hcju/isabelleps huggingface dataset:
bind_eq_join_pmf: fixes M :: "'e pmf"   and f :: "'e \<Rightarrow> 'd pmf" shows "M \<bind> f = join_pmf (map_pmf f M)"
remdups_eq_nil_iff: fixes x :: "'g list" shows "(remdups x = []) = (x = [])"
rbt_inorder_sorted: fixes t :: "('a, 'b) rbt" shows "rbt_sorted t = Lists_Ex.strict_sorted (map fst (rbt_in_traverse_pairs t))"
exists_index: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and ev :: "('d, 'e, 'f) event"   and j :: "nat"   and i :: "nat" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "ev \<in> set (take (j - i) (drop i t))" shows "\<exists>k\<ge>i. k < j \<and> ev = t ! k"
wellformed_def: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp" shows "wellformed \<Gamma> = wf {(q, p). p \<leadsto>\<^bsub>\<Gamma>\<^esub> q}"
stim_idem: fixes x :: "'a" shows "x \<oplus> x = x"
compact_frontier: fixes S :: "'b set" assumes "compact S" shows "compact (frontier S)"
SC_full_to_mini: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "image_mset to_mini_formula \<Gamma> \<Rightarrow> image_mset to_mini_formula \<Delta>"
strongly_connected_spanning_imp_strongly_connected: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "spanning H G"   and "strongly_connected H" shows "strongly_connected G"
l_zero: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "0\<^sub>v n + x = x"
continuous_map_pullback': fixes T1 :: "'a topology"   and T2 :: "'b topology"   and f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and A :: "'c set" assumes "continuous_map T1 T2 (f \<circ> g)"   and "topspace T1 \<subseteq> g -` A" shows "continuous_map T1 (pullback_topology A f T2) g"
cases: fixes x :: "('a \<times> 'a) set \<times> ('b \<Rightarrow> 'a) \<times> 'a set \<times> 'b list" obtains uu_ :: "('a \<times> 'a) set" and uv_ :: "'b \<Rightarrow> 'a" and U :: "'a set" where "x = (uu_, uv_, U__, [])"   | I :: "('a \<times> 'a) set" and D :: "'b \<Rightarrow> 'a" and U :: "'a set" and xs :: "'b list" and xa :: "'b" where "x = (I, D, U, xs @ [xa])"
maximal_exists: fixes P :: "'b set \<Rightarrow> bool"   and N :: "nat"   and B :: "'b set" assumes "\<And>A. P A \<Longrightarrow> finite A \<and> card A \<le> N"   and "P B" shows "\<exists>A. finite A \<and> maximal A P"
REF_R_DONE: fixes \<sigma> :: "'\<sigma>"   and \<sigma>' :: "'\<sigma>a" assumes "(({}, \<sigma>), {}, \<sigma>') \<in> R" shows "(\<sigma>, \<sigma>') \<in> R'"
grey_reachable_def: fixes y :: "'d" shows "grey_reachable y = (\<lambda>s. \<exists>x. grey x s \<and> (x reaches y) s)"
ts'_def_2: shows "\<C> \<turnstile> [C v2] : ts' _> t2s"
hyps_1: shows "atom ` domA \<Delta> \<sharp>* \<Gamma>"
residue_of_sum: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" shows "(a \<oplus> b) k = a k \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> b k"
append_ap_distrib: fixes fs :: "('b \<Rightarrow> 'a) list"   and gs :: "('b \<Rightarrow> 'a) list"   and xs :: "'b list" shows "(fs @ gs) \<diamondop> xs = fs \<diamondop> xs @ gs \<diamondop> xs"
quit_iteration: fixes m :: "'a \<Rightarrow> 'b option"   and p :: "('a, 'b) hashtable"   and m' :: "'a \<Rightarrow> 'b option"   and it :: "nat \<times> ('a \<times> 'b) list \<times> ('a, 'b) hashtable" shows "hm_is_it m p m' it \<Longrightarrow>\<^sub>A is_hashmap m p * true"
atLeast_Suc_lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "prod g {Suc m..<Suc n} = prod (g \<circ> Suc) {m..<n}"
