hcju/isabelleps huggingface dataset:
invariant_iff_pos_invariant_and_compl_pos_invariant: fixes M :: "'a set" shows "rev.invariant M = (rev.positively_invariant M \<and> rev.positively_invariant (X - M))"
component_of_map_component: fixes f :: "'k \<Rightarrow> 'k"   and v :: "'t" shows "component_of_term (map_component f v) = f (component_of_term v)"
lower_3_tendsto: fixes x :: "real_alg_3" shows "lower_3 x \<longlonglongrightarrow> real_of_3 x"
inj_idom_hom_axioms: shows "inj_idom_hom (map_poly hom)"
word_and_full_mask_simp: fixes x :: "'a word" shows "x && mask LENGTH('a) = x"
bury_correct: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int"   and X :: "char list set"   and t :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> s'"   and "s = t on L c X" shows "\<exists>t'. (bury c X, t) \<Rightarrow> t' \<and> s' = t' on X"
compl_structI: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" assumes "\<And>d gs bs ps sps data. \<lbrakk>dickson_grading d; sps \<noteq> []; set sps \<subseteq> set ps\<rbrakk> \<Longrightarrow> dgrad_p_set_le d (fst ` set (fst (compl gs bs (ps -- sps) sps data))) (args_to_set (gs, bs, ps))"   and "\<And>gs bs ps sps data. \<lbrakk>sps \<noteq> []; set sps \<subseteq> set ps\<rbrakk> \<Longrightarrow> component_of_term ` Keys (fst ` set (fst (compl gs bs (ps -- sps) sps data))) \<subseteq> component_of_term ` Keys (args_to_set (gs, bs, ps))"   and "\<And>gs bs ps sps data. \<lbrakk>sps \<noteq> []; set sps \<subseteq> set ps\<rbrakk> \<Longrightarrow> 0 \<notin> fst ` set (fst (compl gs bs (ps -- sps) sps data))"   and "\<And>gs bs ps sps h b data. \<lbrakk>sps \<noteq> []; set sps \<subseteq> set ps; h \<in> set (fst (compl gs bs (ps -- sps) sps data)); b \<in> set gs \<union> set bs; fst b \<noteq> 0\<rbrakk> \<Longrightarrow> \<not> lt (fst b) adds\<^sub>t lt (fst h)" shows "compl_struct compl"
sumComm: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs"   and Q :: "ccs" assumes "Id \<subseteq> Rel" shows "P \<oplus> Q \<leadsto>[Rel] Q \<oplus> P"
set_mult_is_times: fixes A :: "('a \<Rightarrow> 'b) set"   and B :: "('a \<Rightarrow> 'b) set" shows "set_mult A B = A * B"
arity_def: fixes fundef :: "'a fundef" shows "arity fundef = (case fundef of Fundef x1 x2 \<Rightarrow> x2)"
idb_3: fixes k :: "nat" assumes "i_b \<le> k"   and "k < i_e" shows "A $$ (k, Suc k) = (1::'b)"
Step1: shows "(\<lambda>g. \<lambda>H\<in>{H. H \<subseteq> carrier G}. g <# H #> inv g) \<in> carrier G \<rightarrow> carrier (BijGroup {H. H \<subseteq> carrier G})"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop4 n (l, r) = (\<exists>i j k t. i + j = n \<and> 0 < i \<and> 0 < j \<and> k + t = j \<and> (l, r) = (Oc \<up> k @ Bk \<up> Suc j @ Oc \<up> i, Oc \<up> t))"
equivalent_objects_def: fixes a :: "'a"   and b :: "'a" shows "C.equivalent_objects a b \<equiv> \<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright> \<and> C.equivalence_map f"
simps_3: fixes oneC :: "int \<times> (real \<Rightarrow> real \<times> real)"   and v :: "int \<times> (real \<Rightarrow> real \<times> real)"   and va :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" shows "valid_chain_list (oneC # v # va) = (pathfinish (coeff_cube_to_path oneC) = pathstart (rec_join (v # va)) \<and> valid_chain_list (v # va))"
