hcju/isabelleps huggingface dataset:
gfp_unfold: fixes f :: "'d \<Rightarrow> 'd" assumes "mono f" shows "\<mu> f = f (\<mu> f)"
assms: shows "\<guillemotleft>g \<cdot> f : a \<rightarrow> c\<guillemotright>"
distinct_77: fixes x5 :: "load_store_type"   and x6 :: "arith_type" shows "load_store_type x5 \<noteq> arith_type x6"
image_eqI: fixes b :: "'d"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e"   and A :: "'e set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
l2_secrecy_def: shows "l2_secrecy \<equiv> {s. dy_fake_msg (l2_state.bad s) (ik s) (chan s) \<inter> secret s = {}}"
subprob_measure_le_1: fixes M :: "'h pmf"   and X :: "'h set" shows "measure_pmf.prob M X \<le> 1"
assms_3: shows "\<not> FactoredSystem.inj last (state_set (state_list s as)) (reachable_s PROB s)"
not_single_rt_fresh_asI2: fixes rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "\<not> the (rt2 dip) \<sqsubseteq> the (rt1 dip)" shows "\<not> rt1 \<approx>\<^bsub>dip\<^esub> rt2"
asms_3: fixes a :: "'a"   and b :: "'a" shows "Cf__ a b \<and> Q"
iWeakSince_iSince_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t'. t' \<B> t I. Q t) = ((P t'. t' \<S> t I. Q t) \<or> (\<box> t I. P t))"
advantage_def: fixes \<A> :: "(bool, bool list, bool list) gpv" shows "local.prf.advantage \<A> = \<bar>spmf (local.prf.game_0 \<A>) True - spmf (local.prf.game_1 \<A>) True\<bar>"
proj_exec_proj_eq_exec_proj': fixes vs :: "'d set"   and s :: "('d, 'e) fmap"   and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" shows "fmrestrict_set vs (exec_plan (fmrestrict_set vs s) (as_proj as vs)) = fmrestrict_set vs (exec_plan s (as_proj as vs))"
adjust_preserves_span: fixes w :: "'a vec"   and us :: "'a vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "(w \<in> local.span (set us)) = (adjuster n w us + w \<in> local.span (set us))"
assms: shows "x \<in>\<^sub>\<circ> vdomain A"
sp_Cons: fixes g :: "('d, 'e) rgraph_scheme"   and v1 :: "'d"   and e :: "'d edge"   and es :: "'d edge list"   and v2 :: "'d" shows "Graph.subpath g v1 (e # es) v2 = (src e = v1 \<and> e \<in> edges g \<and> Graph.subpath g (tgt e) es v2)"
GcdI: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and b :: "'b" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times b a"   and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times c a) \<Longrightarrow> dvd.dvd times c b"   and "normalize b = b" shows "b = Gcd A"
abelian_group_axioms: shows "abelian_group B addB zeroB"
a_ensure_pre_insertion_validity_def: fixes node :: "(_) node_ptr"   and parent :: "(_) object_ptr"   and child_opt :: "(_) node_ptr option" shows "a_ensure_pre_insertion_validity node parent child_opt = Heap_Error_Monad.bind (if is_character_data_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r parent then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind (get_ancestors parent) (\<lambda>ancestors. Heap_Error_Monad.bind (if cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node \<in> set ancestors then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind (case child_opt of None \<Rightarrow> return () | Some child \<Rightarrow> Heap_Error_Monad.bind (get_parent child) (\<lambda>child_parent. if child_parent \<noteq> Some parent then error NotFoundError else return ())) (\<lambda>_. Heap_Error_Monad.bind (get_child_nodes parent) (\<lambda>children. Heap_Error_Monad.bind (if children \<noteq> [] \<and> is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r parent then error HierarchyRequestError else return ()) (\<lambda>_. if is_character_data_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node \<and> is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r parent then error HierarchyRequestError else return ()))))))"