freshFrameDest_1: fixes xvec :: "name list"   and A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b" assumes "xvec \<sharp>* (\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>)"   and "xvec \<sharp>* A\<^sub>F" shows "xvec \<sharp>* \<Psi>\<^sub>F"
map_gpv_conv_map_gpv': fixes f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h" shows "map_gpv f g = map_gpv' f g id"
LeftDerives1_splits_at_is_word: fixes \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and s :: "symbol list"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" assumes "LeftDerives1 \<delta> i r s"   and "splits_at \<delta> i \<alpha> N \<beta>" shows "is_word \<alpha>"
that_2: assumes "(u, v1) \<in> E\<^sup>*"   and "(v2, w) \<in> E\<^sup>*" shows "thesis"
Suc_rtos_i_mod_r_3: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0"   and "i div s < r - Suc 0" shows "Suc (rtos r s i) mod r = Suc (i div s)"
skolemize2: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<forall>x. P x \<longrightarrow> (\<exists>y z. Q x y z)" shows "\<exists>f g. \<forall>x. P x \<longrightarrow> Q x (f x) (g x)"
one_endpoint: fixes I :: "'b set"   and a :: "'b"   and b :: "'b"   and x :: "'b" assumes "Q I a b"   and "x = a" shows "P x I"
preserves_iso: fixes f :: "'a" assumes "C.iso f" shows "D.iso (F f)"
nonzero: shows "\<forall>xa. insertion (nth_default 0 (xs @ xa # \<Gamma>)) (isolate_variable_sparse p x 2) \<noteq> 0 \<or> insertion (nth_default 0 (xs @ xa # \<Gamma>)) (isolate_variable_sparse p x 1) \<noteq> 0 \<or> insertion (nth_default 0 (xs @ xa # \<Gamma>)) (isolate_variable_sparse p x 0) \<noteq> 0"
msubset_mset_set_iff: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "finite B" shows "(mset_set A \<subseteq># mset_set B) = (A \<subseteq> B)"
convolution_def: fixes M :: "'a measure"   and N :: "'a measure" shows "(M \<star> N) = distr (M \<Otimes>\<^sub>M N) borel (\<lambda>(x, y). x + y)"
edg_simps_3: fixes v :: "'v vertex"   and x :: "'v" shows "edg v (Vertex x) = (\<exists>xa. v = Edge xa x \<and> edge \<Delta> xa x)"
measure_pmf_zero_iff: fixes p :: "'b pmf"   and s :: "'b set" shows "(measure_pmf.prob p s = 0) = (set_pmf p \<inter> s = {})"
len_takeWhile_lt: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "length (takeWhile ((\<noteq>) x) xs) < length xs"
comp_FSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'h, 'i, 'j) model_scheme"   and h :: "'h \<Rightarrow> 'k"   and hA :: "'i \<Rightarrow> 'l"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'k, 'l, 'm) model_scheme"   and h' :: "'k \<Rightarrow> 'n"   and hA' :: "'l \<Rightarrow> 'o"   and MOD'' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'n, 'o, 'p) model_scheme" assumes "igWlsAbsIsInBar MOD"   and "FSbImorph h hA MOD MOD'"   and "FSbImorph h' hA' MOD' MOD''" shows "FSbImorph (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
permutations_of_multiset_lists: fixes A :: "'a multiset" shows "permutations_of_multiset A \<subseteq> lists (set_mset A)"
facet_of_convex_hull_affine_independent_alt: fixes S :: "'b set"   and T :: "'b set" assumes "\<not> affine_dependent S" shows "(T facet_of convex hull S) = (2 \<le> card S \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u})))"
xs_def: shows "xs = \<Oplus>Ns\<leftarrow> x"
equalityCE: fixes A :: "'d set"   and B :: "'d set"   and c :: "'d" assumes "A = B" obtains "c \<in> A" and "c \<in> B"   | "c \<notin> A" and "c \<notin> B"
