hcju/isabelleps huggingface dataset:
code_blue_dfs_hash_correct: fixes G :: "'a b_graph_rec"   and Gi :: "('a \<Rightarrow> bool, 'a \<Rightarrow> 'a list, 'a list, 'a \<Rightarrow> bool) gen_bg_impl" assumes "b_graph G"   and "finite ((g_E G)\<^sup>* `` g_V0 G)"   and "(Gi, G) \<in> bg_impl_rel_ext unit_rel Id" shows "RETURN (code_blue_dfs_hash Gi) \<le> SPEC (\<lambda>r. case r of None \<Rightarrow> \<forall>prpl. \<not> b_graph.is_lasso_prpl G prpl | Some L \<Rightarrow> b_graph.is_lasso_prpl G L)"
root_list_def: fixes p :: "real Polynomial.poly" shows "root_list p \<equiv> {x. poly p x = 0}"
unitary_def: fixes A :: "'b mat" shows "unitary A = (A \<in> carrier_mat (dim_row A) (dim_row A) \<and> inverts_mat A (adjoint A))"
tan_total_pos: fixes y :: "real" assumes "0 \<le> y" shows "\<exists>x\<ge>0. x < pi / 2 \<and> tan x = y"
t_twice_even: shows "2 * (length t_twice div 2) = length t_twice"
find_consistent_signs_at_roots_copr: fixes p :: "real poly"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "\<And>q. q \<in> set qs \<Longrightarrow> algebraic_semidom_class.coprime p q" shows "set (find_consistent_signs_at_roots p qs) = set (characterize_consistent_signs_at_roots_copr p qs)"
Max_subset: fixes A :: "'a set"   and B :: "'a set" assumes "A \<noteq> {}"   and "A \<subseteq> B"   and "finite B" shows "Max A \<le> Max B"
a_not: shows "(\<forall>chain. a \<noteq> Goto chain) \<and> a \<noteq> Unknown"
tendsto_compose_filtermap: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'd"   and T :: "'c"   and F :: "'b filter" shows "((g \<circ> f) \<longlongrightarrow> T) F = (g \<longlongrightarrow> T) (filtermap f F)"
relpow_commute: fixes R :: "('a \<times> 'a) set"   and n :: "nat" shows "R O R ^^ n = R ^^ n O R"
assms: shows "\<forall>(j, p)\<in>set A \<union> B. \<forall>(k, q)\<in>set A \<union> B. p = q \<longrightarrow> j = k"
groupParts_same_fw_wi0: assumes "B \<in> set (groupWIs c rs)" shows "\<forall>w\<in>set (map wordinterval_to_set B). \<forall>a1\<in>w. \<forall>a2\<in>w. same_fw_behaviour_one a1 a2 c rs"
typeop_type_det: fixes op :: "call_kind"   and k :: "typeop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type"   and \<rho>\<^sub>1 :: "'a OCL_Types.type"   and \<rho>\<^sub>2 :: "'a OCL_Types.type" assumes "typeop_type op k \<tau> \<sigma> \<rho>\<^sub>1"   and "typeop_type op k \<tau> \<sigma> \<rho>\<^sub>2" shows "\<rho>\<^sub>1 = \<rho>\<^sub>2"
stalk_axioms: shows "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str {U. is_open U \<and> identity S x \<in> U} (identity S x)"
prems_4: shows "set (xs @ [x]) \<subseteq> set ops"
increase_i_1: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_invariant upw i fs"   and "i < m"   and "upw \<Longrightarrow> i = 0"   and "i \<noteq> 0 \<Longrightarrow> \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) (i - 1)\<parallel>\<^sup>2 \<le> \<alpha> * \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2" shows "LLL_invariant True (Suc i) fs"
sets_scylinder: fixes xs :: "'b set list"   and S :: "'b measure" assumes "\<forall>x\<in>set xs. x \<in> sets S" shows "scylinder (space S) xs \<in> sets (stream_space S)"
acyclic_2_2a: fixes x :: "'a" assumes "symmetric x" shows "acyclic_2 x = acyclic_2a x"
components_full_gb: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "component_of_term ` Keys (fst ` set (full_gb bs)) = component_of_term ` Keys (fst ` set bs)"
