hcju/isabelleps huggingface dataset:
lea_distincts: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F" shows "A \<noteq> B \<and> C \<noteq> B \<and> D \<noteq> E \<and> F \<noteq> E"
underS_notIn: fixes a :: "'c"   and r :: "('c \<times> 'c) set" shows "a \<notin> underS r a"
abs_eq: fixes x :: "('a, 'b) pfun" assumes "eq_onp (\<lambda>f. finite (pdom f)) x x" shows "ffun_graph (ffun.Abs_pfun x) = pfun_graph x"
prems_3: shows "\<forall>k<length (ops # \<pi>). are_all_operators_applicable (trace_parallel_plan_strips I (ops # \<pi>) ! k) ((ops # \<pi>) ! k) \<and> are_all_operator_effects_consistent ((ops # \<pi>) ! k) \<and> are_all_operators_non_interfering ((ops # \<pi>) ! k)"
chain: fixes i :: "nat" shows "M i \<tturnstile> M (Suc i)"
P8_b: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x r\<rightarrow> y) l\<rightarrow> (z r\<rightarrow> x) r\<rightarrow> z r\<rightarrow> y = (1::'a)"
f: shows "(\<Sum>x\<in>UNIV. f x *s Y $ x) = v"
lowlink_path_def: fixes s :: "('v, 'a) state_scheme"   and v :: "'v"   and p :: "'v list"   and w :: "'v" shows "lowlink_path s v p w \<equiv> path E v p w \<and> p \<noteq> [] \<and> (last p, w) \<in> cross_edges s \<union> back_edges s \<and> (1 < length p \<longrightarrow> p ! 1 \<in> dom (finished s) \<and> (\<forall>k<length p - 1. (p ! k, p ! Suc k) \<in> tree_edges s))"
le_TC_refl: fixes x :: "V" shows "x \<sqsubseteq> x"
ISO: shows "\<forall>j\<in>J. |A j| =o |B j|"
step: shows "LeftDerives1 (b1 @ [X] @ suffix) (Suc (e + length b1)) r (b1 @ [X] @ b2) \<and> LeftDerivation (b1 @ [X] @ b2) D c"
val_Zp_not_equal_imp_notequal_2: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "val_Zp x \<noteq> val_Zp y" shows "x \<ominus> y \<noteq> \<zero>"
induct: fixes P :: "int \<Rightarrow> nat \<Rightarrow> bool"   and p :: "int"   and a0 :: "int"   and a1 :: "nat" assumes "\<And>x n. (\<And>xa xb y. \<lbrakk>n \<noteq> 0; xa = divmod_nat n 2; (xb, y) = xa\<rbrakk> \<Longrightarrow> P (mult_p p x x) xb) \<Longrightarrow> P x n" shows "P a0 a1"
span_eq_iff: fixes s :: "'a set" assumes "s \<subseteq> S" shows "(span s = s) = subspace s"
filter_IFS_no_violations_accu_complies: fixes M :: "('b SecurityInvariant \<times> 'b SecurityInvariant_configured) list"   and G :: "'b list_graph"   and Es :: "('b \<times> 'b) list"   and accu :: "('b \<times> 'b) list" assumes "\<forall>(m_impl, m_spec)\<in>set M. SecurityInvariant_complies_formal_def m_impl m_spec"   and "wf_list_graph G"   and "set Es \<subseteq> set (edgesL G)"   and "set accu \<subseteq> set (edgesL G)"   and "distinct (Es @ accu)" shows "TopoS_Stateful_Policy_impl.filter_IFS_no_violations_accu G (get_impl M) accu Es = TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations_accu (list_graph_to_graph G) (get_spec M) accu Es"
SMP_singleton_ex_2: fixes m :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "m \<in> M"   and "t \<in> SMP {m}" shows "t \<in> SMP M"