base: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k" shows "rtrancl_path r x [] x"
concat_append: fixes xs :: "'g list list"   and ys :: "'g list list" shows "concat (xs @ ys) = concat xs @ concat ys"
AE_measure_pmf_iff: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c pmf" shows "almost_everywhere (measure_pmf M) P = (\<forall>y\<in>set_pmf M. P y)"
ps_intro: shows "child p (Grid_Point.inv dir) d = child ps dir d"
cs_single_nocd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat"   and x :: "'n" assumes "cs\<^bsup>\<pi>\<^esup> i = [x]" shows "\<forall>k. \<not> i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
Q_def: shows "Q = {q. f -` q \<in> a}"
X_sub: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" assumes "UP_cring R"   and "p \<in> carrier (UP R)" shows "Cring_Poly.compose R p (X_poly R) = p"
devExt_same: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and n' :: "nat"   and n'' :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a"   and n\<^sub>s :: "nat"   and n''' :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "\<forall>n'''>n'. n''' \<le> n'' \<longrightarrow> Option.is_none (Blockchain.devBC active cmp bc mining t n nid n''')"   and "n\<^sub>s \<le> n'"   and "n''' \<le> n''"   and "n' \<le> n'''" shows "Blockchain.devExt active cmp bc mining t n nid n\<^sub>s (n''' - n\<^sub>s) = Blockchain.devExt active cmp bc mining t n nid n\<^sub>s (n' - n\<^sub>s)"
conforms_locals: fixes a :: "abopt"   and b :: "st"   and G :: "prog"   and L :: "lenv"   and x :: "lname"   and T :: "ty" assumes "(a, b)\<Colon>\<preceq>(G, L)"   and "L x = Some T"   and "locals b x \<noteq> None" shows "G,b\<turnstile>the (locals b x)\<Colon>\<preceq>T"
big_mono: fixes f :: "'i \<Rightarrow> 'j"   and g :: "'i \<Rightarrow> 'j"   and F :: "'i filter" assumes "\<forall>\<^sub>F x in F. norm (f x) \<le> norm (g x)" shows "f \<in> O[F](g)"
add_left_cancel: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and a :: "'g"   and b :: "'g"   and c :: "'g" assumes "class.cancel_semigroup_add plus" shows "(plus a b = plus a c) = (b = c)"
match_type_1: fixes p :: "pat"   and T\<^sub>1 :: "POPLmarkRecord.type"   and \<Delta> :: "binding list"   and \<Gamma>\<^sub>2 :: "binding list"   and t\<^sub>1 :: "trm"   and \<Gamma>\<^sub>1 :: "binding list"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmarkRecord.type"   and ts :: "trm list" assumes "\<turnstile> p : T\<^sub>1 \<Rightarrow> \<Delta>"   and "\<Gamma>\<^sub>2 \<turnstile> t\<^sub>1 : T\<^sub>1"   and "\<Gamma>\<^sub>1 @ \<Delta> @ \<Gamma>\<^sub>2 \<turnstile> t\<^sub>2 : T\<^sub>2"   and "\<turnstile> p \<rhd> t\<^sub>1 \<Rightarrow> ts" shows "\<down>\<^sub>e \<parallel>\<Delta>\<parallel> 0 \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<turnstile> t\<^sub>2[\<parallel>\<Gamma>\<^sub>1\<parallel> \<mapsto>\<^sub>s ts] : \<down>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> \<parallel>\<Gamma>\<^sub>1\<parallel> T\<^sub>2"
option_bind_None: fixes f :: "'d \<Rightarrow> 'c option" shows "option_bind None f = None"
ipurge_ref_aux_single_dom: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'b list"   and X :: "'b set" shows "ipurge_ref_aux I D {u} xs X = ipurge_ref I D u xs X"
weakCongruence_def: fixes P :: "pi"   and Q :: "pi" shows "P \<simeq> Q \<equiv> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q \<and> Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P"
