hcju/isabelleps huggingface dataset:
vsubset_vsingleton_rightI: fixes A :: "V"   and a :: "V" assumes "A = set {a} \<or> A = 0" shows "A \<subseteq>\<^sub>\<circ> set {a}"
abel_semigroup_axioms: shows "abel_semigroup dbm_add"
preserves_section_retraction: fixes e :: "'c"   and m :: "'c" assumes "C.ide (e \<cdot>\<^sub>C m)" shows "C.ide (C.L e \<cdot>\<^sub>C C.L m)"
LIM_offset_zero_cancel: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and L :: "'b" assumes "(\<lambda>h. f (a + h)) \<midarrow>0::'a\<rightarrow> L" shows "f \<midarrow>a\<rightarrow> L"
field_le_epsilon: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "\<And>e. less zero e \<Longrightarrow> less_eq x (plus y e)" shows "less_eq x y"
finite_dimensional_vector_space_with_def: fixes plus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S :: "'d"   and minus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set" shows "finite_dimensional_vector_space_with plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale basis \<equiv> vector_space_with plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale \<and> finite_dimensional_vector_space_with_axioms plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S scale basis"
integer_mod_group_1: fixes n :: "nat" shows "(1 \<in> carrier (Z n)) = (n \<noteq> 1)"
leK: fixes n :: "nat" assumes "n \<le> N + K" shows "B (Suc n) \<le> C * Max {B i |i. i \<le> n}"
simps: fixes \<Gamma> :: "(Vars.var \<times> 'b) list"   and S :: "stack_elem list" shows "heap_upds_ok (\<Gamma>, S) = (domA \<Gamma> \<inter> upds S = {} \<and> distinct (upds_list S))"
list_all_def: shows "list_all \<equiv> \<lambda>P x. Ball (set x) P"
ordertype_eq_ordertype_iso_ll: fixes A :: "nat list set"   and B :: "nat list set" assumes "Field (Restr (lenlex less_than) A) = A"   and "Field (Restr (lenlex less_than) B) = B" shows "(ordertype A (lenlex less_than) = ordertype B (lenlex less_than)) = (\<exists>f. iso_ll A B f)"
sblit_def: fixes a :: "'b array"   and s :: "nat"   and d :: "nat"   and l :: "nat" shows "sblit a s d l \<equiv> blit a s a d l"
r_alt_def: fixes i :: "nat" shows "r i = (\<lambda>q. state_rank q i, w i, \<lambda>q. state_rank q (Suc i))"
dbm_entry_val_add_1: fixes u :: "'c \<Rightarrow> 'd"   and c :: "'c"   and d :: "'c"   and a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "dbm_entry_val u (Some c) (Some d) a"   and "dbm_entry_val u (Some d) None b" shows "dbm_entry_val u (Some c) None (dbm_add a b)"
Longest_common_prefix_longest: fixes L :: "'c list set"   and ps :: "'c list" assumes "L \<noteq> {}"   and "\<forall>xs\<in>L. prefix ps xs" shows "length ps \<le> length (Longest_common_prefix L)"
pickFT_theFTOne: fixes cl :: "('d, 'e, 'f) cmd list"   and dl :: "('g, 'h, 'i) cmd list" assumes "WtFT cl = 1" shows "pickFT cl \<in> theFTOne cl dl"
simps_6: fixes t :: "'c local.term" shows "\<lbrace>\<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>]\<rbrace> = \<ll>'.map \<lbrace>t\<rbrace>"
lc_Fvar: fixes x :: "char list" shows "lc (Fvar x)"
distinct_vars_list: fixes p :: "linear_poly" shows "distinct (Abstract_Linear_Poly.vars_list p)"