dghm_const_components_2: fixes \<CC> :: "V"   and \<DD> :: "V"   and a :: "V"   and f :: "V" shows "dghm_const \<CC> \<DD> a f\<lparr>ArrMap\<rparr> = vconst_on (\<CC>\<lparr>Arr\<rparr>) f"
confTs_confT_sup: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and loc :: "val list"   and LT :: "ty err list"   and n :: "nat"   and T :: "ty"   and T' :: "ty" assumes "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT"   and "n < length LT"   and "LT ! n = OK T"   and "subtype P T T'" shows "P,h \<turnstile> loc ! n :\<le> T'"
le_finfun_def: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" shows "(f \<le> g) = (\<forall>x. f $ x \<le> g $ x)"
nondegenerate: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real"   and S :: "(real \<times> real) set" assumes "distinct3 p q r"   and "in3 S p q r" shows "ccw p q r \<or> ccw p r q"
bind_map_spmf: fixes f :: "'i \<Rightarrow> 'h"   and p :: "'i spmf"   and g :: "'h \<Rightarrow> 'g spmf" shows "map_spmf f p \<bind> g = p \<bind> g \<circ> f"
Domainp_mod_ring_rel: shows "Domainp mod_ring_rel = (\<lambda>v. v \<in> {0..<p})"
alpha_tAct: fixes \<alpha>1 :: "'c"   and t1 :: "('a, 'b, 'c) Formula.Tree"   and \<alpha>2 :: "'c"   and t2 :: "('a, 'b, 'c) Formula.Tree" shows "(Formula.Tree.tAct \<alpha>1 t1 =\<^sub>\<alpha> Formula.Tree.tAct \<alpha>2 t2) = (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2))"
CD_on_def2: fixes ds :: "'b set"   and A :: "'a set" shows "CD_on ds A = (\<Union>d\<in>ds. Cd d (A \<inter> Field (Pd d)))"
singl_def: fixes h :: "nat \<Rightarrow> nat option"   and x :: "nat"   and y :: "nat" shows "[x \<mapsto> y] = (dom h = {x} \<and> h x = Some y)"
sel_1: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "shd (siterate f x) = x"
2: shows "\<forall>x\<in>filter_sup F G. \<forall>y\<in>filter_sup F G. \<exists>z\<in>filter_sup F G. z \<le> x \<and> z \<le> y"
true_clss_def: fixes I :: "'a set"   and CC :: "'a literal multiset set" shows "(I \<Turnstile>s CC) = (\<forall>C\<in>CC. I \<Turnstile> C)"
interaction_any_bounded_converter_parallel_converter2': fixes conv1 :: "('i, 'j, 'k, 'l) converter"   and n :: "enat"   and conv2 :: "('m, 'n, 'o, 'p) converter"   and m :: "enat" assumes "interaction_any_bounded_converter conv1 n"   and "interaction_any_bounded_converter conv2 m" shows "interaction_any_bounded_converter (conv1 |\<^sub>= conv2) (max n m)"
partition_on_partition_on_unique: fixes A :: "'b set"   and P :: "'b set set"   and x :: "'b" assumes "partition_on A P"   and "x \<in> A" shows "\<exists>!X. x \<in> X \<and> X \<in> P"
path_targetnode: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "as \<noteq> []" shows "last (CFG.targetnodes targetnode as) = n'"
assms_3: shows "\<forall>\<phi>2\<in>set \<phi>2s. \<exists>\<phi>1\<in>set \<phi>1s. prv (imp \<phi>1 \<phi>2)"
pderiv_sum: fixes f :: "'b \<Rightarrow> 'a poly"   and A :: "'b set" shows "pderiv (sum f A) = (\<Sum>x\<in>A. pderiv (f x))"
