hcju/isabelleps huggingface dataset:
simply_connected_eq_frontier_properties: fixes S :: "complex set" assumes "open S" shows "simply_connected S = (connected S \<and> (if bounded S then connected (frontier S) else \<forall>C\<in>components (frontier S). \<not> bounded C))"
append_lists_of_length_alt_def: fixes T :: "'e list set"   and n :: "nat"   and X :: "'e set" shows "append_sets T (lists_of_length X (Suc n)) = append_set (append_sets T (lists_of_length X n)) X"
simps_2: fixes \<rho> :: "'b \<Rightarrow> ('a, 'b) tm"   and s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" shows "subst \<rho> (App s t) = App (subst \<rho> s) (subst \<rho> t)"
B_def: shows "B \<equiv> \<phi> ` elts (ZFC_in_HOL.succ \<gamma>)"
case_rews_2: fixes f1 :: "'a"   and f2 :: "'b \<rightarrow> [:'b:] \<rightarrow> 'a" shows "slist_case\<cdot>f1\<cdot>f2\<cdot>[::] = f1"
oppositeLiteralListIdempotency: fixes literalList :: "Literal list" shows "oppositeLiteralList (oppositeLiteralList literalList) = literalList"
openin_subtopology_Int: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" assumes "openin X S" shows "openin (subtopology X T) (S \<inter> T)"
ideal_prodp_ideal_prod_eq: fixes R :: "('c, 'd) ring_scheme"   and I :: "'c set"   and J :: "'c set" shows "ideal_prodp R (\<lambda>x. x \<in> I) (\<lambda>x. x \<in> J) = (\<lambda>x. x \<in> I \<cdot>\<^bsub>R\<^esub> J)"
derives_trans: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and b :: "symbol list"   and c :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.derives \<NN> \<TT> \<RR> a b"   and "CFG.derives \<NN> \<TT> \<RR> b c" shows "CFG.derives \<NN> \<TT> \<RR> a c"
flow_to_path_def: fixes x :: "'a"   and t :: "real"   and t' :: "real" shows "flow_to_path x t t' = flow0 x \<circ> linepath t t'"
preserves_ide: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "functor A B F"   and "partial_magma.ide A a" shows "partial_magma.ide B (F a)"
T0: fixes a :: "letter"   and b :: "letter"   and ws :: "letter list list"   and zs :: "letter list list"   and w :: "letter list" assumes "a \<noteq> b"   and "R b ws zs" shows "T a (w # zs) ((a # w) # zs)"
XA: fixes x :: "'a" assumes "x \<in> X" shows "Domainp A x"
fc_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "top \<sqinter> top \<le> top \<sqinter> top"
lex_ext_cong: fixes xs1 :: "'c list"   and ys1 :: "'d list"   and m1 :: "nat"   and m2 :: "nat"   and xs2 :: "'c list"   and ys2 :: "'d list"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> bool \<times> bool"   and g :: "'d \<Rightarrow> 'd \<Rightarrow> bool \<times> bool" assumes "length xs1 = length ys1"   and "m1 = m2"   and "length xs2 = length ys2"   and "\<And>i. \<lbrakk>i < length ys1; i < length ys2\<rbrakk> \<Longrightarrow> f (xs1 ! i) (xs2 ! i) = g (ys1 ! i) (ys2 ! i)" shows "lex_ext f m1 xs1 xs2 = lex_ext g m2 ys1 ys2"
silent_moves_slp: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (m # ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "valid_node m"   and "\<forall>mx\<in>set ms. \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "\<forall>mx\<in>set ms'. \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "m -as\<rightarrow>\<^bsub>sl\<^esub>* m' \<and> ms = ms'"
trg_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg (trg \<mu>) = trg \<mu>"
andalso_thms_4: fixes y :: "tr" shows "(y andalso TT) = y"
Carr: shows "{A, C, P * P', Q' * Q} \<subseteq> carrier_mat n n"
path2_not_Nil3: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" shows "1 \<le> length ns"