elims: fixes x :: "'b hp list"   and y :: "'b hp option" assumes "pass\<^sub>2 x = y" obtains "x = []" and "y = None"   | h :: "'b hp" and hs :: "'b hp list" where "x = h # hs" and "y = Some (case pass\<^sub>2 hs of None \<Rightarrow> h | Some x \<Rightarrow> link h x)"
witness_submatrix_def: fixes f :: "nat \<Rightarrow> real" shows "witness_submatrix f = submatrix (A' f) rows_with_1 rows_with_1"
le_funI: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. f x \<le> g x" shows "f \<le> g"
hext_objD: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a :: "nat"   and C :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set" assumes "h \<unlhd> h'"   and "h a = \<lfloor>(C, S)\<rfloor>" shows "\<exists>S'. h' a = \<lfloor>(C, S')\<rfloor>"
measure_finite_Union: fixes S :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "finite S"   and "A ` S \<subseteq> sets M"   and "disjoint_family_on A S"   and "\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>" shows "Sigma_Algebra.measure M (\<Union> (A ` S)) = (\<Sum>i\<in>S. Sigma_Algebra.measure M (A i))"
dim_eq_card_independent: fixes B :: "('c, 'd) vec set" assumes "vec.independent B" shows "vec.dim B = card B"
null_sets_distr_iff: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and A :: "'e set" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "(A \<in> null_sets (distr M N f)) = (f -` A \<inter> space M \<in> null_sets M \<and> A \<in> sets N)"
assms_2: shows "x \<equiv> real_of_1 plr"
prefix_sdrop_prefix_eq: fixes xs :: "[:'r:]"   and ys :: "[:'r:]" assumes "prefix\<cdot>xs\<cdot>ys = TT" shows "eq\<cdot>(sdrop\<cdot>(slength\<cdot>xs)\<cdot>ys)\<cdot>[::] = eq\<cdot>ys\<cdot>xs"
case_1: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c, 'b) tllist \<Rightarrow> 'a"   and x1 :: "'b" shows "(case TNil x1 of TNil x \<Rightarrow> f1 x | TCons x xa \<Rightarrow> f2 x xa) = f1 x1"
dom_image_ran: fixes f :: "'d \<Rightarrow> 'c option" shows "f ` dom f = Some ` ran f"
is_ground_subst_cls: fixes C :: "'a literal multiset"   and \<sigma> :: "'s" assumes "is_ground_cls C" shows "C \<cdot> \<sigma> = C"
induct: fixes P :: "'a prog"   and x1 :: "ty"   and x2 :: "ty"   and Pa :: "ty \<Rightarrow> ty \<Rightarrow> bool" assumes "P \<turnstile> x1 \<le> x2"   and "\<And>T. Pa T T"   and "\<And>C D. P \<turnstile> C \<preceq>\<^sup>* D \<Longrightarrow> Pa (Class C) (Class D)"   and "\<And>C. Pa NT (Class C)"   and "\<And>A. Pa NT (A\<lfloor>\<rceil>)"   and "\<And>A. Pa (A\<lfloor>\<rceil>) (Class Object)"   and "\<And>A B. \<lbrakk>P \<turnstile> A \<le> B; Pa A B\<rbrakk> \<Longrightarrow> Pa (A\<lfloor>\<rceil>) (B\<lfloor>\<rceil>)" shows "Pa x1 x2"
tiTable_ack_connection_ok: fixes req :: "nat \<Rightarrow> reqType list"   and x :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st_in :: "nat \<Rightarrow> GatewayStatus"   and b :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and st_out :: "nat \<Rightarrow> GatewayStatus"   and t :: "nat" assumes "tiTable_SampleT req x stop lose st_in b ack i1 vc st_out"   and "ack t = [connection_ok]"   and "msg (Suc 0) x"   and "ts lose"   and "msg (Suc 0) stop" shows "(st_in t = call \<or> st_in t = connection_ok \<and> req t \<noteq> [send]) \<and> lose t = [False]"
split: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and prod :: "'d \<times> 'e" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<forall>x1 x2. prod = (x1, x2) \<longrightarrow> P (f x1 x2))"