less_trans: fixes x :: "'c word"   and y :: "'c word"   and z :: "'c word" assumes "x <s y"   and "y <s z" shows "x <s z"
simps_3: fixes uu :: "'b list" shows "uu \<and>\<^sub>p \<epsilon> = \<epsilon>"
sup0: fixes a :: "'c" shows "a \<equiv> a \<squnion> \<bottom>"
assms_3: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
mask_twice2: fixes n :: "nat"   and m :: "nat"   and x :: "'b word" assumes "n \<le> m" shows "(x AND mask m) AND mask n = x AND mask n"
R_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "R x y"   and "R y z" shows "R x z"
h2: shows "|~ G \<and> Unchanged v \<longrightarrow> \<circle>G"
merge_R1: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option" assumes "merge_on UNIV h bo m" shows "merge_on UNIV (list_R1.map_list_R1 h) (list_R1.rel_list_R1 bo) (list_R1.merge_R1 m)"
fbox_arden: fixes x :: "'a"   and q :: "'a"   and p :: "'a" assumes "\<nabla>x = (zero_class.zero::'a)"   and "d q \<cdot>  |x] p \<le> d p" shows " |x\<^sup>\<star>] q \<le> d p"
extensible_def: fixes R :: "('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph"   and G :: "('b, 'd) labeled_graph"   and f :: "('c \<times> 'd) set" shows "extensible R G f \<equiv> \<exists>g. graph_homomorphism (snd R) G g \<and> agree_on (fst R) f g"
zip_truncate_left: fixes xs :: "'a list"   and ys :: "'b list" shows "zip xs ys = zip (take (length ys) xs) ys"
fmapU_llist_simps_2: fixes f :: "udom \<rightarrow> udom" shows "fmapU\<cdot>f\<cdot>LNil = LNil"
no_excess_imp_maxflow: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" assumes "Labeling c s t f l"   and "\<forall>u\<in>Graph.V c - {s, t}. Network.excess c f u = (0::'a)" shows "Network.isMaxFlow c s t f"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "'b option" shows "P (case option of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> (\<exists>x2. option = \<lfloor>x2\<rfloor> \<and> \<not> P (f2 x2))))"
adm_edges_leaving_u: fixes u' :: "nat"   and v :: "nat" assumes "(u', v) \<in> adm_edges f l - adm_edges fo lo" shows "u' = u"
cInf_singleton: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Inf {x} = x"
sound_bounded: fixes P :: "'b \<Rightarrow> real" assumes "sound P" shows "Expectations.bounded P"
exp_prod1: fixes b :: "nat"   and x :: "nat" assumes "1 < b"   and "2 ^ x = 2 * b" shows "even b"
lift_None: fixes h :: "'e \<Rightarrow> 'd"   and inp :: "'f \<Rightarrow> 'e option"   and i :: "'f" shows "(lift h inp i = None) = (inp i = None)"
lq_self: fixes u :: "'b list" shows "u\<inverse>\<^sup>>u = \<epsilon>"
inline_sel: fixes c :: "('c, 'call, 'ret) gpv"   and s :: "'s" shows "the_gpv (local.inline c s) = map_spmf (\<lambda>result. case result of Inl xs \<Rightarrow> Pure xs | Inr (out, oracle, rpv) \<Rightarrow> IO out (\<lambda>input. oracle input \<bind> (\<lambda>(x, s'). local.inline (rpv x) s'))) (local.inline1 c s)"
env_restr_below_subset: fixes S :: "'d set"   and S' :: "'d set"   and m1 :: "'d \<Rightarrow> 'e"   and m2 :: "'d \<Rightarrow> 'e" assumes "S \<subseteq> S'"   and "m1 f|` S' \<sqsubseteq> m2 f|` S'" shows "m1 f|` S \<sqsubseteq> m2 f|` S"
