hcju/isabelleps huggingface dataset:
complemented_regular: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.pd_algebra inf less_eq less sup bot top uminus"   and "\<exists>y. inf x y = bot \<and> sup x y = top" shows "x = uminus (uminus x)"
en_eq_2: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" shows "[\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>\<lbrace>x,F\<rbrace> in v]"
sh': shows "sh' = sh(C \<mapsto> (sfs, Error))"
l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs \<equiv> type_wf = DocumentClass.type_wf \<and> get_tag_name = l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name \<and> get_tag_name_locs = l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name_locs"
insert_somewhere: fixes ref :: "'c option"   and r :: "'c"   and list :: "'c list"   and oid :: "'c" assumes "ref = None \<or> ref = Some r \<and> r \<in> set list" shows "\<exists>xs ys. list = xs @ ys \<and> insert_spec list (oid, ref) = xs @ oid # ys"
nn_integral_eq_integrable: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure"   and x :: "real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "0 \<le> x" shows "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal x) = (integrable M f \<and> integral\<^sup>L M f = x)"
lookup_times_zero: fixes p :: "'c \<Rightarrow>\<^sub>0 'b"   and q :: "'c \<Rightarrow>\<^sub>0 'b" shows "lookup (p * q) (0::'c) = lookup p (0::'c) * lookup q (0::'c)"
naturality: fixes f :: "'d" assumes "D.arr f" shows "\<eta>.map (D.cod f) \<cdot>\<^sub>D D.map f = FG.map f \<cdot>\<^sub>D \<eta>.map (D.dom f)"
p'_grid: shows "p' \<in> grid (child p (Grid_Point.inv dir) d) {d}"
g2: shows "L\<lbrakk>Vx : A \<turnstile> e2 : B\<rbrakk> \<rightarrow> IMor g2"
inf_sup_aci_3: fixes inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'f"   and y :: "'f"   and z :: "'f" assumes "class.lattice inf less_eq less sup" shows "inf x (inf y z) = inf y (inf x z)"
surjec_def: fixes F :: "('e, 'f) aGroup_scheme"   and G :: "('g, 'h) aGroup_scheme"   and f :: "'e \<Rightarrow> 'g" shows "surjec\<^bsub>F,G\<^esub> f = (f \<in> aHom F G \<and> surj_to f (carrier F) (carrier G))"
extended_Gromov_distance_PInf_boundary_2: fixes x :: "'b Gromov_completion"   and y :: "'b Gromov_completion" assumes "x \<in> Gromov_boundary" shows "extended_Gromov_distance y x = \<infinity>"
simps_1: fixes f :: "char list \<Rightarrow> aexp \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "bexp \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'a"   and Q :: "'a" shows "Step f g S (SKIP {Q}) = SKIP {S}"
nat_pow_eone: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier (UP R)" shows "[1] \<cdot>\<^bsub>UP R\<^esub> x = x"
Brouwer_degree2_unique: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and d :: "int" assumes "continuous_map (nsphere p) (nsphere p) f"   and "\<And>x. x \<in> carrier (reduced_homology_group (int p) (nsphere p)) \<Longrightarrow> hom_induced (int p) (nsphere p) {} (nsphere p) {} f x = x [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> d" shows "Brouwer_degree2 p f = d"
add_closed: fixes R :: "'a set"   and g :: "'a"   and h :: "'a" assumes "Ring1 R"   and "g \<in> R"   and "h \<in> R" shows "g + h \<in> R"
scale_minus_left_on: fixes x :: "'b"   and a :: "'a" assumes "x \<in> U\<^sub>M" shows "- a *s x = - (a *s x)"
is_suffix_def: fixes u :: "'a list"   and v :: "'a list" shows "is_suffix u v = (\<exists>w. w @ u = v)"
