hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'a literal multiset" assumes "ord_resolve a1 a2 a3 a4 a5" obtains CAs :: "'a literal multiset list" and n :: "nat" and Cs :: "'a literal multiset list" and AAs :: "'a multiset list" and As :: "'a list" and D :: "'a literal multiset" where "a1 = CAs" and "a2 = D + negs (mset As)" and "a3 = AAs" and "a4 = As" and "a5 = \<Sum>\<^sub># (mset Cs) + D" and "length CAs = n" and "length Cs = n" and "length AAs = n" and "length As = n" and "n \<noteq> 0" and "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)" and "\<forall>i<n. AAs ! i \<noteq> {#}" and "\<forall>i<n. \<forall>A\<in>#AAs ! i. A = As ! i" and "eligible As (D + negs (mset As))" and "\<forall>i<n. strictly_maximal_wrt (As ! i) (Cs ! i)" and "\<forall>i<n. S (CAs ! i) = {#}"
distributed_def: fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> ennreal" shows "distributed M N X f = (distr M N X = density N f \<and> f \<in> borel_measurable N \<and> X \<in> M \<rightarrow>\<^sub>M N)"
bind_put: fixes s :: "'s"   and m :: "'m"   and f :: "'a \<Rightarrow> 'm" shows "bind (put s m) f = put s (bind m f)"
R\<^sub>C_of_def: fixes \<R>\<^sub>A :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) set"   and \<R> :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set"   and P :: "((((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set" shows "R\<^sub>C_of \<R>\<^sub>A \<R> P \<equiv> {(x, y). \<exists>x\<^sub>A y\<^sub>A. (x\<^sub>A, x) \<in> \<R> \<and> (y\<^sub>A, y) \<in> \<R> \<and> (x\<^sub>A, y\<^sub>A) \<in> \<R>\<^sub>A \<and> snd (fst x) = snd (fst y) \<and> conc.low_mds_eq (snd (fst x)) (snd x) (snd y) \<and> (x, y) \<in> P}"
cyclic_on_edge_succ: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d"   and HM :: "'d pre_map"   and x :: "'c" assumes "subdiv1_contr G rev_G H rev_H u v w uv uw vw HM"   and "x \<in> verts G"   and "out_arcs G x \<noteq> {}" shows "cyclic_on (edge_succ (subdiv1_contr.GM rev_G rev_H uv uw vw HM)) (out_arcs G x)"
sanity_wf_ruleset_def: fixes \<Gamma> :: "(char list \<times> 'c rule list) list" shows "sanity_wf_ruleset \<Gamma> \<equiv> distinct (map fst \<Gamma>) \<and> (\<forall>rs\<in>ran (map_of \<Gamma>). \<forall>r\<in>set rs. case get_action r of Call chain \<Rightarrow> chain \<in> dom (map_of \<Gamma>) | Goto chain \<Rightarrow> chain \<in> dom (map_of \<Gamma>) | Unknown \<Rightarrow> False | _ \<Rightarrow> True)"
iT_div_4: fixes k :: "nat"   and m :: "nat"   and r :: "nat" assumes "0 < k"   and "m mod k = 0" shows "[ r, mod m ] \<oslash> k = [ r div k, mod m div k ]"
HomDom_def: shows "HomDom = 2\<^sub>\<nat>"
branch_length: fixes b :: "bool list"   and n :: "nat" assumes "branch b (deeptree n)" shows "length b = n"
simps_1: fixes q :: "hf" shows "nextl q [] = q"
set_spmf_def: fixes p :: "'c spmf" shows "set_spmf p = set_pmf p \<bind> set_option"
pred_prod_beta: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool"   and xy :: "'d \<times> 'e" shows "pred_prod P Q xy = (P (fst xy) \<and> Q (snd xy))"
create_opt_sol_satisfies_leq: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "y \<bullet> b \<le> x \<bullet> c"
assms: shows "compile_env \<A> \<O> F1 = Ok F2"
rx: shows "r x = (A *v x) $h j / x $h j"
seqE: fixes g :: "'a"   and f :: "'a" assumes "S.seq g f" obtains "S.arr f" and "S.arr g" and "S.dom g = S.cod f"
tautology_Rf: fixes A :: "'b"   and C :: "'b literal multiset"   and N :: "'b literal multiset set" assumes "Pos A \<in># C"   and "Neg A \<in># C" shows "C \<in> standard_redundancy_criterion.Rf N"
