hcju/isabelleps huggingface dataset:
simps_2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and y :: "'d"   and ys :: "'d list" assumes "class.linorder less_eq less" shows "linorder.insort_key less_eq f x (y # ys) = (if less_eq (f x) (f y) then x # y # ys else y # linorder.insort_key less_eq f x ys)"
subst_clss_empty: fixes \<sigma> :: "'s" shows "{} \<cdot>cs \<sigma> = {}"
us_le: fixes y :: "nat list"   and i :: "nat" assumes "0 < length y" shows "\<^bold>u y i \<le>\<^sub>v y"
antisym: fixes x :: "'a multiset"   and y :: "'a multiset" assumes "x \<subseteq># y"   and "y \<subseteq># x" shows "x = y"
dom12_d_U: fixes x :: "'a" shows "d x = 1\<^sub>\<sigma> \<sqinter> x \<cdot> 1\<^sub>\<pi> \<parallel> U"
one_plus_square_neq_zero: fixes x :: "real" shows "1 + (cor x)\<^sup>2 \<noteq> 0"
is_weak_composition: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" assumes "horizontal_composition V H src trg" shows "weak_composition V H"
graph_antimono: fixes A :: "('b, 'a) nba"   and w :: "'b stream" shows "decseq (graph A w)"
lemma1': fixes ts :: "dB list"   and ts' :: "dB list"   and r :: "dB"   and r' :: "dB" assumes "ts [\<rightarrow>\<^sub>s] ts'"   and "r \<rightarrow>\<^sub>s r'" shows "r \<degree>\<degree> ts \<rightarrow>\<^sub>s r' \<degree>\<degree> ts'"
simps_1: fixes xs :: "'b list"   and oid :: "'b" shows "insert_spec xs (oid, None) = oid # xs"
induct: fixes P :: "'b tree list \<Rightarrow> 'b list \<Rightarrow> 'b tree list \<Rightarrow> bool"   and a0 :: "'b tree list"   and a1 :: "'b list"   and a2 :: "'b tree list" assumes "\<And>l ls x xs r rs. P ls xs rs \<Longrightarrow> P (l # ls) (x # xs) (r # rs)"   and "\<And>l ls x xs. P ls xs [] \<Longrightarrow> P (l # ls) (x # xs) []"   and "\<And>x xs r rs. P [] xs rs \<Longrightarrow> P [] (x # xs) (r # rs)"   and "\<And>x xs. P [] xs [] \<Longrightarrow> P [] (x # xs) []"   and "\<And>ls rs. P ls [] rs" shows "P a0 a1 a2"
per_lt: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B"   and "Per A B C" shows "A B Lt A C \<and> C B Lt A C"
Abs1_eq_3: fixes a :: "'b"   and x :: "'a"   and y :: "'a" shows "([[atom a]]lst. x = [[atom a]]lst. y) = (x = y)"
SubstAtomicP_unique: fixes v :: "tm"   and tm :: "tm"   and x :: "tm"   and y :: "tm"   and y' :: "tm" shows "{SubstAtomicP v tm x y, SubstAtomicP v tm x y'} \<turnstile> y' EQ y"
rtrancl_trans: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set"   and z :: "'a" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>*"
multiset_inter_single: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b" shows "{#a#} \<inter># {#b#} = {#}"
lossless_sample_set_excl: fixes Q :: "nat set"   and P :: "nat" assumes "finite Q"   and "2 < card Q" shows "lossless_spmf (sample_set_excl Q {P})"
simps: fixes a :: "'d stream"   and A :: "'d set" shows "(a \<in> streams A) = (\<exists>aa s. a = aa ## s \<and> aa \<in> A \<and> s \<in> streams A)"
comp_ide_arr: fixes b :: "'a"   and f :: "'a" assumes "C.ide b"   and "C.seq b f" shows "b \<cdot> f = f"
sel_1: fixes x1 :: "'a"   and x2 :: "'a msstream" shows "un_MSSCons1 (MSSCons x1 x2) = x1"
cases: fixes x :: "(('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" obtains bs :: "('t \<Rightarrow>\<^sub>0 'b) list" and ss :: "'t list" and z :: "nat" where "x = ((bsa__, ssa__, []), za__)"   | bs :: "('t \<Rightarrow>\<^sub>0 'b) list" and ss :: "'t list" and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" and z :: "nat" where "x = ((bs, ss, p # ps), z)"
