hcju/isabelleps huggingface dataset:
that_2: fixes dd :: "nat"   and vh :: "nat" assumes "d = Suc dd"   and "dist v dd vh"   and "(vh, v') \<in> E" shows "thesis"
hom_minus: fixes x :: "'a"   and y :: "'a" shows "hom (x - y) = hom x - hom y"
poly_split_def: fixes m :: "'d monom"   and p :: "('d monom \<times> 'c) list" shows "poly_split m p \<equiv> case List.extract (\<lambda>(n, uu_). m = n) p of None \<Rightarrow> (0::'c, p) | Some (p1, (x, c), p2) \<Rightarrow> (c, p1 @ p2)"
le_lang_equiv: shows "(\<sim>) \<le> (\<sim>\<^sub>L)"
transfer: shows "rel_fun pcr_real pcr_real (\<lambda>X. if vanishes X then \<lambda>n. 0 else (\<lambda>n. inverse (X n))) inverse"
alphaGood_def: shows "alphaGood \<equiv> \<lambda>qX qY. qGood qX \<and> qGood qY \<and> qX #= qY"
is_invarI: fixes I :: "('v, 'es) state_scheme \<Rightarrow> bool" assumes "\<And>e. I (empty_state e)"   and "\<And>s s' v0 e. \<lbrakk>I s; cond s; DFS_invar G param s; DFS_invar G param s'; stack s = []; v0 \<notin> dom (discovered s); v0 \<in> V0; s' = new_root v0 s\<lparr>state.more := e\<rparr>\<rbrakk> \<Longrightarrow> I s'"   and "\<And>s s' u e. \<lbrakk>I s; cond s; DFS_invar G param s; DFS_invar G param s'; stack s \<noteq> []; pending s `` {u} = {}; u = hd (stack s); s' = finish u s\<lparr>state.more := e\<rparr>\<rbrakk> \<Longrightarrow> I s'"   and "\<And>s s' u v e c b. \<lbrakk>I s; cond s; DFS_invar G param s; DFS_invar G param s'; stack s \<noteq> []; v \<in> pending s `` {u}; v \<in> dom (discovered s); u = hd (stack s); cross_edges s \<subseteq> c; back_edges s \<subseteq> b; s' = s\<lparr>pending := pending s - {(u, v)}, state.more := e, cross_edges := c, back_edges := b\<rparr>\<rbrakk> \<Longrightarrow> I s'"   and "\<And>s s' u v e. \<lbrakk>I s; cond s; DFS_invar G param s; DFS_invar G param s'; stack s \<noteq> []; v \<in> pending s `` {u}; v \<notin> dom (discovered s); u = hd (stack s); s' = discover u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<lparr>state.more := e\<rparr>\<rbrakk> \<Longrightarrow> I s'" shows "is_invar I"
real_interval_Ipoly: fixes p :: "float interval poly"   and xs :: "float interval list" assumes "num_params p \<le> length xs" shows "real_interval (Ipoly xs p) = Ipoly (map real_interval xs) (map_poly real_interval p)"
force_enabled_in_set_pmf: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and \<omega> :: "'a stream" shows "shd (MC_syntax.force_enabled K x \<omega>) \<in> set_pmf (K x)"
fin: fixes s :: "'a rexp" shows "finite {fold pderiv_set w {s} |w. True}"
cos_coeff_Suc: fixes n :: "nat" shows "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
divide_self_if: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a a = (if a = zero then zero else one)"
inj_on_map_of: fixes xs :: "('c \<times> 'b) list" assumes "distinct (map snd xs)" shows "inj_on (map_of xs) (set (map fst xs))"
less_eq_fun_def: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" shows "(f \<lesssim> g) = (\<exists>c>0. \<exists>n. \<forall>m>n. f m \<le> c * g m)"
expand_powers: fixes f :: "'c \<Rightarrow> 'b"   and n_as :: "(nat \<times> 'c) list" shows "(\<Prod>(n, a)\<leftarrow>n_as. f a ^ n) = prod_list (map f (expand_powers n_as))"
elims: fixes x :: "'b list"   and xa :: "'b list"   and y :: "'b list" assumes "splice x xa = y" obtains ys :: "'b list" where "x = []" and "xa = ysa__" and "y = ysa__"   | xb :: "'b" and xs :: "'b list" and ys :: "'b list" where "x = xb # xs" and "xa = ys" and "y = xb # splice ys xs"