prio_match_matcher_alt2: fixes flow_entries :: "('d, 'e) flow_entry_match list"   and \<gamma> :: "'d set \<Rightarrow> 'f \<Rightarrow> bool"   and packet :: "'f" shows "(let matching = {f \<in> set flow_entries. \<gamma> (ofe_fields f) packet} in {f \<in> matching. \<forall>fo\<in>matching. ofe_prio fo \<le> ofe_prio f}) = set (let matching = filter (\<lambda>f. \<gamma> (ofe_fields f) packet) flow_entries in filter (\<lambda>f. \<forall>fo\<in>set matching. ofe_prio fo \<le> ofe_prio f) matching)"
set_cases: fixes e :: "'f"   and a :: "'f option" assumes "e \<in> set_option a" obtains "a = Some e"
group_disjoint_sum_axioms: shows "group_disjoint_sum G A B"
map_spmf_conv_bind_spmf: fixes f :: "'s \<Rightarrow> 'r"   and p :: "'s spmf" shows "map_spmf f p = p \<bind> (\<lambda>x. return_spmf (f x))"
star_subset_lists: fixes A :: "'a list set"   and S :: "'a set" assumes "A \<subseteq> lists S" shows "star A \<subseteq> lists S"
degree1_coeffs: fixes p :: "'a poly" assumes "degree p = 1" shows "\<exists>a b. p = [:b, a:] \<and> a \<noteq> (0::'a)"
card_ord_of_nat: fixes m :: "nat" shows "card (elts (m\<^sub>\<nat>)) = m"
cmp_in_hom: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'b \<Rightarrow> 'c"   and \<Phi>\<^sub>F :: "'b \<times> 'b \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<mu>\<nu> :: "'b \<times> 'b" assumes "composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G"   and "partial_magma.arr (subcategory.comp (product_category.comp V\<^sub>B V\<^sub>B) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>B (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>B (snd \<mu>\<nu>) \<and> src\<^sub>B (fst \<mu>\<nu>) = trg\<^sub>B (snd \<mu>\<nu>))) \<mu>\<nu>" shows "partial_magma.in_hom V\<^sub>D (composite_pseudofunctor.cmp V\<^sub>B H\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<mu>\<nu>) (((\<lambda>\<mu>\<nu>. H\<^sub>D (fst \<mu>\<nu>) (snd \<mu>\<nu>)) \<circ> weak_arrow_of_homs.FF V\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D src\<^sub>D trg\<^sub>D (G \<circ> F)) (partial_magma.dom (subcategory.comp (product_category.comp V\<^sub>B V\<^sub>B) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>B (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>B (snd \<mu>\<nu>) \<and> src\<^sub>B (fst \<mu>\<nu>) = trg\<^sub>B (snd \<mu>\<nu>))) \<mu>\<nu>)) ((G \<circ> F \<circ> (\<lambda>\<mu>\<nu>. H\<^sub>B (fst \<mu>\<nu>) (snd \<mu>\<nu>))) (partial_magma.cod (subcategory.comp (product_category.comp V\<^sub>B V\<^sub>B) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>B (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>B (snd \<mu>\<nu>) \<and> src\<^sub>B (fst \<mu>\<nu>) = trg\<^sub>B (snd \<mu>\<nu>))) \<mu>\<nu>))"
guard_def: fixes P :: "heap \<Rightarrow> bool"   and f :: "heap \<Rightarrow> 'a \<times> heap" shows "guard P f = Heap (\<lambda>h. if P h then Some (f h) else None)"
lhd_ldropWhile: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "\<exists>x\<in>lset xs. \<not> P x" shows "\<not> P (lhd (ldropWhile P xs))"
T_F: fixes s :: "'b event list"   and P :: "'b process" assumes "s \<in> T P" shows "(s, {}) \<in> F P"
subgroupE_1: fixes G :: "('b, 'c) monoid_scheme"   and H :: "'b set" assumes "Group.group G"   and "subgroup H G" shows "H \<subseteq> carrier G"
cut_greater_singleton: fixes a :: "'a"   and t :: "'a" shows "{a} \<down>> t = (if t < a then {a} else {})"
n_intersect_number_def: fixes b1 :: "'b set"   and n :: "nat"   and b2 :: "'b set" shows "b1 |\<inter>|\<^sub>n b2 \<equiv> int (card {x \<in> Pow (b1 \<inter> b2). card x = n})"