S_sym: fixes S :: "'b set"   and T :: "'b set" assumes "PreCoxeterSystem S"   and "order.greater_eq S T" shows "order.greater_eq T (uminus ` T)"
splus_pp_term: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.nat_pp_term plus zero rep_nat_term splus" shows "splus = plus"
code: fixes z :: "gauss_int" shows "gauss2complex z = Complex (real_of_int (ReZ z)) (real_of_int (ImZ z))"
SF2: fixes N :: "state \<times> state \<Rightarrow> bool"   and B :: "state \<times> state \<Rightarrow> bool"   and f :: "state \<Rightarrow> 'a"   and M :: "state \<times> state \<Rightarrow> bool"   and g :: "state \<Rightarrow> 'b"   and P :: "state \<Rightarrow> bool"   and A :: "state \<times> state \<Rightarrow> bool"   and F :: "'c \<Rightarrow> bool" assumes "\<turnstile> N \<and> <B>_f \<longrightarrow> <M>_g"   and "\<turnstile> $P \<and> P$ \<and> <N \<and> A>_f \<longrightarrow> B"   and "\<turnstile> P \<and> Enabled (<M>_g) \<longrightarrow> Enabled (<A>_f)"   and "\<turnstile> \<box>(N \<and> [\<not> B]_f) \<and> SF(A)_f \<and> \<box>F \<and> \<box>\<diamond>Enabled (<M>_g) \<longrightarrow> \<diamond>\<box>P" shows "\<turnstile> \<box>N \<and> SF(A)_f \<and> \<box>F \<longrightarrow> SF(M)_g"
length_drop_mask_inner: fixes n :: "nat"   and n' :: "nat"   and ip :: "'a word" assumes "n \<le> LENGTH('a) - n'" shows "length (dropWhile Not (to_bl (ip && (mask n << n') >> n'))) \<le> n"
tendsto_eventually: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and net :: "'c filter" assumes "class.topological_space open"   and "\<forall>\<^sub>F x in net. f x = l" shows "filterlim f (topological_space.nhds open l) net"
stable_pair_on_def: fixes ds :: "'b set" shows "stable_pair_on ds = (\<lambda>(XD, XH). XD = - Rf CH XH \<and> XH = - Rf (CD_on ds) XD)"
hyps_2: assumes "0 < lead_coeff p" shows "\<exists>n. \<forall>x\<ge>n. lead_coeff p \<le> poly p x"
nonsimp: fixes us :: "'b list list"   and G :: "'b list set" assumes "us \<in> lists G\<^sub>+"   and " concat us \<in>B G " shows "us = [concat us]"
axioms_8: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and insert_before :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> (_) node_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and insert_before_locs :: "(_) object_ptr \<Rightarrow> (_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool" assumes "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent get_parent_locs get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_ancestors adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_owner_document insert_before insert_before_locs type_wf known_ptr known_ptrs" shows "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs"
fSup: shows "feasible (Sup_trans (range (iterates body G)))"
MGU_nil: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" shows "MGU Var s t = (s = t)"
dom_lc3: fixes x :: "'a"   and y :: "'a" shows "ad (ad x) \<cdot> ad (ad (x \<cdot> y)) = ad (ad (x \<cdot> y))"
true_lit_simps_2: fixes I :: "'b set"   and A :: "'b" shows "(I \<Turnstile>l Neg A) = (A \<notin> I)"
in_set_conv_decomp_last: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = append ys (Cons x zs) \<and> x \<notin> set zs)"