homotopic_from_subtopology: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a topology"   and X' :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and s :: "'a set" assumes "homotopic_with P X X' f g" shows "homotopic_with P (subtopology X s) X' f g"
stream_rel_snth: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and u :: "'c stream"   and v :: "'d stream" shows "stream_all2 P u v = (\<forall>i. P (u !! i) (v !! i))"
ipcidr_to_interval_def: fixes cidr :: "'a word \<times> nat" shows "ipcidr_to_interval cidr \<equiv> (ipcidr_to_interval_start cidr, ipcidr_to_interval_end cidr)"
relative_to_inc: fixes P :: "'d set \<Rightarrow> bool"   and S :: "'d set"   and U :: "'d set" assumes "P S" shows "(P relative_to U) (U \<inter> S)"
sel: fixes x2 :: "'f" shows "the \<lfloor>x2\<rfloor> = x2"
subformulae_self: fixes F :: "'b formula" shows "F \<in> set (subformulae F)"
simps: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list" shows "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems = (length cms\<^sub>1 = length cms\<^sub>2 \<and> length cms\<^sub>1 = length mems \<and> (\<forall>i<length cms\<^sub>1. (\<forall>\<sigma>. dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i \<longrightarrow> (cms\<^sub>1 ! i, subst \<sigma> (fst (mems ! i))) \<approx> (cms\<^sub>2 ! i, subst \<sigma> (snd (mems ! i)))) \<and> (\<forall>x. mem\<^sub>1 x = mem\<^sub>2 x \<or> dma mem\<^sub>1 x = High \<or> x \<in> \<C> \<longrightarrow> x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i)) \<and> (length cms\<^sub>1 = 0 \<and> mem\<^sub>1 =\<^sup>l mem\<^sub>2 \<or> (\<forall>x. \<exists>i<length cms\<^sub>1. x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i)))"
items_le_Complete: fixes I :: "item set"   and k :: "nat" assumes "wellformed_items I" shows "items_le k (Complete k I) = Complete k (items_le k I)"
substn_subst_n: fixes t :: "dB"   and s :: "dB"   and n :: "nat" shows "substn t s n = t[liftn n s 0/n]"
count_le_replicate_mset_subset_eq: fixes n :: "nat"   and M :: "'k multiset"   and x :: "'k" shows "(n \<le> count M x) = (replicate_mset n x \<subseteq># M)"
rep_abs_rsp_left: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and x1 :: "'a"   and x2 :: "'a" assumes "Quotient3 R Abs Rep"   and "R x1 x2" shows "R (Rep (Abs x1)) x2"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "expr list"   and a3 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a4 :: "expr list"   and a5 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,a1 \<turnstile> \<langle>a2,a3\<rangle> [\<Rightarrow>] \<langle>a4,a5\<rangle>" obtains E :: "char list \<Rightarrow> ty option" and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" where "a1 = Eb__" and "a2 = []" and "a3 = s" and "a4 = []" and "a5 = s"   | E :: "char list \<Rightarrow> ty option" and e :: "expr" and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and v :: "val" and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and es :: "expr list" and es' :: "expr list" and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" where "a1 = Ec__" and "a2 = eb__ # esb__" and "a3 = s\<^sub>0'__" and "a4 = Val v # es'" and "a5 = s\<^sub>2" and "P,Ec__ \<turnstile> \<langle>eb__,s\<^sub>0'__\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1''__\<rangle>" and "P,Ec__ \<turnstile> \<langle>esb__,s\<^sub>1''__\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>"   | E :: "char list \<Rightarrow> ty option" and e :: "expr" and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and e' :: "expr" and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and es :: "expr list" where "a1 = E" and "a2 = e # es" and "a3 = s\<^sub>0" and "a4 = throw e' # es" and "a5 = s\<^sub>1" and "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