assms_3: fixes p :: "'val"   and q :: "'val" assumes "r p q" shows "phiArg g p q"
strict_mono_less_eq: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "(f x \<le> f y) = less_eq x y"
Suc_diff: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n"   and "1 \<le> m" shows "Suc (n - m) = n - (m - 1)"
right_inv_add: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "c \<ominus> a \<ominus> b = c \<ominus> (a \<oplus> b)"
cSup_insert_If: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 X :: "'c set"   and a :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_above less_eq X" shows "Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
\<psi>\<^sub>2_is_state: shows "state 2 (Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[real (1 - f 0) / 2 - real (f 0) / 2, real (f 0) / 2 - real (1 - f 0) / 2, real (1 - f 1) / 2 - real (f 1) / 2, real (f 1) / 2 - real (1 - f 1) / 2]]))"
awp_Int_eq: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set" shows "awp F (A \<inter> B) = awp F A \<inter> awp F B"
lookup_update_self: fixes i :: "'a"   and m :: "'a \<Rightarrow>\<^sub>0 'b" shows "Poly_Mapping.update i (lookup m i) m = m"
simps: fixes a :: "lam" shows "val a = ((\<exists>x e. a = LAM [x].e) \<or> (\<exists>n. a = NUM n) \<or> a = FALSE \<or> a = TRUE)"
saturate_finite: fixes C :: "'a literal multiset"   and CC :: "'a literal multiset set" assumes "C \<in> saturate CC" shows "\<exists>DD\<subseteq>CC. finite DD \<and> C \<in> saturate DD"
add_to_nat_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "plus x y = mod_type.from_nat Abs (mod_type.to_nat Rep x + mod_type.to_nat Rep y)"
oreachable_other': fixes \<sigma> :: "'a"   and p :: "'b"   and A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<sigma>' :: "'a" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "U \<sigma> \<sigma>'" shows "(\<sigma>', p) \<in> oreachable A S U"
order_char_1: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<le> y \<squnion> - x)"
negligible_subset: fixes s :: "'b set"   and t :: "'b set" assumes "negligible s"   and "t \<subseteq> s" shows "negligible t"
inv_inv_f_eq: shows "inv (inv (inv (inv hom))) = inv (inv hom)"
predicate1I: fixes P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool" assumes "\<And>x. P x \<Longrightarrow> Q x" shows "P \<le> Q"
coeff_hom': fixes p :: "'a list"   and i :: "nat" assumes "set p \<subseteq> carrier R" shows "h (R.coeff p i) = S.coeff (map h p) i"
elims_1: fixes x :: "'a Point"   and xa :: "'a Cone"   and y :: "bool" assumes "onCone x xa = y" obtains p :: "'a Point" and cone :: "'a Cone" where "x = p" and "xa = cone" and "y = (space2 (vertex cone) p = slope cone * slope cone * time2 (vertex cone) p)"
less_QF_same: fixes i' :: "nat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i' < i"   and "j \<le> k" shows "QF k (j, i') \<lless> QF k (j, i)"
leg0_composite_2: shows "HH\<mu>\<nu>\<pi>.leg0 = \<pi>.leg0 \<cdot> \<mu>\<nu>\<pi>.Prj\<^sub>0"
HStartBallot_HInv4a_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HStartBallot s s' p"   and "HInv4a s p"   and "Inv2a_inner s' p" shows "HInv4a s' p"
applicableBackjumpCharacterization: fixes stateA :: "State" shows "applicableBackjump stateA = (\<exists>l level. getConflictFlag stateA = True \<and> isBackjumpLevel level l (getC stateA) (getM stateA))"