lastMarkedIsMemberOfItsMarkedElementsTo: fixes M :: "('b \<times> bool) list" assumes "uniq (elements M)"   and "markedElements M \<noteq> []" shows "lastMarked M \<in> set (markedElementsTo (lastMarked M) M)"
product_language_state: fixes A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and q1 :: "'f"   and q2 :: "'g" shows "LS (product A B) (q1, q2) = LS A q1 \<inter> LS B q2"
induct: fixes x :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "gdelta x"   and "\<And>F. (\<And>n. open (F n)) \<Longrightarrow> P (\<Inter> (range F))" shows "P x"
open_W: fixes i :: "nat" assumes "i \<in> N" shows "open (W i)"
Derivation_append: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and E :: "(nat \<times> symbol \<times> symbol list) list"   and c :: "symbol list" shows "Derivation a (D @ E) c = (\<exists>b. Derivation a D b \<and> Derivation b E c)"
abs_less_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (abs a) b = (less a b \<and> less (uminus a) b)"
simps_22: fixes e :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (throw e) A = \<D> e A"
nprv_prvI: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv {} \<phi>"   and "\<phi> \<in> fmla" shows "prv \<phi>"
left_tree_initial_edge: fixes s :: "'c"   and t :: "'c" assumes "s \<rightarrow>\<^bsub>T\<^esub> t" shows "t \<notin> T.left_tree s t"
ivp1: shows "Y\<^sub>1 \<in> Sols f U S t\<^sub>0 s"
\<Sigma>_protocols_to_commitments_def: fixes init :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and response :: "'c \<Rightarrow> 'b \<Rightarrow> 'e \<Rightarrow> 'f spmf"   and check :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and Rel :: "('a \<times> 'b) set"   and S_raw :: "'a \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'f) spmf"   and \<A>ss :: "'a \<Rightarrow> 'd \<times> 'e \<times> 'f \<Rightarrow> 'd \<times> 'e \<times> 'f \<Rightarrow> 'b spmf"   and challenge_space :: "'e set"   and valid_pub :: "'a set"   and G :: "('a \<times> 'b) spmf" shows "\<Sigma>_protocols_to_commitments init response check Rel S_raw \<A>ss challenge_space valid_pub G \<equiv> \<Sigma>_protocols_base Rel valid_pub \<and> \<Sigma>_protocols_to_commitments_axioms init response check Rel S_raw \<A>ss challenge_space valid_pub G"
no_real_zero: shows "\<forall>x\<in>proots pp. Im x \<noteq> 0"
nn_integral_real_affine_lebesgue: fixes f :: "real \<Rightarrow> ennreal"   and c :: "real"   and t :: "real" assumes "f \<in> borel_measurable lebesgue"   and "c \<noteq> 0" shows "integral\<^sup>N lebesgue f = ennreal \<bar>c\<bar> * \<integral>\<^sup>+ x. f (t + c * x) \<partial>lebesgue"
norm_f: fixes p :: "nat"   and n :: "nat" assumes "prime p" shows "cmod (f n p) = ln (real p) / real p / real n powr Re s"
assms_2: shows "\<forall>n. f n \<in> s"
push_pres_dis_invar: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and u :: "nat"   and fo :: "nat \<times> nat \<Rightarrow> 'a"   and lo :: "nat \<Rightarrow> nat"   and no :: "nat \<Rightarrow> nat set"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and v :: "nat" assumes "discharge_invar c s t u fo lo no f l n"   and "Network.push_precond c f l (u, v)" shows "discharge_invar c s t u fo lo no (Network.push_effect c f (u, v)) l n"