induct: fixes P :: "pexpr \<Rightarrow> nat \<Rightarrow> pexpr \<Rightarrow> bool"   and a0 :: "pexpr"   and a1 :: "nat"   and a2 :: "pexpr" assumes "\<And>e1 e2 n e. \<lbrakk>P e1 n e; \<And>x xa y xb ya. \<lbrakk>x = split_aux e1 n e; (xa, y) = x; (xb, ya) = y\<rbrakk> \<Longrightarrow> P e2 n ya\<rbrakk> \<Longrightarrow> P (PExpr2 (PMul e1 e2)) n e"   and "\<And>e' m n e. (m \<noteq> 0 \<Longrightarrow> P e' (m * n) e) \<Longrightarrow> P (PExpr2 (PPow e' m)) n e"   and "\<And>e' n e. P (PExpr1 e') n e" shows "P a0 a1 a2"
that: fixes args' :: "'a list" assumes "t = name $$ args'"   and "list_all2 P args' args" shows "thesis"
gso_carrier: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat" assumes "gram_schmidt_fs_Rn n fs"   and "i < length fs" shows "gram_schmidt_fs.gso n fs i \<in> carrier_vec n"
weak_matrix_match2_def: fixes A1 :: "int poly fract list list"   and B1 :: "int poly fract list list"   and B2 :: "int poly fract list list" shows "weak_matrix_match2 A1 B1 B2 \<equiv> mat (rat_poly.row_length A1) 1 A1 \<and> mat (rat_poly.row_length B1) (length B1) B1 \<and> mat (rat_poly.row_length B2) (length B2) B2 \<and> A1 \<noteq> [] \<and> B1 \<noteq> [] \<and> B2 \<noteq> [] \<and> length B1 = rat_poly.row_length B2"
mor_Rep_JF: shows "mor UNIV UNIV dtor1 dtor2 car_final1 car_final2 str_final1 str_final2 Rep_JF1 Rep_JF2"
gla: fixes y :: "'a"   and x :: "'a" assumes "test y" shows "(y \<le> ! (d x)) = (y \<sqinter> x \<le> bot)"
hyps_1: shows "\<Gamma>\<turnstile> (r, s) \<rightarrow> (r'', s'')"
zin_zexplode_eq: fixes x :: "ZF"   and A :: "ZF" shows "zin x (zexplode A) = Elem x A"
1: shows "AE x in M. (x \<in> C) = (x \<in> A \<or> x \<in> B)"
AssertionStatEqClosed: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and p :: "(name \<times> name) list" assumes "\<Psi> \<simeq> \<Psi>'" shows "p \<bullet> \<Psi> \<simeq> p \<bullet> \<Psi>'"
uint32_shiftl_def: fixes x :: "uint32"   and n :: "integer" shows "uint32_shiftl x n = (if n < 0 \<or> 32 \<le> n then undefined push_bit x n else push_bit (nat_of_integer n) x)"
pmf_le_1: fixes p :: "'c pmf"   and x :: "'c" shows "pmf p x \<le> 1"
in_set_concat_map: fixes i :: "nat"   and ls :: "'a list"   and x :: "'b"   and f :: "'a \<Rightarrow> 'b list" assumes "i < length ls"   and "x \<in> set (f (ls ! i))" shows "x \<in> set (concat (map f ls))"
f_components: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "f\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Arr\<rparr>"
Actevent_def: fixes L :: "(('e, 'd) expr \<times> ('f data \<Rightarrow> bool) \<times> 'd set \<times> 'f pupdate) set" shows "Actevent L = actevent ` L"
asset_ii_spied: fixes s :: "(agent \<times> msg) set"   and B :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "PriKey B \<in> spied s"   and "(Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> s" shows "Auth_PriKey n \<in> spied s \<and> (\<exists>C SK. (Asset n, Token n (Auth_PriK n) B C SK) \<in> s)"