IH: fixes p1 :: "'c"   and p2 :: "'d" assumes "productF A B FAIL AB"   and "well_formed A"   and "well_formed B"   and "path A (ws || r1s) p1 \<and> path B (ws || r2s) p2 \<or> target (ws || r1s || r2s) (p1, p2) = FAIL \<and> 0 < length ws \<and> path A (butlast (ws || r1s)) p1 \<and> path B (butlast (ws || r2s)) p2 \<and> succ A (last ws) (target (butlast (ws || r1s)) p1) = {} \<and> succ B (last ws) (target (butlast (ws || r2s)) p2) \<noteq> {}"   and "p1 \<in> nodes A"   and "p2 \<in> nodes B" shows "path AB (ws || r1s || r2s) (p1, p2)"
linear_conv_closed: fixes x :: "'a" assumes "linear x" shows "linear (x\<^sup>T)"
strict_mono_setsD: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b set"   and x :: "'a"   and y :: "'a" assumes "strict_mono_sets A f"   and "x < y"   and "x \<in> A"   and "y \<in> A" shows "f x \<lless> f y"
maximal_ends_on_deadend: fixes G :: "('c, 'd) Graph_scheme"   and P :: "'c llist" assumes "Digraph G"   and "Digraph.maximal_path G P"   and "lfinite P"   and "\<not> lnull P" shows "\<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. llast P \<rightarrow>\<^bsub>G\<^esub> w)"
bool_split: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "\<tau> \<Turnstile> x \<triangleq> invalid \<or> \<tau> \<Turnstile> x \<triangleq> null \<or> \<tau> \<Turnstile> x \<triangleq> true \<or> \<tau> \<Turnstile> x \<triangleq> false"
unat_of_bl_128_16_less_helper: shows "unat (of_bl (to_bl b)) < 2 ^ 16"
const_0_sum_mod_2: assumes "const 0" shows "(f 0 + f 1) mod 2 = 0"
has_bochner_integral_integrable: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M f" shows "has_bochner_integral M f (integral\<^sup>L M f)"
col_col_per_per: fixes A :: "'p"   and X :: "'p"   and C :: "'p"   and U :: "'p"   and V :: "'p" assumes "A \<noteq> X"   and "C \<noteq> X"   and "Col U A X"   and "Col V C X"   and "Per A X C" shows "Per U X V"
6: shows "((f \<sqinter> - q)\<^sup>\<star> * e)\<^sup>+ = (f \<sqinter> - q)\<^sup>\<star> * e"
subreachableE_pair: fixes A :: "(('a \<Rightarrow> 'b) \<times> 'c, 'd) automaton"   and U :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and J :: "'a set"   and \<sigma> :: "'a \<Rightarrow> 'b"   and s :: "'c"   and I :: "'d \<Rightarrow> bool" assumes "subreachable A U J"   and "(\<sigma>, s) \<in> oreachable A (\<lambda>s s'. I) U" shows "\<exists>\<zeta>. (\<forall>j\<in>J. \<zeta> j = \<sigma> j) \<and> (\<zeta>, s) \<in> reachable A I"
bex_triv_one_point2: fixes A :: "'e set"   and a :: "'e" shows "(\<exists>x\<in>A. a = x) = (a \<in> A)"
Procs_defined: shows "set Procs \<subseteq> dom \<Gamma>"
MergeGuardsI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P merge_guards c Q,A"
right_cancel: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "x \<in> carrier (UP INTEG)"   and "y \<in> carrier (UP INTEG)"   and "z \<in> carrier (UP INTEG)" shows "(y \<oplus>\<^bsub>UP INTEG\<^esub> x = z \<oplus>\<^bsub>UP INTEG\<^esub> x) = (y = z)"
zero_right: fixes a :: "'b \<Rightarrow>\<^sub>L 'a" shows "blinfun_apply a (0::'b) = (0::'a)"
guarantees_Join_I1: fixes F :: "'b program"   and X :: "'b program set"   and Y :: "'b program set"   and G :: "'b program" assumes "F \<in> X guarantees Y"   and "F ok G" shows "F \<squnion> G \<in> X guarantees Y"