HAInitStates_def: fixes HA :: "('a, 'b, 'c) hierauto" shows "HAInitStates HA \<equiv> \<Union>A\<in>SAs HA. {InitState A}"
Rep_process: fixes x :: "'a process" shows "Rep_process x \<in> process_set"
admissible_subst: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c" assumes "ccpo.admissible luba orda P"   and "mcont lubb ordb luba orda f" shows "ccpo.admissible lubb ordb (\<lambda>x. P (f x))"
exec_gpv_oracle_bisim_bad: fixes bad2 :: "'b \<Rightarrow> bool"   and s2 :: "'b"   and X_bad :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and s1 :: "'d"   and X :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and bad1 :: "'d \<Rightarrow> bool"   and oracle1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and oracle2 :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b) spmf"   and \<I> :: "('e, 'f) \<I>"   and gpv :: "('g, 'e, 'f) gpv"   and R :: "'g \<times> 'd \<Rightarrow> 'g \<times> 'b \<Rightarrow> bool" assumes "if bad2 s2 then X_bad s1 s2 else X s1 s2"   and "bad1 s1 = bad2 s2"   and "\<And>s1 s2 x. X s1 s2 \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else a = b \<and> X s1' s2')) (oracle1 s1 x) (oracle2 s2 x)"   and "\<And>s2. bad2 s2 \<Longrightarrow> callee_invariant_on oracle1 (\<lambda>s1. bad1 s1 \<and> X_bad s1 s2) \<I>"   and "\<And>s1. bad1 s1 \<Longrightarrow> callee_invariant_on oracle2 (\<lambda>s2. bad2 s2 \<and> X_bad s1 s2) \<I>"   and "\<And>s1 x. bad1 s1 \<Longrightarrow> lossless_spmf (oracle1 s1 x)"   and "\<And>s2 x. bad2 s2 \<Longrightarrow> lossless_spmf (oracle2 s2 x)"   and "lossless_gpv \<I> gpv"   and "\<And>s1. \<I> \<turnstile>c oracle1 s1 \<surd>"   and "\<And>s2. \<I> \<turnstile>c oracle2 s2 \<surd>"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>a s1 b s2. \<lbrakk>bad1 s1 = bad2 s2; \<not> bad2 s2 \<Longrightarrow> a = b \<and> X s1 s2; bad2 s2 \<Longrightarrow> X_bad s1 s2\<rbrakk> \<Longrightarrow> R (a, s1) (b, s2)" shows "rel_spmf R (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
fsubset_trans: fixes A :: "'b fset"   and B :: "'b fset"   and C :: "'b fset" assumes "A |\<subseteq>| B"   and "B |\<subseteq>| C" shows "A |\<subseteq>| C"
I_lift_eq_qe: fixes as :: "'a list"   and qe :: "'a list \<Rightarrow> 'a fm"   and xs :: "'b list" assumes "\<forall>a\<in>set as. depends\<^sub>0 a"   and "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<and> \<not> solvable\<^sub>0 a \<Longrightarrow> I (qe as) xs = (\<exists>x. \<forall>a\<in>set as. I\<^sub>a a (x # xs))" shows "I (lift_eq_qe qe as) xs = (\<exists>x. \<forall>a\<in>set as. I\<^sub>a a (x # xs))"
lowner_is_lub_def: fixes M :: "complex mat" shows "local.lowner_is_lub M = ((\<forall>n. f n \<le>\<^sub>L M) \<and> (\<forall>M'. (\<forall>n. f n \<le>\<^sub>L M') \<longrightarrow> M \<le>\<^sub>L M'))"
vreal_of_real_impl''_def: shows "vreal_of_real_impl'' = vreal_of_real_impl' \<circ> vreal_of_real_impl"
Val: 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 v :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" shows "P \<turnstile> \<langle>Val v,s\<rangle> \<Rightarrow> \<langle>Val v,s\<rangle>"
inc: fixes n :: "nat" shows "f n \<le>\<^sub>L f (Suc n)"
hyps_8: assumes "list_all is_Assignment (unlabel A)" shows "list_all is_Assignment (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
atLeastLessThan_subseteq_atLeastAtMost_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.dense_linorder less_eq less" shows "(ord.atLeastLessThan less_eq less a b \<subseteq> ord.atLeastAtMost less_eq c d) = (less a b \<longrightarrow> less_eq c a \<and> less_eq b d)"