is_dghm_op: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
strict_mono_onI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>r s. \<lbrakk>r \<in> A; s \<in> A; r < s\<rbrakk> \<Longrightarrow> f r < f s" shows "strict_mono_on f A"
withdraw_claim_def: fixes ts :: "traffic"   and c :: "cars"   and ts' :: "traffic" shows "ts \<^bold>\<midarrow>wdc( c ) \<^bold>\<rightarrow> ts' \<equiv> pos ts' = pos ts \<and> res ts' = res ts \<and> dyn ts' = dyn ts \<and> physical_size ts' = physical_size ts \<and> braking_distance ts' = braking_distance ts \<and> clm ts' = (clm ts)(c := bot)"
Sigma_mset_def: fixes A :: "'c multiset"   and B :: "'c \<Rightarrow> 'd multiset" shows "Sigma_mset A B \<equiv> \<Sum>a\<in>#A. image_mset (Pair a) (B a)"
get_host_wf_is_l_get_host_wf: shows "l_get_host_wf Shadow_DOM.heap_is_wellformed ShadowRootClass.known_ptr ShadowRootClass.known_ptrs ShadowRootClass.type_wf get_host"
set_simps_2: fixes x21 :: "'a"   and x22 :: "'a list" shows "set (x21 # x22) = insert x21 (set x22)"
almost_all_eventually_provable_restrict_to_G_depth: fixes \<phi> :: "'b ltl"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set" assumes "\<PP>\<^sub>\<infinity> \<phi> \<G> w"   and "Only_G \<G>" shows "\<PP>\<^sub>\<infinity> \<phi> (\<G> \<inter> {\<psi>. G_depth \<psi> \<le> G_depth \<phi>}) w"
ih: fixes T' :: "tree"   and Cs :: "fterm literal set set" assumes "treesize T' < treesize T"   and "closed_tree T' Cs"   and "\<forall>C\<in>Cs. finite C" shows "\<exists>Cs'. resolution_deriv Cs Cs' \<and> {} \<in> Cs'"
chine_composite: shows "\<mu>\<nu>.chine = \<langle>\<mu>.chine \<cdot> \<p>\<^sub>1[\<mu>.dom.leg0, \<nu>.dom.leg1] \<lbrakk>\<mu>.cod.leg0, \<nu>.cod.leg1\<rbrakk> \<nu>.chine \<cdot> \<p>\<^sub>0[\<mu>.dom.leg0, \<nu>.dom.leg1]\<rangle>"
card_of_Plus_cong: fixes A :: "'a set"   and B :: "'b set"   and C :: "'c set"   and D :: "'d set" assumes "ordIso2 (card_of A) (card_of B)"   and "ordIso2 (card_of C) (card_of D)" shows "ordIso2 (card_of (A <+> C)) (card_of (B <+> D))"
neutral: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "\<forall>x\<in>A. g x = (0::'e)" shows "sum g A = (0::'e)"
simps: fixes v1 :: "'vertex"   and p1 :: "('form, 'var) out_port"   and v2 :: "'vertex"   and p2 :: "('form, 'var) antecedent" shows "edge_end ((v1, p1), v2, p2) = v2"
neg_le_divide_eq: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c < (0::'a)" shows "(a \<le> b / c) = (b \<le> a * c)"
arith_simps_11: fixes n :: "num" shows "num.One * n = n"
msg_is_cm_safe: fixes c :: "('b, 't, 'loc) Combined.configuration"   and q :: "'b"   and p :: "'b" assumes "safe (prop_config c q)"   and "InvGlobPointstampsEq c"   and "cri.InvMsgInGlob (exchange_config c)"   and "c_msg (exchange_config c) p q \<noteq> []" shows "\<forall>(loc, t)\<in>#\<^sub>zhd (c_msg (exchange_config c) p q). \<exists>t'. t' \<in>\<^sub>A frontier (c_imp (prop_config c q) loc) \<and> t' \<le> t"
antilinearI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>c x. f (c *\<^sub>C x) = cnj c *\<^sub>C f x" shows "antilinear f"
I: fixes idn :: "variable"   and T :: "typ" assumes "idn \<in> set fresh_idns" shows "(idn, T) \<notin> fst ` set insts"
open_iff: fixes m :: "'a set" shows "(m open\<^bsub>T\<^esub>) = (m = {})"
tell_env_def: fixes tell :: "'i \<Rightarrow> 'h \<Rightarrow> 'h"   and w :: "'i"   and m :: "('g, 'h) envT" shows "tell_env tell w m = EnvT (\<lambda>r. tell w (run_env m r))"
components_are_iso: fixes a :: "'b" assumes "D.ide a" shows "D.iso (src\<^sub>D a)"