expand_body_mono: shows "trimono (\<lambda>expand (n, ns). if new n = {} then if \<exists>n'\<in>ns. old n' = old n \<and> next n' = next n then RETURN (name n, upd_incoming n ns) else expand (\<lparr>name = expand_new_name (name n), incoming = {name n}, new = next n, old = {}, next = {}\<rparr>, {n} \<union> ns) else SPEC (\<lambda>x. x \<in> new n) \<bind> (\<lambda>\<phi>. let n = n\<lparr>new := new n - {\<phi>}\<rparr> in if \<exists>q. \<phi> = prop\<^sub>r(q) \<or> \<phi> = nprop\<^sub>r(q) then if not\<^sub>r \<phi> \<in> old n then RETURN (name n, ns) else expand (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) else if \<phi> = true\<^sub>r then expand (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) else if \<phi> = false\<^sub>r then RETURN (name n, ns) else if \<exists>\<nu> \<mu>. \<phi> = \<nu> and\<^sub>r \<mu> \<or> \<phi> = X\<^sub>r \<nu> then expand (n\<lparr>new := new1 \<phi> \<union> new n, old := {\<phi>} \<union> old n, next := next1 \<phi> \<union> next n\<rparr>, ns) else expand (n\<lparr>new := new1 \<phi> \<union> new n, old := {\<phi>} \<union> old n, next := next1 \<phi> \<union> next n\<rparr>, ns) \<bind> (\<lambda>(nm, nds). expand (n\<lparr>name := nm, new := new2 \<phi> \<union> new n, old := {\<phi>} \<union> old n\<rparr>, nds))))"
shiftr_and_eq_shiftl: fixes w :: "'a word"   and n :: "nat"   and x :: "'a word"   and y :: "'a word" assumes "(w >> n) && x = y" shows "w && (x << n) = y << n"
b': shows "\<forall>r'\<in>R. leftPrincipal r' (F \<nabla> [x].A) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')"
i_le_n: shows "mod_type_class.to_nat i \<le> n"
upd_const_same: fixes a :: "'a"   and b :: "'b" shows "upd a b (cnst b) = cnst b"
derived_points_order: shows "derived_incidence_sys.\<v> = \<k>"
comp_apply: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(f \<circ> g) x = f (g x)"
r_sub: fixes a :: "nat"   and b :: "nat" shows "eval r_sub [a, b] \<down>= a - b"
assms: shows "know P (kKS m) \<or> know Q (kKS m)"
minus_one_in_totatives: fixes n :: "nat" assumes "2 \<le> n" shows "n - 1 \<in> totatives n"
scf_alts: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "lowered.is_pref_profile R" shows "lowered R \<subseteq> alts'"
boundsu_def: fixes s :: "('d, 'c) state" shows "\<B>\<^sub>u s = map_option snd \<circ> Mapping.lookup (\<B>\<^sub>i\<^sub>u s)"
quotients_butlast: assumes "1 < length \<GG>" shows "butlast comp\<GG>.quotients = normal_series.quotients (G\<lparr>carrier := \<GG> ! (length \<GG> - 1 - 1)\<rparr>) (take (length \<GG> - 1) \<GG>)"
inv_is_inverse: fixes f :: "'b" assumes "D.iso f" shows "D.inverse_arrows f (D.inv f)"
assms_2: shows "\<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) n"
p_3: fixes K :: "'a set" assumes "K \<in> p" shows "K \<noteq> {}"
h1: shows "|~ \<langle>N \<and> B\<rangle>_f \<longrightarrow> \<langle>M\<rangle>_g"
minimal_unsat_core_tabl_atoms_def: fixes I :: "'a set"   and t :: "(nat \<times> linear_poly) list"   and as :: "('a \<times> 'b atom) set" shows "minimal_unsat_core_tabl_atoms I t as = (I \<subseteq> fst ` as \<and> (\<nexists>v. v \<Turnstile>\<^sub>t t \<and> (I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as) \<and> (distinct_indices_atoms as \<longrightarrow> (\<forall>J\<subset>I. \<exists>v. v \<Turnstile>\<^sub>t t \<and> (J, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>e\<^sub>s as)))"
assms_2: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_cauchy s" shows "is_Zp_cauchy (f \<circ> s)"
