hcju/isabelleps huggingface dataset:
cscc_invar_loc_def: fixes G :: "('a, 'b) graph_rec_scheme"   and v0 :: "'a"   and D0 :: "'a set"   and l :: "'a set list"   and p :: "'a set list"   and D :: "'a set"   and pE :: "('a \<times> 'a) set" shows "cscc_invar_loc G v0 D0 l p D pE \<equiv> invar_loc G v0 D0 p D pE \<and> cscc_invar_ext G l D"
normalize_sort_removed_elem_irrelevant: fixes c :: "String.literal"   and s :: "String.literal set" assumes "sort_ex cs (insert c s)"   and "c \<notin> normalize_sort cs (insert c s)" shows "normalize_sort cs (insert c s) = normalize_sort cs s"
multE: fixes r :: "('b \<times> 'b) set"   and M :: "'b multiset"   and N :: "'b multiset" assumes "trans r"   and "(M, N) \<in> mult r" obtains I :: "'b multiset" and J :: "'b multiset" and K :: "'b multiset" where "M = I + K" and "N = I + J" and "J \<noteq> {#}" and "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r"
augment_rels_shared_exchange: fixes R :: "nat set"   and S' :: "nat set"   and S :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "\<forall>a\<in>R. (a \<in> S') = (a \<in> S)" shows "augment_rels S R \<R> = augment_rels S' R \<R>"
sel_2: fixes x21 :: "'d"   and x22 :: "'f" shows "generat.output (IO x21 x22) = x21"
high_proc_no_low_output_change_def: fixes s\<^sub>H :: "stmt" shows "high_proc_no_low_output_change s\<^sub>H \<equiv> \<forall>\<sigma> \<sigma>'. match \<sigma> \<Gamma>\<^sub>0 \<and> s\<^sub>H \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<longrightarrow> the (\<sigma>' ''lowOut'') = the (\<sigma> ''lowOut'')"
simps_1: fixes proto :: "protocol" shows "simple_proto_conjunct ProtoAny proto = Some proto"
ord_Suc_0: fixes n :: "nat" shows "ord (Suc 0) n = 1"
assms_2: shows "A \<Delta> B \<in> null_sets M"
3: shows "coreflexive (x\<^sup>T\<^sup>\<star> \<sqinter> x\<^sup>T * x)"
insertCI: fixes a :: "'k"   and B :: "'k set"   and b :: "'k" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> insert b B"
MGFn_NormalI: fixes G :: "prog"   and A :: "state triple set"   and n :: "nat"   and t :: "Term.term" assumes "G,A\<turnstile>{Normal ((\<lambda>Y' s' s. s' = s \<and> normal s) \<and>. G\<turnstile>init\<le>n)} t\<succ> {\<lambda>Y s' s. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y, s')}" shows "G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>}"
tauChainPar1: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P" shows "\<Psi> \<rhd> P \<parallel> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<parallel> Q"
pre_nts: shows "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)..<j'] nts"
exists_component_superset: fixes t :: "'a set"   and s :: "'a set" assumes "t \<subseteq> s"   and "s \<noteq> {}"   and "connected t" shows "\<exists>c. c \<in> components s \<and> t \<subseteq> c"
vector_derivative_diff_chain_within: fixes f :: "real \<Rightarrow> 'b"   and f' :: "'b"   and x :: "real"   and S :: "real set"   and g :: "'b \<Rightarrow> 'c"   and g' :: "'b \<Rightarrow> 'c" assumes "(f has_vector_derivative f') (at x within S)"   and "(g has_derivative g') (at (f x) within f ` S)" shows "(g \<circ> f has_vector_derivative g' f') (at x within S)"
assms_2: shows "\<not> isRes \<sigma> \<or> protCl \<sigma>"
inject_4: fixes x4 :: "db"   and y4 :: "db" shows "(DBAbsV x4 = DBAbsV y4) = (x4 = y4)"