dagger_slide_var1_eq: fixes x :: "'a" shows "x\<^sup>\<dagger> \<cdot> x = x \<cdot> x\<^sup>\<dagger>"
Id_O_R: fixes R :: "('e \<times> 'f) set" shows "Id O R = R"
in_notinI: fixes j :: "'a"   and i :: "'a" assumes "(j, i) \<notin> r \<or> j = i"   and "i \<in> Field r"   and "j \<in> Field r" shows "(i, j) \<in> r"
mig_real_abs_le: fixes x :: "real"   and a :: "real"   and b :: "real" assumes "x \<in> {a..b}" shows "\<bar>mig_real a b\<bar> \<le> \<bar>x\<bar>"
invertible_def: fixes u :: "'a" assumes "u \<in> N" shows "sub.invertible u = (\<exists>v\<in>N. u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>)"
fg1: fixes x :: "'n"   and k :: "nat" assumes "x \<in> cbox a b" shows "f k x \<le> g x"
tagged_division_split_right_inj_content: fixes \<D> :: "('a \<times> 'a set) set"   and S :: "'a set"   and x1 :: "'a"   and K1 :: "'a set"   and x2 :: "'a"   and K2 :: "'a set"   and c :: "real"   and k :: "'a" assumes "\<D> tagged_division_of S"   and "(x1, K1) \<in> \<D>"   and "(x2, K2) \<in> \<D>"   and "K1 \<noteq> K2"   and "K1 \<inter> {x. c \<le> x \<bullet> k} = K2 \<inter> {x. c \<le> x \<bullet> k}"   and "k \<in> Basis" shows "content (K1 \<inter> {x. c \<le> x \<bullet> k}) = 0"
proper_map_from_compact: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and K :: "'b set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "compact S"   and "closedin (top_of_set T) K" shows "compact (S \<inter> f -` K)"
absorb2: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "sup a b = b"
card_Collect_le_nat: fixes n :: "nat" shows "card {i. i \<le> n} = Suc n"
defAss'Uses_def: fixes g :: "'a" shows "CFG2.defAss'Uses g \<equiv> \<forall>m\<in>set (\<alpha>n2 g). \<forall>v\<in>uses2 g m. CFG2.defAss' g m v"
obs_intraE: fixes n' :: "'node"   and n :: "'node"   and S :: "'node set" assumes "n' \<in> obs_intra n S" obtains as :: "'edge list" where "n -as\<rightarrow>\<^sub>\<iota>* n'" and "\<forall>nx\<in>set (sourcenodes as). nx \<notin> S" and "n' \<in> S"
l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.known_ptr DocumentClass.type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel"
b: shows "steps0 (st, l, r) (adjust0 ap) stp = (st'', l'', r'')"
assms: shows "path_entry (g_E G) (la @ lb) w"
scalar_prod_right_unit: fixes i :: "nat"   and n :: "nat"   and v :: "'a Matrix.vec" assumes "i < n" shows "v \<bullet> unit_vec n i = v $v i"
sup_comute: fixes a :: "'a"   and b :: "'a" shows "a \<squnion> b = b \<squnion> a"
exch_alw_InvMsgInGlob: fixes s :: "('b, 't, 'loc) Combined.configuration stream" assumes "FullSpec s" shows "alw (\<lambda>s. cri.InvMsgInGlob (exchange_config (shd s))) s"
gbm_edges_list_it_def: fixes g :: "'m1" shows "gbm_edges_list_it g \<equiv> set_iterator_image (\<lambda>((v1, m1), (v2, m2), w). (v1, w, v2)) (set_iterator_product (m1.iteratei g) (\<lambda>(v, m2). set_iterator_product (m2.iteratei m2) (\<lambda>(w, y). s3.iteratei y)))"
size_correct: fixes s :: "'a"   and s' :: "'b list" assumes "(s, s') \<in> arel" shows "sz s = length s'"
not_Some_eq: fixes x :: "'b option" shows "(\<forall>y. x \<noteq> \<lfloor>y\<rfloor>) = (x = \<bottom>)"