exec_gpv_oracle_bisim: fixes X :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and s1 :: "'b"   and s2 :: "'d"   and oracle1 :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b) spmf"   and oracle2 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and gpv :: "('g, 'e, 'f) gpv"   and R :: "'g \<times> 'b \<Rightarrow> 'g \<times> 'd \<Rightarrow> bool" assumes "X s1 s2"   and "\<And>s1 s2 x. X s1 s2 \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). a = b \<and> X s1' s2') (oracle1 s1 x) (oracle2 s2 x)"   and "\<And>x s1' s2'. \<lbrakk>X s1' s2'; (x, s1') \<in> set_spmf (exec_gpv oracle1 gpv s1); (x, s2') \<in> set_spmf (exec_gpv oracle2 gpv s2)\<rbrakk> \<Longrightarrow> R (x, s1') (x, s2')" shows "rel_spmf R (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
prod_eqI: fixes p :: "'g \<times> 'h"   and q :: "'g \<times> 'h" assumes "fst p = fst q"   and "snd p = snd q" shows "p = q"
destrmi'_def: fixes ni :: "nat"   and s :: "bdd" shows "destrmi' ni s \<equiv> oassert (bdd_sane s \<and> bdd_node_valid s ni) \<bind> (\<lambda>_. Some (destrmi ni s))"
Int: fixes S :: "'b set"   and B :: "'b set set"   and U :: "'b set"   and V :: "'b set" assumes "generated_topology S B U"   and "generated_topology S B V" shows "generated_topology S B (U \<inter> V)"
le_less: fixes x :: "'b"   and y :: "'b" shows "(x \<le> y) = (x < y \<or> x = y)"
att_base: fixes s :: "'a set \<times> 'a set" shows "(\<turnstile>\<N>\<^bsub>s\<^esub>) = (\<forall>x\<in>fst s. Bex (snd s) ((\<rightarrow>\<^sub>i) x))"
combine_take: fixes Us :: "'a list"   and Ks :: "'a list" shows "combine (take (length Us) Ks) Us = combine Ks Us"
Na_def: fixes x1 :: "name"   and x2 :: "trm" shows "x1:x2 \<equiv> Abs_ntrm (Na_Rep ([x1].x2))"
homotopic_with_compose_continuous_right: fixes p :: "('c \<Rightarrow> 'b) \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'a"   and X :: "'a set"   and Y :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and W :: "'c set" assumes "homotopic_with_canon (\<lambda>f. p (f \<circ> h)) X Y f g"   and "continuous_on W h"   and "h ` W \<subseteq> X" shows "homotopic_with_canon p W Y (f \<circ> h) (g \<circ> h)"
cases: fixes a1 :: "cld"   and a2 :: "meth_def list" assumes "class_methods a1 a2" obtains dcl :: "char list" and cl :: "cl" and fds :: "fd list" and meth_defs :: "meth_def list" where "a1 = cld_def dcl cl fds meth_defs" and "a2 = meth_defs"
cnt_remove_all_cycles: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x \<in> set xs" shows "cnt x (remove_all_cycles xs ys) \<le> 1"
external_call_hconf: fixes t :: "'thread_id"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap"   and U :: "ty" assumes "compP2 P,t \<turnstile> \<langle>a\<bullet>M(vs),h\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>"   and "compP2 P,h \<turnstile> a\<bullet>M(vs) : U"   and "hconf h" shows "hconf h'"
noStuck: shows "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
inj_map_eq_map: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list" assumes "inj f" shows "(map f xs = map f ys) = (xs = ys)"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "D.mkCone F f = C.null"
IntE: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A \<inter> B" obtains "c \<in> A" and "c \<in> B"
bind_const: fixes m :: "'m"   and m' :: "'m" shows "bind m (\<lambda>_. m') = m'"
hyps: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat"   and D :: "'a mat" assumes "A \<in> carrier_mat m m"   and "B = 0\<^sub>m m (n - m)"   and "C = 0\<^sub>m (n - m) m"   and "D = 1\<^sub>m (n - m)"   and "m < n" shows "Determinant.det (four_block_mat A B C D) = Determinant.det A"
