hcju/isabelleps huggingface dataset:
maintained_standard_noconstants: fixes C :: "'b set"   and L :: "'b Standard_Constant set"   and G' :: "('b Standard_Constant, 'c) labeled_graph" assumes "maintainedA (standard_rules C L) G'"   and "graph G'"   and "fst ` edges G' \<subseteq> L"   and "getRel S_Bot G' = {}" shows "\<exists>f g G. G = map_graph_fn G (f \<circ> g) \<and> G = map_graph_fn G' f \<and> subgraph (map_graph_fn G g) G' \<and> standard' C G \<and> (\<forall>r. maintained r G' \<longrightarrow> maintained r G) \<and> (\<forall>x y e. x \<in> vertices G' \<longrightarrow> y \<in> vertices G' \<longrightarrow> (g (f x), g (f y)) \<in> :map_graph_fn G g:\<lbrakk>e\<rbrakk> \<longrightarrow> (x, y) \<in> :G':\<lbrakk>e\<rbrakk>)"
fimage_funion: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e fset"   and B :: "'e fset" shows "f |`| (A |\<union>| B) = f |`| A |\<union>| f |`| B"
ct_suffix_def: fixes xs :: "ct list"   and ys :: "ct list" shows "ct_suffix xs ys = (\<exists>as bs. ys = as @ bs \<and> ct_list_eq bs xs)"
inner_node_Exit_edge: fixes n :: "'b" assumes "inner_node n" obtains a :: "'a" where "valid_edge a" and "intra_kind (kind a)" and "inner_node (sourcenode a)" and "targetnode a = Exit"
rep_eq: fixes x :: "'e"   and xa :: "'e['g, 'f]" shows "fromQ (x *\<^sub>Q xa) = \<lparr>mag = x \<cdot> mag (fromQ xa), dim = QD('g), unit_sys = unit\<rparr>"
sorted_list_of_set_insert_cons: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "{a} \<lless> A" shows "list_of (insert a A) = a # list_of A"
PDataSpace_def: shows "PDataSpace = snd \<circ> Rep_pdata"
log2_abs_int_add_less_half_sgn_eq: fixes b :: "real"   and ai :: "int" assumes "\<bar>b\<bar> \<le> 1 / 2"   and "ai \<noteq> 0" shows "\<lfloor>log 2 \<bar>real_of_int ai + b\<bar>\<rfloor> = \<lfloor>log 2 \<bar>real_of_int ai + sgn b / 2\<bar>\<rfloor>"
mult_1s_2: fixes a :: "'a" shows "a * Numeral1 = a"
simps_1: fixes uu :: "'a infty" shows "uu + Infty = Infty"
inf_import_p: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "inf x (uminus (inf x y)) = inf x (uminus y)"
restZero: fixes j :: "'sv" assumes "j \<in> UNIV"   and "j \<noteq> i" shows "f j = 0"
routing_def: shows "MAC.routing \<equiv> (1\<^sub>C |\<^sub>= lassocr\<^sub>C) \<odot> swap_lassocr \<odot> (1\<^sub>C |\<^sub>= (1\<^sub>C |\<^sub>= swap_lassocr) \<odot> swap_lassocr) \<odot> rassocl\<^sub>C"
UN_le_add_shift_strict: fixes M :: "nat \<Rightarrow> 'b set"   and k :: "nat"   and n :: "nat" shows "(\<Union>i<n. M (i + k)) = \<Union> (M ` {k..<n + k})"
seq_ns: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and t :: "state"   and rs\<^sub>2 :: "'c rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow>\<^sub>s t"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow>\<^sub>s t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ rs\<^sub>2, Undecided\<rangle> \<Rightarrow>\<^sub>s t'"
base_pair: fixes x :: "state \<Rightarrow> 'a"   and y :: "state \<Rightarrow> 'b" assumes "basevars (x, y)" shows "basevars x \<and> basevars y"
V_axiom_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "V_axiom_on A f = (\<forall>B\<subseteq>A. \<forall>y\<in>B. (\<forall>x\<in>B. (x, y) \<in> rwp_on A f) \<longrightarrow> y \<in> f B)"
tame_def: fixes g :: "graph" shows "tame g \<equiv> tame9a g \<and> tame10 g \<and> tame11a g \<and> tame11b g \<and> tame12o g \<and> tame13a g"