interchange: fixes h :: "'d"   and g :: "'d"   and h' :: "'d"   and g' :: "'d" assumes "seq h g"   and "seq h' g'" shows "(h \<otimes> h') \<cdot> (g \<otimes> g') = h \<cdot> g \<otimes> h' \<cdot> g'"
times_int_raw_fst: fixes x :: "nat \<times> nat"   and z :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "x \<approx> z" shows "times_int_raw x y \<approx> times_int_raw z y"
dfs_eq_reachable: fixes g :: "(node \<times> node) list"   and xs :: "node list" shows "set (dfs g xs []) = reachable g xs"
Zero_Subset_I: fixes H :: "fm set"   and t :: "tm" shows "H \<turnstile> Zero SUBS t"
induct: fixes x1 :: "state"   and x2 :: "com"   and x3 :: "state"   and P :: "state \<Rightarrow> com \<Rightarrow> state \<Rightarrow> bool" assumes "x1 -x2\<rightarrow> x3"   and "\<And>t f s. t \<in> f s \<Longrightarrow> P s (Do f) t"   and "\<And>s0 c1 s1 c2 s2. \<lbrakk>s0 -c1\<rightarrow> s1; P s0 c1 s1; s1 -c2\<rightarrow> s2; P s1 c2 s2\<rbrakk> \<Longrightarrow> P s0 (c1; c2) s2"   and "\<And>b s c1 t c2. \<lbrakk>b s; s -c1\<rightarrow> t; P s c1 t\<rbrakk> \<Longrightarrow> P s (IF b THEN c1 ELSE c2) t"   and "\<And>b s c2 t c1. \<lbrakk>\<not> b s; s -c2\<rightarrow> t; P s c2 t\<rbrakk> \<Longrightarrow> P s (IF b THEN c1 ELSE c2) t"   and "\<And>b s c. \<not> b s \<Longrightarrow> P s (WHILE b DO c) s"   and "\<And>b s c t u. \<lbrakk>b s; s -c\<rightarrow> t; P s c t; t -WHILE b DO c\<rightarrow> u; P t (WHILE b DO c) u\<rbrakk> \<Longrightarrow> P s (WHILE b DO c) u"   and "\<And>s t. \<lbrakk>s -body\<rightarrow> t; P s body t\<rbrakk> \<Longrightarrow> P s CALL t"   and "\<And>f s c t g. \<lbrakk>f s -c\<rightarrow> t; P (f s) c t\<rbrakk> \<Longrightarrow> P s (LOCAL f; c; g) (g s t)" shows "P x1 x2 x3"
A2: fixes x :: "i" shows "cod (dom x) \<cong> dom x"
setprod_eq_zero_2: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" shows "(prod f {..n} = 0) = (\<exists>i\<le>n. f i = 0)"
assms_3: shows "callee_invariant_on callee2 I2 \<I>"
iso_tuple_UNIV_I: fixes x :: "'f" shows "x \<in> UNIV \<equiv> True"
ramsey0: fixes q1 :: "nat"   and q2 :: "nat" shows "\<exists>N. partn_lst {..<N} [q1, q2] 0"
atLeast0_lessThan_Suc: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "sum g {0..<Suc n} = sum g {0..<n} + g n"
Cons_2: fixes i' :: "nat"   and j' :: "nat" assumes "i' < j'"   and "i' < length (x # xs)"   and "j' < length (x # xs)"   and "filter P (x # xs) ! i = (x # xs) ! i'"   and "filter P (x # xs) ! j = (x # xs) ! j'" shows "thesis"
trans_less_add1_hmset: fixes i :: "hmultiset"   and j :: "hmultiset"   and m :: "hmultiset" assumes "i < j" shows "i < j + m"
chine_assoc'_props_2: shows "\<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1\<^sub>1 \<cdot> \<mu>_\<nu>_\<pi>\<rho>.chine_assoc' = \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1"
filter_shuffles_disjoint1_1: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "set xs \<inter> set ys = {}"   and "zs \<in> shuffles xs ys" shows "filter (\<lambda>x. x \<in> set xs) zs = xs"
adjoint: shows "dist_over_sup (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)"
iteratei_def: fixes S :: "'m1" shows "m1.iteratei S \<equiv> it_to_it (m1.list_it S)"
widen_st_def: shows "(\<nabla>) \<equiv> map_fun rep_st (map_fun rep_st St) (map2_st_rep (\<nabla>))"
delete_edge_commute: fixes a1 :: "'b"   and a2 :: "'b"   and b1 :: "'b"   and b2 :: "'b"   and G :: "'b list_graph" shows "FiniteListGraph.delete_edge a1 a2 (FiniteListGraph.delete_edge b1 b2 G) = FiniteListGraph.delete_edge b1 b2 (FiniteListGraph.delete_edge a1 a2 G)"