add_right_imp_eq: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and b :: "'k"   and a :: "'k"   and c :: "'k" assumes "class.cancel_semigroup_add plus"   and "plus b a = plus c a" shows "b = c"
INF_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c"   and Inf :: "'c set \<Rightarrow> 'c" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
wfP_induct_rule: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "wfP r"   and "\<And>x. (\<And>y. r y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
sel_7: fixes x1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x3 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x4 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x5 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x6 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x7 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x8 :: "((nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping) \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and x9 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x10 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "Direction.UI (Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) = x7"
cong: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat" shows "Optimal_BST2.opt_bst2 w \<equiv> Optimal_BST2.opt_bst2 w"
fold1_o_ctor1: fixes s1 :: "('d, 'e, 'f) F1 \<Rightarrow> 'e"   and s2 :: "('d, 'e, 'f) F2 \<Rightarrow> 'f" shows "fold1 s1 s2 \<circ> ctor1 = s1 \<circ> F1map id (fold1 s1 s2) (fold2 s1 s2)"
LeftDerivationFix_from_singleton: fixes A :: "symbol"   and i :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list" assumes "LeftDerivationFix [A] i D j \<gamma>" shows "D = []"
bounded_linear_minus: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_linear f" shows "bounded_linear (\<lambda>x. - f x)"
finite_Union: fixes X :: "'c set set"   and M :: "'c measure" assumes "finite X"   and "X \<subseteq> sets M" shows "\<Union> X \<in> sets M"
that: fixes y :: "'a" assumes "y \<noteq> (0::'a)"   and "orthogonal x y" shows "thesis"
simps_8: fixes x1 :: "'c"   and x2 :: "'c stream" shows "sset (x1 ## x2) = insert x1 (sset x2)"
i_append_i_take: fixes n :: "nat"   and xs :: "'d list"   and f :: "nat \<Rightarrow> 'd" shows "(xs \<frown> f) \<Down> n = xs \<down> n @ f \<Down> (n - length xs)"
distinct_17: fixes x3 :: "bool"   and x5 :: "nat \<times> char list list" shows "Bool x3 \<noteq> Ref x5"
strict_sorted_acc_lengths: fixes ls :: "'a list list"   and acc :: "nat" assumes "ls \<in> lists (- {[]})" shows "strict_sorted (acc_lengths acc ls)"
trnr\<^sub>\<epsilon>_def: fixes u :: "'a"   and \<nu> :: "'a" shows "T0.trnr\<^sub>\<epsilon> u \<nu> \<equiv> \<r>[u] \<cdot> (u \<star> \<epsilon>) \<cdot> \<a>[u, f, f\<^sup>*] \<cdot> (\<nu> \<star> f\<^sup>*)"
word_upto_distinct_hlp: fixes a :: "'a word"   and b :: "'a word" assumes "a \<le> b"   and "a \<noteq> b" shows "b \<notin> set (word_upto a (b - 1))"
exhaust: fixes y :: "('a, 'b, 'c, 'd) Tree" obtains x1 :: "('a, 'b, 'c, 'd) Tree set['a]" where "y = tConj x1"   | x2 :: "('a, 'b, 'c, 'd) Tree" where "y = tNot x2"   | x31 :: "'d" and x32 :: "'b" where "y = tPred x31 x32"   | x41 :: "'d" and x42 :: "'c" and x43 :: "('a, 'b, 'c, 'd) Tree" where "y = tAct x41 x42 x43"
subst_bop: fixes \<sigma> :: "'f \<Rightarrow> 'g"   and f :: "'h \<Rightarrow> 'i \<Rightarrow> 'e"   and u :: "('h, 'g) uexpr"   and v :: "('i, 'g) uexpr" shows "\<sigma> \<dagger> bop f u v = bop f (\<sigma> \<dagger> u) (\<sigma> \<dagger> v)"