rAct: fixes \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* Q"   and "\<alpha> \<noteq> \<tau>"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C"   and "yvec \<sharp>* \<alpha>"   and "yvec \<sharp>* Q'"   and "yvec \<sharp>* \<Psi>'" shows "\<exists>P''. \<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'' \<and> (\<exists>P'. \<Psi> \<otimes> \<Psi>' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi> \<otimes> \<Psi>', P', Q') \<in> Rel)"
hs_dfs_def: fixes \<Sigma>i :: "('a, unit) HashMap.hashmap"   and post :: "'a \<Rightarrow> 'a dlist" shows "hs_dfs \<Sigma>i post \<equiv> fst (while hs_dfs_cond (hs_dfs_step post) (hs_dfs_initial \<Sigma>i))"
orthogonal_to_subspace_exists: fixes S :: "'a set" assumes "dim S < DIM('a)" obtains x :: "'a" where "x \<noteq> (0::'a)" and "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y"
map_apply_def: shows "Map_To_Mapping.map_apply = (\<lambda>m. m)"
exhaust_sel: fixes list :: "'e list" obtains "list = []"   | "list = hd list # tl list"
gen_subalgebra_sigma_sets: fixes G :: "'c set set"   and M :: "'c measure" assumes "G \<subseteq> sets M"   and "sigma_algebra (space M) G" shows "sets (gen_subalgebra M G) = G"
rm_set: fixes t :: "'b Tree"   and h :: "'b \<Rightarrow> int" shows "t \<noteq> Tip \<and> sortedTree h t \<longrightarrow> rm h t \<in> setOf t"
C1l: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> y \<le> y \<cdot> z" shows "x\<^sup>\<star> \<cdot> y \<le> y \<cdot> z\<^sup>\<star>"
Arr_in_Hom_2: fixes t :: "'a local.term" assumes "Arr t" shows "t \<in> VHom (Dom t) (Cod t)"
spanning_tree_imp_tree: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "spanning_tree H G" shows "tree H"
norm_real: fixes x :: "(real, 1) vec" shows "norm x = \<bar>x $ 1\<bar>"
id_nec: fixes v :: "i"   and \<alpha> :: "'a"   and \<beta> :: "'a" shows "[\<alpha> \<^bold>= \<beta> \<^bold>\<equiv> \<^bold>\<box>\<alpha> \<^bold>= \<beta> in v]"
pd_trans: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "IFC prob"   and "IFC_def.is_pd prob y x"   and "IFC_def.is_pd prob z y" shows "IFC_def.is_pd prob z x"
splitAt_dist_ram20: fixes vs :: "'a list"   and a :: "'a list"   and ram1 :: "'a"   and b :: "'a list"   and ram2 :: "'a"   and c :: "'a list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "c = snd (splitAt ram2 vs)"
path_connectedin_discrete_topology: fixes U :: "'b set"   and S :: "'b set" shows "path_connectedin (discrete_topology U) S = (S \<subseteq> U \<and> (\<exists>a. S \<subseteq> {a}))"
simps_2: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> order option"   and ky :: "'a"   and vy :: "'e"   and ys :: "('a \<times> 'e) list" shows "lex_ord_pair f [] ((ky, vy) # ys) = (let aux = f ky (0::'d) vy in if aux = Some Eq then lex_ord_pair f [] ys else aux)"
cnj_cnj: fixes q :: "octo" shows "Octonions.cnj (Octonions.cnj q) = q"
init_state_def: fixes src :: "nat" shows "init_state src \<equiv> return (False, [src \<mapsto> src], {src}, {}, 0)"
wt_compP2: fixes P :: "'a J1_prog" assumes "wf_J1_prog P" shows "wf_jvm_prog (compP2 P)"
join_sub: fixes L :: "nat set"   and R :: "nat set"   and n :: "nat"   and t1 :: "'a option list set"   and t2 :: "'a option list set"   and pos :: "bool" assumes "L \<subseteq> R"   and "table n L t1"   and "table n R t2" shows "join t2 pos t1 = {as \<in> t2. proj_tuple_in_join pos (join_mask n L) as t1}"