select_convs_10: fixes p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'b word"   and p_dst :: "'b word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'a" shows "simple_packet.more \<lparr>p_iiface = p_iiface::char list, p_oiface = p_oiface::char list, p_src = p_src::'b word, p_dst = p_dst::'b word, p_proto = p_proto::8 word, p_sport = p_sport::16 word, p_dport = p_dport::16 word, p_tcp_flags = p_tcp_flags::tcp_flag set, p_payload = p_payload::char list, \<dots> = more::'a\<rparr> = more"
edge_obs_subset: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and a :: "'a"   and S :: "'b set" assumes "CFG sourcenode targetnode valid_edge Entry"   and "valid_edge a"   and "sourcenode a \<notin> S" shows "CFG.obs sourcenode targetnode valid_edge (targetnode a) S \<subseteq> CFG.obs sourcenode targetnode valid_edge (sourcenode a) S"
compute_bound_tm_mono: fixes t :: "taylor_model"   and I :: "float interval list"   and J :: "float interval list"   and a :: "float interval list"   and prec :: "nat" assumes "num_params (tm_poly t) \<le> length I"   and "I all_subset J"   and "a all_subset I" shows "set_of (compute_bound_tm prec I a t) \<subseteq> set_of (compute_bound_tm prec J a t)"
C: shows "C = 0\<^sub>m (n - m) m"
swap_nested_emb_step_at: fixes p :: "dir list"   and q :: "dir list"   and d2 :: "dir"   and d1 :: "dir"   and t :: "('a, 'b) tm" shows "emb_step_at (p @ q) d2 (emb_step_at p d1 t) = emb_step_at p d1 (emb_step_at (p @ [d1] @ q) d2 t)"
cases: fixes x :: "'d tree" obtains "x = \<langle>\<rangle>"   | l :: "'d tree" and a :: "'d" and r :: "'d tree" where "x = \<langle>l, a, r\<rangle>"
map_of_zip_map_Some: fixes v :: "nat"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "v < length xs" shows "map_of (zip [0..<length xs] (map f xs)) v = Some (f (xs ! v))"
lists_iff_set: fixes w :: "'e list"   and A :: "'e set" shows "(w \<in> lists A) = (set w \<subseteq> A)"
inject_1: fixes x3 :: "subject"   and x1 :: "name"   and x2 :: "pi"   and y3 :: "subject"   and y1 :: "name"   and y2 :: "pi" shows "(x3\<guillemotleft>x1\<guillemotright> \<prec> x2 = y3\<guillemotleft>y1\<guillemotright> \<prec> y2) = (x3 = y3 \<and> [x1].x2 = [y1].y2)"
finiteHom_def: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and f :: "'c \<Rightarrow> 'd" shows "finiteHom I A f = (f \<in> carr_prodag I A \<and> (\<exists>H\<subseteq>I. finite H \<and> (\<forall>j\<in>I - H. f j = \<zero>\<^bsub>A j\<^esub>)))"
bhv2cnf_lActive: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "lfinite (\<pi>\<^bsub>c\<^esub>inf_llist t)" shows "\<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) = Suc \<langle>c \<and> t\<rangle>"
subgraph_del_vert: fixes u :: "'a" shows "subgraph (H.del_vert u) (del_vert u)"
admissible_chfin: fixes Sup :: "'d set \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool" assumes "class.ccpo Sup less_eq less"   and "\<forall>S. Complete_Partial_Order.chain less_eq S \<longrightarrow> finite S" shows "ccpo.admissible Sup less_eq P"
entails_member: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'a Literal set"   and S :: "'a Literal set set" assumes "propositional_atoms atom_ordering"   and "C \<in> S" shows "propositional_atoms.entails S C"
SEQ_def: fixes A :: "'a set" shows "SEQ A = {f. \<forall>i. f i \<in> A}"
summable: shows "summable (\<lambda>k. real_of_int (p k) / real_of_int q ^ fact (Suc k))"