assms_5: shows "ptr \<notin> set |h \<turnstile> get_scdom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)|\<^sub>r"
indicator_eq_1_iff: fixes A :: "'c set"   and x :: "'c" shows "(indicator A x = (1::'b)) = (x \<in> A)"
assms: shows "eval cid t t' n \<gamma> \<and> eval cid t t' n \<gamma>'"
card_of_Plus1: fixes A :: "'a set"   and B :: "'b set" shows "|A| \<le>o |A <+> B|"
zero: fixes f :: "'c \<Rightarrow> 'd" assumes "additive f" shows "f (0::'c) = (0::'d)"
sig_ext_def: fixes name :: "mname"   and parTs :: "ty list"   and more :: "'a" shows "\<lparr>name = name, parTs = parTs, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons sig_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple name parTs) more"
assms: shows "(x0, x1) \<in> Domain Rel_OR"
add_opp: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p :: "'c point" assumes "ell_field R"   and "ell_field.on_curve R a b p" shows "ell_field.add R a p (ell_field.opp R p) = Infinity"
simps_1: fixes f :: "'a \<Rightarrow> 'a"   and s :: "'a" shows "pred \<Up>f s = True"
snoc_1: fixes ts :: "t list"   and ts' :: "t list" assumes "const_list xs"   and "\<S>\<bullet>\<C> \<turnstile> xs : ts _> ts'" shows "\<exists>tvs. ts' = ts @ tvs \<and> length xs = length tvs \<and> \<S>\<bullet>\<C>' \<turnstile> xs : [] _> tvs"
shortest_path_inf: fixes u :: "'a"   and v :: "'a"   and f :: "'b \<Rightarrow> real" assumes "\<not> u \<rightarrow>\<^sup>* v" shows "\<mu> f u v = \<infinity>"
elims: fixes x :: "'b"   and xa :: "'b list"   and y :: "'b list" assumes "ins_list x xa = y" obtains x :: "'b" where "xa__ = xb__" and "xa = []" and "y = [xb__]"   | x :: "'b" and a :: "'b" and xs :: "'b list" where "xa__ = x" and "xa = a # xs" and "y = (if x < a then x # a # xs else if x = a then a # xs else a # ins_list x xs)"
intro: fixes a :: "nat list"   and b :: "nat list" assumes "0 \<notin> set a"   and "0 \<notin> set b" shows "hlde a b"
l11_31_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B"   and "D \<noteq> E"   and "F \<noteq> E"   and "Bet D E F" shows "A B C LeA D E F"
empty_neutral_2: fixes x :: "'b multiset" shows "x + {#} = x"
s2: shows "set (labels \<tau>3'') \<subseteq> r \<down>s {fst \<beta>_step, fst \<alpha>_step}"
sub_def: fixes as :: "'b iarray"   and n :: "nat" shows "as !! n = IArray.list_of as ! n"
DBM_reset'_complete_empty: fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and cs :: "'c list"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>set cs. v c \<le> n"   and "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> M" shows "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> reset' (FW M n) n cs v d"
keys_map_scale: fixes k :: "'d"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "keys (k \<cdot> t) = (if k = (0::'d) then {} else keys t)"
polymul_degreen: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "m \<le> min n0 n1" shows "degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)"
lv_aforms_rel_comp_br_Affine_le: shows "lv_aforms_rel O br Affine top \<subseteq> \<langle>lv_rel\<rangle>aforms_relp"
Cod_Ide: fixes t :: "'a local.term" assumes "Ide t" shows "Cod t = t"
zero_order_3: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and n :: "'f" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
comp: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'f \<Rightarrow> 'd" shows "(-`) f \<circ> (-`) g = (-`) (g \<circ> f)"