Nmlize_Src_1: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Src t\<^bold>\<rfloor> = Src \<^bold>\<lfloor>t\<^bold>\<rfloor>"
power_0_Suc: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero (Suc n) = zero"
hyps: assumes "\<And>x. x \<in> vars_ctxt C \<Longrightarrow> \<sigma> x = \<tau> x" shows "C \<cdot>\<^sub>c \<sigma> = C \<cdot>\<^sub>c \<tau>"
simps_2: fixes x :: "'d"   and xs :: "'d list" shows "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
IH2_1: shows "factor_node_info (product_factor_tree p r) = Mp (\<Prod>\<^sub># (factors_of_factor_tree (product_factor_tree p r)))"
simps_2: fixes c :: "'d" shows "get_const (Lt c) = c"
inverse_prod_list: fixes xs :: "'a list" shows "inverse (prod_list xs) = prod_list (map inverse xs)"
ta_upd_proj_simps_3: fixes ta :: "('c, 'a, 'd, 'e, 'b, 'f) thread_action"   and obs :: "'f" shows "\<lbrace>ta_update_obs ta obs\<rbrace>\<^bsub>w\<^esub> = \<lbrace>ta\<rbrace>\<^bsub>w\<^esub>"
lessThan_iff: fixes i :: "'a Word.word"   and k :: "'a Word.word" shows "(i \<in> signed.lessThan k) = (i <s k)"
normalI: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "\<And>f. OrdinalInduct.strict_mono f \<Longrightarrow> F (oLimit f) = oLimit (\<lambda>n. F (f n))"   and "\<And>x. F x < F (oSuc x)" shows "normal F"
product_over_sum: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "O x y \<and> O x z" shows "x \<otimes> (y \<oplus> z) = x \<otimes> y \<oplus> x \<otimes> z"
mult_pCons_left: fixes a :: "'c"   and p :: "'c poly"   and q :: "'c poly" shows "pCons a p * q = smult a q + pCons (0::'c) (p * q)"
distincts_grouped_arcs: fixes G_list :: "'b list \<times> ('b \<times> 'b) list" assumes "distinct (snd G_list)" shows "distincts (grouped_out_arcs G_list)"
simps_5: fixes b :: "nat"   and t1 :: "trm"   and t2 :: "trm" shows "fvi b (LessEq t1 t2) = fvi_trm b t1 \<union> fvi_trm b t2"
simp: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations" shows "complex_rotationsp r"
rel_map_2: fixes Sa :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b list"   and g :: "'d \<Rightarrow> 'c"   and y :: "'d list" shows "list_all2 Sa x (map g y) = list_all2 (\<lambda>x y. Sa x (g y)) x y"
asm: shows "(as, x) \<approx>res (=) supp p (as', x')"
simps_1: fixes l :: "linear_poly"   and r :: "QDelta"   and v :: "nat" shows "qdelta_constraint_to_atom (LEQ_ns l r) v = (if is_monom l then monom_to_atom (LEQ_ns l r) else Leq v r)"
fmupd_reorder_neq: fixes a :: "'a"   and b :: "'a"   and x :: "'b"   and y :: "'b"   and m :: "('a, 'b) fmap" assumes "a \<noteq> b" shows "m(b $$:= y)(a $$:= x) = m(a $$:= x)(b $$:= y)"
valid_cases: fixes X :: "msg" assumes "X \<in> sym.valid" obtains A :: "agent" and B :: "agent" and M :: "msg" where "X = sym.implInsec Aa__ Ba__ Ma__" and "Ma__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = sym.implConfid Ab__ Bb__ Mb__" and "Mb__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = sym.implAuth Ac__ Bc__ Mc__" and "Mc__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = sym.implSecure A B M" and "M \<in> payload"
that: fixes lb :: "real" assumes "\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> lb < x"   and "\<forall>x\<le>lb. \<forall>p\<in>set ps. sgn (poly p x) = sgn_neg_inf p"   and "lb < b" shows "thesis"
assms: fixes s :: "'a" shows "run_state m s = run_state n s"
unique_moebius_pt_conjugate: fixes M1 :: "moebius"   and M2 :: "moebius" assumes "moebius_pt M1 \<circ> conjugate = moebius_pt M2 \<circ> conjugate" shows "M1 = M2"