codomains_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.seq g f" shows "CC.codomains (CC.comp g f) = CC.codomains g"
nonTickFree_n_frontTickFree: fixes s :: "'c event list" assumes "\<not> tickFree s"   and "front_tickFree s" shows "\<exists>t. s = t @ [tick]"
id: shows "of_int_hom.mat_hom (mat_of_cols n fs) * A = of_int_hom.mat_hom (mat_of_cols n gs)"
vdim: shows "dim_vec V = dim_vec (b @\<^sub>v c)"
rep_eq: fixes x :: "('a, 'b) pp"   and xa :: "('a, 'b) pp" shows "drlex_pp x xa = drlex_pm (pp.mapping_of x) (pp.mapping_of xa)"
leadsETo_UN: fixes I :: "'d set"   and F :: "'e program"   and A :: "'d \<Rightarrow> 'e set"   and CC :: "'e set set"   and B :: "'e set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i leadsTo[CC] B" shows "F \<in> \<Union> (A ` I) leadsTo[CC] B"
row_rw: shows "row (local.berlekamp_mat u) k = vec_of_list (map (\<lambda>cs. coeffs cs @ replicate (degree u - length (coeffs cs)) 0) (power_polys (power_poly_f_mod u [:0, 1:] (nat (int CARD('a)))) u 1 (degree u)) ! k)"
nth_map_eq: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and g :: "'e \<Rightarrow> 'c"   and ys :: "'e list"   and i :: "nat" assumes "map f xs = map g ys"   and "i < length xs" shows "f (xs ! i) = g (ys ! i)"
pderiv_char_poly: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "pderiv (char_poly A) = (\<Sum>i<n. char_poly (mat_delete A i i))"
dom_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "local.dom (mkarr t) = mkarr (Dom t)"
assms_4: shows "rk {A', C', \<beta>} = 2"
ample_set_def: fixes q :: "'state"   and A :: "'action set" shows "ample_set q A \<equiv> A \<subseteq> {a. en a q} \<and> (A \<subset> {a. en a q} \<longrightarrow> A \<noteq> {}) \<and> (\<forall>a. A \<subset> {a. en a q} \<longrightarrow> a \<in> A \<longrightarrow> src (ex a q) q) \<and> (A \<subset> {a. en a q} \<longrightarrow> A \<subseteq> invisible) \<and> (\<forall>w. A \<subset> {a. en a q} \<longrightarrow> path w q \<longrightarrow> A \<inter> set w = {} \<longrightarrow> Ind A (set w))"
F_Un_D_subset: shows "F \<union> D \<subseteq> A"
listset_Cons_mem_conv: fixes a :: "'b"   and vs :: "'b list"   and AVS :: "'b set list" shows "(a # vs \<in> listset AVS) = (\<exists>A VS. AVS = A # VS \<and> a \<in> A \<and> vs \<in> listset VS)"
c_graph_f_is_ce: shows "c_graph f \<in> ce_sets"
set_filter_diff: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "{a \<in> A. P a} - {x} = {a \<in> A - {x}. P a}"
add_diff_cancel: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "minus (plus a b) b = a"
RES_rule: fixes S :: "'b set"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "\<And>x. x \<in> S \<Longrightarrow> \<Phi> x" shows "RES S \<le> SPEC \<Phi>"
coboundedI: fixes A :: "'c set"   and a :: "'c" assumes "finite A"   and "a \<in> A" shows "a \<le> Max A"
fold_graph_closed: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and z :: "'e"   and B :: "'e set"   and y :: "'e" assumes "comp_fun_commute_on f A"   and "fold_graph f z B y"   and "B \<subseteq> A"   and "z \<in> A" shows "y \<in> A"
ind_valid_graph: fixes E' :: "(nat \<times> int \<times> nat) set"   and L :: "(nat \<times> int \<times> nat) list" assumes "E' \<subseteq> set L" shows "valid_graph \<lparr>nodes = fst ` set L \<union> (snd \<circ> snd) ` set L, edges = E'\<rparr>"
