hcju/isabelleps huggingface dataset:
del_list_split: fixes ts :: "('a btree \<times> 'a) list"   and x :: "'a"   and ls :: "('a btree \<times> 'a) list"   and rs :: "('a btree \<times> 'a) list"   and t :: "'a btree" assumes "split ts x = (ls, rs)"   and "sorted_less (inorder (Node ts t))" shows "del_list x (inorder (Node ts t)) = inorder_list ls @ del_list x (inorder_list rs @ inorder t)"
card_of_Times_singleton: fixes A :: "'c set"   and x :: "'d" shows "|A \<times> {x}| =o |A|"
refl_ge_eq: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. R x x" shows "(=) \<le> R"
swap: fixes g :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and B :: "'d set"   and A :: "'c set" shows "(\<Sum>i\<in>A. sum (g i) B) = (\<Sum>j\<in>B. \<Sum>i\<in>A. g i j)"
simps: fixes bs :: "atom set"   and x :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'a" shows "(bs, x) \<approx>res R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y)"
assms_2: fixes x :: "real" assumes "x \<in> S" shows "f x = g x"
h2_2: shows "r2 = h2 \<otimes> g"
simps_6: fixes p :: "nat"   and i :: "nat"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Var i) vs = (if i < length vs then Some (vs ! i) else None)"
seq_char: fixes f :: "(bool, unit) concrete_category.arr"   and g :: "(bool, unit) concrete_category.arr" shows "partial_magma.arr (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ())) (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ()) f g) = (partial_magma.arr (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ())) f \<and> f = g)"
2: shows "interp_ins op_list = yxs @ y # yys @ [z] @ yzs"
elem: fixes x :: "'b"   and xs :: "'b list" shows "ListMem x (x # xs)"
real_root_strict_decreasing: fixes n :: "nat"   and N :: "nat"   and x :: "real" assumes "0 < n"   and "n < N"   and "1 < x" shows "root N x < root n x"
ring_record_simps_12: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and add :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'c" shows "(\<oplus>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub>) = add"
ccBindsExtra_def: fixes ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "CoCallAnalysis.ccBindsExtra ccExp \<Gamma> = (\<Lambda> i. snd i \<squnion> CoCallAnalysis.ccBinds ccExp \<Gamma>\<cdot>i \<squnion> (\<Squnion>x\<mapsto>e\<in>map_of \<Gamma>. fv e G\<times> ccNeighbors x (snd i)))"
assms_5: shows "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"
complex_inner_i_left: fixes x :: "complex" shows "inner \<i> x = Im x"
\<G>_semantics': fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<G> \<phi> w = subformulas\<^sub>\<nu> \<phi> \<inter> {\<psi>. w \<Turnstile>\<^sub>n G\<^sub>n \<psi>}"
hom_char: fixes a :: "'a"   and b :: "'a" shows "R.hom a b = hom a b \<inter> Collect (right a)"
path2_snoc: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and m' :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "m \<in> set (predecessors g m')" shows "g \<turnstile> n-ns @ [m']\<rightarrow>m'"
assms: shows "\<tau> \<Turnstile> \<delta> X"
Bex_set: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "Bex (set xs) P = list_ex P xs"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "web_flow \<Gamma> f" obtains "current \<Gamma> f" and "\<And>x. \<lbrakk>x \<in> \<^bold>V\<^bsub>\<Gamma>\<^esub>; x \<notin> A \<Gamma>; x \<notin> B \<Gamma>\<rbrakk> \<Longrightarrow> KIR f x"
plugInLinear: fixes var :: "nat"   and L :: "real list"   and B :: "real"   and b :: "real mpoly"   and c :: "real mpoly"   and C :: "real" assumes "var < length L"   and "B \<noteq> 0"   and "\<forall>v. insertion (nth_default 0 (L[var := v])) b = B"   and "\<forall>v. insertion (nth_default 0 (L[var := v])) c = C" shows "aEval (Eq (b * Var var + c)) (L[var := - C / B])"