pr_naturality_1: fixes g :: "'a"   and b :: "'a"   and d :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "arr g"   and "local.dom g = b"   and "cod g = d"   and "arr f"   and "local.dom f = a"   and "cod f = c" shows "pr0 c d \<cdot> local.prod f g = g \<cdot> pr0 a b"
insert_absorb2: fixes x :: "'b"   and A :: "'b set" shows "x \<triangleright> x \<triangleright> A = x \<triangleright> A"
lower_one_diff_uppert: fixes i :: "nat"   and n :: "nat"   and k :: "nat"   and B :: "'b mat" assumes "i < n"   and "lower_one k B i k" shows "??.Jordan_Normal_Form_Existence.diff_ev B i k \<and> ??.Jordan_Normal_Form_Existence.uppert B i k"
inf_bot_right: fixes x :: "'d multiset" shows "x \<inter># {#} = {#}"
ipresFresh_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'f, 'g, 'var, 'c, 'h, 'i) Iteration.model_scheme" shows "ipresFresh h MOD \<equiv> \<forall>ys y s X. wls s X \<longrightarrow> fresh ys y X \<longrightarrow> igFresh MOD ys y (h X)"
sum_upto_\<psi>_x_over_n_asymptotics: shows "(\<lambda>x. sum_upto (\<lambda>xa. \<psi> (x / real xa)) x - x * ln x + x) \<in> O(ln)"
relax_outgoing_def: fixes w :: "'b \<times> 'b \<Rightarrow> enat"   and u :: "'b"   and D :: "'b \<Rightarrow> enat" shows "Dijkstra.relax_outgoing w u D \<equiv> \<lambda>v. min (D v) (D u + w (u, v))"
inj_fun: fixes f :: "'a \<Rightarrow> 'b" assumes "inj f" shows "inj (\<lambda>x y. f x)"
set_remdups_wrt: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "f ` set (remdups_wrt f xs) = f ` set xs"
arcs_cases': fixes a :: "nat"   and b :: "nat"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "(a, b) \<in> set (arcs i j xs)"   and "xs \<noteq> []" shows "(\<exists>ys. xs = b # ys \<and> a = i) \<or> (\<exists>ys. xs = ys @ [a] \<and> b = j) \<or> (\<exists>ys zs. xs = ys @ a # b # zs)"
rel_def: shows "rel \<equiv> {(s, s'). (ss_stack s, ss_stack s') \<in> \<langle>Id \<times>\<^sub>r {(U, U'). U - R = U'}\<rangle>list_rel \<and> on_stack s = on_stack s' \<and> visited s = visited s' \<union> R \<and> visited s' \<inter> R = {} \<and> simple_state.more s = simple_state.more s'}"
simplicial_chain_of: fixes p :: "nat"   and S :: "('c \<Rightarrow> real) set"   and c :: "(nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "simplicial_chain p S (frag_of c) = simplicial_simplex p S c"
rep_eq: fixes x :: "complex"   and xa :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(*\<^sub>V) (x *\<^sub>C xa) = (\<lambda>xb. x *\<^sub>C (xa *\<^sub>V xb))"
MatchF: fixes r :: "formula Regex.regex"   and I :: "\<I>" assumes "safe_regex Futu Strict r"   and "\<forall>\<phi>\<in>atms r. P \<phi>" shows "P (MatchF I r)"
merge_Nil: fixes c :: "'a list"   and pc :: "nat"   and x :: "'a" shows "merge c pc [] x = x"
stable_onI: fixes ds :: "D2 set"   and X :: "X4 set" assumes "StableNoDecomp.individually_rational_on ds X"   and "StableNoDecomp.stable_no_blocking_on ds X" shows "StableNoDecomp.stable_on ds X"
pdevs_apply_One_pdevs: fixes i :: "nat" shows "pdevs_apply One_pdevs i = (if i < length Basis_list then Basis_list ! i else (0::'d))"
listsp_lists_eq: fixes A :: "'b set" shows "listsp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> lists A)"
eq_None_iff_not_dom: fixes f :: "'d \<Rightarrow> 'c option"   and x :: "'d" shows "(f x = None) = (x \<notin> dom f)"
empty_set_k_words: fixes k :: "nat" assumes "0 < k" shows "set_of_k_words_length_m k 0 = {}"