assms_2: shows "x \<in> set (unlabel (transaction_strand T))"
evaln_InsInitV: fixes G :: "prog"   and s :: "st"   and c :: "stmt"   and w :: "Term.var"   and n :: "nat"   and v :: "vals"   and s' :: "state" shows "G\<turnstile>Norm s \<midarrow>In2 (InsInitV c w)\<succ>\<midarrow>n\<rightarrow> (v, s') = False"
gbinomial_Suc_Suc: fixes a :: "'a"   and k :: "nat" shows "a + (1::'a) gchoose Suc k = a gchoose k + (a gchoose Suc k)"
minit_safe_minit: fixes \<phi> :: "'b MFOTL.formula" assumes "mmonitorable \<phi>" shows "minit_safe \<phi> = minit \<phi>"
inj_f: fixes f :: "'a \<Rightarrow> 'b" assumes "injective f" shows "inj f"
hyps: assumes "trace init t final"   and "channel cid = Some (p, q)" shows "Marker \<notin> set (msgs (S t i) cid) \<or> (\<exists>!j. j < length (msgs (S t i) cid) \<and> msgs (S t i) cid ! j = Marker)"
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier L; y \<in> carrier L\<rbrakk> \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> y \<or> y \<sqsubseteq>\<^bsub>L\<^esub> x" shows "total_order_axioms L"
simps_4: fixes f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a" shows "(case 0 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f1"
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"
shadow_root_eq2_h3: fixes ptr' :: "(_) element_ptr" shows "|h3 \<turnstile> get_shadow_root ptr'|\<^sub>r = |h' \<turnstile> get_shadow_root ptr'|\<^sub>r"
intros: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" assumes "r a b" shows "Rangep r b"
lower_bound_def: fixes A :: "'a set" shows "lower_bound A = {a. \<forall>x\<in>A. a \<le> x}"
False: shows "\<not> v\<^sub>e \<noteq> 0"
cong_iff_lin_nat: fixes a :: "nat"   and b :: "nat"   and m :: "nat" shows "[a = b] (mod m) = (\<exists>k1 k2. b + k1 * m = a + k2 * m)"
rtranclp_into_tranclp1: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
bind_pure_returns_result_I: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and x :: "'c"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog"   and y :: "'d" assumes "pure f h"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>r y"
REL_FORCE_ID_def: fixes R :: "('g \<times> 'g) set" shows "REL_FORCE_ID R \<equiv> R = Id"
index_add_mat_1: fixes i :: "nat"   and B :: "'a mat"   and j :: "nat"   and A :: "'a mat" assumes "i < dim_row B"   and "j < dim_col B" shows "(A + B) $$ (i, j) = A $$ (i, j) + B $$ (i, j)"
ai: fixes ai :: "'a vec" assumes "ai \<in> A" shows "(0::'a) \<le> c \<bullet> ai"
iMin_imp_iUntil: fixes I :: "nat set"   and Q :: "nat \<Rightarrow> bool"   and P :: "nat \<Rightarrow> bool" assumes "I \<noteq> {}"   and "Q (iMin I)" shows "P t'. t' \<U> t I. Q t"
nfa_accepting'_set_of_bv: fixes l :: "bool list"   and r :: "bool list" shows "nfa_accepting' l r = (set_of_bv l \<inter> set_of_bv r \<noteq> {})"
IdNatTransCompDefDom: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTrans f" shows "IdNatTrans (NTDom f) \<approx>>\<bullet> f"
merge_winning_strategies: fixes p :: "Player" shows "\<exists>\<sigma>. strategy p \<sigma> \<and> (\<forall>v\<in>winning_region p. winning_strategy p \<sigma> v)"
node_ptr_kinds_eq2_h2: shows "|h2 \<turnstile> node_ptr_kinds_M|\<^sub>r = |h3 \<turnstile> node_ptr_kinds_M|\<^sub>r"
all_dd: shows "\<forall>D\<in>DD. D < old_main_prem_of \<gamma>"
divide_Quantity_ext_def: fixes x :: "('a, 'b, 'c) Quantity_scheme"   and y :: "('a, 'b, 'c) Quantity_scheme" shows "x / y = \<lparr>mag = mag x / mag y, dim = Quantity.dim x / Quantity.dim y, \<dots> = Quantity.more x / Quantity.more y\<rparr>"