pinduct: fixes a0 :: "'a set"   and a1 :: "'a set"   and P :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" assumes "local.mk_linear_orders_dom (a0, a1)"   and "\<And>C B. \<lbrakk>local.mk_linear_orders_dom (C, B); \<And>x xa xb. \<lbrakk>f B \<noteq> {}; f B \<subseteq> C; x = xa # xb; xa \<in> f B\<rbrakk> \<Longrightarrow> P {} (B - {xa}); \<And>x xa xb. \<lbrakk>f B \<noteq> {}; \<not> f B \<subseteq> C; x = (SOME x. x \<in> f B - C); xa = x # xb\<rbrakk> \<Longrightarrow> P C (B - {x})\<rbrakk> \<Longrightarrow> P C B" shows "P a0 a1"
cSUP_upper: fixes x :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e multiset" assumes "x \<in> A"   and "subset_mset.bdd_above (f ` A)" shows "f x \<subseteq># \<Squnion> (f ` A)"
adm_ball': fixes A :: "'b \<Rightarrow> 'a set"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>y. adm (\<lambda>x. y \<in> A x \<longrightarrow> P x y)" shows "adm (\<lambda>x. \<forall>y\<in>A x. P x y)"
elims_3: fixes x :: "('fsym, 'tp) efsym" assumes "\<not> TE_wtFsym x" obtains f :: "'fsym" where "x = Oldf f" and "\<not> wtFsym f"   | \<sigma> :: "'tp" where "x = Wit \<sigma>" and "\<not> \<not> isRes \<sigma>"
Sup_eq_0_iff: fixes A :: "V set" shows "(\<Squnion> A = 0) = (A \<subseteq> {0} \<or> \<not> small A)"
fin_support_def: fixes z :: "'e"   and A :: "'d set" shows "fin_support z A = {f. finite (support z A f)}"
simps_1: fixes l :: "nat"   and i :: "nat" shows "(_ l _) \<oplus> i = (_ l + i _)"
good_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'c" shows "good P f = (\<exists>i j. i < j \<and> P (f i) (f j))"
lfp_to_gfp: shows "lfp = \<partial> \<circ> gfp \<circ> \<partial>\<^sub>F"
cases: fixes x :: "call \<times> (nat \<Rightarrow> contour option) \<times> ((nat \<times> char list) \<times> contour \<Rightarrow> d option) \<times> contour" obtains c :: "call" and \<beta> :: "nat \<Rightarrow> contour option" and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option" and b :: "contour" where "x = (c, \<beta>, ve, b)"
simps_6: fixes n :: "nat"   and p :: "pf" shows "dfa_of_pf' n (Exist p) = min_dfa (rquot (det_nfa (quantify_nfa 0 (nfa_of_dfa (dfa_of_pf' (Suc n) p)))) n)"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> derived G (carrier G)"   and "y \<in> derived G (carrier G)" shows "x \<otimes> y \<in> derived G (carrier G)"
rcOpen: fixes P' :: "pi" assumes "P \<longmapsto> a[y] \<prec> P'"   and "a \<noteq> y" shows "F ([(x, y)] \<bullet> P')"
compare_Maybe_simps_4: fixes x :: "'b"   and y :: "'b" shows "compare\<cdot>(Just\<cdot>x)\<cdot>(Just\<cdot>y) = compare\<cdot>x\<cdot>y"
norm_assertion_simps_17: fixes R :: "assn"   and Q :: "'a \<Rightarrow> assn" shows "R * (\<exists>\<^sub>Ax. Q x) = (\<exists>\<^sub>Ax. R * Q x)"
assigns_expr_list_simp: fixes es :: "expr list" shows "assigns (In3 es) = assignsEs es"
Span_append_eq_set_add: fixes Us :: "'a list"   and Vs :: "'a list" assumes "set Us \<subseteq> carrier R"   and "set Vs \<subseteq> carrier R" shows "Span K (Us @ Vs) = Span K Us <+> Span K Vs"
is_path_undir_simps_1: fixes G :: "('c, 'd) graph"   and v :: "'c" assumes "valid_graph G" shows "is_path_undir G v [] v = (v \<in> nodes G)"
wf_ruleset_def: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs :: "'b rule list" shows "wf_ruleset \<gamma> p rs \<equiv> \<forall>r\<in>set rs. \<not> matches \<gamma> (get_match r) (get_action r) p \<or> (\<nexists>chain. get_action r = Call chain) \<and> get_action r \<noteq> Return \<and> (\<nexists>chain. get_action r = Goto chain) \<and> get_action r \<noteq> Unknown"
comm_mult: fixes a :: "'b"   and b :: "'b" assumes "a \<in> B"   and "b \<in> B" shows "multB a b = multB b a"
return_def: fixes a :: "'c" shows "return a = State (Pair a)"
ys_def: shows "ys \<equiv> map ((!) ys' \<circ> f) [0..<length ys']"