isArrLenLoc_def: fixes l :: "Location" shows "isArrLenLoc l = (case l of arrLenLoc T a \<Rightarrow> True | _ \<Rightarrow> False)"
section_is_mono: fixes g :: "'b" assumes "\<phi>.B.section g" shows "\<phi>.B.mono g"
rep_eq: fixes x :: "'b zmultiset" shows "zcount x = (case rep_zmultiset x of (Mp, Mn) \<Rightarrow> \<lambda>x. int (count Mp x) - int (count Mn x))"
nat_pow_inv: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and i :: "nat" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> i = inv\<^bsub>G\<^esub> (x [^]\<^bsub>G\<^esub> i)"
ordered_nsets_3_eq: fixes A :: "'a set" shows "[A]\<^bsup>3\<^esup> = {{x, y, z} |x y z. x \<in> A \<and> y \<in> A \<and> z \<in> A \<and> x < y \<and> y < z}"
last_fin_make_untimed_append: fixes z :: "'d list list"   and a :: "'d" shows "last (fin_make_untimed (z @ [[a]])) = a"
L_wf_state: fixes s :: "'s" assumes "wf_state s" shows "L s \<subseteq> lists (set (\<sigma> n))"
baIA: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and \<phi> :: "'cmp \<Rightarrow> bool"   and t' :: "nat \<Rightarrow> 'cmp" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<phi> (\<sigma>\<^bsub>c\<^esub>t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>)" shows "eval c t t' n [\<phi>]\<^sub>b"
proper_map_fst: fixes T :: "'a set"   and K :: "'b set"   and S :: "'b set" assumes "compact T"   and "K \<subseteq> S"   and "compact K" shows "compact (S \<times> T \<inter> fst -` K)"
remove1_insort_same_eq: fixes x :: "'b"   and cmp :: "'b comparator"   and xs :: "'b list" shows "remove1 x (Sorting_Algorithms.insort cmp x xs) = xs"
\<delta>\<^sub>p_def_1: shows "p\<^sub>1 = find_closest_bf p\<^sub>0 (take 7 (p\<^sub>2 # ps))"
prob_proj_comm: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs :: "'d set"   and vs' :: "'d set" shows "prob_proj (prob_proj PROB vs) vs' = prob_proj (prob_proj PROB vs') vs"
pwD_2: fixes S :: "'b nres"   and S' :: "'b nres"   and x :: "'b" assumes "S \<le> S'"   and "inres S x" shows "inres S' x"
Cn_conj: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<and>. q) = (Cn {p} \<union> Cn {q} \<subseteq> Cn A)"
SeqAbstFormP_Neg: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and k1 :: "tm"   and x :: "tm"   and x' :: "tm"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, s1, k1, x, x', v, i)"   and "atom k \<sharp> (s1, k1, x, x', v, i)" shows "{SeqAbstFormP v i x x' s1 k1, OrdP i, VarP v} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqAbstFormP v i (Q_Neg x) (Q_Neg x') (Var s) (Var k)))"
list2FWpolicys_eq: fixes Filter :: "((int \<times> int) set set, int) Combinators list" assumes "Filter \<noteq> []" shows "C (list2policyR Filter) = C (list2FWpolicy (rev Filter))"
that: fixes g :: "'a \<Rightarrow> 'b" assumes "polynomial_function g"   and "g ` S \<subseteq> T"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x - g x) < e" shows "thesis"
alpha_l_Suc_l_derive: fixes l :: "nat" shows "2 * (alpha_l l * complex_of_real (cos (\<theta> / 2)) - beta_l l * complex_of_real (sin (\<theta> / 2))) * complex_of_real (cos (\<theta> / 2)) - alpha_l l = alpha_l (l + 1)"
distrib_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b + c) = a * b + a * c"
renaming_preserves_subsumption: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'a Literal set"   and D :: "'a Literal set"   and Sel :: "'a set" assumes "propositional_atoms atom_ordering"   and "subsumes C D" shows "subsumes (propositional_atoms.rename_clause Sel C) (propositional_atoms.rename_clause Sel D)"