showsp_poly_def: fixes p :: "nat"   and x :: "'b poly" shows "showsp_poly p x = shows_string (show_poly x)"
BinOp: fixes G :: "prog"   and s0 :: "st"   and e1 :: "expr"   and n :: "nat"   and v1 :: "val"   and s1 :: "state"   and binop :: "binop"   and e2 :: "expr"   and v2 :: "val"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e1-\<succ>v1\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>(if need_second_arg binop v1 then In1l e2 else In1r Skip)\<succ>\<midarrow>n\<rightarrow> (In1 v2, s2)" shows "G\<turnstile>Norm s0 \<midarrow>BinOp binop e1 e2-\<succ>eval_binop binop v1 v2\<midarrow>n\<rightarrow> s2"
project_Constrains_I: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set" assumes "extend h F \<squnion> G \<in> extend_set h A Co extend_set h B" shows "F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> A Co B"
sorted_sublist_wrt_cons: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and lo :: "nat"   and hi :: "nat" assumes "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"   and "sorted_sublist_wrt R xs (lo + 1) hi"   and "lo \<le> hi"   and "hi < length xs"   and "R (xs ! lo) (xs ! (lo + 1))" shows "sorted_sublist_wrt R xs lo hi"
lprefix_lnull: fixes ys :: "'c llist"   and xs :: "'c llist" assumes "lnull ys" shows "xs \<sqsubseteq> ys = lnull xs"
swing_def: fixes x :: "wall"   and y :: "wall" shows "swing x y \<equiv> swing_pos x y \<or> swing_neg x y"
icard_Un_Int: fixes A :: "'a set"   and B :: "'a set" shows "icard A + icard B = icard (A \<union> B) + icard (A \<inter> B)"
abs_eq: fixes xa :: "'b set"   and x :: "'b set" assumes "eq_onp ord_class.filter xa xa"   and "eq_onp ord_class.filter x x" shows "Filters.filter.Abs_filter xa \<squnion> Filters.filter.Abs_filter x = Filters.filter.Abs_filter (filter_sup xa x)"
is_extensional: fixes f :: "'c \<times> 'c" assumes "\<not> C.CC.arr f" shows "FoT\<^sub>C.map f = D.null"
simps: fixes t1 :: "('b \<times> nat) tree"   and t2 :: "('b \<times> nat) tree" shows "single t1 t2 = (\<Gamma>2 * (size_wbt t1 + 1) < \<Gamma>1 * (size_wbt t2 + 1))"
zmod_zmult2_eq: fixes c :: "int"   and a :: "int"   and b :: "int" assumes "0 \<le> c" shows "a mod (b * c) = b * (a div b mod c) + a mod b"
expand_singleton: fixes t :: "('a, 'sz) trm" shows "singleton t = (\<lambda>i. if i = vid1 then t else Const 0)"
cases: fixes x :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list \<times> ('a, 'b) rose_tree\<^sub>m" obtains p :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list" and a :: "'a" and ts :: "('a, 'b) rose_tree\<^sub>m list" where "x = (p, Tree\<^sub>m (Unblinded (a, ts)))"   | v :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list" and vc :: "('b \<times> 'b rose_tree\<^sub>h list) blindable\<^sub>h" where "x = (v, Tree\<^sub>m (Blinded vc))"
refl_rtrancl: fixes r :: "('b \<times> 'b) set" shows "refl (r\<^sup>*)"
subdist_2_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x + y \<cdot> z \<le> x + y"
inter_sgs: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and K :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> K " shows "G \<guillemotright> H \<inter> K "