diff_is_0_eq': fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "m - n = 0"
vars_iff_occseq: fixes x :: "'b"   and t :: "'b trm" shows "(x \<in> vars_of t) = (Var x \<prec> t \<or> Var x = t)"
midpoint_preserves_per: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and M :: "'p"   and A1 :: "'p"   and B1 :: "'p"   and C1 :: "'p" assumes "Per A B C"   and "M Midpoint A A1"   and "M Midpoint B B1"   and "M Midpoint C C1" shows "Per A1 B1 C1"
weight_w0_unary_1: fixes w :: "'b \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and pr_weak :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and least :: "'b \<Rightarrow> bool"   and scf :: "'b \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and t :: "('b, 'c) Term.term"   and f :: "'b"   and ts :: "('b, 'c) Term.term list"   and t1 :: "('b, 'c) Term.term"   and ts' :: "('b, 'c) Term.term list" assumes "admissible_kbo w w0 pr_strict pr_weak least scf"   and "weight_fun.weight w w0 scf t = w0"   and "t = Fun f ts"   and "ts = t1 # ts'" shows "ts' = []"
guarantees_INT_right_iff: fixes F :: "'b program"   and X :: "'b program set"   and Y :: "'c \<Rightarrow> 'b program set"   and I :: "'c set" shows "(F \<in> X guarantees \<Inter> (Y ` I)) = (\<forall>i\<in>I. F \<in> X guarantees Y i)"
pseudo_proj_True_preimage_stake_space: fixes w :: "bool stream"   and n :: "nat" assumes "w = pseudo_proj_True n w" shows "pseudo_proj_True n -` {w} \<inter> space M = {x \<in> space M. stake n x = stake n w}"
find_sig_reducer_SomeD_red: fixes t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and i :: "nat" assumes "t \<in> keys (rep_list p)"   and "find_sig_reducer (map spp_of bs) (lt p) t 0 = Some i" shows "sig_red (\<prec>\<^sub>t) (\<preceq>) (set bs) p (p - monom_mult (lookup (rep_list p) t / punit.lc (rep_list (bs ! i))) (t - punit.lt (rep_list (bs ! i))) (bs ! i))"
ltl_prop_unfold_equiv_equivp: shows "equivp (\<sim>\<^sub>Q)"
eff_enabled_def: fixes s :: "nat \<Rightarrow> nat option" shows "eff_enabled s \<equiv> \<lambda>(pres, uu_, uu_, uu_). s \<in> subsuming_states (map_of pres)"
hom_inv: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g"   and x :: "'e" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h (inv\<^bsub>G\<^esub> x) = inv\<^bsub>H\<^esub> h x"
times_pnat_def: fixes x :: "pnat"   and y :: "pnat" shows "x * y \<equiv> pnat_plus x y"
ev: fixes v :: "'a"   and e :: "'a" assumes "v \<sqinter> top = v"   and "e \<le> v \<sqinter> - id v" shows "e \<sqinter> v = bot"
in_hom_def: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright> \<equiv> a \<in> C.domains f \<and> b \<in> C.codomains f"
stable_imp_Stable: fixes F :: "'b program"   and A :: "'b set" assumes "F \<in> stable A" shows "F \<in> Stable A"
comp_inv_arr': fixes f :: "'a" assumes "S.iso f" shows "S.inv f \<cdot>\<^sub>S f = S.dom f"
funpow_times_power: fixes f :: "'a \<Rightarrow> nat"   and x :: "'a" shows "(*) x ^^ f x = (*) (x ^ f x)"
assms_5: shows "P,shp s \<turnstile>\<^sub>b (es,b) \<surd>"
ex_equiv: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'c \<Longrightarrow> 'b"   and P :: "'b upred" assumes "x \<approx>\<^sub>L y" shows "(\<exists> x \<bullet> P) = (\<exists> y \<bullet> P)"