Cons_2: shows "set (l # q) \<subseteq> carrier R"
S'_LIMSEQ_Digamma: fixes x :: "real" assumes "x \<noteq> 0" shows "(\<lambda>n. ln (real n) - S' n x - 1 / (2 * x)) \<longlonglongrightarrow> Digamma x"
block_designI: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and k :: "nat" assumes "proper_design point_set block_collection"   and "\<And>bl. bl \<in># block_collection \<Longrightarrow> card bl = k" shows "block_design point_set block_collection (int k)"
simps: fixes R1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a1 :: "'e + 'g"   and a2 :: "'f + 'h" shows "rel_sum R1 R2 a1 a2 = ((\<exists>a c. a1 = Inl a \<and> a2 = Inl c \<and> R1 a c) \<or> (\<exists>b d. a1 = Inr b \<and> a2 = Inr d \<and> R2 b d))"
rerename_subst_noop: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and "from" :: "nat"   and V :: "'a set"   and s :: "'c"   and to :: "nat"   and f :: "'a \<Rightarrow> 'a" assumes "Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"   and "freshenLC from ` V \<inter> subst_lconsts s = {}" shows "subst_renameLCs (Abstract_Formulas.rerename freshenLC V from to f) s = subst_renameLCs f s"
accepting_pair\<^sub>G\<^sub>R_LTS_def: fixes \<delta> :: "('d \<times> 'e \<times> 'd) set"   and q\<^sub>0 :: "'d"   and P :: "('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set set"   and w :: "nat \<Rightarrow> 'e" shows "accepting_pair\<^sub>G\<^sub>R_LTS \<delta> q\<^sub>0 P w \<equiv> \<exists>r. LTS_is_inf_run \<delta> w r \<and> r 0 = q\<^sub>0 \<and> limit (\<lambda>i. (r i, w i, r (Suc i))) \<inter> fst P = {} \<and> (\<forall>I\<in>snd P. limit (\<lambda>i. (r i, w i, r (Suc i))) \<inter> I \<noteq> {})"
continuity_lem_gt0: fixes c :: "real"   and b :: "real"   and a :: "real"   and r :: "real" assumes "0 < poly [:c, b, a:] r" shows "\<exists>y'>r. \<forall>x\<in>{r<..y'}. 0 < poly [:c, b, a:] x"
Max_le_Min_imp_singleton: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}"   and "Max A \<le> Min A" shows "A = {Min A}"
stream_all2_weaken: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a stream"   and ys :: "'b stream"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "stream_all2 P xs ys"   and "\<And>x y. P x y \<Longrightarrow> Q x y" shows "stream_all2 Q xs ys"
Restr_Field: fixes r :: "('d \<times> 'd) set" shows "Restr r (Field r) = r"
FG_reducible: assumes "of_nat (card G) \<noteq> (0::'d)" shows "\<exists>Us. (\<forall>U\<in>set Us. IrrFinGroupRepresentation G (*) U) \<and> 0 \<notin> set Us \<and> ActingGroup.group_ring = (\<Oplus>U\<leftarrow>Us. U)"
se_star_one: fixes c1 :: "('c, 'd) conf"   and l :: "('c, 'd) label"   and c2 :: "('c, 'd) conf" shows "se_star c1 [l] c2 = se c1 l c2"
simple_block_size_eq_card: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "simple_incidence_system point_set block_collection" shows "int (size block_collection) = int (card (incidence_system.design_support block_collection))"
subdag_eq_def: fixes t\<^sub>1 :: "dag"   and t\<^sub>2 :: "dag" shows "subdag_eq t\<^sub>1 t\<^sub>2 = (t\<^sub>1 = t\<^sub>2 \<or> subdag t\<^sub>1 t\<^sub>2)"
len_def: shows "len = Cn 4 r_length [recf.Id 4 3]"
take_hd_drop: fixes n :: "nat"   and xs :: "'d list" assumes "n < length xs" shows "take n xs @ [hd (drop n xs)] = take (Suc n) xs"