cod_mkcan: fixes a :: "'c local.term set"   and b :: "'c local.term set" assumes "ide a"   and "ide b"   and "\<^bold>\<lfloor>DOM a\<^bold>\<rfloor> = \<^bold>\<lfloor>COD b\<^bold>\<rfloor>" shows "cod (mkcan a b) = b"
convert_obs_initial_empty_TA: shows "convert_obs_initial \<lbrace>\<rbrace> = \<lbrace>\<rbrace>"
trss_c_cases_s: fixes s :: "'f list"   and c :: "'f list multiset"   and e :: "('g, 'h) label"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "((s, c), e, s', c') \<in> trss fg" obtains "c' = c"   | p :: "'g" and u :: "'f" and v :: "'f" where "e = LSpawn p" and "(u, Spawn p, v) \<in> edges fg" and "hd s = u" and "hd s' = v" and "c' = {#[entry fg p]#} + c"
simps_1: fixes f :: "nat \<Rightarrow> nat" shows "map_ticket f INIT = INIT"
inv_in_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "S.iso f"   and "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>" shows "\<guillemotleft>S.inv f : b \<Rightarrow>\<^sub>S a\<guillemotright>"
independent_card_le_dim: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and Basis :: "'e set"   and B :: "'e set"   and V :: "'e set" assumes "finite_dimensional_vector_space scale Basis"   and "B \<subseteq> V"   and "\<not> module.dependent scale B" shows "card B \<le> vector_space.dim scale V"
scaleC_one: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC 1 x = x"
connected_refl: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a" assumes "Graph G"   and "v \<in> V\<^bsub>G\<^esub>" shows "Graph.connected G v v"
abs_eq: shows "ii\<^sub>h = abs_complex_homo ii\<^sub>h\<^sub>c"
minus_divide_divide: 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 a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (uminus a) (uminus b) = divide a b"
intvs_ie_incdec_1: fixes l :: "int"   and h :: "int" assumes "l \<le> h" shows "{l..<h + 1} = Set.insert h {l..<h}"
steps_Stuck: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com" shows "\<Gamma>\<turnstile> (c, Stuck) \<rightarrow>\<^sup>* (Skip, Stuck)"
hyps_1: shows "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s')"
create_graph_old_finite: fixes \<phi> :: "'b ltlr" shows "create_graph \<phi> \<le> SPEC (\<lambda>nds. \<forall>nd\<in>nds. finite (old nd))"
signed_take_bit_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = or (take_bit n a) (times (zero_neq_one.of_bool one zero (bit a n)) (not (mask n)))"
subs: fixes C :: "'a set"   and T :: "'a set" assumes "compactin X C"   and "path_connectedin X C"   and "T \<in> \<U>"   and "\<not> disjnt C T" shows "C \<subseteq> T"
obj_trg: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "C.obj (trg\<^sub>C \<mu>)"
simps_1: fixes i :: "nat"   and s :: "dB"   and k :: "nat" shows "Var i[s/k] = (if k < i then Var (i - 1) else if i = k then s else Var i)"
lstrict_prefix_lappend_conv: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lstrict_prefix xs (lappend xs ys) = (lfinite xs \<and> \<not> lnull ys)"
c2: assumes "ip \<in> kD rt"   and "sqn rt ip < \<pi>\<^sub>2 r" shows "P (rt(ip \<mapsto> r))"
is_process_REP_STOP: shows "is_process ({(s, X). s = []}, {})"
has_disc_alist_and'_append: fixes disc' :: "'b \<Rightarrow> bool"   and ls1 :: "'b negation_type list"   and ls2 :: "'b negation_type list" shows "has_disc disc' (alist_and' (ls1 @ ls2)) = (has_disc disc' (alist_and' ls1) \<or> has_disc disc' (alist_and' ls2))"