all_shared_takeWhile_subset: fixes P :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" shows "all_shared (takeWhile P sb) \<subseteq> all_shared sb"
\<chi>_def: shows "\<chi> = subst \<phi> t x"
filter_raw_def: fixes P :: "'c \<Rightarrow> bool"   and g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "filter_raw P g s = (case g s of Done \<Rightarrow> Done | Skip s' \<Rightarrow> Skip s' | Yield a s' \<Rightarrow> if P a then Yield a s' else Skip s')"
simps_5: fixes f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
P_valid: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "vm_path G P v0" shows "Digraph.valid_path G P"
eventually_sup: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and F' :: "'b filter" shows "eventually P (sup F F') = (eventually P F \<and> eventually P F')"
emptyNotInRange: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> allAllocations N G" shows "{} \<notin> Range a"
lossless_R2: fixes M :: "'grp \<times> 'grp"   and \<sigma> :: "bool" shows "lossless_spmf (R2 M \<sigma>)"
simps_7: fixes va :: "int"   and vb :: "int"   and vc :: "int list" shows "inf\<^sub>- (Atom (Dvd va vb vc)) = Atom (Dvd va vb vc)"
block: fixes vs :: "e list"   and n :: "nat"   and t1s :: "t list"   and t2s :: "t list"   and m :: "nat"   and es :: "b_e list" assumes "const_list vs"   and "length vs = n"   and "length t1s = n"   and "length t2s = m" shows "\<lparr>vs @ [$Block (t1s _> t2s) es]\<rparr> \<leadsto> \<lparr>[Label m [] (vs @ ($* es))]\<rparr>"
mset_le_add_mset_decr_left1: fixes c :: "'a"   and a :: "'a multiset"   and b :: "'a multiset" assumes "add_mset c a \<subseteq># b" shows "a \<subseteq># b"
Abs_process_inverse: fixes y :: "('a event list \<times> 'a event set) set \<times> 'a event list set" assumes "y \<in> {p. is_process p}" shows "Rep_process (Abs_process y) = y"
components_are_equivalences: fixes a :: "'a" assumes "C.obj a" shows "D.equivalence_map (\<sigma>\<^sub>0 a)"
conforms_Jump: fixes j :: "jump"   and s :: "st"   and G :: "prog"   and L :: "lenv" assumes "j = Ret \<longrightarrow> locals s Result \<noteq> None" shows "(Some (Jump j), s)\<Colon>\<preceq>(G, L) = Norm s\<Colon>\<preceq>(G, L)"
inj_lassocr_swap_sum: shows "inj lassocr_swap_sum"
inject_2: fixes x7 :: "CTypeId"   and y7 :: "CTypeId" shows "(CClassT x7 = CClassT y7) = (x7 = y7)"
extends_closed: fixes T :: "ty"   and \<Gamma> :: "binding list"   and \<Delta> :: "binding list" assumes "T closed_in \<Gamma>"   and "\<Delta> extends \<Gamma>" shows "T closed_in \<Delta>"
sublens_pres_indep: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and Z :: "'g \<Longrightarrow> 'e" assumes "X \<subseteq>\<^sub>L Y"   and "Y \<bowtie> Z" shows "X \<bowtie> Z"
conj_disj_distr: fixes P :: "'d upred"   and Q :: "'d upred"   and R :: "'d upred" shows "(P \<and> (Q \<or> R)) = (P \<and> Q \<or> P \<and> R)"
inf_prefix_length_NilL: fixes ys :: "'b list" shows "inf_prefix_length [] ys = 0"
constant_in_carrier: fixes a :: "'a mod_ring" shows "[:a:] \<in> carrier R"
hp: shows "\<forall>x\<in>H. h x \<le> p x"
strict_sorted_list_of_set: fixes A :: "'a set" shows "strict_sorted (sorted_list_of_set A)"
assms: shows "B \<inter>\<^sub>\<circ> A \<noteq> 0"
LyndonD: fixes w :: "'a list"   and n :: "nat" assumes "Lyndon w"   and "0 < n"   and "n < \<^bold>|w\<^bold>|" shows "w <lex rotate n w"
dom_def: fixes m :: "'c \<Rightarrow> 'd option" shows "dom m = {a. m a \<noteq> None}"