union_le_mono1: fixes B :: "'k multiset"   and D :: "'k multiset"   and C :: "'k multiset" assumes "B < D" shows "B + C < D + C"
predict_imp_adapt: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and X :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "disc_filtr_prob_space M F"   and "predict_stoch_proc F X N" shows "adapt_stoch_proc F X N"
CD_on_domain: fixes d :: "'d"   and ds :: "'d set"   and A :: "'x set" assumes "d \<in> ds" shows "CD_on ds A \<inter> Field (Pd d) = Cd d (A \<inter> Field (Pd d))"
reciprocal_0: fixes p :: "nat" shows "reciprocal_poly p 0 = 0"
silent_moves_equal_length_2: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s')" shows "length ms' = length s'"
strand_vars_split_1: fixes S :: "('b, 'a) strand_step list"   and S' :: "('b, 'a) strand_step list" shows "vars\<^sub>s\<^sub>t (S @ S') = vars\<^sub>s\<^sub>t S \<union> vars\<^sub>s\<^sub>t S'"
init_fin_descend_thr_None_conv: fixes ts :: "'f \<Rightarrow> (('c \<times> 'd) \<times> 'e \<Rightarrow>f nat) option"   and t :: "'f" assumes "ts t = None" shows "init_fin_descend_thr ts t = None"
C'leqC: shows "P \<turnstile> C' \<prec>\<^sup>1 C"
sdrop_neg: fixes i :: "Integer"   and xs :: "[:'a:]" assumes "le\<cdot>i\<cdot>0 = TT" shows "sdrop\<cdot>i\<cdot>xs = xs"
parts_insertI: fixes c :: "msg"   and G :: "msg set"   and a :: "msg" assumes "c \<in> parts G" shows "c \<in> parts (insert a G)"
id_Rel_components_2: fixes A :: "V" shows "id_Rel A\<lparr>ArrDom\<rparr> = A"
single_add: fixes k :: "'b"   and a :: "'c"   and b :: "'c" shows "monomial (a + b) k = monomial a k + monomial b k"
trace_eq'_parallel_resource: fixes A :: "'a set"   and res1 :: "('a, 'b) resource"   and res1' :: "('a, 'b) resource"   and B :: "'c set"   and res2 :: "('c, 'd) resource"   and res2' :: "('c, 'd) resource" assumes "A \<turnstile>\<^sub>R res1 \<approx> res1'"   and "B \<turnstile>\<^sub>R res2 \<approx> res2'" shows "(A <+> B) \<turnstile>\<^sub>R res1 \<parallel> res2 \<approx> res1' \<parallel> res2'"
eq_fold: fixes M :: "'b multiset" shows "\<Prod>\<^sub># M = fold_mset (*) (1::'b) M"
axioms_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b"   and \<eta> :: "'b \<Rightarrow> 'b"   and \<epsilon> :: "'a \<Rightarrow> 'a" assumes "equivalence_of_categories C D F G \<eta> \<epsilon>" shows "functor D C F"
MapOps_emptyD: fixes \<alpha> :: "'c \<Rightarrow> 'b"   and k :: "'c"   and d :: "'b set"   and ops :: "('d, 'c, 'e) MapOps" assumes "\<alpha> k \<in> d"   and "MapOps \<alpha> d ops" shows "lookup ops (MapOps.empty ops) k = None"
less_eq_scene_def: fixes X :: "'b scene"   and Y :: "'b scene" shows "X \<subseteq>\<^sub>S Y = (\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 on Y \<oplus>\<^sub>S s\<^sub>3 on X = s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 \<oplus>\<^sub>S s\<^sub>3 on X on Y)"
assms_2: shows "Units (R\<lparr>carrier := K\<rparr>) = K - {\<zero>}"
dgrad_p_set_varnum: fixes X :: "'x set" shows "punit.dgrad_p_set (varnum X) 0 = P[X]"
B: fixes i :: "'n"   and j :: "'n" shows "0 \<le> B $h i $h j \<and> B $h i $h j \<le> A $h i $h j"
rBangActE: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and S' :: "('a, 'b, 'c) psi" assumes "\<Psi>' \<rhd> !S \<longmapsto> \<alpha> \<prec> S'"   and "guarded S"   and "bn \<alpha> \<sharp>* S"   and "\<alpha> \<noteq> \<tau>"   and "bn \<alpha> \<sharp>* subject \<alpha>" shows "\<exists>T. \<Psi>' \<rhd> S \<longmapsto> \<alpha> \<prec> T \<and> (\<one>, S', T \<parallel> !S) \<in> Rel'"