Prf_Stars: fixes vs :: "'b val list"   and r :: "'b rexp" assumes "\<forall>v\<in>set vs. \<turnstile> v : r" shows "\<turnstile> Stars vs : Star r"
subst_pp_def: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'d \<Rightarrow>\<^sub>0 nat" shows "subst_pp f t = (\<Prod>x\<in>keys t. f x ^ lookup t x)"
same_proc_call_unique_target: fixes a :: "'edge"   and a' :: "'edge"   and Q\<^sub>1 :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r\<^sub>1 :: "'ret"   and p :: "'pname"   and fs\<^sub>1 :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and Q\<^sub>2 :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r\<^sub>2 :: "'ret"   and fs\<^sub>2 :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "valid_edge a"   and "valid_edge a'"   and "kind a = Q\<^sub>1:r\<^sub>1\<hookrightarrow>\<^bsub>p\<^esub>fs\<^sub>1"   and "kind a' = Q\<^sub>2:r\<^sub>2\<hookrightarrow>\<^bsub>p\<^esub>fs\<^sub>2" shows "targetnode a = targetnode a'"
red_single_mult_const: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and f :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a"   and c :: "'c" assumes "punit.red_single p q f t"   and "c \<noteq> (0::'c)" shows "punit.red_single p q (local.punit.monom_mult c (0::'a) f) t"
nodes_add_edge: fixes v :: "'c"   and e :: "'d"   and v' :: "'c"   and g :: "('c, 'd, 'e) graph_scheme" shows "nodes (add_edge v e v' g) = insert v (insert v' (nodes g))"
units_of_mult: fixes G :: "('e, 'f) monoid_scheme" shows "(\<otimes>\<^bsub>units_of G\<^esub>) = (\<otimes>\<^bsub>G\<^esub>)"
compat_append_right: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and xs :: "'a list"   and zs :: "'a list" assumes "ext_compat_snoc ext"   and "ext gt ys xs" shows "ext gt (ys @ zs) (xs @ zs)"
pr0_simps_2: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "local.dom (hfsetcat.pr0 a b) = hfsetcat.prod a b"
neg_0_equal_iff_equal: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.group_add minus plus zero uminus" shows "(zero = uminus a) = (zero = a)"
simps_2: fixes u :: "dbtm"   and i :: "name"   and name :: "name" shows "subst_dbtm u i (DBVar name) = (if i = name then u else DBVar name)"
y: fixes i :: "'a" assumes "i \<in> Basis" shows "bound_intersect_2d_ud p (inner2_aform Z n i) g = Some (y i)"
D_Mprefix: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" shows "D (Mprefix A P) = {d. d \<noteq> [] \<and> hd d \<in> ev ` A \<and> (\<exists>a. ev a = hd d \<and> tl d \<in> D (P a))}"
trail_app_isoI: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "trail u p v"   and "digraph_isomorphism hom" shows "pre_digraph.trail (app_iso hom G) (iso_verts hom u) (map (iso_arcs hom) p) (iso_verts hom v)"
strict_coboundedI2: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c < b" shows "c < max a b"
flow_in_ivp_sols_ex_ivl: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" assumes "picard_lindeloef f T S t\<^sub>0"   and "s \<in> S" shows "ll_on_open.flow T f S t\<^sub>0 s \<in> Sols f (ll_on_open.existence_ivl T f S t\<^sub>0) S t\<^sub>0 s"
in_diff_count: fixes a :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" shows "(a \<in># M - N) = (count N a < count M a)"
AbstrLevels_A8_A82: fixes i :: "AbstrLevelsID" assumes "sA8 \<in> AbstrLevel i" shows "sA82 \<notin> AbstrLevel i"
un_LCons1_def: fixes llist :: "'c llist" shows "un_LCons1 llist = (case llist of LCons x21 x22 \<Rightarrow> x21)"