span_add1: fixes U :: "'c set"   and v :: "'c"   and w :: "'c" assumes "U \<subseteq> carrier V"   and "v \<in> span U"   and "w \<in> span U" shows "v \<oplus>\<^bsub>V\<^esub> w \<in> span U"
dgrad_p_set_closed_red_rtrancl: fixes d :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow> nat"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and m :: "nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "F \<subseteq> punit.dgrad_p_set d m"   and "p \<in> punit.dgrad_p_set d m"   and "(punit.red F)\<^sup>*\<^sup>* p q" shows "q \<in> punit.dgrad_p_set d m"
strict_implies_order: fixes b :: "'d DBMEntry"   and a :: "'d DBMEntry" assumes "b \<prec> a" shows "b \<preceq> a"
fls_residue_deriv: fixes f :: "'a fls" shows "fls_residue (fls_deriv f) = (0::'a)"
distinct_adj_singleton: fixes x :: "'g" shows "distinct_adj [x]"
simps: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" shows "has_bochner_integral M f x = (\<exists>s. f \<in> borel_measurable M \<and> (\<forall>x. simple_bochner_integrable M (s x)) \<and> (\<lambda>i. \<integral>\<^sup>+ x. ennreal (norm (f x - s i x)) \<partial>M) \<longlonglongrightarrow> 0 \<and> (\<lambda>i. simple_bochner_integral M (s i)) \<longlonglongrightarrow> x)"
count_mult: shows "count (A \<times># B) x = count A (fst x) * count B (snd x)"
reductive_inference_system_axioms: shows "reductive_inference_system \<Gamma>"
add_mono_thms_linordered_field_4: fixes i :: "'b"   and j :: "'b"   and k :: "'b"   and l :: "'b" assumes "i \<le> j \<and> k < l" shows "i + k < j + l"
r8: fixes x :: "name"   and s :: "trm"   and t :: "trm" assumes "x \<sharp> s" shows "[s] to x in t \<mapsto> t[x::=s]"
assms: fixes x :: "'a" assumes "x = v" shows "f x \<le> SPEC \<Phi>"
Cn_conj: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(conj_PL p q \<in> Cn A) = (Cn {p} \<union> Cn {q} \<subseteq> Cn A)"
set_less_eq_empty: fixes A :: "'a set" shows "(A \<sqsubseteq> {}) = (A = {})"
map_add_ac_1: fixes m1 :: "'c \<Rightarrow> 'd option"   and m2 :: "'c \<Rightarrow> 'd option"   and m3 :: "'c \<Rightarrow> 'd option" shows "m1 ++ m2 ++ m3 = m1 ++ (m2 ++ m3)"
maxcoeff_nz: fixes t :: "MIR.num" assumes "nozerocoeff t"   and "maxcoeff t = 0" shows "t = C 0"
unfold_4: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "(P \<U> Q) \<sigma> = (Q \<sigma> \<or> P \<sigma> \<and> (\<circle>(P \<U> Q)) \<sigma>)"
int_pow_1: fixes x :: "'a" assumes "x \<in> carrier R" shows "[1] \<cdot> x = x"
flush_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "flush_instr instr \<equiv> let op_list = snd instr in do addr \<leftarrow> gets (get_addr op_list); modify flush_cache_all; return () od"
norm_19: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<diamond>\<diamond>P = \<diamond>P"
limpt_of_closure: fixes x :: "'a"   and S :: "'a set" shows "x islimpt closure S = x islimpt S"
treesize_cutoff: fixes red :: "bool list \<Rightarrow> bool"   and ds :: "bool list"   and T :: "tree" shows "treesize (cutoff red ds T) \<le> treesize T"
da_Lit: fixes A :: "assigned"   and B :: "lname set"   and Env :: "env"   and v :: "val" assumes "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>" shows "Env\<turnstile> B \<guillemotright>\<langle>Lit v\<rangle>\<guillemotright> A"
map\<^sub>1_simps_2: fixes f :: "'a" assumes "C.ide f" shows "src\<^sub>D (\<tau>\<^sub>1 f) = F.map\<^sub>0 (src\<^sub>C f)"