tighten_poly_bounds_binary: fixes p1 :: "int poly"   and l1 :: "rat"   and r1 :: "rat"   and p2 :: "int poly"   and l2 :: "rat"   and r2 :: "rat"   and I :: "rat \<Rightarrow> bool"   and bnd :: "(rat \<times> rat \<times> rat) \<times> rat \<times> rat \<times> rat \<Rightarrow> rat \<times> rat"   and f :: "real \<Rightarrow> real \<Rightarrow> real"   and sr1 :: "rat"   and sr2 :: "rat" assumes "\<exists>!x. root_cond (p1, l1, r1) x"   and "\<exists>!x. root_cond (p2, l2, r2) x"   and "poly_cond2 p1"   and "poly_cond2 p2"   and "\<And>l1a r1a l2a r2a l r sr1 sr2. \<lbrakk>I l1a; I l2a; root_cond (p1, l1a, r1a) (real_of_1 (p1, l1, r1)); root_cond (p2, l2a, r2a) (real_of_1 (p2, l2, r2)); bnd ((l1a, r1a, sr1), l2a, r2a, sr2) = (l, r)\<rbrakk> \<Longrightarrow> real_of_rat l \<le> f (real_of_1 (p1, l1, r1)) (real_of_1 (p2, l2, r2)) \<and> f (real_of_1 (p1, l1, r1)) (real_of_1 (p2, l2, r2)) \<le> real_of_rat r"   and "\<And>l1 r1 l2 r2 l1' r1' l2' r2' l l' r r' sr1 sr2 sr1' sr2'. \<lbrakk>I l1; I l2; l1 \<le> r1; l2 \<le> r2; (l, r) = bnd ((l1, r1, sr1), l2, r2, sr2); (l', r') = bnd ((l1', r1', sr1'), l2', r2', sr2'); (l1', r1') \<in> {(l1, (l1 + r1) / 2), ((l1 + r1) / 2, r1)}; (l2', r2') \<in> {(l2, (l2 + r2) / 2), ((l2 + r2) / 2, r2)}\<rbrakk> \<Longrightarrow> r' - l' \<le> 3 / 4 * (r - l) \<and> l \<le> l' \<and> r' \<le> r"   and "\<And>l l'. \<lbrakk>I l; l \<le> l'\<rbrakk> \<Longrightarrow> I l'"   and "I l1"   and "I l2"   and "sr1 = sgn (ipoly p1 r1)"   and "sr2 = sgn (ipoly p2 r2)" shows "converges_to (\<lambda>i. bnd ((tighten_poly_bounds_binary p1 p2 ^^ i) ((l1, r1, sr1), l2, r2, sr2))) (f (real_of_1 (p1, l1, r1)) (real_of_1 (p2, l2, r2)))"
mod_cut_greater_aux_t_less: fixes m :: "nat"   and r :: "nat"   and t :: "nat" assumes "0 < m"   and "r \<le> t" shows "t < t + m - (t - r) mod m"
bind_altc1: fixes C :: "'c cset"   and g :: "'c \<Rightarrow> 'm"   and f :: "'a \<times> 's \<Rightarrow> 'm" shows "bind (altc C g) f = altc C (\<lambda>c. bind (g c) f)"
map_comp: fixes g1 :: "'i \<Rightarrow> 'g"   and g2 :: "'j \<Rightarrow> 'h"   and f1 :: "'k \<Rightarrow> 'i"   and f2 :: "'l \<Rightarrow> 'j"   and v :: "('k, 'l) Op" shows "map_Op g1 g2 (map_Op f1 f2 v) = map_Op (g1 \<circ> f1) (g2 \<circ> f2) v"
Collapse_dfa_def: shows "Collapse.Collapse_dfa = \<lparr>states = HF ` states Collapse_dfa // Collapse.eq_right_lang, init = HF (Collapse.eq_right_lang `` {init Collapse_dfa}), final = {HF (Collapse.eq_right_lang `` {q}) |q. q \<in> final Collapse_dfa}, nxt = \<lambda>Q x. HF (\<Union>q\<in>hfset Q. Collapse.eq_right_lang `` {nxt Collapse_dfa q x})\<rparr>"
pf: fixes t :: "real" assumes "t \<in> {0..1}" shows "pathfinish (p t) = pathstart (q t)"
less_imp_le: fixes y :: "'a"   and x :: "'a" assumes "y < x" shows "y \<le> x"