biQuots_rQuots_lQuots: fixes x :: "'b list"   and y :: "'b list"   and A :: "'b list set" shows "biQuots x y A = rQuots y (lQuots x A)"
star_ext_aux: fixes x :: "'a" shows "x \<le> x \<cdot> x\<^sup>\<star>"
coerce_stateT_abs: fixes x :: "'f \<rightarrow> ('d \<times> 'f)\<cdot>'e" shows "COERCE('d\<cdot>('e, 'f) stateT, 'a\<cdot>('b, 'c) stateT)\<cdot>(stateT_abs\<cdot>x) = stateT_abs\<cdot>(COERCE('f \<rightarrow> ('d \<times> 'f)\<cdot>'e, 'c \<rightarrow> ('a \<times> 'c)\<cdot>'b)\<cdot>x)"
spec_ops_add_any: fixes xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set"   and oid :: "'c"   and oper :: "'d" assumes "spec_ops (xs @ ys) deps"   and "\<forall>i\<in>set (map fst xs). i < oid"   and "\<forall>i\<in>set (map fst ys). oid < i"   and "\<forall>ref\<in>deps oper. ref < oid" shows "spec_ops (xs @ [(oid, oper)] @ ys) deps"
order_trans_rules_24: fixes x :: "'g"   and y :: "'g" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
exhaust: fixes y :: "'k list" obtains "y = []"   | x21 :: "'k" and x22 :: "'k list" where "y = x21 # x22"
distinct_1: fixes x1 :: "nat" shows "ONat x1 \<noteq> OFun"
connected_component_of_set: fixes X :: "'b topology"   and x :: "'b" shows "connected_component_of_set X x = {y. \<exists>T. connectedin X T \<and> x \<in> T \<and> y \<in> T}"
transrec3_Limit: fixes i :: "V"   and a :: "V"   and b :: "V \<Rightarrow> V \<Rightarrow> V"   and c :: "V \<Rightarrow> (V \<Rightarrow> V) \<Rightarrow> V" assumes "Limit i" shows "transrec3 a b c i = c i (restrict (transrec3 a b c) (elts i))"
Abs_BetaE: fixes x :: "name"   and s :: "Lambda_Z.term"   and t :: "Lambda_Z.term" assumes "Abs x s \<rightarrow>\<^sub>\<beta> t" obtains u :: "Lambda_Z.term" where "t = Abs x u" and "s \<rightarrow>\<^sub>\<beta> u"
chf_lm_3: fixes x :: "nat"   and A :: "nat set" shows "(x \<notin> A) = (chf A x = 1)"
ord_spmf_eqD_emeasure: fixes p :: "'c spmf"   and q :: "'c spmf"   and A :: "'c set" assumes "ord_spmf (=) p q" shows "emeasure (measure_spmf p) A \<le> emeasure (measure_spmf q) A"
rcs_def: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and a :: "'c" shows "H \<bullet>\<^bsub>G\<^esub> a = {b. \<exists>h\<in>H. h \<cdot>\<^bsub>G\<^esub> a = b}"
safe_reach_upto_le: fixes n :: "nat"   and safe :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option) \<Rightarrow> bool"   and c\<^sub>0 :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and m :: "nat" assumes "safe_reach_upto n safe c\<^sub>0"   and "m \<le> n" shows "safe_reach_upto m safe c\<^sub>0"
sel_4: fixes x21 :: "'a"   and x22 :: "('a, 'b) Term.term list" shows "args (Fun x21 x22) = x22"
substEnv_def: fixes xs :: "'varSort"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and x :: "'var" shows "substEnv xs X x \<equiv> psubstEnv (idEnv [x \<leftarrow> X]_xs)"
wordinterval_to_set_ipcidr_tuple_to_wordinterval_uncurry: fixes ipcidr :: "'a word \<times> nat" shows "wordinterval_to_set (ipcidr_tuple_to_wordinterval ipcidr) = Hs_Compat.uncurry ipset_from_cidr ipcidr"
tuple_pr: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and h :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "\<guillemotleft>h : local.dom h \<rightarrow> local.prod a b\<guillemotright>" shows "tuple (local.comp (pr1 a b) h) (local.comp (pr0 a b) h) = h"
simps_3: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \<union> (#) y ` shuffles (x # xs) ys"