Beta_App_cong_aux_2: fixes t1 :: "lam"   and t2 :: "lam"   and s :: "lam" assumes " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2" shows " App s t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* App s t2"
numeral_uexpr_rep_eq: fixes x :: "num"   and b :: "'b" shows "\<lbrakk>numeral x\<rbrakk>\<^sub>e b = numeral x"
B_2: shows "\<forall>x. norm (f x) \<le> B * norm x"
heap_is_wellformed_def: fixes 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 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_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs" shows "heap_is_wellformed = (\<lambda>h. heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h \<and> acyclic (parent_child_rel h \<union> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_host_shadow_root_rel get_shadow_root h \<union> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_ptr_disconnected_node_rel get_disconnected_nodes h) \<and> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_all_ptrs_in_heap get_shadow_root h \<and> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_distinct_lists get_shadow_root h \<and> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_shadow_root_valid get_shadow_root get_tag_name h)"
powr_add_minus_powr_asymptotics: fixes a :: "complex" shows "(\<lambda>z. ((1 + z) powr a - 1) / z) \<midarrow>0\<rightarrow> a"
s_def: shows "s = (\<lambda>x. Some (Suc (e_hd x)))"
ode_d_fa_def: fixes n :: "nat"   and xs :: "floatarith list"   and ds :: "floatarith list" shows "ode_d_fa n xs ds = map (subst_floatarith ((!) (xs @ ds @ ds))) (ode_d_expr (length xs) n)"
column_transpose: fixes i :: "'d"   and A :: "(('b, 'c) vec, 'd) vec" shows "column i (Finite_Cartesian_Product.transpose A) = row i A"
n_add_idem: fixes x :: "'a" shows "!x \<oplus> !x = !x"
tendsto_cblinfun_of_matrix: fixes b :: "'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> complex"   and a :: "'a \<Rightarrow> 'b \<Rightarrow> complex"   and F :: "'c filter" assumes "\<And>i j. \<lbrakk>i \<in> CBasis; j \<in> CBasis\<rbrakk> \<Longrightarrow> ((\<lambda>n. b n i j) \<longlongrightarrow> a i j) F" shows "((\<lambda>n. cblinfun_of_matrix (b n)) \<longlongrightarrow> cblinfun_of_matrix a) F"
converse_trancl_induct: fixes a :: "'k"   and b :: "'k"   and r :: "('k \<times> 'k) set"   and P :: "'k \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "\<And>y. (y, b) \<in> r \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(y, z) \<in> r; (z, b) \<in> r\<^sup>+; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
rel_spmf_weightD: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf" assumes "rel_spmf A p q" shows "weight_spmf p = weight_spmf q"
pmf_not_neg: fixes p :: "'b pmf"   and x :: "'b" shows "\<not> pmf p x < 0"
set_spmf_sample_coprime: fixes N :: "nat" shows "set_spmf (sample_coprime N) = {x. coprime x N \<and> 1 < x \<and> x < N}"
fall2_def: fixes F :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream \<Rightarrow> (('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream \<Rightarrow> (('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list \<Rightarrow> bool" shows "fall2 F \<equiv> fall (\<lambda>\<pi>. fall (F \<pi>))"