Int_Collect: fixes x :: "'c"   and A :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "(x \<in> A \<inter> rel_ext P) = (x \<in> A \<and> P x)"
take_rews_2: fixes n :: "nat"   and discr :: "bool discr" shows "\<psi>\<^sup>D\<^bsub>Suc n\<^esub>\<cdot>(B\<cdot>discr) = B\<cdot>discr"
cases: fixes outPorts :: "'a \<Rightarrow> 'b fset"   and vertices :: "'c fset"   and nodeOf :: "'c \<Rightarrow> 'a"   and edges :: "(('c \<times> 'b) \<times> 'c \<times> 'd) set"   and ps :: "'c \<times> 'd"   and a :: "'c" assumes "Pre_Port_Graph.scopep outPorts vertices nodeOf edges ps a" obtains v :: "'c" where "a = v" and "v |\<in>| vertices" and "\<And>pth v'. \<lbrakk>Pre_Port_Graph.path edges v v' pth; Vertex_Graph.terminal_vertex outPorts vertices nodeOf v'\<rbrakk> \<Longrightarrow> ps \<in> snd ` set pth"
suma_middle_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I" shows "Tarski_neutral_dimensionless.SumA Bet Cong A B C F E D G H I"
codomains_comp: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.codomains (C g f) = C.codomains g"
sumind: shows "\<forall>x. (\<lambda>i. indpre i x) sums indicat_real (Y -` (A \<inter> range Y) \<inter> space M) x"
simps_6: fixes v :: "char list"   and va :: "aexp"   and vb :: "'a"   and vc :: "'a" shows "(v ::= va {vb} \<le> SKIP {vc}) = False"
mpop_min_def: fixes \<sigma> :: "('V \<Rightarrow> 'W infty option) \<times> ('V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option)" shows "mpop_min \<sigma> \<equiv> let (wl, res) = \<sigma> in prio_pop_min wl \<bind> (\<lambda>(v, w, wl'). RETURN (v, w, wl', res))"
simps_2: fixes "next" :: "bool list \<Rightarrow> bool list"   and n :: "nat" shows "buildchain next (Suc n) = next (buildchain next n)"
set_less_eq_refl: fixes A :: "'a set" shows "A \<sqsubseteq> A"
ivp_1: fixes t :: "real"   and s :: "'a" assumes "t \<in> T"   and "s \<in> S" shows " D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0--t}"
simps: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set" shows "(a \<in> least_map f S) = (\<exists>x. a = x \<and> x \<in> S \<and> (\<forall>x'\<in>S. f x \<le> f x'))"
case_8: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f6 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f7 :: "'b ltln \<Rightarrow> 'a"   and f8 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f9 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f10 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f11 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and x81 :: "'b ltln"   and x82 :: "'b ltln" shows "(case x81 U\<^sub>n x82 of true\<^sub>n \<Rightarrow> f1 | false\<^sub>n \<Rightarrow> f2 | prop\<^sub>n(x) \<Rightarrow> f3 x | nprop\<^sub>n(x) \<Rightarrow> f4 x | x and\<^sub>n xa \<Rightarrow> f5 x xa | x or\<^sub>n xa \<Rightarrow> f6 x xa | X\<^sub>n x \<Rightarrow> f7 x | x U\<^sub>n xa \<Rightarrow> f8 x xa | x R\<^sub>n xa \<Rightarrow> f9 x xa | x W\<^sub>n xa \<Rightarrow> f10 x xa | x M\<^sub>n xa \<Rightarrow> f11 x xa) = f8 x81 x82"
fresh_abs_fun_iff: fixes x :: "'a"   and b :: "'b"   and a :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp x)" shows "b \<sharp> [a].x = (b = a \<or> b \<sharp> x)"
is_universal: fixes a' :: "'c"   and \<chi>' :: "J.Arr \<Rightarrow> 'c" assumes "D.cone a' \<chi>'" shows "\<exists>!f. \<guillemotleft>f : a' \<rightarrow> C.dom p0\<guillemotright> \<and> D.cones_map f (D.mkCone p0 p1) = \<chi>'"