rcos_generate: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "principalideal I R"   and "cring R" shows "\<exists>x\<in>I. I = carrier R #>\<^bsub>R\<^esub> x"
node_total_order_irrefl: fixes e :: "'a event"   and i :: "nat" shows "\<not> e \<sqsubset>\<^sup>i e"
ah_interleavable1: fixes w :: "'a list"   and w1 :: "'a list"   and \<alpha> :: "'a \<Rightarrow> 'b set \<times> 'b set"   and w2 :: "'a list" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" shows "\<alpha>ah (map \<alpha> w1) [*] \<alpha>ah (map \<alpha> w2)"
unbind_lambda_lambda2: fixes x :: "name"   and y :: "name"   and z :: "name" shows "Lam [x].Lam [x].Var x \<mapsto> [y, z],Var z"
S2: shows "A = {z. \<exists>x. z = c_pair x (univ_for_pr x)}"
last_half_segments: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "half_segments_of_aform X \<noteq> []" shows "snd (last (half_segments_of_aform X)) = mirror_point (fst X) (lowest_vertex (fst X, nlex_pdevs (snd X)))"
times_ereal_1: fixes x :: "ereal" shows "x * ereal 1 = x"
l_span_closed3: fixes A :: "'b set"   and H :: "'a set" assumes "ideal R A"   and "generator R M H"   and "A \<odot>\<^bsub>R\<^esub> M = carrier M" shows "linear_span R M A H = carrier M"
div_mult2_eq: fixes m :: "nat"   and n :: "nat"   and q :: "nat" shows "m div (n * q) = m div n div q"
cases: fixes x :: "('l, 'm) result" obtains uu_ :: "'l" where "x = Error uu_"   | xa :: "'m" where "x = Ok xa"
select_convs_6: fixes ob :: "'c set"   and ar :: "'b set"   and dom :: "'b \<Rightarrow> 'c"   and cod :: "'b \<Rightarrow> 'c"   and id :: "'c \<Rightarrow> 'b"   and comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'d" shows "(\<bullet>\<^bsub>\<lparr>ob = ob, ar = ar, dom = dom, cod = cod, id = id, comp = comp, \<dots> = more\<rparr>\<^esub>) = comp"
inorder_insert: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (insert x t) = ins_list x (Tree2.inorder t)"
assms_4: shows "v \<noteq> 0\<^sub>h"
continuous_on_open_invariant: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_on s f = (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
fromFull_toFull: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "intT \<sigma> a" shows "fromFull \<sigma> (toFull \<sigma> a) = a"
GuardStripSwap: 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 R :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P :: "'e set"   and f :: "'f"   and g :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c Q,A"   and "P \<subseteq> R"   and "f \<in> F" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Guard f g c Q,A"
sup_least: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "x \<le> z" shows "x \<le> y \<sqinter> z"
distributed_def: fixes M :: "'d measure"   and N :: "'e measure"   and X :: "'d \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> ennreal" shows "distributed M N X f = (distr M N X = density N f \<and> f \<in> borel_measurable N \<and> X \<in> M \<rightarrow>\<^sub>M N)"
os2__inangle: fixes B :: "'p"   and A :: "'p"   and C :: "'p"   and P :: "'p" assumes "B A OS C P"   and "B C OS A P" shows "P InAngle A B C"
1: shows "A - rat_poly_times (rat_poly_times B B) B = A - B ^ 3"
list_relE1: fixes l :: "'a list"   and R :: "('a \<times> 'b) set" assumes "(l, []) \<in> \<langle>R\<rangle>list_rel" obtains "l = []"
set_help: fixes v :: "'a"   and as :: "'a list"   and bs :: "'a list" assumes "v \<in> set (as @ bs)" shows "v \<in> set as \<or> v \<in> set bs"
move: shows "\<tau>Exec_movet ci P t body h (stk, loc, pc, xcp) (stk', loc', pc', xcp')"