sset_szip_finite: fixes u :: "'c stream"   and v :: "'d stream" shows "finite (sset (u ||| v)) = (finite (sset u) \<and> finite (sset v))"
Rep_real_int_inverse: fixes x :: "real_int" shows "Abs_real_int (Rep_real_int x) = x"
elims_2: fixes x :: "'a uprod set"   and xa :: "'a"   and xb :: "'a uprod list"   and xc :: "'a" assumes "epath x xa xb xc" obtains E :: "'a uprod set" and u :: "'a" and v :: "'a" where "x = E__" and "xa = u__" and "xb = []" and "xc = v__" and "u__ = v__"   | E :: "'a uprod set" and u :: "'a" and xd :: "'a uprod" and xs :: "'a uprod list" and v :: "'a" where "x = E" and "xa = u" and "xb = xd # xs" and "xc = v" and "(\<exists>w. u \<noteq> w \<and> Upair u w = xd \<and> epath E w xs v) \<and> xd \<in> E"
lesss_def: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and xs :: "'b list" shows "lesss R x xs = filter (\<lambda>y. (y, x) \<in> R \<and> (x, y) \<notin> R) xs"
elims_3: fixes x :: "('b \<times> nat) tree"   and xa :: "('b \<times> nat) tree" assumes "\<not> single x xa" obtains t1 :: "('b \<times> nat) tree" and t2 :: "('b \<times> nat) tree" where "x = t1" and "xa = t2" and "\<not> 10 * (size_wbt t1 + 1) < 14 * (size_wbt t2 + 1)"
inner_node_Entry_edge: fixes n :: "'node" assumes "inner_node n" obtains a :: "'edge" where "valid_edge a" and "inner_node (targetnode a)" and "sourcenode a = (_Entry_)"
fdj: shows "\<forall>p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))"
xs'_def: shows "xs' \<equiv> map (\<lambda>(j, y). if j = i then (j, x) else (j, y)) xs"
power: fixes n :: "nat"   and m :: "nat" shows "f (n ^ m) = f n ^ m"
prems_5: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S'))"
lcs_def: fixes G :: "('e, 'f) Group_scheme"   and a :: "'e"   and H :: "'e set" shows "a \<diamondsuit>\<^bsub>G\<^esub> H = {b. \<exists>h\<in>H. a \<cdot>\<^bsub>G\<^esub> h = b}"
perm1: fixes p :: "perm" assumes "supp p \<sharp>* c" shows "p \<bullet> f as x c = f (p \<bullet> as) (p \<bullet> x) c"
calc_x_axis_intersection_mono: fixes b1 :: "real"   and a1 :: "real"   and b2 :: "real"   and a2 :: "real" assumes "1 < - b1 / a1"   and "a1 \<noteq> 0"   and "- b1 / a1 \<le> - b2 / a2"   and "a2 \<noteq> 0" shows "(- b2 + sgn b2 * sqrt (b2\<^sup>2 - a2\<^sup>2)) / a2 \<le> (- b1 + sgn b1 * sqrt (b1\<^sup>2 - a1\<^sup>2)) / a1"
completeness_def: shows "local.\<Sigma>_protocols_base.completeness \<equiv> \<forall>h w e. (h, w) \<in> R \<longrightarrow> e \<in> challenge_space \<longrightarrow> spmf (local.\<Sigma>_protocols_base.completeness_game h w e) True = 1"
insR': fixes x :: "'a"   and t :: "('a \<times> real) tree" assumes "x \<in> A"   and "t \<in> space (tree_sigma (count_space A \<Otimes>\<^sub>M lborel))" shows "insR x t A = uniform_measure lborel {0..1} \<bind> (\<lambda>u. return (tree_sigma (count_space A \<Otimes>\<^sub>M lborel)) (ins x u t))"
finite_image_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "finite (f ` A) = finite A"
is_min2_mergeI: fixes x :: "'d \<times> 'e"   and xs :: "('d \<times> 'e) set"   and y :: "'d \<times> 'e"   and ys :: "('d \<times> 'e) set" assumes "is_min2 x xs"   and "is_min2 y ys" shows "is_min2 (min2 x y) (xs \<union> ys)"
adv_P1_def: fixes m1 :: "bool \<times> bool"   and m2 :: "bool"   and D :: "'v_OT121 \<Rightarrow> bool spmf" shows "OT_12_sim.adv_P1 m1 m2 D \<equiv> \<bar>spmf (R1_OT12 m1 m2 \<bind> D) True - spmf (funct_OT_12 m1 m2 \<bind> (\<lambda>(o1, o2). S1_OT12 m1 o1 \<bind> D)) True\<bar>"
comp1: fixes prod :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'c" assumes "bounded_bilinear prod"   and "bounded_linear g" shows "bounded_bilinear (\<lambda>x. prod (g x))"