cases: fixes a :: "'a"   and r :: "('a \<times> 'b) set" assumes "a \<in> Domain r" obtains aa :: "'a" and b :: "'b" where "a = aa" and "(aa, b) \<in> r"
RelT_def: shows "RelT = _Collect (T1, T2) (\<exists>S1 S2. (S1, S2) \<in> RelS \<and> T1 = \<lbrakk>S1\<rbrakk> \<and> T2 = \<lbrakk>S2\<rbrakk>)"
merge_monoms_poly_def: shows "merge_monoms_poly = msort (\<le>)"
simps_2: fixes x :: "gauss_int" shows "ImZ (- x) = - ImZ x"
simple_path_endless: fixes c :: "real \<Rightarrow> 'b" assumes "simple_path c" shows "path_image c - {pathstart c, pathfinish c} = c ` {0<..<1}"
fold_insert_remove: fixes A :: "'a set"   and z :: "'a option"   and x :: "'a" assumes "finite A" shows "Finite_Set.fold MaxR_f z (insert x A) = MaxR_f x (Finite_Set.fold MaxR_f z (A - {x}))"
replicate_0: fixes x :: "'b" shows "replicate 0 x = []"
unprot_nvC_tC: fixes x :: "Preliminaries.var"   and c :: "('fsym, 'a) lit list" assumes "unprot (TE.tpOfV x)" shows "(x \<in> nvC (tC c)) = (x \<in> nvC c)"
dist_of_nat: fixes m :: "nat"   and n :: "nat" shows "dist (of_nat m) (of_nat n) = real_of_int \<bar>int m - int n\<bar>"
undef_mat_def: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" shows "undef_mat nr nc f \<equiv> \<lambda>(i, j). map (\<lambda>i. map (\<lambda>j. f (i, j)) [0..<nc]) [0..<nr] ! i ! j"
a_get_child_nodes_def: fixes ptr :: "(_) object_ptr" shows "Shadow_DOM.get_child_nodes ptr = invoke (Core_DOM_Functions.a_get_child_nodes_tups @ Shadow_DOM.a_get_child_nodes_tups) ptr ()"
contractible_space_euclideanreal: shows "contractible_space euclideanreal"
alphaAbs_preserves_qGoodAbs2: fixes A :: "('c, 'd, 'varSort, 'var, 'e) qAbs"   and A' :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "A $= A'"   and "qGoodAbs A'" shows "qGoodAbs A"
s2: shows "ccw'.sortedP 0 (u # (v + w) # y # ys)"
refl_on_restrictPI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set" assumes "refl_onP A r" shows "refl_onP (A \<inter> B) (r |` B)"
pentagon': fixes f :: "'d"   and g :: "'d"   and h :: "'d"   and k :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "D.ide k"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h"   and "src\<^sub>D h = trg\<^sub>D k" shows "((\<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] \<star>\<^sub>D k) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[f, g \<star>\<^sub>D h, k]) \<cdot>\<^sub>D (f \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[g, h, k]) = \<a>\<^sub>D\<^sup>-\<^sup>1[f \<star>\<^sub>D g, h, k] \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h \<star>\<^sub>D k]"
\<G>_semantics: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G> \<phi> w \<equiv> {\<psi> \<in> subformulas\<^sub>\<nu> \<phi>. w \<Turnstile>\<^sub>n G\<^sub>n \<psi>}"
inverse_arrow_unique: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and g' :: "'a \<times> 'a" assumes "VV.inverse_arrows f g"   and "VV.inverse_arrows f g'" shows "g = g'"
rel_refl: fixes Ra :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c list" assumes "\<And>x. Ra x x" shows "list_all2 Ra x x"
less_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<prec> b"   and "b \<prec> c" shows "a \<prec> c"
is_hashmap_def: fixes ht :: "('a, 'b) hashtable" shows "is_hashmap m ht = (\<exists>\<^sub>Al. is_hashmap' m l ht)"
Fract_less_zero_iff: fixes b :: "'b"   and a :: "'b" assumes "(0::'b) < b" shows "(Fract a b < 0) = (a < (0::'b))"
less_eq_def: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
single_eq_add_mset: fixes a :: "'a"   and b :: "'a"   and M :: "'a multiset" shows "(a, {#} = b, M) = (b = a \<and> M = {#})"