oinvariant_step_anyact: fixes p :: "('a \<times> 'b, 'c) automaton"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool" assumes "p \<Turnstile>\<^sub>A (act TT, U \<rightarrow>) P" shows "p \<Turnstile>\<^sub>A (S, U \<rightarrow>) P"
help'''_notin: fixes ram :: "'c"   and b :: "'c list"   and xs :: "'c list" assumes "ram \<notin> set b" shows "fst (splitAtRec ram xs b) = xs @ fst (splitAtRec ram [] b)"
B: shows "(xs @ y # ys @ [y'], Y) \<in> failures P"
nested_prop_atoms\<^sub>\<nu>_card: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "card (nested_prop_atoms\<^sub>\<nu> \<phi> X) \<le> card (nested_prop_atoms \<phi>)"
assms_2: assumes "irreducible p"   and "0 < lead_coeff p"   and "root_free p"   and "square_free p"   and "p \<noteq> 0" shows "thesis"
split_paired_all: fixes P :: "'h \<times> 'i \<Rightarrow> prop" shows "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))"
ModelIkPolMcalc2C_axioms: shows "ModelIkPolMcalc2C wtFsym wtPsym arOf resOf parOf \<Phi> infTp pol grdOf intT intF intP"
ProcModifyReturn_sound: 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 init :: "'e \<Rightarrow> 'e"   and p :: "'d"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and ModifAbr :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P call init p return' c Q,A"   and "\<forall>\<sigma>. \<Gamma>,\<Theta>\<Turnstile>\<^bsub>/UNIV\<^esub> {\<sigma>} Call p (Modif \<sigma>),(ModifAbr \<sigma>)"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P call init p return c Q,A"
inf_notin_disc_o_x_axis: shows "\<infinity>\<^sub>h \<notin> disc o_x_axis"
word_le_0_iff: fixes x :: "'b word" shows "(x \<le> 0) = (x = 0)"
leq_refl: fixes x :: "'e nres" shows "x \<le> x"
awalk_last_in_verts: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "H.awalk u p v" shows "v \<in> verts H"
Union_image_empty: fixes A :: "'a set"   and f :: "'b \<Rightarrow> 'a set" shows "A \<union> \<Union> (f ` {}) = A"
Enat: fixes n :: "nat"   and \<rho> :: "(nat \<times> val) list" shows "E (ENat n) \<rho> = {v. v = VNat n}"
has_ell2_norm_infsetsum: fixes x :: "'c \<Rightarrow> complex" shows "has_ell2_norm x = ((\<lambda>i. (cmod (x i))\<^sup>2) abs_summable_on UNIV)"
sumZero: fixes P :: "pi" shows "P \<oplus> \<zero> \<approx> P"
I: fixes x :: "'a"   and S :: "'b" assumes "\<not> memb x S"   and "isNode x"   and "invariant S"   and "I S" shows "I (ins x S)"
fw_int_aux_c: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and a :: "nat"   and b :: "nat"   and c :: "nat"   and i :: "nat"   and j :: "nat" assumes "dbm_int M n"   and "a \<le> n"   and "b \<le> n"   and "c \<le> n"   and "i \<le> n"   and "j \<le> n"   and "fw M n 0 0 c i j \<noteq> \<infinity>" shows "get_const (fw M n 0 0 c i j) \<in> \<int>"
simps_3: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "psums (x # y # xs) = x # psums ((x + y) # xs)"
suminf_split_initial_segment: fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "summable f" shows "suminf f = (\<Sum>n. f (n + k)) + sum f {..<k}"
valid_path_imp_reverse: fixes g :: "real \<Rightarrow> 'b" assumes "valid_path g" shows "valid_path (reversepath g)"
aux0_6: fixes xs :: "('a, 'b) Combinators list"   and x :: "('a, 'b) Combinators" assumes "singleCombinators xs"   and "\<nexists>a b. x = a \<oplus> b" shows "singleCombinators (x # xs)"
ceiling_le_zero: fixes x :: "'a" shows "(\<lceil>x\<rceil> \<le> 0) = (x \<le> (0::'a))"