preserves_comp_2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'a \<Rightarrow> 'c"   and G :: "'a \<Rightarrow> 'c"   and \<tau> :: "'a \<Rightarrow> 'c"   and f' :: "'a"   and f :: "'a" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A (A f' f)" shows "\<tau> (A f' f) = B (\<tau> f') (F f)"
add_canc2: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.cancellative_partial_semigroup times D"   and "partial_semigroup.gR_rel times D x y" shows "times x (cancellative_partial_semigroup.rquot times D y x) = y"
igSwapInp_termMOD: shows "igSwapInp termMOD = swapInp"
assms_1: shows "poincare_collinear {a, a1, a2}"
not_eq_extremum: fixes a :: "nat" shows "(a \<noteq> 0) = (0 < a)"
pbdom_def: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" shows "Maps.pbdom f g \<equiv> Maps.dom (Maps.PRJ\<^sub>0 f g)"
diff_strict_right_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less a b" shows "less (minus a c) (minus b c)"
goal3_4: shows "Rmi_g ni2 n2 bdd"
simps_3: fixes x :: "'g"   and y :: "'g"   and xs :: "'g list" shows "remdups_adj (x # y # xs) = (if x = y then remdups_adj (x # xs) else x # remdups_adj (y # xs))"
distinct_11: fixes x31 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x32 :: "'c"   and x33 :: "'d"   and x34 :: "(('a \<Rightarrow> 'b option) \<Rightarrow> 'b option) list"   and x41 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x42 :: "'d"   and x43 :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option" shows "x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34 \<noteq> x41\<hookleftarrow>\<^bsub>x42\<^esub>x43"
singleton_insert_inj_eq: fixes b :: "'d"   and a :: "'d"   and A :: "'d set" shows "({b} = insert a A) = (a = b \<and> A \<subseteq> {b})"
compactin_subtopology: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "compactin (subtopology X S) T = (compactin X T \<and> T \<subseteq> S)"
notStuck_CallD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "'d"   and s :: "'e"   and bdy :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"   and "\<Gamma> p = Some bdy" shows "\<Gamma>\<turnstile> \<langle>bdy,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
seqE: fixes g :: "'b"   and f :: "'b" assumes "\<tau>.B.seq g f" obtains "\<tau>.B.arr f" and "\<tau>.B.arr g" and "\<tau>.B.dom g = \<tau>.B.cod f"
sync_with_def: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and a :: "nat"   and a' :: "nat" shows "(P,E \<turnstile> a \<le>sw a') = (P,E \<turnstile> a \<le>so a' \<and> P \<turnstile> (action_tid E a, action_obs E a) \<leadsto>sw (action_tid E a', action_obs E a'))"
module_hom_axioms: shows "module_hom (*a) (*b) f"
map_cong0: fixes x :: "('d, 'e, 'f) F1"   and f1 :: "'d \<Rightarrow> 'g"   and g1 :: "'d \<Rightarrow> 'g"   and f2 :: "'e \<Rightarrow> 'h"   and g2 :: "'e \<Rightarrow> 'h"   and f3 :: "'f \<Rightarrow> 'i"   and g3 :: "'f \<Rightarrow> 'i" assumes "\<And>z1. z1 \<in> F1set1 x \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> F1set2 x \<Longrightarrow> f2 z2 = g2 z2"   and "\<And>z3. z3 \<in> F1set3 x \<Longrightarrow> f3 z3 = g3 z3" shows "F1map f1 f2 f3 x = F1map g1 g2 g3 x"
restrict_relp_False1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" shows "R \<upharpoonleft> (\<lambda>_. False) \<otimes> Q = \<bottom>"