strict_lunit: fixes f :: "'a" assumes "ide f" shows "\<l>[f] = f"
ladder_\<gamma>_def: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" shows "ladder_\<gamma> a D L index = Derive a (take (ladder_n L index) D)"
to_bl_mask: fixes n :: "nat" shows "to_bl (mask n) = replicate (LENGTH('b) - n) False @ replicate (min LENGTH('b) n) True"
br'_invar_def: fixes \<delta> :: "('d, 'e) ta_rule set" shows "br'_invar \<delta> \<equiv> br'_invar_add \<delta> \<inter> {\<Sigma>. br'_\<alpha> \<Sigma> \<in> br_invar \<delta>}"
r_enumdom: fixes i :: "nat"   and y :: "nat" shows "eval r_enumdom [i, y] = (if \<exists>z. eval r_dovedelay [z, i, y] \<down>= 0 then Some (pdec1 (LEAST z. eval r_dovedelay [z, i, y] \<down>= 0)) else None)"
vars_poly_of_vec: fixes v :: "rat vec" shows "vars (poly_of_vec v) \<subseteq> {0..<dim_vec v}"
cases: fixes a1 :: "'a list set"   and a2 :: "'a list set" assumes "bisimilar a1 a2" obtains K :: "'a list set" and L :: "'a list set" where "a1 = K" and "a2 = L" and "([] \<in> K) = ([] \<in> L)" and "\<And>x. bisimilar (Deriv x K) (Deriv x L)"
sorted_wrt_mono_rel: fixes xs :: "'f list"   and P :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "sorted_wrt P xs" shows "sorted_wrt Q xs"
elims: fixes x :: "('fun, 'var) extstrand_step list"   and y :: "('fun, 'var) strand_step list" assumes "to_st x = y" obtains "x = []" and "y = []"   | xa :: "('fun, 'var) strand_step" and S :: "('fun, 'var) extstrand_step list" where "x = Step xa # S__" and "y = xa # to_st S__"   | t :: "('fun, 'var) Term.term" and S :: "('fun, 'var) extstrand_step list" where "x = Decomp t # S" and "y = decomp t @ to_st S"
distrib_left: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a * (b + c) = a * b + a * c"
size_1_singleton_mset: fixes M :: "'a multiset" assumes "size M = 1" shows "\<exists>a. M = {#a#}"
elims_2: fixes x :: "('a, 'b) Combinators list" assumes "wellformed_policy1_strong x" obtains xa :: "('a, 'b) Combinators" and xs :: "('a, 'b) Combinators list" where "x = xa # xs" and "xa = DenyAll \<and> DenyAll \<notin> set xs"
csols_empty_iff: fixes t0 :: "real"   and x0 :: "'a" shows "(csols t0 x0 = {}) = (t0 \<notin> T \<or> x0 \<notin> X)"
tensor_mat_scale1: fixes m1 :: "'a mat"   and m2 :: "'a mat"   and a :: "'a" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d2 d2" shows "tensor_mat (a \<cdot>\<^sub>m m1) m2 = a \<cdot>\<^sub>m tensor_mat m1 m2"
exhaust: fixes y :: "atom" obtains x11 :: "int" and x12 :: "int list" where "y = Le x11 x12"   | x21 :: "int" and x22 :: "int" and x23 :: "int list" where "y = Dvd x21 x22 x23"   | x31 :: "int" and x32 :: "int" and x33 :: "int list" where "y = NDvd x31 x32 x33"
X: fixes x :: "'a set" assumes "x \<in> X" shows "x \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
assms_4: shows "AE x\<in>A in M. f x \<le> g x"
WHILE_unfold: fixes b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a nres"   and s :: "'a" shows "WHILE b f s = (if b s then f s \<bind> WHILE b f else RETURN s)"
fps_const_lr_inverse_2: fixes b :: "'a"   and y :: "'a" shows "fps_right_inverse (fps_const b) y = fps_const y"
finite_linorder_min_induct: fixes A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "finite A"   and "P {}"   and "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. b < a; P A\<rbrakk> \<Longrightarrow> P (insert b A)" shows "P A"
simps_20: fixes x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'d"   and x21 :: "nat"   and x22 :: "nat" shows "x31\<not>x32:arrive(x33) \<noteq> x21:deliver(x22)"