simps_1: fixes M :: "nat"   and inputs :: "real vec list" shows "evaluate_net (Input M) inputs = hd inputs"
cltn2_inverse_id: shows "cltn2_inverse cltn2_id = cltn2_id"
finite_imp_foldSetD: fixes B :: "'c set"   and D :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'c set"   and e :: "'d" assumes "LCD B D f"   and "finite A"   and "A \<subseteq> B"   and "e \<in> D" shows "\<exists>x. (A, x) \<in> foldSetD D f e"
fps_integral0_add: fixes a :: "'a fps"   and b :: "'a fps" shows "fps_integral0 (a + b) = fps_integral0 a + fps_integral0 b"
simps_1: fixes i :: "nat"   and fs :: "int vec list" shows "basis_reduction_add_rows_loop i fs 0 = fs"
distinct_concat_map_I: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b list" assumes "distinct xs"   and "\<And>x. x \<in> set xs \<Longrightarrow> distinct (f x)"   and "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs; x \<noteq> y\<rbrakk> \<Longrightarrow> set (f x) \<inter> set (f y) = {}" shows "distinct (concat (map f xs))"
valid_edges: shows "\<forall>(ps1, ps2)\<in>edges. valid_out_port ps1 \<and> valid_in_port ps2"
that: fixes C :: "'a fset" assumes "C |\<subseteq>| A"   and "C \<in> min_set B" shows "thesis"
absorb1: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "inf a b = a"
is_projection_on_in_image: fixes \<pi> :: "'a \<Rightarrow> 'a"   and M :: "'a set"   and h :: "'a" assumes "is_projection_on \<pi> M" shows "\<pi> h \<in> M"
1: shows "rawpsubst (subst \<phi> (Var u) x) (zip (map Var vs) xs) = subst (rawpsubst \<phi> (zip (map Var vs) xs)) (Var u) x"
domains_simp: fixes f :: "(bool, bool list) concrete_category.arr" assumes "arr f" shows "domains f = {local.dom f}"
case_1: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd llist \<Rightarrow> 'c" shows "(case LNil of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = f1"
elims_3: fixes x :: "'v"   and xa :: "('v \<times> 'v) list"   and xb :: "'v" assumes "\<not> is_path x xa xb" obtains v :: "'v" and v' :: "'v" where "x = va__" and "xa = []" and "xb = v'a__" and "\<not> (va__ = v'a__ \<and> v'a__ \<in> V)"   | v :: "'v" and v1 :: "'v" and v2 :: "'v" and p :: "('v \<times> 'v) list" and v' :: "'v" where "x = v" and "xa = (v1, v2) # p" and "xb = v'" and "\<not> (v = v1 \<and> (v1, v2) \<in> E \<and> is_path v2 p v')"
jstart_eq: shows "jumpF_pathstart (part_circlepath z r st tt) z0 = jumpF h (at_right st)"
fps_times_divide_eq: fixes g :: "'a fps"   and f :: "'a fps" assumes "g \<noteq> 0"   and "subdegree g \<le> subdegree f" shows "f / g * g = f"
map_comp: fixes g :: "'c \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'c"   and v :: "'d set['b]" shows "map_bset g (map_bset f v) = map_bset (g \<circ> f) v"
par_onel: fixes x :: "'a" shows "(1::'a) * x = x"
inf_continuous_mono: fixes F :: "'d \<Rightarrow> 'e" assumes "inf_continuous F" shows "mono F"
cts_wp_SetDC: fixes S :: "'b \<Rightarrow> 'a set"   and a :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>x s. x \<in> S s \<Longrightarrow> bd_cts (wp (a x))"   and "\<And>x s. x \<in> S s \<Longrightarrow> healthy (wp (a x))"   and "\<And>s. finite (S s)"   and "\<And>s. S s \<noteq> {}" shows "bd_cts (wp (SetDC a S))"
compower_stepsD: fixes n :: "nat"   and s :: "'a"   and s' :: "'a" assumes "(E ^^ n) s s'" obtains xs :: "'a list" where "local.steps xs" and "hd xs = s" and "last xs = s'" and "length xs = n + 1"
square_rat_def: fixes x :: "rat" shows "square_rat x = x * x"
typ_ok_sig_imp_wf_type: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and T :: "typ" assumes "typ_ok_sig \<Sigma> T" shows "wf_type \<Sigma> T"