bs: shows "set bs = {(A i, i) |i. i < m \<and> A i \<noteq> 1}"
inf_action: fixes P :: "('a, 'b) action"   and Q :: "('a, 'b) action" shows "inf P Q \<equiv> action_of (action.relation_of P \<or> action.relation_of Q)"
scaleR_zero_left: fixes x :: "'c" shows "0 *\<^sub>R x = (0::'c)"
pre_f: shows "pre_splitFace g' ram1 ram2 f []"
map_comp: fixes g1 :: "'g \<Rightarrow> 'd"   and g2 :: "'h \<Rightarrow> 'e"   and g3 :: "'i \<Rightarrow> 'f"   and f1 :: "'j \<Rightarrow> 'g"   and f2 :: "'k \<Rightarrow> 'h"   and f3 :: "'l \<Rightarrow> 'i"   and v :: "('j, 'k, 'l) generat" shows "map_generat g1 g2 g3 (map_generat f1 f2 f3 v) = map_generat (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) v"
fatom_of_nat_def: shows "fatom_of_nat = (\<lambda>n. fatom_of_hatom (hatom_of_nat n))"
dims_def: fixes A :: "'b tensor" shows "dims A = fst (Rep_tensor A)"
induct: fixes P :: "'b set list \<Rightarrow> bool"   and a0 :: "'b set list" assumes "P []"   and "\<And>x xs. (find_sub_in x xs True = None \<Longrightarrow> P xs) \<Longrightarrow> P (x # xs)" shows "P a0"
zmset_subseteq_add_iff1: fixes j :: "nat"   and i :: "nat"   and u :: "'b zmultiset"   and m :: "'b zmultiset"   and n :: "'b zmultiset" assumes "j \<le> i" shows "(repeat_zmset i u + m \<subseteq>#\<^sub>z repeat_zmset j u + n) = (repeat_zmset (i - j) u + m \<subseteq>#\<^sub>z n)"
Jeroslow_Diagonalization_def: 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 num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and Ops :: "('b \<Rightarrow> 'b) set"   and tenc :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv enc Ops tenc F encF N ssap \<equiv> (Deduct_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv \<and> Encode var trm fmla Var FvarsT substT Fvars subst num enc) \<and> TermEncode var trm fmla Var FvarsT substT Fvars subst num Ops tenc \<and> Jeroslow_Diagonalization_axioms var fmla Fvars subst eql imp fls num prv enc Ops tenc F encF N ssap"
fset_def: shows "fset \<equiv> (rep_fset ---> id) set"
rel_envT_simps: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and M :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" shows "local.rel_envT R M (EnvT f) (EnvT g) = (R ===> M) f g"
map_inj_on: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c list" assumes "map f xs = map f ys"   and "inj_on f (ran\<^sub>l xs \<union> ran\<^sub>l ys)" shows "xs = ys"
LC: shows "\<forall>ts'. ts \<^bold>\<Rightarrow> ts' \<longrightarrow> (\<forall>x. (\<exists>xa. (\<forall>v'. move ts ts' v = xa > v' \<longrightarrow> xa \<noteq> x \<and> (\<exists>v u. v'=v\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> ((0 < \<parallel>ext v\<parallel> \<and> len v ts' x = ext v \<and> restrict v (clm ts') x = lan v \<and> |lan v| = 1) \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts' xa = ext v \<and> restrict v (res ts') xa = lan v \<and> |lan v| = 1 \<or> 0 < \<parallel>ext v\<parallel> \<and> len v ts' xa = ext v \<and> restrict v (clm ts') xa = lan v \<and> |lan v| = 1)) \<and> True)) \<and> True))) \<or> (\<forall>v'. move ts ts' v = x > v' \<longrightarrow> xa \<noteq> x \<and> (\<exists>v u. v'=v\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> ((0 < \<parallel>ext v\<parallel> \<and> len v ts' x = ext v \<and> restrict v (clm ts') x = lan v \<and> |lan v| = 1) \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts' xa = ext v \<and> restrict v (res ts') xa = lan v \<and> |lan v| = 1 \<or> 0 < \<parallel>ext v\<parallel> \<and> len v ts' xa = ext v \<and> restrict v (clm ts') xa = lan v \<and> |lan v| = 1)) \<and> True)) \<and> True)))) \<longrightarrow> (\<forall>ts'a. ts' \<^bold>\<midarrow>r( x ) \<^bold>\<rightarrow> ts'a \<longrightarrow> False))"