Units_closed: fixes x :: "int \<times> int" assumes "x \<in> Units R" shows "x \<in> carrier R"
differing_lists_finite: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat" shows "finite (differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i)"
generalized_sfwSomeD: fixes fw :: "('a simple_match \<times> 'b) list"   and p :: "('a, 'c) simple_packet_scheme"   and r :: "'a simple_match"   and d :: "'b" assumes "generalized_sfw fw p = Some (r, d)" shows "(r, d) \<in> set fw \<and> simple_matches r p"
prv_fls: fixes \<phi> :: "'fmla" shows "bprv (imp fls \<phi>)"
chain_map_ident: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "chain_map p id c = c"
a2_map_keys: shows "Mapping.keys a2_map = {tp - len..tp}"
fresh_var_eqE: fixes e :: "exp"   and x :: "Vars.var" assumes "fresh_var e = x" shows "x \<notin> fv e"
paq: shows "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
tarjan_disc_def: fixes v :: "'v"   and s :: "'v tarjan_state" shows "tarjan_disc v s = RETURN \<lparr>sccs = sccs s, lowlink = lowlink s(v \<mapsto> \<delta> s v), tj_stack = v # tj_stack s\<rparr>"
v1: fixes u :: "'t" assumes "v1 \<prec>\<^sub>t u" shows "lookup a u = lookup b u"
between_last: fixes f :: "face"   and u :: "nat" assumes "distinct (vertices f)"   and "u \<in> \<V> f" shows "between (vertices f) u (last (verticesFrom f u)) = butlast (tl (verticesFrom f u))"
\<tau>move2_FAssD1: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and F :: "String.literal"   and D :: "String.literal"   and e' :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (e\<bullet>F{D} := e') pc xcp"   and "pc < length (compE2 e)" shows "\<tau>move2 P h stk e pc xcp"
simps_14: fixes ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and m1 :: "'a common_primitive match_expr"   and m2 :: "'a common_primitive match_expr" shows "iiface_constrain ipassmt (MatchAnd m1 m2) = MatchAnd (iiface_constrain ipassmt m1) (iiface_constrain ipassmt m2)"
fls_integral_sub: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_integral (f - g) = fls_integral f - fls_integral g"
cos_periodic: fixes x :: "real" shows "cos (x + 2 * pi) = cos x"
elem_set_implies_incl_in: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b"   and b :: "'b" assumes "set_category S \<AA>"   and "a \<in> set_category_data.set S (set_category.some_img S \<AA>) b" shows "set_category.incl_in S \<AA> a b"
f: fixes i :: "'c" assumes "i \<in> I" shows "set_integrable M (A i) f"
snoc_8: shows "set ys \<subseteq> set (ops @ [a])"
intro: fixes \<G> :: "'d cyclic_group" assumes "Cyclic_Group.cyclic_group \<G>" shows "ddh_ext \<G>"
integrable_diff: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "f integrable_on S"   and "g integrable_on S" shows "(\<lambda>x. f x - g x) integrable_on S"
MIN: shows "\<forall>v. v \<notin> S \<longrightarrow> D u \<le> D v"
power_float_interval_mono: fixes A :: "float interval"   and B :: "float interval"   and prec :: "nat"   and n :: "nat" assumes "set_of A \<subseteq> set_of B" shows "set_of (power_float_interval prec n A) \<subseteq> set_of (power_float_interval prec n B)"
inj: shows "inj_on (\<lambda>M. M + {#x#}) {M. set_mset M \<subseteq> insert x A \<and> size M = k}"