fmlookup_filter: fixes P :: "'f \<Rightarrow> bool"   and m :: "'f \<rightharpoonup>\<^sub>f 'e"   and x :: "'f" shows "fmlookup (fmfilter P m) x = (if P x then fmlookup m x else None)"
le_of_int_ceiling: fixes x :: "'c" shows "x \<le> of_int \<lceil>x\<rceil>"
dist_triangle_side_middle: fixes G :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "geodesic_segment_between G x y" shows "dist z (geodesic_segment_param G x (Gromov_product_at x z y)) \<le> Gromov_product_at z x y + 2 * deltaG TYPE('a)"
ge_less_neq_conv: fixes a :: "'a"   and n :: "'a" shows "(a \<le> n) = (\<forall>x<a. n \<noteq> x)"
permute_set_eq_image: fixes p :: "perm"   and X :: "'e set" shows "p \<bullet> X = permute p ` X"
factorization_def: fixes P :: "'a eclause"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" shows "factorization P C \<sigma> k C' = (\<exists>L1 L2 L' t s u v Cl_P Cl_C trms_C. eligible_literal L1 P \<sigma> \<and> L1 \<in> cl_ecl P \<and> L2 \<in> cl_ecl P - {L1} \<and> Cl_C = cl_ecl C \<and> Cl_P = cl_ecl P \<and> orient_lit_inst L1 t s pos \<sigma> \<and> orient_lit_inst L2 u v pos \<sigma> \<and> t \<lhd> \<sigma> \<noteq> s \<lhd> \<sigma> \<and> t \<lhd> \<sigma> \<noteq> v \<lhd> \<sigma> \<and> ck_unifier t u \<sigma> k \<and> L' = Neg (Eq s v) \<and> (C = Ecl Cl_C trms_C \<and> trms_C = get_trms Cl_C (dom_trms Cl_C (subst_set (trms_ecl P \<union> proper_subterms_of t) \<sigma>)) k) \<and> Cl_C = subst_cl C' \<sigma> \<and> C' = Cl_P - {L2} \<union> {L'})"
disjoint_image_disjoint_family_on: fixes A :: "'c \<Rightarrow> 'b set"   and I :: "'c set" assumes "disjoint (A ` I)"   and "inj_on A I" shows "disjoint_family_on A I"
const_on1: fixes x :: "Vars.var" shows "const_on (FBinds \<Delta>\<cdot>(Aheap \<Delta> e\<cdot>a)) (carrier ((FBinds \<Gamma>\<cdot>ae) x)) TTree.empty"
B_def: shows "B = isolate_variable_sparse p2 var 1"
set_cases: fixes e :: "'x"   and a :: "'x list" assumes "e \<in> set a" obtains z2 :: "'x list" where "a = e # z2__"   | z1 :: "'x" and z2 :: "'x list" where "a = z1 # z2" and "e \<in> set z2"
sgn_poly_def: fixes x :: "'a poly" shows "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
h1: shows "\<turnstile>\<^sub>p {adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1)} hadamard_n n {Q2}"
useful_tautologies_4: fixes v :: "i"   and \<psi> :: "\<o>"   and \<phi> :: "\<o>" shows "[\<^bold>\<not>\<psi> \<^bold>\<rightarrow> \<^bold>\<not>\<phi> \<^bold>\<rightarrow> (\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
star_inductr_var: fixes y :: "'a"   and x :: "'a" assumes "y ; x \<le> y" shows "y ; x\<^sup>\<star> \<le> y"
prems_1: shows "strict_mono_sets {..<Suc k} A"
simps_2: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and n :: "nat" shows "DBM_upd M f (Suc i) 0 n = DBM_update (DBM_upd M f i n n) (Suc i) 0 (f (Suc i) 0)"
asIMOD_igSwapAllIPresIGWlsAll: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSwapAllIPresIGWlsAll (asIMOD SEM)"
principal_domain_def: fixes R :: "('c, 'd) ring_scheme" shows "principal_domain R \<equiv> domain R \<and> principal_domain_axioms R"
dDia_def: fixes A :: "'b hrel"   and \<Phi> :: "'b upred" shows "\<^bold><A\<^bold>>\<Phi> = (\<not> \<^bold>[A\<^bold>](\<not> \<Phi>))"
parabolic_isometry_def: fixes f :: "'b \<Rightarrow> 'b" shows "parabolic_isometry f = (isometry f \<and> \<not> elliptic_isometry f \<and> stable_translation_length f = 0)"