less_cSUP_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<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 A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq (f ` A)" shows "less a (Sup (f ` A)) = (\<exists>x\<in>A. less a (f x))"
semiring_0_ow_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semiring_0_ow U plus zero times \<equiv> semiring_ow U plus times \<and> comm_monoid_add_ow U plus zero \<and> mult_zero_ow U times zero"
if_eval_false: fixes mem :: "'Var \<Rightarrow> 'Val"   and e :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set" assumes "\<not> ev\<^sub>B mem e" shows "\<langle>Stmt.If e c\<^sub>1 c\<^sub>2 \<otimes> annos, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>2, mds \<oplus> annos, mem\<rangle>"
spmf_obsf_distinguisher_obsf_resource_True: fixes \<A> :: "('a, 'b) distinguisher"   and res :: "('a, 'b) resource" shows "spmf (connect_obsf (obsf_distinguisher \<A>) (obsf_resource res)) True = spmf (connect \<A> res) True"
cs_3: shows "\<pi> =\<^bsub>k\<^esub> \<pi>'"
b: shows " t \<longrightarrow>\<^sub>1 t2"
parallel_handler_left: fixes left :: "'r \<Rightarrow> 't \<Rightarrow> 'r spmf"   and right :: "'s \<Rightarrow> 'u \<Rightarrow> 's spmf" shows "map_fun id (map_fun Inl id) (parallel_handler left right) = (\<lambda>(s_l, s_r) q. map_spmf (\<lambda>s_l'. (s_l', s_r)) (left s_l q))"
rcos_mult_step3: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "derived G (carrier G) <#> (derived G (carrier G) #> x) #> y = derived G (carrier G) #> x \<otimes> y"
exec_gpv_fixp_induct: fixes P :: "(('h, 'i, 'j) gpv \<Rightarrow> 'k \<Rightarrow> ('h \<times> 'k) spmf) \<Rightarrow> bool"   and callee :: "'k \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'k) spmf" assumes "spmf.admissible (\<lambda>f. P (\<lambda>c s. f (c, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>exec_gpv. P exec_gpv \<Longrightarrow> P (\<lambda>c s. the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). exec_gpv (c x) y)))" shows "P (exec_gpv callee)"
union_add_left_zmset: fixes a :: "'b"   and A :: "'b zmultiset"   and B :: "'b zmultiset" shows "add_zmset a A + B = add_zmset a (A + B)"
ord_less_eq_trans: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "less a b"   and "b = c" shows "less a c"
Abs_ffilter: fixes f :: "'b \<Rightarrow> bool"   and s :: "'b fset"   and s' :: "'b fset" shows "(ffilter f s = s') = ({e \<in> fset s. f e} = fset s')"
intro: fixes empty :: "'b"   and is_empty :: "'b \<Rightarrow> bool"   and insert :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and get_min :: "'b \<Rightarrow> 'c"   and del_min :: "'b \<Rightarrow> 'b"   and invar :: "'b \<Rightarrow> bool"   and mset :: "'b \<Rightarrow> 'c multiset"   and merge :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "Priority_Queue empty is_empty insert get_min del_min invar mset"   and "Priority_Queue_Merge_axioms invar mset merge" shows "Priority_Queue_Merge is_empty insert get_min del_min invar mset empty merge"
wf_trancl: fixes r :: "('k \<times> 'k) set" assumes "wf r" shows "wf (r\<^sup>+)"
\<S>_V: fixes l :: "'s"   and R :: "('c \<Rightarrow> real) set" assumes "(l, R) \<in> \<S>" shows "R \<in> \<R>"
lnth_LCons': fixes x :: "'a"   and xs :: "'a llist"   and n :: "nat" shows "lnth (LCons x xs) n = (if n = 0 then x else lnth xs (n - 1))"