atS_simps_4: fixes s :: "('h, 'g, 'i, 'j, 'k) system_state"   and p :: "'i"   and ls :: "'g set" assumes "\<forall>l. at p l s \<longrightarrow> l \<notin> ls" shows "\<not> atS p ls s"
inf_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "inf (inf x y) z = inf x (inf y z)"
poly_pfd_simple: fixes x :: "'a poly"   and cs :: "('a \<times> nat) list" shows "partial_fraction_decomposition x (map (\<lambda>(c, n). ([:1::'a, - c:], n)) cs) = (fst (poly_pfd_simple x cs), map (map (\<lambda>c. [:c:])) (snd (poly_pfd_simple x cs)))"
index_mat_addrow_mat_3: fixes n :: "nat"   and a :: "'c"   and k :: "nat"   and l :: "nat" shows "dim_col (addrow_mat n a k l) = n"
intro: fixes bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and h :: "'e \<Rightarrow> 'f" assumes "bo \<le> vimage2p h h (=)" shows "blinding_respects_hashes h bo"
Star_eq_map_rexp_iff_2: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e rexp"   and r :: "'d rexp" shows "(map_rexp f x = Star r) = (\<exists>r'. x = Star r' \<and> map_rexp f r' = r)"
r_zero: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<oplus> \<zero> = x"
nesting_lemma: fixes P :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and mgt_call :: "pname \<Rightarrow> 'b"   and mgt :: "com \<Rightarrow> 'b"   and wt :: "com \<Rightarrow> bool"   and U :: "pname set"   and uG :: "'b set"   and G :: "'b set"   and n :: "nat"   and c :: "com" assumes "\<And>G ts. ts \<subseteq> G \<Longrightarrow> P G ts"   and "\<And>G pn. P (insert (mgt_call pn) G) {mgt (the (body pn))} \<Longrightarrow> P G {mgt_call pn}"   and "\<And>G c. \<lbrakk>wt c; \<And>pn. pn \<in> U \<Longrightarrow> P G {mgt_call pn}\<rbrakk> \<Longrightarrow> P G {mgt c}"   and "\<And>pn. pn \<in> U \<Longrightarrow> wt (the (body pn))"   and "finite U"   and "uG = mgt_call ` U"   and "G \<subseteq> uG"   and "n \<le> card uG"   and "card G = card uG - n"   and "wt c" shows "P G {mgt c}"
count_union: fixes M :: "'k multiset"   and N :: "'k multiset"   and a :: "'k" shows "count (M + N) a = count M a + count N a"
2: shows "d_repr \<in> getOneIp ` set (build_ip_partition c rs)"
field_lbound_gt_zero: fixes d1 :: "'a"   and d2 :: "'a" assumes "(0::'a) < d1"   and "(0::'a) < d2" shows "\<exists>e>0::'a. e < d1 \<and> e < d2"
min: fixes y :: "'b" assumes "(tz, y) \<in> R" shows "y \<notin> range t"
domo_afilter: fixes a :: "aexp"   and X :: "char list set"   and S :: "'av Abs_State.st option"   and i :: "'av" assumes "vars a \<subseteq> X"   and "domo S \<subseteq> X" shows "domo (afilter a i S) \<subseteq> X"
real_alg_2': fixes q :: "int poly"   and l :: "rat"   and r :: "rat"   and ri :: "root_info" assumes "\<exists>!x. root_cond (q, l, r) x"   and "poly_cond q"   and "root_info_cond ri q" shows "invariant_2 (real_alg_2' ri q l r) \<and> real_of_2 (real_alg_2' ri q l r) = real_of_1 (q, l, r)"
horizontal_composition_axioms: shows "horizontal_composition (\<cdot>\<^sub>D) (\<star>\<^sub>D) src\<^sub>D trg\<^sub>D"
nz: shows "\<forall>i\<in>A. m i \<noteq> 0"
cSUP_upper: fixes x :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset" assumes "x \<in> A"   and "subset_mset.bdd_above (f ` A)" shows "f x \<subseteq># Sup (f ` A)"
cBallE: fixes A :: "'d cset"   and P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "cBall A P" obtains "P x"   | "\<not> cin x A"
distinct_3: fixes x11 :: "name"   and x12 :: "value list"   and x31 :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and x32 :: "name"   and x33 :: "(name, value) fmap" shows "Vconstr x11 x12 \<noteq> Vrecabs x31 x32 x33"
AtSuc: fixes x0 :: "'a"   and At :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and Follows :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and n :: "nat" assumes "Chain x0 At Follows"   and "At x n" shows "\<exists>x'. At x' (Suc n) \<and> Follows x' x"