that: shows "c \<subseteq>\<^sub>g Cond b c1 c2"
degree_map_poly_hom: fixes p :: "'c poly poly" shows "degree (map_poly (pCons (0::'c)) p) = degree p"
shiftr_eq_drop_bit: 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 a :: "'c"   and n :: "nat" assumes "class.semiring_bit_syntax plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_bit_syntax.shiftr drop_bit a n = drop_bit n a"
step_step': fixes S :: "'av st option"   and C :: "'av st option acom" shows "step (local.\<gamma>\<^sub>o S) (local.\<gamma>\<^sub>c C) \<le> local.\<gamma>\<^sub>c (local.step' S C)"
unfold: shows "rep2 = (\<Lambda> pat usvs. Node\<cdot>usvs\<cdot>(left2\<cdot>pat\<cdot>usvs)\<cdot>(right2\<cdot>pat\<cdot>usvs))"
dickson_leD1: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "dickson_le d m s t" shows "d s \<le> m"
degree_sum_le: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c poly"   and n :: "nat" assumes "finite S"   and "\<And>p. p \<in> S \<Longrightarrow> degree (f p) \<le> n" shows "degree (sum f S) \<le> n"
sb_def: fixes ys :: "'b"   and y1 :: "'a"   and y2 :: "'a"   and xs :: "'b"   and x :: "'a" shows "x @xs[y1 / y2]_ys \<equiv> if ys = xs then if x = y2 then y1 else x else x"
is_confluent_def: shows "is_confluent \<equiv> is_confluent_on UNIV"
has_max_lideal: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "local_ring R addition multiplication zero unit" shows "\<exists>\<ww>. max_lideal \<ww> R addition multiplication zero unit"
map\<^sub>1_simps_5: fixes f :: "'a" assumes "C.ide f" shows "D.cod (\<tau>\<^sub>1 f) = \<tau>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f"
cobounded2: fixes a :: "'a upred"   and b :: "'a upred" shows "b \<sqsubseteq> (a \<and> b)"
sum_g: shows "(\<Sum>x\<in>UNIV. g x *s cart_basis_Rm $ x) = f (cart_basis_Rn $ j)"
inject_1: fixes x1 :: "rat"   and y1 :: "rat" shows "(Rat_Info x1 = Rat_Info y1) = (x1 = y1)"
errorT_abs_def: shows "errorT_abs \<equiv> PRJ('e\<cdot>('g, 'f) errorT) oo EMB('e\<cdot>'f error\<cdot>'g)"
path_join_imp: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "path g1"   and "path g2"   and "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2)"
hn_ctxt_list: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a list"   and y :: "'b list"   and z :: "assn" assumes "list_assn A x y = z" shows "hn_ctxt (list_assn A) x y = z"
absorb_iff1: fixes b :: "'a"   and a :: "'a" shows "(b \<sqsubseteq> a) = (ord.max (\<sqsubseteq>) a b = a)"
is_empty_as_list: fixes e :: "'c Tree" assumes "hs_is_empty e" shows "Heap.multiset e = {#}"
convex_closure_rel_interior: fixes S :: "'b set" assumes "convex S" shows "closure (rel_interior S) = closure S"
simps_3: fixes c :: "int"   and e :: "Cooper.num" shows "iszlfm (Eq (CN 0 c e)) = (0 < c \<and> numbound0 e)"
onll_def: fixes \<Gamma> :: "'h \<Rightarrow> ('f, 'g, 'h, 'i) seqp"   and P :: "('e \<times> 'i) \<times> 'j \<times> 'e \<times> 'i \<Rightarrow> bool" shows "onll \<Gamma> P \<equiv> \<lambda>((\<xi>, p), a, \<xi>', p'). \<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')"
lexord_irreflexive: fixes r :: "('g \<times> 'g) set"   and xs :: "'g list" assumes "\<forall>x. (x, x) \<notin> r" shows "(xs, xs) \<notin> lexord r"
unl_con: fixes c :: "'c"   and w :: "'d" shows "(#c) w \<equiv> c"
