and ts :: "('c \<Rightarrow> ('d, 'c) Term.term) list" shows "compose (ss @ ts) = compose ss \<circ>\<^sub>s compose ts"
AbGroup_set_plus: fixes H :: "'a set"   and G :: "'a set" assumes "AbGroup H"   and "AbGroup G" shows "AbGroup (H + G)"
ShBaseclass_isBaseclass: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Bs :: "base list"   and fs :: "(char list \<times> ty) list"   and ms :: "(char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, fs, ms)\<rfloor>"   and "Shares D \<in> set Bs" shows "D \<in> baseClasses Bs"
lcm_factorial_def: fixes a :: "'a"   and b :: "'a" shows "lcm_factorial a b = (if a = (0::'a) \<or> b = (0::'a) then 0::'a else normalize (\<Prod>\<^sub># (prime_factorization a \<union># prime_factorization b)))"
elem_Sup_llist_imp_Sup_upto_llist: fixes x :: "'b"   and Xs :: "'b set llist" assumes "x \<in> Sup_llist Xs" shows "\<exists>j. enat j < llength Xs \<and> x \<in> Sup_upto_llist Xs (enat j)"
EndPhase1_HInv3_R: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase1 s s' t"   and "HInv3_R s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "HInv3_R s' p q d"
bts_augment_v_union_1: fixes t :: "('d, 'c) bintree"   and rs :: "('d, 'c) bintree list"   and q :: "('c, 'd) pq" shows "set |bt_augment t (bts_augment rs q)| = set |bt_augment t q| \<union> set |bts_augment rs q|"
ground_n_entails_bot: shows "entails_q q (lifted_q_calc.\<G>_Fset N) (lifted_q_calc.\<G>_Fset {B})"
d_test_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y" shows "d (x \<sqinter> y) = x \<sqinter> y"
thetaSSeqI_def: shows "thetaSSeqI \<equiv> {(e ;; c, e ;; d) |e c d. siso e \<and> c \<approx>s d}"
ferrack01_def: fixes p :: "fm" shows "ferrack01 p \<equiv> let p' = rlfm (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p); U = remdups (map simp_num_pair (map (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs (\<Upsilon> p')))) in decr (evaldjf (\<upsilon> p') U)"
mult_lift: fixes x :: "V"   and y :: "V"   and z :: "V" shows "x * lift y z = lift (x * y) (x * z)"
coeff_zero: fixes x :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff 0 x = (0::'a)"
abc: shows "reduce_system_R p ([1], base_case_info_R) = (a, b, c)"
compare_Eq_is_eq: fixes x :: "'a"   and y :: "'a" shows "(compare x y = Eq) = (x = y)"
included_project: fixes V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set" shows "included V (New_max.projectQuery V Q)"
d_guard_def: fixes r_decisions :: "process \<Rightarrow> val option"   and r_votes :: "process \<Rightarrow> val option" shows "d_guard r_decisions r_votes \<equiv> \<forall>p v. r_decisions p = Some v \<longrightarrow> locked_in_vf r_votes v"
wf_syscls_is_class_sub_Throwable: fixes P :: "'a prog"   and C :: "String.literal" assumes "wf_syscls P"   and "P \<turnstile> C \<preceq>\<^sup>* Throwable" shows "is_class P C"
equivp_reflp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "equivp R" shows "R x x"
strict_mono_imp_inj_on: fixes f :: "'a word \<Rightarrow> 'c"   and A :: "'a word set" assumes "signed.strict_mono f" shows "inj_on f A"
subspace_inter: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and B :: "'d set" assumes "module scale"   and "module.subspace scale A"   and "module.subspace scale B" shows "module.subspace scale (A \<inter> B)"