progress_le_gen: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<phi> :: "Formula.formula" assumes "pred_mapping (\<lambda>x. x \<le> 0) P" shows "Monitor.progress \<sigma> P \<phi> 0 \<le> 0"
cands_def: shows "cands = filter (\<lambda>j. j \<notin> snd ` set (pivot_positions A)) [0..<nc]"
pathin_const: fixes X :: "'a topology"   and a :: "'a" shows "pathin X (\<lambda>x. a) = (a \<in> topspace X)"
eq_set_tree_empty: fixes t :: "'a tree" shows "(set_tree t = {}) = (t = \<langle>\<rangle>)"
first_def: fixes ps :: "ref list" shows "first ps = (case ps of [] \<Rightarrow> Null | p # rs \<Rightarrow> p)"
assms_2: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) list"   and t :: "'thread_id"   and ln :: "'l \<Rightarrow>f nat" shows "ta_hb_consistent P E (llist_of (map (Pair t) (convert_RA ln)))"
Image_Collect_case_prod: fixes P :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'d set" shows "{(x, y). P x y} `` A = {y. \<exists>x\<in>A. P x y}"
Ball_Collect: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "Ball A P = (A \<subseteq> Collect P)"
cyclic_group_def: fixes G :: "('c, 'd) monoid_scheme"   and gen :: "'c" shows "Finite_And_Cyclic_Groups.cyclic_group G gen \<equiv> Group.group G \<and> cyclic_group_axioms G gen"
erf_conv_erfc: fixes z :: "'a" shows "erf z = (1::'a) - erfc z"
F'_def: shows "F' = (\<Union>h\<in>hom_components p. fst (q0 h))"
is_LTLGlobal_def: fixes ltl :: "'b ltl" shows "is_LTLGlobal ltl = (\<exists>x8. ltl = G x8)"
simps: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "methodDef list"   and a3 :: "nat" shows "a1 \<turnstile>+ a2 OK IN a3 = ((\<exists>CT C. a1 = CT \<and> a2 = [] \<and> a3 = C) \<or> (\<exists>CT m C ms. a1 = CT \<and> a2 = m # ms \<and> a3 = C \<and> CT \<turnstile> m OK IN C \<and> CT \<turnstile>+ ms OK IN C))"
run_stake: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and V :: "(nat \<times> 'd) set"   and r :: "(nat \<times> 'd) stream"   and p :: "nat \<times> 'd"   and k :: "nat" assumes "grun A w V r p" shows "gpath A w V (stake k r) p"
surjective: fixes r :: "('e, 'f) monoid_scheme" shows "r = \<lparr>carrier = carrier r, mult = (\<otimes>\<^bsub>r\<^esub>), one = \<one>\<^bsub>r\<^esub>, \<dots> = monoid.more r\<rparr>"
Max_equality: fixes k :: "'b"   and A :: "'b set" assumes "k \<in> A"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<le> k" shows "Max A = k"
Zleast_def: fixes A :: "int set" shows "Zleast A = (THE a. a \<in> A \<and> (\<forall>x\<in>A. a \<le> x))"
prod_in_hom: fixes f :: "'a"   and a :: "'a"   and c :: "'a"   and g :: "'a"   and b :: "'a"   and d :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> c\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> d\<guillemotright>" shows "\<guillemotleft>local.prod f g : local.prod a b \<rightarrow> local.prod c d\<guillemotright>"
run_limit_not_empty: fixes \<phi> :: "'a ltl" shows "limit (run\<^sub>t (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) w) \<noteq> {}"
abs_ge_bnd: shows "bnd < \<bar>M * v $ i\<bar>"
fv_Var: fixes x :: "Vars.var" shows "fv (Var x) = {x}"
Pi_pmf_geometric_filter: fixes A :: "'b set"   and p :: "real"   and dflt :: "bool" assumes "finite A"   and "p \<in> {0<..1}" shows "Pi_pmf A 0 (\<lambda>_. geometric_pmf p) = Pi_pmf A dflt (\<lambda>_. bernoulli_pmf p) \<bind> (\<lambda>fb. Pi_pmf {x \<in> A. \<not> fb x} 0 (\<lambda>_. map_pmf Suc (geometric_pmf p)))"
ord_of_nonzero_2: fixes p :: "int"   and x :: "nat \<Rightarrow> int" assumes "padic_integers p"   and "x \<in> carrier (padic_int p)"   and "0 \<le> padic_integers.ord_Zp p x" shows "x \<in> nonzero (padic_int p)"
pmf_eq_0_set_pmf: fixes M :: "'c pmf"   and p :: "'c" shows "(pmf M p = 0) = (p \<notin> set_pmf M)"
fundamental_theorem_of_calculus_strong: fixes S :: "real set"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a" assumes "finite S"   and "a \<le> b"   and "\<And>x. x \<in> {a..b} - S \<Longrightarrow> (f has_vector_derivative f' x) (at x)"   and "continuous_on {a..b} f" shows "(f' has_integral f b - f a) {a..b}"