stake_cycle_le: fixes u :: "'d list"   and n :: "nat" assumes "u \<noteq> []"   and "n < length u" shows "stake n (cycle u) = take n u"
f_on2_def: fixes fa :: "'e \<Rightarrow> 'g pmf"   and fb :: "'f \<Rightarrow> 'h pmf" shows "f_on2 fa fb = (\<lambda>a. case a of Inl e \<Rightarrow> map_pmf Inl (fa e) | Inr e \<Rightarrow> map_pmf Inr (fb e))"
not_different_AKcryptSK: fixes authK :: "nat"   and servK :: "nat"   and evs :: "event list"   and authK' :: "nat" assumes "AKcryptSK authK servK evs"   and "authK' \<noteq> authK"   and "evs \<in> kerbV" shows "\<not> AKcryptSK authK' servK evs \<and> servK \<in> symKeys"
nfoldli_upt_rule: fixes lb :: "nat"   and ub :: "nat"   and \<sigma>0 :: "nat list"   and c :: "nat list \<Rightarrow> bool"   and f :: "nat \<Rightarrow> nat list \<Rightarrow> nat list nres"   and P :: "nat list \<Rightarrow> bool" assumes "lb \<le> ub"   and "length \<sigma>0 = M + 1 \<and> (\<forall>i<lb. \<sigma>0 ! i = lcount i)"   and "\<And>i \<sigma>. \<lbrakk>lb \<le> i; i < ub; length \<sigma> = M + 1 \<and> (\<forall>ia<i. \<sigma> ! ia = lcount ia); c \<sigma>\<rbrakk> \<Longrightarrow> f i \<sigma> \<le> SPEC (\<lambda>c. length c = M + 1 \<and> (\<forall>ia<i + 1. c ! ia = lcount ia))"   and "\<And>i \<sigma>. \<lbrakk>lb \<le> i; i \<le> ub; length \<sigma> = M + 1 \<and> (\<forall>ia<i. \<sigma> ! ia = lcount ia); \<not> c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma>. \<lbrakk>length \<sigma> = M + 1 \<and> (\<forall>i<ub. \<sigma> ! i = lcount i); c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "nfoldli [lb..<ub] c f \<sigma>0 \<le> SPEC P"
D_WHILE_H: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and b :: "BExpr" assumes "(G, c, HighSec) \<in> Deriv" shows "(G, While b c, HighSec) \<in> Deriv"
simps_2: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f" shows "cond_action_ok s t Yield = True"
induct: fixes x1 :: "char list exp list"   and x2 :: "val list"   and x3 :: "char list exp list"   and P :: "char list exp list \<Rightarrow> val list \<Rightarrow> char list exp list \<Rightarrow> bool" assumes "map_val2 x1 x2 x3"   and "P [] [] []"   and "\<And>xs ys zs y. \<lbrakk>map_val2 xs ys zs; P xs ys zs\<rbrakk> \<Longrightarrow> P (Val y # xs) (y # ys) zs"   and "\<And>e xs. P (throw e # xs) [] (throw e # xs)" shows "P x1 x2 x3"
inject_3: fixes x3 :: "dB"   and y3 :: "dB" shows "(Abs x3 = Abs y3) = (x3 = y3)"
convex_scaling: fixes S :: "'b set"   and c :: "real" assumes "convex S" shows "convex ((*\<^sub>R) c ` S)"
sum_list_concat_map: fixes f :: "'e \<Rightarrow> 'd list"   and xs :: "'e list" shows "sum_list (concat (map f xs)) = (\<Sum>x\<leftarrow>xs. sum_list (f x))"
Exp_Exp_Gen_synth: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "Exp (Exp Gen X) Y \<in> synth H" shows "Exp (Exp Gen X) Y \<in> H \<or> X \<in> synth H \<or> Y \<in> synth H"
simps: fixes x :: "Integer" shows "intsFrom\<cdot>x = seq\<cdot>x\<cdot>(x : intsFrom\<cdot>(x + 1))"
Gcd_fract_def: shows "Gcd \<equiv> normalization_euclidean_semiring_class.Gcd"
sup_definite: fixes M :: "('c \<times> real) set set"   and E :: "'c set"   and p :: "'c \<Rightarrow> real"   and F :: "'c set"   and f :: "'c \<Rightarrow> real"   and c :: "('c \<times> real) set set"   and x :: "'c"   and y :: "real"   and z :: "real" assumes "M = norm_pres_extensions E p F f"   and "c \<in> chains M"   and "(x, y) \<in> \<Union> c"   and "(x, z) \<in> \<Union> c" shows "z = y"