finfun_eq_finite_UNIV: assumes "finite UNIV" shows "finfun = UNIV"
sublist_lt: fixes hi :: "nat"   and lo :: "nat"   and xs :: "'c list" assumes "hi < lo" shows "WB_Sort.sublist xs lo hi = []"
rel_pred_pos_distr: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "rel_pred_pos_distr_cond A B" shows "rel_pred A OO rel_pred B \<le> rel_pred (A OO B)"
rep_eq: fixes x :: "'b" shows "Rep_ttree (single x) = {[], [x]}"
C: shows "set (local.compute_set_of_jordan_blocks A ev) = {(k, ev) |k. local.compute_nr_of_jordan_blocks A ev k \<noteq> 0 \<and> k \<noteq> 0 \<and> k < Suc kk}"
mult_row_row_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c"   and q :: "'a"   and i :: "'c" shows "mult_row_row A a q i = (\<chi>j. if i = a then q * A $ a $ j else A $ i $ j)"
row_echelon_pivot_fun: fixes A :: "'c mat" obtains f :: "nat \<Rightarrow> nat" where "pivot_fun (row_echelon A) f (dim_col (row_echelon A))"
these_insert_None: fixes A :: "'a option set" shows "Option.these (insert None A) = Option.these A"
subseq_to_smaller: fixes j :: "nat"   and i :: "nat"   and w :: "nat \<Rightarrow> 'b" assumes "j \<le> i" shows "w [i \<rightarrow> j] = []"
inorder_del: fixes t :: "'b bro"   and h :: "nat"   and x :: "'b" assumes "t \<in> T h"   and "Sorted_Less.sorted (inorder t)" shows "inorder (del x t) = del_list x (inorder t)"
divide_right_mono_neg: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and 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 a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less_eq c zero" shows "less_eq (divide b c) (divide a c)"
simE: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and \<alpha> :: "act"   and Q' :: "ccs" assumes "P \<leadsto>[Rel] Q"   and "Q \<longmapsto>\<alpha> \<prec> Q'" obtains P' :: "ccs" where "P \<longmapsto>\<alpha> \<prec> P'" and "(P', Q') \<in> Rel"
wa_refine_intro: fixes WAA :: "'a while_algo"   and invar_addc :: "'b set"   and condc :: "'b set"   and \<alpha> :: "'b \<Rightarrow> 'a"   and stepc :: "('b \<times> 'b) set"   and initialc :: "'b set" assumes "while_algo WAA"   and "\<And>s s'. \<lbrakk>s \<in> invar_addc; s \<in> condc; \<alpha> s \<in> wa_invar WAA; (s, s') \<in> stepc\<rbrakk> \<Longrightarrow> s' \<in> invar_addc"   and "initialc \<subseteq> invar_addc"   and "\<And>s. \<lbrakk>s \<in> invar_addc; \<alpha> s \<in> wa_invar WAA; s \<in> condc\<rbrakk> \<Longrightarrow> \<alpha> s \<in> wa_cond WAA"   and "\<And>s s'. \<lbrakk>s \<in> invar_addc; s \<in> condc; \<alpha> s \<in> wa_invar WAA; (s, s') \<in> stepc\<rbrakk> \<Longrightarrow> (\<alpha> s, \<alpha> s') \<in> wa_step WAA"   and "\<alpha> ` initialc \<subseteq> wa_initial WAA" shows "while_algo \<lparr>wa_cond = condc, wa_step = stepc, wa_initial = initialc, wa_invar = invar_addc \<inter> {s. \<alpha> s \<in> wa_invar WAA}\<rparr> \<and> wa_refine \<lparr>wa_cond = condc, wa_step = stepc, wa_initial = initialc, wa_invar = invar_addc \<inter> {s. \<alpha> s \<in> wa_invar WAA}\<rparr> WAA \<alpha>"