pigeonhole_infinite_rel: fixes A :: "'a set"   and B :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "infinite A"   and "finite B"   and "\<forall>a\<in>A. \<exists>b\<in>B. R a b" shows "\<exists>b\<in>B. infinite {a \<in> A. R a b}"
simulation1: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and tl1 :: "'t \<times> ('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s1' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2"   and "r1.redT s1 tl1 s1'"   and "\<not> m\<tau>move1 s1 tl1 s1'" shows "\<exists>s2' s2'' tl2. \<tau>mRed2 s2 s2' \<and> r2.redT s2' tl2 s2'' \<and> \<not> m\<tau>move2 s2' tl2 s2'' \<and> s1' \<approx>m s2'' \<and> tl1 \<sim>T tl2"
subtensor_decomposition: fixes A :: "'b tensor" assumes "dims A \<noteq> []" shows "listsum (dims A) (map (\<lambda>i. unit_vec (hd (dims A)) i \<otimes> subtensor A i) [0..<hd (dims A)]) = A"
simps_2: fixes f :: "('b \<Rightarrow> ('c \<times> ('b, 'c, 'd, 'e) converter, 'd, 'e) gpv) \<Rightarrow> 'a"   and x :: "'b \<Rightarrow> ('c \<times> ('b, 'c, 'd, 'e) converter, 'd, 'e) gpv" shows "(case Converter x of Converter x \<Rightarrow> f x) = f x"
monotone_lfilter: fixes P :: "'c \<Rightarrow> bool" shows "monotone (\<sqsubseteq>) (\<sqsubseteq>) (lfilter P)"
d_min: fixes C :: "'a literal multiset" assumes "C \<in> N"   and "\<not> INTERP N \<Turnstile> C" shows "DA \<le> C"
permutation_mat_right: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and p :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat nr n"   and "p permutes {..<n}" shows "A * permutation_mat n p = Matrix.mat nr n (\<lambda>(i, j). A $$ (i, p j))"
trnl\<^sub>\<epsilon>_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c"   and u :: "'c"   and \<nu> :: "'c" assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "adjunction_in_bicategory.trnl\<^sub>\<epsilon> V H \<a> \<i> src trg f g \<epsilon> u \<nu> \<equiv> V (bicategory.lunit V H \<a> \<i> src trg u) (V (H \<epsilon> u) (V (bicategory.\<a>' V H \<a> src trg f g u) (H f \<nu>)))"
\<phi>_naturality: fixes f :: "'c"   and x :: "'c"   and x' :: "'c"   and g :: "'d"   and y' :: "'d"   and y :: "'d"   and h :: "'c" assumes "\<guillemotleft>f : x \<rightarrow>\<^sub>C x'\<guillemotright>"   and "\<guillemotleft>g : y' \<rightarrow>\<^sub>D y\<guillemotright>"   and "\<guillemotleft>h : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<phi> y' (f \<cdot>\<^sub>C h \<cdot>\<^sub>C F g) = G f \<cdot>\<^sub>D \<phi> y h \<cdot>\<^sub>D g"
remdups_adj_append': fixes xs :: "'a list"   and ys :: "'a list" assumes "xs = [] \<or> ys = [] \<or> last xs \<noteq> hd ys" shows "remdups_adj (xs @ ys) = remdups_adj xs @ remdups_adj ys"
assms_2: fixes \<eta> :: "nat" shows "local.ind_cpa.ibounded_by (\<A> \<eta>) (enat (q \<eta>))"
simple_distinct_map': fixes r :: "Vertex \<Rightarrow> Vertex"   and a :: "Vertex"   and b :: "Vertex"   and c :: "Vertex"   and d :: "Vertex"   and f :: "Vertex" assumes "r \<in> simple_rotations"   and "distinct [a, b, c, d]"   and "\<forall>e\<in>{a, b, c}. r f \<noteq> e" shows "r f = d"
enum_all_prot_atom_def: fixes P :: "'b prot_atom \<Rightarrow> bool" shows "enum_class.enum_all P \<equiv> list_all P (map prot_atom.Atom enum_class.enum @ [Value, SetType, AttackType, Bottom, OccursSecType])"
case_col: shows "h ` {..<Suc l} \<subseteq> col -` {j} \<inter> {a..b}"
simAction: shows "\<forall>a ec t. t \<in> jkbpC \<and> simAbs ec = sim_equiv_class a t \<longrightarrow> set (simAction a ec) = set (jAction MC t a)"
Times_mset_single_left: fixes a :: "'b"   and B :: "'c multiset" shows "{#a#} \<times># B = image_mset (Pair a) B"
assms: shows "s' = snd (fst (flush_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