closed_put_unis: fixes m :: "nat"   and k :: "nat"   and p :: "('a, 'b) form" shows "closed m (put_unis k p) = closed (m + k) p"
divide_p_integer_def: fixes p :: "integer"   and x :: "integer"   and y :: "integer" shows "divide_p_integer p x y = mult_p_integer p x (inverse_p_integer p y)"
nonempty_J: shows "{j \<in> {Max {i. postrecording_event t i \<and> 0 < card {j \<in> {i + 1..<length t}. prerecording_event t j}} + 1..<length t}. prerecording_event t j} \<noteq> {}"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "C.arr (src\<^sub>C f)"
middle_rel_a: fixes a :: "'a Bonk_Schramm_extension_unfolded"   and b :: "'a Bonk_Schramm_extension_unfolded" shows "(map_aux a, map_aux (middle a b)) \<in> wo_aux - Id"
semiring_norm_167: fixes v :: "num"   and w :: "num"   and y :: "'a" shows "- numeral v + (numeral w + y) = neg_numeral_class.sub w v + y"
ln_fact_bounds_2: fixes n :: "nat" assumes "0 < n" shows "ln (fact n) \<le> ln (2 * pi * real n) / 2 + real n * ln (real n) - real n + 1 / (12 * real n)"
length_0_conv: fixes xs :: "['b]" shows "(length\<cdot>xs = MkI\<cdot>0) = (xs = [])"
strong_consensus_cluster_def: fixes I :: "'node set" shows "strong_consensus_cluster I \<equiv> I \<subseteq> W \<and> (\<forall>p\<in>I. \<exists>Q\<subseteq>I. quorum_of p Q) \<and> (\<forall>Q Q'. quorum_of_set I Q \<and> quorum_of_set I Q' \<longrightarrow> I \<inter> Q \<inter> Q' \<noteq> {})"
subst_restrict_closed: fixes t :: "'a"   and S :: "name fset"   and env :: "(name, 'a) fmap" assumes "closed_except t S" shows "subst t (fmrestrict_fset S env) = subst t env"
simps: fixes prec :: "nat"   and x :: "float" shows "lb_sqrt prec x = (if 0 < x then float_divl prec x (sqrt_iteration prec prec x) else if x < 0 then - ub_sqrt prec (- x) else 0)"
necessary_condition_two: shows "\<v> * (\<Lambda> * (\<v> - 1) div (\<k> - 1)) = \<b> * \<k>"
set_def: fixes a :: "'s" shows "S.set a = S.some_img ` S.hom S.unity a"
jw_Lam: fixes x :: "Syntax.var"   and \<Gamma> :: "(Syntax.var, ty) fmap"   and \<tau>\<^sub>1 :: "ty"   and e :: "Syntax.term"   and \<tau>\<^sub>2 :: "ty" assumes "atom x \<sharp> \<Gamma>"   and "\<Gamma>(x $$:= \<tau>\<^sub>1) \<turnstile>\<^sub>W e : \<tau>\<^sub>2" shows "\<Gamma> \<turnstile>\<^sub>W Lam x e : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"
F_strict_imp_F_inj: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> local.F_strict" shows "f \<in> local.F_inj"
nat_mirror_def: fixes n :: "nat"   and l :: "nat"   and r :: "nat" shows "nat_mirror n l r \<equiv> l + r - n"
rank_Some_time: fixes x :: "nat"   and n :: "nat"   and j :: "nat" assumes "\<MM>.rank x n = Some j" shows "x \<le> n"
varSort_lt_var_INNER: shows "|UNIV| <o |UNIV|"
simplex_insert_dimplus1: fixes n :: "int"   and S :: "'b set"   and a :: "'b" assumes "n simplex S"   and "a \<notin> affine hull S" shows "n + 1 simplex convex hull insert a S"
Krein_Milman_Minkowski_aux: fixes S :: "'b set"   and n :: "nat" assumes "dim S = n"   and "compact S"   and "convex S"   and "(0::'b) \<in> S" shows "(0::'b) \<in> convex hull {x. x extreme_point_of S}"
ub: fixes n :: "nat" assumes "0 < n" shows "k n < real (n + 1)"
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_all_front_tickfree_traces1: fixes A :: "'a set" shows "T (CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) = {s. front_tickFree s \<and> set s \<subseteq> ev ` A \<union> {tick}}"