fps_one_over_one_minus_fps_X_squared': assumes "inverse (1::'a) = (1::'a)" shows "inverse ((1 - fps_X)\<^sup>2) = Abs_fps (\<lambda>n. of_nat (n + 1))"
ctr_transfer_1: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_err R Err Err"
ev: shows "\<rho> \<turnstile> e \<Down> v"
DBPass_enabled: shows "\<turnstile> Enabled (<DBPass>_(inp, mid, out, q1, q2)) = (q1 \<noteq> #[])"
assms: shows "SN (S\<^sup>* O R O S\<^sup>*)"
natural_transformation_axioms: shows "natural_transformation J C \<chi>.A.map D' \<tau>o\<chi>.map"
subset_unI1: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<subseteq> B" shows "A \<subseteq> B \<union> C"
uedges_edge_ugraph: fixes es :: "nat set set" shows "uedges (edge_ugraph es) = es \<inter> S_edges"
ask_writer_def: fixes ask :: "('g \<Rightarrow> 'f) \<Rightarrow> 'f"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) writerT" shows "ask_writer ask f = WriterT (ask (\<lambda>r. run_writer (f r)))"
space_subprob_algebra: fixes A :: "'a measure" shows "space (subprob_algebra A) = {M. subprob_space M \<and> sets M = sets A}"
absolutely_integrable_absolutely_integrable_lbound: fixes f :: "'a \<Rightarrow> real"   and S :: "'a set"   and g :: "'a \<Rightarrow> real" assumes "f integrable_on S"   and "g absolutely_integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> g x \<le> f x" shows "f absolutely_integrable_on S"
INT_insert: fixes B :: "'c \<Rightarrow> 'b set"   and a :: "'c"   and A :: "'c set" shows "\<Inter> (B ` insert a A) = B a \<inter> \<Inter> (B ` A)"
not_iTILL_iMODb_subset: fixes n' :: "nat"   and m :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < n'"   and "m \<noteq> Suc 0" shows "\<not> [\<dots>n'] \<subseteq> [ r, mod m, c ]"
zcount_update_zmultiset: fixes M :: "'c zmultiset"   and t :: "'c"   and n :: "int"   and t' :: "'c" shows "zcount (update_zmultiset M t n) t' = zcount M t' + (if t = t' then n else 0)"
iMin_in_idem: fixes n :: "'a"   and I :: "'a set" assumes "n \<in> I" shows "min n (iMin I) = iMin I"
pred_subset_eq2: fixes R :: "('a \<times> 'b) set"   and S :: "('a \<times> 'b) set" shows "((\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S)) = (R \<subseteq> S)"
3: shows "gcs (pp_of_term v) t = (0::'a)"
hyp: fixes p :: "trm"   and q :: "trm"   and k :: "stack" assumes "\<And>p'. p \<mapsto> p' \<Longrightarrow> P p' q k"   and "\<And>q' k. q \<mapsto> q' \<Longrightarrow> P p q' k"   and "\<And>k'. |k'| < |k| \<Longrightarrow> P p q k'" shows "P p q k"
distinguishing_weak_formula_is_weak: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and P :: "'c"   and Q :: "'c" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition"   and "\<not> indexed_weak_nominal_ts.weakly_logically_equivalent TYPE('a) \<tau> satisfies transition P Q" shows "indexed_weak_nominal_ts.weak_formula \<tau> (indexed_weak_nominal_ts.distinguishing_weak_formula \<tau> satisfies transition P Q)"
moebius_translation_zero: shows "moebius_translation 0 = id_moebius"
simps_50: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Stop \<noteq> x41 ;; x42"
contlub_lambda: fixes S :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>x. chain (\<lambda>i. S i x)" shows "(\<lambda>x. \<Squnion>i. S i x) = (\<Squnion>i. S i)"
holomorphic_on_empty: fixes f :: "complex \<Rightarrow> complex" shows "f holomorphic_on {}"