power_up_even_nonneg: fixes n :: "nat"   and p :: "nat"   and x :: "real" assumes "even n" shows "0 \<le> power_up p x n"
ortho_span': fixes W :: "'a vec set"   and X :: "'a vec set"   and w :: "'a vec"   and x :: "'a vec" assumes "W \<subseteq> carrier_vec n"   and "X \<subseteq> carrier_vec n"   and "\<And>w x. \<lbrakk>w \<in> W; x \<in> X\<rbrakk> \<Longrightarrow> x \<bullet> w = (0::'a)"   and "w \<in> local.span W"   and "x \<in> X" shows "x \<bullet> w = (0::'a)"
simps_1: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "funpower f 0 x = x"
face_cycle_pred_def: fixes M :: "'e pre_map" shows "pre_digraph_map.face_cycle_pred M \<equiv> edge_rev M \<circ> edge_pred M"
secrecy_lemma: fixes A :: "agent"   and NA :: "msg"   and B :: "agent"   and K :: "nat"   and evs :: "event list" assumes "Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>\<rbrace>) \<in> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> ns_shared" shows "(\<forall>NB. Notes Spy \<lbrace>NA, NB, Key K\<rbrace> \<notin> set evs) \<longrightarrow> Key K \<notin> analz (knows Spy evs)"
preserves_cod: fixes f :: "'a \<times> 'a" assumes "C.VV.arr f" shows "C.cod (fst f \<star>\<^sub>C snd f) = fst (C.VV.cod f) \<star>\<^sub>C snd (C.VV.cod f)"
ide_cod: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "ide (cod f)"
assms_6: shows "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr)|\<^sub>r"
that: fixes heap :: "'a"   and x :: "'b"   and heap' :: "'a" assumes "P heap"   and "run_state f heap = (x, heap')" shows "P heap'"
set_iterator_map_rev_linord_def: fixes it :: "('f \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'g \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f"   and S0 :: "('a \<times> 'g) set" shows "set_iterator_map_rev_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>(k, uu_) (k', uu_). k' \<le> k)"
All_I: fixes H :: "fm set"   and A :: "fm"   and i :: "name" assumes "H \<turnstile> A"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "H \<turnstile> SyntaxN.All i A"
increasing: fixes x :: "'a" assumes "x \<in> Field leq" shows "(x, f x) \<in> leq"
sba3_inf_associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> z) = x \<sqinter> y \<sqinter> z"
fin_Par_noWhileL: fixes cl :: "('a, 'b, 'c) cmd list" shows "noWhile (Par cl) = noWhileL cl"
aallE: fixes F :: "(state \<Rightarrow> 'a) \<Rightarrow> behavior \<Rightarrow> bool"   and x :: "state \<Rightarrow> 'a" shows "\<turnstile> (\<forall>\<forall> x. F x) \<longrightarrow> F x"
fun_lens_def: fixes x :: "'b" shows "fun_lens x = \<lparr>lens_get = \<lambda>f. f x, lens_put = \<lambda>f. fun_upd f x\<rparr>"
assms_1: shows "injective (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)"
v: fixes x :: "'a" shows "(\<lambda>n. blinfun_apply (X n) x) \<longlonglongrightarrow> v x"
history_consistent_hd_prog: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and p' :: "'b"   and xs :: "'b memref list"   and p :: "'b" assumes "program.history_consistent program_step \<theta> p' xs" shows "program.history_consistent program_step \<theta> (hd_prog p xs) xs"
F1: shows "\<forall>u v. (v, u) \<in> CLF_set \<longrightarrow> u \<in> pset v \<rightarrow> pset v \<and> Tarski.monotone u (pset v) (order v)"
induct: fixes x :: "'b"   and r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool" assumes "x \<in> Wellfounded.acc r"   and "\<And>x. \<lbrakk>\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r; \<And>y. (y, x) \<in> r \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P x"