lemma5': fixes r :: "nat \<Rightarrow> nat \<Rightarrow> b"   and p :: "nat \<times> nat list \<times> nat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "is_path r p (Suc i) j k"   and "\<not> is_path r p i j k" shows "\<not> (\<forall>q. \<not> is_path r q i j i) \<and> \<not> (\<forall>q'. \<not> is_path r q' i i k)"
shiftr': shows "effect (shiftr a key i) h\<^sub>1 h\<^sub>2 ()"
prod_tOp_func: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" assumes "\<forall>k\<in>I. Ring (A k)" shows "prod_tOp I A \<in> carr_prodag I A \<rightarrow> carr_prodag I A \<rightarrow> carr_prodag I A"
permutations_of_set_def: fixes A :: "'a set" shows "permutations_of_set A = {xs. set xs = A \<and> distinct xs}"
bij_betw_imageI: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "inj_on f A"   and "f ` A = B" shows "bij_betw f A B"
InvariantClCurrentLevel_def: fixes Cl :: "Literal"   and M :: "(Literal \<times> bool) list" shows "InvariantClCurrentLevel Cl M \<equiv> elementLevel Cl M = currentLevel M"
abs_eq: shows "0 = Interval (0::'a, 0::'a)"
extend_transient_slice: fixes F :: "'a program"   and A :: "'c set"   and y :: "'b" assumes "extend h F \<in> transient A" shows "F \<in> transient (slice A y)"
closed_othersD: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and c\<^sub>A :: "'Com\<^sub>A"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>C :: "'Com\<^sub>C"   and mem\<^sub>C' :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "closed_others \<R>"   and "(\<langle>c\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R>"   and "\<And>x. mem\<^sub>C' x \<noteq> mem\<^sub>C x \<or> dma\<^sub>C mem\<^sub>C' x \<noteq> dma\<^sub>C mem\<^sub>C x \<Longrightarrow> \<not> var_asm_not_written mds\<^sub>C x" shows "(\<langle>c\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>C'\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C'\<rangle>\<^sub>C) \<in> \<R>"
locally_def: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set" shows "locally P S \<equiv> \<forall>w x. openin (top_of_set S) w \<and> x \<in> w \<longrightarrow> (\<exists>u v. openin (top_of_set S) u \<and> P v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)"
IH: assumes "t \<in> trms\<^sub>s\<^sub>s\<^sub>t S"   and "subst_domain \<theta> \<inter> bvars\<^sub>s\<^sub>s\<^sub>t S = {}" shows "fv (t \<cdot> \<theta>) \<subseteq> vars\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
intro: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and R' :: "'d set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" assumes "Set_Theory.map \<eta> R R'"   and "ring R addition multiplication zero unit"   and "ring R' addition' multiplication' zero' unit'"   and "group_homomorphism \<eta> R addition zero R' addition' zero'"   and "monoid_homomorphism \<eta> R multiplication unit R' multiplication' unit'" shows "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'"
simps: fixes I :: "'c set"   and v :: "nat \<Rightarrow> 'd"   and as :: "('c \<times> 'd atom) set" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>e\<^sub>s as = v \<Turnstile>\<^sub>a\<^sub>e\<^sub>s (snd ` (as \<inter> I \<times> UNIV))"
rotate_Suc2: fixes n :: "nat"   and xs :: "'a list" shows "rotate n (rotate1 xs) = rotate (Suc n) xs"