rewrite_negated_dst_ports: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'b common_primitive match_expr"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m" shows "matches (\<beta>, \<alpha>) (rewrite_negated_dst_ports m) a p = matches (\<beta>, \<alpha>) m a p"
DF_in_N: fixes j :: "nat"   and i :: "nat" assumes "j \<le> k" shows "enum (DF k i) j \<in> N"
has_ticket_def: fixes s :: "thread"   and k :: "nat" shows "has_ticket s k \<equiv> s = WAIT k \<or> s = HOLD k \<or> s = REL k"
assms: shows "\<forall>x\<in>S. \<forall>y\<in>S. x = y \<or> x \<notin> T \<or> y \<notin> T"
that: fixes c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c, s) \<rightarrow>*c (c', s')"   and "s' \<approx> t'"   and "c' \<approx>T d'" shows "thesis"
exp_add: fixes x :: "'a"   and y :: "'a" shows "exp (x + y) = exp x * exp y"
assms_2: shows "\<forall>\<^sub>F x in at_top. g x < f x"
poly_add_is_polynomial: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list" assumes "subring K R"   and "set p1 \<subseteq> K"   and "set p2 \<subseteq> K" shows "polynomial K (R.poly_add p1 p2)"
assms_3: shows "C \<in> carrier_mat nr2 nc1"
okClocks_sym: fixes p :: "nat"   and q :: "nat"   and i :: "nat" assumes "okClocks p q i" shows "okClocks q p i"
fix_is_fix_of_limit: fixes f :: "'a set \<Rightarrow> 'a set"   and I :: "'a set" assumes "f I = I" shows "limit f I = I"
interior_complement: fixes S :: "'b set" shows "interior (- S) = - closure S"
dim_hyp2: shows "matr_option (dim_row (M_mat signs subsets)) (mat_inverse (M_mat signs subsets)) \<in> carrier_mat (length signs) (length signs)"
simps_1: fixes "is" :: "'b set"   and t :: "'b" shows "interrupt_action_ok is (Interrupt t) = True"
map_snd_annotate_rlen: fixes l :: "'b list" shows "map snd (annotate_rlen l) = l"
subgroup_is_group: assumes "Group.group \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "Group.group (\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<lparr>carrier := Collect is_K2_isometry\<rparr>)"
f1: shows "Deliver (i, Append i e1) \<in> set (history j)"
has_subprob_density_imp_subprob_space: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "is_subprob_density M f" shows "subprob_space (density M f)"
direct_decomp_repeated_eq_zero: fixes A :: "'c set"   and ss :: "'c set list"   and X :: "'c set" assumes "direct_decomp A ss"   and "1 < count_list ss X"   and "\<And>s. s \<in> set ss \<Longrightarrow> (0::'c) \<in> s" shows "X = {0::'c}"
unowned_shared: shows "- (\<Union>(uu_, uu_, uu_, uu_, uu_, \<O>, uu_)\<in>set ts. \<O>) \<subseteq> dom \<S>"
A32_DAcc_level1: shows "DAcc level1 sA32 = {sA41}"
fMax_finsert: fixes x :: "'b"   and A :: "'b fset" shows "fMax (finsert x A) = (if A = {||} then x else max x (fMax A))"
last_conv_nth_default: fixes xs :: "'a list"   and dflt :: "'a" assumes "xs \<noteq> []" shows "last xs = nth_default dflt xs (length xs - 1)"
simps_1: fixes e :: "'b"   and a :: "'a" shows "FingerTreeStruc.gmn (Tip e a) = a"
split: fixes P :: "'g \<Rightarrow> bool"   and f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'h list \<Rightarrow> 'g"   and list :: "'h list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = ((list = [] \<longrightarrow> P f1) \<and> (\<forall>x21 x22. list = x21 # x22 \<longrightarrow> P (f2 x21 x22)))"
countable_classI: fixes f :: "'a \<Rightarrow> nat" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "OFCLASS('a, countable_class)"
E_validD_1: fixes G :: "('a, 'b) graph"   and v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "valid_graph G"   and "(v, e, v') \<in> edges G" shows "v \<in> nodes G"
rep_eq: shows "fis_singleton = (\<lambda>A. is_singleton (fset A))"