Seq: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e\<^sub>2,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>0;; e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>2,s\<^sub>2\<rangle>"
iso_runit': fixes f :: "'d" assumes "D.ide f" shows "D.iso \<r>\<^sub>D\<^sup>-\<^sup>1[f]"
verts_app_inv_iso: fixes hom :: "('a, 'a \<times> nat \<times> 'a, 'c, 'd) digraph_isomorphism" assumes "H.digraph_isomorphism hom" shows "iso_verts (H.inv_iso hom) ` iso_verts hom ` verts (app_iso iso_f G) = verts (app_iso iso_f G)"
fbox_arden_iff: fixes x :: "'a"   and p :: "'a"   and q :: "'a" assumes "\<nabla>x = (zero_class.zero::'a)" shows "(d p = d q \<cdot>  |x] p) = (d p =  |x\<^sup>\<star>] q)"
edges_makeFaceFinal: fixes f :: "face"   and g :: "graph" shows "\<E> (makeFaceFinal f g) = \<E> g"
norm_sum_half: fixes e :: "real"   and a :: "'a"   and b :: "'a" assumes "e \<le> norm (a + b)" shows "e / 2 \<le> norm a \<or> e / 2 \<le> norm b"
assms_8: shows "z \<in> circline_set (poincare_line u v) \<inter> circline_set H"
infinite_imp_nonempty: fixes S :: "'f set" assumes "infinite S" shows "S \<noteq> {}"
cols_submatrix_subset: fixes A :: "'b mat"   and J :: "nat set" shows "set (cols (submatrix A UNIV J)) \<subseteq> set (cols A)"
BIT_init_def: fixes init :: "'b list" shows "BIT_init init = map_pmf (\<lambda>l. (l, init)) (bv (length init))"
relcompEpair: fixes a :: "'b"   and c :: "'c"   and r :: "('b \<times> 'd) set"   and s :: "('d \<times> 'c) set" assumes "(a, c) \<in> r O s" obtains b :: "'d" where "(a, b) \<in> r" and "(b, c) \<in> s"
is_Prim\<^sub>0_def: fixes "term" :: "'a local.term" shows "is_Prim\<^sub>0 term = (\<exists>x1. term = \<^bold>\<langle>x1\<^bold>\<rangle>\<^sub>0)"
new_pairs_sorted_def: fixes rel :: "nat \<times> 'g \<Rightarrow> bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" shows "new_pairs_sorted rel gs bs hs data = fold (add_pairs_single_sorted (rel data) True gs bs) hs (pairs (add_pairs_single_sorted (rel data)) False hs)"
fdia_def: fixes x :: "'a"   and y :: "'a" shows "antidomain_semiringl.fdia a (\<cdot>) x y = a (a (x \<cdot> y))"
Lower_def: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" shows "Lower L A = {l. \<forall>x. x \<in> A \<inter> carrier L \<longrightarrow> l \<sqsubseteq>\<^bsub>L\<^esub> x} \<inter> carrier L"
store_extension_imp_store_typing: fixes s :: "s"   and s' :: "s"   and \<S> :: "s_context" assumes "store_extension s s'"   and "store_typing s \<S>" shows "store_typing s' \<S>"
rel_intros_2: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x21 :: "'a"   and y21 :: "'b"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x22 :: "('a, 'c) tllist"   and y22 :: "('b, 'd) tllist" assumes "R1 x21 y21"   and "tllist_all2 R1 R2 x22 y22" shows "tllist_all2 R1 R2 (TCons x21 x22) (TCons y21 y22)"
generate_topology_on_Inter: fixes \<F> :: "'c set set"   and \<S> :: "'c set set" assumes "finite \<F>"   and "\<And>K. K \<in> \<F> \<Longrightarrow> generate_topology_on \<S> K"   and "\<F> \<noteq> {}" shows "generate_topology_on \<S> (\<Inter> \<F>)"