type_max_dma_type: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and x :: "'a"   and mem :: "'a \<Rightarrow> 'b" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.type_max ev\<^sub>B (dma_type x) mem = dma mem x"
complex_eq_cvec_symp: shows "symp (\<approx>\<^sub>v)"
emb: shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
SubobjsR_Subobjs: 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 Cs :: "char list list"   and C' :: "char list"   and Cs' :: "char list list" assumes "Subobjs\<^sub>R P (hd (Cs @ C' # Cs')) (Cs @ C' # Cs')" shows "Subobjs P C' (C' # Cs')"
inputSuppPres: fixes P :: "pi"   and Q :: "pi"   and x :: "name"   and Rel :: "(pi \<times> pi) set"   and a :: "name" assumes "\<And>y. y \<in> supp (P, Q, x) \<Longrightarrow> (P[x::=y], Q[x::=y]) \<in> Rel"   and "eqvt Rel" shows "a<x>.P \<leadsto>[Rel] a<x>.Q"
execute_serial_plan_split_i: fixes s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator"   and \<pi> :: "'a strips_operator list" assumes "are_all_operators_applicable s (op # \<pi>)"   and "are_all_operators_non_interfering (op # \<pi>)" shows "are_all_operators_applicable (s \<then> op) \<pi>"
that: fixes e :: "real" assumes "0 < e"   and "e < 1"   and "z = y - e *\<^sub>R (y - x)" shows "thesis"
norm_triangle_half_r: fixes y :: "'b"   and x1 :: "'b"   and e :: "real"   and x2 :: "'b" assumes "norm (y - x1) < e / 2"   and "norm (y - x2) < e / 2" shows "norm (x1 - x2) < e"
edc_rand: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and e :: "expr"   and f :: "cexpr"   and dst :: "pdf_dist" assumes "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c e \<Rightarrow> f" shows "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c Random dst e \<Rightarrow> \<integral>\<^sub>c map_vars (case_nat 0 (\<lambda>x. x + 2)) f *\<^sub>c dist_dens_cexpr dst (CVar 0) (CVar 1) \<partial>dist_param_type dst"
abs_eq: shows "double_of_Float = (\<lambda>x. if is_finite_Float TYPE((11, 52) IEEE.float) x then Some (Double.of_finite_Float x) else None)"
rem_condless_valid_6: fixes P :: "('c, 'd) fmap \<times> ('c, 'd) fmap \<Rightarrow> bool"   and s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "length (filter P (rem_condless_act s [] as)) \<le> length (filter P as)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "int"   and a3 :: "int list" assumes "\<And>n m c bs. (\<And>x xa y. \<lbrakk>\<not> (m = 0 \<or> n = 0); x = bin_split (min m n) c; (xa, y) = x\<rbrakk> \<Longrightarrow> P n (m - n) xa (y # bs)) \<Longrightarrow> P n m c bs" shows "P a0 a1 a2 a3"
is_run_def: fixes m :: "'b"   and r :: "nat \<Rightarrow> 'Q" shows "graph_defs.is_run (to_gbg_ext m) r \<equiv> r 0 \<in> g_V0 (to_gbg_ext m) \<and> ipath (g_E (to_gbg_ext m)) r"
N_def: shows "N \<equiv> distr M (stock_measure REAL) (lift_RealVal f)"
ext: shows "(r \<in> Ax \<or> r \<in> upRules \<or> r \<in> modRules2) \<and> extendRule S r = (Ps,  \<Gamma> \<Rightarrow>* \<Delta> \<oplus> Modal M Ms)"
equivalence_pseudofunctor_axioms: shows "equivalence_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"
openin_euclidean_subtopology_iff: fixes U :: "'b set"   and S :: "'b set" shows "openin (top_of_set U) S = (S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x' \<in> S))"
join_st_def: fixes F :: "'b Abs_State.st"   and G :: "'b Abs_State.st" shows "F \<squnion> G = FunDom (\<lambda>x. fun F x \<squnion> fun G x) (inter_list (Abs_State.dom F) (Abs_State.dom G))"