red_in_Hom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a monoidal_language.term" assumes "monoidal_language C"   and "monoidal_language.Ide C a" shows "monoidal_language.red C a \<in> {t. monoidal_language.Arr C t \<and> monoidal_language.Dom C t = a \<and> monoidal_language.Cod C t = monoidal_language.Diagonalize C a}"
hconf_new: fixes P :: "'a prog"   and h :: "heap"   and a :: "nat"   and obj :: "obj" assumes "P \<turnstile> h \<surd>"   and "h a = None"   and "P,h \<turnstile> obj \<surd>" shows "P \<turnstile> h(a \<mapsto> obj) \<surd>"
ax_valids2_eq: fixes G :: "prog"   and A :: "'b triple set"   and ts :: "'b triple set" assumes "wf_prog G" shows "G,A|\<Turnstile>\<Colon>ts = G,A|\<Turnstile>ts"
assms: fixes n :: "nat" shows "f n \<in> {0..k}"
intro: fixes 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 set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and create_character_data :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) character_data_ptr) prog"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" assumes "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_val set_val_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_character_data known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr" shows "l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_val set_val_locs create_character_data known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
star_sim2: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z ; x \<le> y ; z" shows "z ; x\<^sup>\<star> \<le> y\<^sup>\<star> ; z"
trace_preserves_length_ts: fixes c :: "nat \<Rightarrow> ('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 n :: "nat"   and k :: "nat"   and l :: "nat"   and x :: "nat" assumes "trace c n k"   and "l \<le> k"   and "x \<le> k" shows "length (fst (c (n + l))) = length (fst (c (n + x)))"
iso_is_arr: fixes f :: "'a" assumes "C.iso f" shows "C.arr f"
abelian_monoid_axioms: shows "abelian_monoid Zp"
assms: shows "set_offending_flows \<lparr>nodes = nodesG, edges = edgesG\<rparr> nP \<noteq> {}"
SystemVCut_def: shows "SystemVCut = (Geq (f0 fid1) (Const 0) \<rightarrow> [[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) (TT && Geq (f0 fid1) (Const 0))]]Geq (f0 fid1) (Const 0))"
croots2_def: fixes p :: "complex poly" shows "croots2 p = (let a = coeff p 2; b = coeff p 1; c = coeff p 0; b2a = b / (2 * a); bac = b2a\<^sup>2 - c / a; e = csqrt bac in remdups [- b2a + e, - b2a - e])"
update_convs_3: fixes p_src' :: "'a word \<Rightarrow> 'a word"   and p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'b" shows "p_src_update p_src' \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr> = \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src' p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr>"
starS_def: fixes S :: "'a rexp set" shows "starS S = {Star (verund xs) |xs. set xs \<subseteq> S}"
cring_class_ops_def: shows "cring_class_ops \<equiv> \<lparr>carrier = UNIV, mult = (*), one = 1::'b, zero = 0::'b, add = (+)\<rparr>"
siftDown_multiset: fixes t :: "'c Tree" shows "Heap.multiset (siftDown t) = Heap.multiset t"
graph_homomorphism_empty: fixes G :: "('e, 'g) labeled_graph"   and f :: "('f \<times> 'g) set" shows "graph_homomorphism (LG {} {}) G f = (f = {} \<and> graph G)"
simps: fixes R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set" shows "saturation R S = {C. \<exists>N. \<forall>M\<ge>N. C \<in> inferred_clause_sets R S M}"
lift_streams_Cons: fixes x :: "'d stream"   and xs :: "'d stream list" shows "lift_streams (x # xs) = smap2 (#) x (lift_streams xs)"
aux_lemma_lose_3: fixes d :: "nat"   and k :: "nat"   and lose :: "nat \<Rightarrow> bool list"   and t :: "nat" assumes "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]" shows "\<forall>ka\<le>Suc d. lose (d + (t + (3 + k)) + ka) = [False]"