induct: fixes x1 :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and x2 :: "com"   and x3 :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "((char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> bool" assumes "\<turnstile> {x1} x2 {x3}"   and "\<And>P. P P SKIP P"   and "\<And>P a x. P (\<lambda>s. P (s[a/x])) (x ::= a) P"   and "\<And>P c\<^sub>1 Q c\<^sub>2 R. \<lbrakk>\<turnstile> {P} c\<^sub>1 {Q}; P P c\<^sub>1 Q; \<turnstile> {Q} c\<^sub>2 {R}; P Q c\<^sub>2 R\<rbrakk> \<Longrightarrow> P P (c\<^sub>1;; c\<^sub>2) R"   and "\<And>P b c\<^sub>1 Q c\<^sub>2. \<lbrakk>\<turnstile> {\<lambda>s. P s \<and> bval b s} c\<^sub>1 {Q}; P (\<lambda>s. P s \<and> bval b s) c\<^sub>1 Q; \<turnstile> {\<lambda>s. P s \<and> \<not> bval b s} c\<^sub>2 {Q}; P (\<lambda>s. P s \<and> \<not> bval b s) c\<^sub>2 Q\<rbrakk> \<Longrightarrow> P P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q"   and "\<And>P b c. \<lbrakk>\<turnstile> {\<lambda>s. P s \<and> bval b s} c {P}; P (\<lambda>s. P s \<and> bval b s) c P\<rbrakk> \<Longrightarrow> P P (WHILE b DO c) (\<lambda>s. P s \<and> \<not> bval b s)"   and "\<And>P' P c Q Q'. \<lbrakk>\<forall>s. P' s \<longrightarrow> P s; \<turnstile> {P} c {Q}; P P c Q; \<forall>s. Q s \<longrightarrow> Q' s\<rbrakk> \<Longrightarrow> P P' c Q'" shows "P x1 x2 x3"
wneq: shows "winding_number \<gamma> z = winding_number p z"
prefixeq_butlast: fixes xs :: "'c list" shows "prefix (butlast xs) xs"
one_plus_numeral_commute: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "num" assumes "class.numeral plus" shows "plus one (numeral.numeral one plus x) = plus (numeral.numeral one plus x) one"
inline1_conv_inline1': fixes gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s" shows "inline1 callee gpv s = map_spmf (map_sum id snd) (local.inline1' gpv s)"
elims_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and x :: "'a list"   and xa :: "'a list"   and y :: "bool" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.same_level_path_aux kind get_return_edges x xa = y" obtains cs :: "'a list" where "x = cs__" and "xa = []" and "y"   | cs :: "'a list" and a :: "'a" and as :: "'a list" where "x = cs" and "xa = a # as" and "y = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> CFG.same_level_path_aux kind get_return_edges (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> False | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> CFG.same_level_path_aux kind get_return_edges cs' as | _ \<Rightarrow> CFG.same_level_path_aux kind get_return_edges cs as)"
assms: fixes n :: "nat" shows "coeff p n \<in> \<real>"
fdisjnt_union_right: fixes M :: "'a fset"   and A :: "'a fset"   and B :: "'a fset" assumes "fdisjnt M A"   and "fdisjnt M B" shows "fdisjnt M (A |\<union>| B)"
tmneg_def: fixes t :: "tm" shows "tmneg t \<equiv> tmmul t (- 1)\<^sub>p"
law2: fixes X :: "'a" assumes "P X" shows "P (g X)"
that: fixes S' :: "('a, 'b) fm list" assumes "set S' \<subseteq> S"   and "p # S' \<turnstile> \<^bold>\<bottom>" shows "thesis"
select_convs_3: fixes break_status :: "bool"   and return_status :: "bool"   and more :: "'b" shows "more \<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = more"
elims: fixes p :: "uint64"   and x :: "uint64"   and xa :: "uint64"   and y :: "uint64" assumes "power_p64 p x xa = y" obtains xb :: "uint64" and n :: "uint64" where "x = xb" and "xa = n" and "y = (if n = 0 then 1 else let rec = power_p64 p (mult_p64 p xb xb) (n >> 1) in if n AND 1 = 0 then rec else mult_p64 p rec xb)"
from_nat_into: fixes A :: "'a set"   and n :: "nat" assumes "A \<noteq> {}" shows "from_nat_into A n \<in> A"