image_ring_zero: fixes f :: "'e \<Rightarrow> 'd"   and R :: "('e, 'f) ring_scheme" shows "\<zero>\<^bsub>image_ring f R\<^esub> = f \<zero>\<^bsub>R\<^esub>"
pw: shows "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
simps: fixes ops :: "'b arith_ops_record"   and nr :: "nat"   and nc :: "nat"   and A :: "'b mat"   and i :: "nat"   and j :: "nat" shows "gauss_jordan_main_i ops nr nc A i j = (if i < nr \<and> j < nc then let aij = A $$ (i, j) in if aij = arith_ops_record.zero ops then case concat (map (\<lambda>i'. if A $$ (i', j) \<noteq> arith_ops_record.zero ops then [i'] else []) [Suc i..<nr]) of [] \<Rightarrow> gauss_jordan_main_i ops nr nc A i (Suc j) | i' # x \<Rightarrow> gauss_jordan_main_i ops nr nc (swaprows i i' A) i j else if aij = arith_ops_record.one ops then let v = \<lambda>i. A $$ (nat_of_integer i, j) in gauss_jordan_main_i ops nr nc (eliminate_entries_i ops v A i j) (Suc i) (Suc j) else let iaij = arith_ops_record.inverse ops aij; A' = multrow_i ops i iaij A; v = \<lambda>i. A' $$ (nat_of_integer i, j) in gauss_jordan_main_i ops nr nc (eliminate_entries_i ops v A' i j) (Suc i) (Suc j) else A)"
sum_Pinfty: fixes f :: "'a \<Rightarrow> ereal"   and I :: "'a set" shows "(sum f I = \<infinity>) = (finite I \<and> (\<exists>i\<in>I. f i = \<infinity>))"
naturality: fixes f :: "'a \<times> 'a \<times> 'a" assumes "CCC.arr f" shows "\<alpha>'.map (CCC.cod f) \<cdot> T.ToCT f = T.ToTC f \<cdot> \<alpha>'.map (CCC.dom f)"
that: fixes X :: "'e set" assumes "k-smooth_on X (apply_chart d2 \<circ> f \<circ> inv_chart d1)"   and "apply_chart d1 y \<in> X"   and "open X"   and "X = codomain d1 \<inter> inv_chart d1 -` (src.carrier \<inter> f -` domain d2)" shows "thesis"
simps_6: fixes \<phi> :: "'aprop dfmla"   and \<psi> :: "'aprop dfmla"   and env :: "pvar \<Rightarrow> nat" shows "sem (Until \<phi> \<psi>) env = until (sem \<phi> env) (sem \<psi> env)"
S_contained_in_set_mult: shows "S \<subseteq> H <#> S"
norm_preserves_Can: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Can C t" shows "monoidal_language.Can C (free_monoidal_category.norm C t)"
is_extensional: fixes f :: "'a" assumes "\<not> V.arr f" shows "local.dom f = V.null"
instance_if_subsumed_and_in_limit: fixes Gs :: "'a literal multiset set llist"   and C :: "'a literal multiset"   and D :: "'a literal multiset"   and i :: "nat" assumes "chain (\<leadsto>) Sts"   and "Gs = lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts"   and "C \<in> Liminf_llist Gs - sr.Rf (Liminf_llist Gs)"   and "D \<in> N_of_state (lnth Sts i) \<union> P_of_state (lnth Sts i) \<union> Q_of_state (lnth Sts i)"   and "enat i < llength Sts"   and "subsumes D C" shows "\<exists>\<sigma>. D \<cdot> \<sigma> = C \<and> is_ground_subst \<sigma>"
p2relD: fixes a :: "'a"   and b :: "'b"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "(a, b) \<in> p2rel R" shows "R a b"