par_skip_left: fixes a :: "'a::refinement_lattice" shows "skip \<parallel> (a::'a::refinement_lattice) = a"
lfinite_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" shows "lfinite (xs @\<^sub>l ys) = (lfinite xs \<and> lfinite ys)"
simple_extension_of_algebraic: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x" shows "Rupt K (Irr K x) \<simeq> R\<lparr>carrier := simple_extension K x\<rparr>"
eqD_16: fixes y :: "cmsg"   and N :: "ltkey" assumes "eq y (cLtK N)" shows "y = cLtK N"
red_preserves_lconf: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id"   and e :: "'addr expr"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and e' :: "'addr expr"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and E :: "String.literal \<Rightarrow> ty option"   and T :: "ty" assumes "extTA,P,t \<turnstile> \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>"   and "P,E,hp s \<turnstile> e : T"   and "P,hp s \<turnstile> lcl s (:\<le>) E" shows "P,hp s' \<turnstile> lcl s' (:\<le>) E"
simps_1: fixes C :: "String.literal" shows "is_Class (Class C) = True"
div_mult_le: fixes m :: "nat"   and n :: "nat" shows "m div n * n \<le> m"
assms: fixes p :: "'a pmf"   and q :: "'a pmf"   and r :: "'a pmf" assumes "p \<in> C"   and "q \<in> C"   and "r \<in> C"   and "p \<succeq>[P] q \<and> q \<succeq>[P] r" shows "\<exists>\<alpha>\<in>{0..1}. mix_pmf \<alpha> p r \<approx>[P] q"
inline1_fixp_induct_strong: fixes P :: "(('a, 'b, 'c) gpv \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'd + 'e \<times> ('c \<times> 'd, 'e, 'f) rpv \<times> ('a, 'b, 'c) rpv) spmf) \<Rightarrow> bool"   and callee :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd, 'e, 'f) gpv" assumes "spmf.admissible (\<lambda>inline1'. P (\<lambda>gpv s. inline1' (gpv, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>inline1'. \<lbrakk>\<And>gpv s. ord_spmf (=) (inline1' gpv s) (inline1 callee gpv s); P inline1'\<rbrakk> \<Longrightarrow> P (\<lambda>gpv s. bind_spmf (the_gpv gpv) (case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. bind_spmf (the_gpv (callee s out)) (case_generat (\<lambda>(x, y). inline1' (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv)))))))" shows "P (inline1 callee)"
finter_fempty_left: fixes B :: "'a fset" shows "{||} |\<inter>| B = {||}"
sum0: shows "sum f (UNIV - {a} - {b}) = (0::'a)"
rearrange_real_euclid_B: fixes y :: "(real, 'b) vec"   and w :: "(real, 'b) vec"   and h :: "real"   and z :: "(real, 'b) vec" shows "(y - w = h *\<^sub>R (z - w)) = (y = h *\<^sub>R z + (1 - h) *\<^sub>R w)"
zero_less_power_eq: fixes 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 times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (power.power one times a n) = (n = 0 \<or> even n \<and> a \<noteq> zero \<or> odd n \<and> less zero a)"
cindex_polyE_def: fixes a :: "real"   and b :: "real"   and q :: "real poly"   and p :: "real poly" shows "cindex_polyE a b q p = jumpF_polyR q p a + real_of_int (cindex_poly a b q p) - jumpF_polyL q p b"
group_reduced_homology_group: fixes p :: "int"   and X :: "'a topology" shows "Group.group (reduced_homology_group p X)"
ideals_set_sum: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "ideal R A"   and "ideal R B"   and "x \<in> A \<minusplus> B" shows "\<exists>h\<in>A. \<exists>k\<in>B. x = h \<plusminus> k"
m\<tau>diverge_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" assumes "s1 \<approx>m s2"   and "r1.mthr.\<tau>diverge s1" shows "r2.mthr.\<tau>diverge s2"