select_convs_15: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "sip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = sip"
rel_open: fixes S :: "'b set" shows "rel_open S = openin (top_of_set (affine hull S)) S"
simps_5: fixes h :: "'a \<Rightarrow> 'b list \<Rightarrow> bool"   and \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm"   and xs :: "'b list" shows "interpret h (Or \<phi>\<^sub>1 \<phi>\<^sub>2) xs = (interpret h \<phi>\<^sub>1 xs \<or> interpret h \<phi>\<^sub>2 xs)"
eq_add_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.ab_group_add plus zero minus uminus" shows "(x = plus x y) = (y = zero)"
face_of_imp_closed: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "closed S"   and "T face_of S" shows "closed T"
zmset_subset_eq_zmultiset_union_diff_commute: fixes A :: "'a zmultiset"   and B :: "'a zmultiset"   and C :: "'a zmultiset" shows "A - B + C = A + C - B"
simps: fixes q1 :: "'a"   and l1 :: "'c"   and qs1 :: "'a list"   and q2 :: "'b"   and l2 :: "'d"   and qs2 :: "'b list" shows "r_prod q1 \<rightarrow> l1 qs1 q2 \<rightarrow> l2 qs2 = (q1, q2) \<rightarrow> l1 zip qs1 qs2"
WT_callee: fixes s :: "'a" shows "\<I> \<turnstile>c callee s \<surd>"
restrict_top_ifex_minimal_invar: fixes i :: "'b ifex"   and a :: "'b"   and val :: "bool" assumes "ifex_minimal i" shows "ifex_minimal (restrict_top i a val)"
in_components_self: fixes s :: "'c set" shows "(s \<in> components s) = (connected s \<and> s \<noteq> {})"
sum_Basis_E: fixes g :: "'a \<Rightarrow> 'c" shows "sum g Basis = (\<Sum>j<DIM('a). g (E ! j))"
l_set_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) 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_set_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs \<equiv> l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs \<and> l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
rb_aux_inv1_D5: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and i :: "nat" assumes "rb_aux_inv1 bs"   and "i < length bs" shows "is_RB_upt dgrad rword (set (drop (Suc i) bs)) (lt (bs ! i))"
IH_1: fixes ds :: "bool list" assumes "anybranch T\<^sub>1 (\<lambda>p. red (ds @ p))" shows "anyinternal (cutoff red ds T\<^sub>1) (\<lambda>p. \<not> red (ds @ p))"
rat_inverse_code: fixes p :: "rat" shows "quotient_of (inverse p) = (let (a, b) = quotient_of p in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
nested_subspaces: fixes W :: "'a vec set"   and X :: "'a vec set" assumes "subspace class_ring W V"   and "subspace class_ring X V"   and "X \<subseteq> W" shows "subspace class_ring X (vs W)"
SAT_def: fixes \<Phi> :: "('fsym, 'psym) lit list set" shows "F.SAT \<Phi> \<equiv> \<forall>\<xi>. F.wtE \<xi> \<longrightarrow> F.satPB \<xi> \<Phi>"
reduc_saturated_def: fixes N :: "'a set" shows "reduc_saturated N = (Inf_from (N - Red_F N) \<subseteq> Red_I N)"
chainI: fixes C :: "'b set set"   and A :: "'b set set" assumes "C \<subseteq> A"   and "\<And>x y. \<lbrakk>x \<in> C; y \<in> C\<rbrakk> \<Longrightarrow> (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x" shows "subset.chain A C"
assms_3: fixes u :: "'a"   and v :: "'b" assumes "u \<in> basisA"   and "v \<in> basisB" shows "is_orthogonal v (F *\<^sub>V u)"
times_rel_fun_def: fixes \<rho> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "rel_magma.times_rel_fun \<rho> f g = \<otimes> \<rho> f g"
hypnat_of_nat_one_not_prime: shows "hypnat_of_nat 1 \<notin> starprime"