intro: fixes f\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" assumes "comm_semigroup f\<^sub>a"   and "comm_semigroup f\<^sub>b" shows "comm_semigroup_pair f\<^sub>a f\<^sub>b"
dvd_mult_imp_div: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times (times a c) b" shows "dvd.dvd times a (divide b c)"
abs_valid_operator_4': fixes name :: "char list"   and pre :: "(nat \<times> nat) list"   and eff :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and cost :: "nat" assumes "wf_operator (name, pre, eff, cost)" shows "list_all (\<lambda>(v, a). abs_range_map v \<noteq> None \<and> ListMem a (the (abs_range_map v))) (map (\<lambda>(uu_, v, uu_, a). (v, a)) eff)"
vle_comparable_def: fixes x :: "V"   and y :: "V" shows "vle_comparable x y \<equiv> x \<unlhd> y \<or> y \<unlhd> x"
mult_of_nat_commute: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "nat"   and y :: "'a" assumes "class.semiring_1 one times plus zero" shows "times (semiring_1.of_nat one plus zero x) y = times y (semiring_1.of_nat one plus zero x)"
rem_def: fixes x :: "'b"   and xs :: "'b list" shows "rem x xs = filter (\<lambda>y. y \<noteq> x) xs"
card_face_cycle_sets_da': shows "card rev.H.face_cycle_sets = card G.face_cycle_sets - 1 + card ({segment G.face_cycle_succ a' rev.a', segment G.face_cycle_succ rev.a' a', {}} - {{}})"
asTerm_pick: fixes X :: "('e, 'f, 'varSort, 'var, 'g) qTerm set" assumes "good X" shows "asTerm (pick X) = X"
freshAbs_psubstAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs :: "'varSort"   and z :: "'var" assumes "goodAbs A"   and "goodEnv rho" shows "freshAbs zs z A $[rho] = (\<forall>ys y. freshAbs ys y A \<or> freshImEnvAt zs z rho ys y)"
simps_3: fixes a :: "'b"   and \<nu> :: "'b set" shows "af_letter prop\<^sub>n(a) \<nu> = (if a \<in> \<nu> then true\<^sub>n else false\<^sub>n)"
U_\<Phi>_def: shows "U_\<Phi> \<equiv> GE_\<Phi>"
beh_non_empty: fixes s :: "'a"   and b :: "'a list"   and T :: "('a, 'b) TS_scheme" assumes "s # b \<in> beh T" obtains "b = []" and "s \<in> init T"   | sa :: "'a" and ba :: "'a list" where "b = sa # ba" and "sa # ba \<in> beh T" and "(sa, s) \<in> TS.trans T"
solution_set_rel_solve: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "consistent A b" shows "solution_set A b = {fst (the (solve A b))} + vec.span (snd (the (solve A b)))"
currentLevelNonMarked: fixes M :: "('b \<times> bool) list"   and l :: "'b" shows "currentLevel M = currentLevel (M @ [(l, False)])"
getEnv_substEnv1: fixes ys :: "'varSort"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "ys \<noteq> xs \<or> y \<noteq> x"   and "rho xs x = None" shows "(rho &[Y / y]_ys) xs x = None"
ComplD: fixes c :: "'e"   and A :: "'e set" assumes "c \<in> - A" shows "c \<notin> A"
wls_Abs_swap_cong: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var"   and x' :: "'var" assumes "wls s X"   and "wls s X'"   and "fresh xs y X"   and "fresh xs y X'"   and "X #[y \<and> x]_xs = X' #[y \<and> x']_xs" shows "Abs xs x X = Abs xs x' X'"
a_star_OrR2: fixes M :: "trm"   and M' :: "trm"   and a :: "coname"   and b :: "coname" assumes "M \<longrightarrow>\<^sub>a* M'" shows "OrR2 <a>.M b \<longrightarrow>\<^sub>a* OrR2 <a>.M' b"
zeroring_no_maximal: assumes "zeroring R" shows "\<nexists>I. maximal_ideal R I"
mem_Collect_eq: fixes a :: "'b::type"   and P :: "'b::type \<Rightarrow> bool" shows "((a::'b::type) \<in> Collect (P::'b::type \<Rightarrow> bool)) = P a"