hcju/isabelleps huggingface dataset:
bounded_clinear_axioms_def: fixes f :: "'b \<Rightarrow> 'c" shows "bounded_clinear_axioms f \<equiv> \<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
inject_5: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) formula"   and y51 :: "'c"   and y52 :: "('a, 'b, 'c) formula" shows "(Exists x51 x52 = Exists y51 y52) = (x51 = y51 \<and> x52 = y52)"
rawpsubst_same_Var: fixes \<phi> :: "'fmla"   and xs :: "'var list" assumes "\<phi> \<in> fmla"   and "set xs \<subseteq> var" shows "rawpsubst \<phi> (map (\<lambda>x. (Var x, x)) xs) = \<phi>"
mresolution_sound: fixes F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool"   and C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set"   and L\<^sub>1 :: "fterm literal set"   and L\<^sub>2 :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "eval\<^sub>c F G C\<^sub>1"   and "eval\<^sub>c F G C\<^sub>2"   and "applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>" shows "eval\<^sub>c F G (mresolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>)"
wf_fourtimes: shows "tm_wf0 t_fourtimes_compile"
prod_topology_subtopology_2: fixes X :: "'d topology"   and Y :: "'e topology"   and T :: "'e set" shows "prod_topology X (subtopology Y T) = subtopology (prod_topology X Y) (topspace X \<times> T)"
rcos_self: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "x \<in> H #>\<^bsub>add_monoid G\<^esub> x"
median_5_aux_def: fixes x1 :: "'c"   and x2 :: "'b"   and x3 :: "'b"   and x4 :: "'b"   and x5 :: "'b" shows "median_5_aux x1 x2 x3 x4 x5 = (if x2 \<le> x3 then if x2 \<le> x4 then min x3 x4 else min x2 x5 else if x4 \<le> x3 then min x3 x5 else min x2 x4)"
select_convs_2: fixes basepoint :: "'b Point"   and direction :: "'b Vector"   and more :: "'c" shows "direction \<lparr>basepoint = basepoint, direction = direction, \<dots> = more\<rparr> = direction"
goal_sinvar_m: assumes "c_offending_flows m \<lparr>nodes = V, edges = E\<rparr> = {}" shows "c_sinvar m (generate_valid_topology_some M Es \<lparr>nodes = V, edges = E\<rparr>)"
sm_AF: fixes k :: "nat" shows "strict_mono_sets UNIV (AF k)"
eq: fixes x :: "'b"   and y :: "'b" shows "(if x = y then 1::'a else (0::'a)) = indicator {y} x"
cases: fixes a1 :: "ord0"   and a2 :: "ord0" assumes "(a1, a2) \<in> ord0_leq" obtains x :: "ord0" and y :: "ord0" where "a1 = x" and "a2 = y" and "\<forall>a. (a, x) \<in> ord0_prec\<^sup>+ \<longrightarrow> (\<exists>b. (b, y) \<in> ord0_prec\<^sup>+ \<and> (a, b) \<in> ord0_leq)"
wp_SetPC_nil: fixes a :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "wp (SetPC a (\<lambda>s a. 0)) = (\<lambda>P s. 0)"
assms: shows "f \<in> space\<^sub>N (\<LL> 1 M)"
of_real_inner_1: fixes x :: "real" shows "of_real x \<bullet> (1::'c) = x"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "exp \<Rightarrow> exp \<Rightarrow> 'b"   and f2 :: "Vars.var \<Rightarrow> 'b"   and f3 :: "Vars.var \<Rightarrow> 'b"   and f4 :: "Vars.var \<Rightarrow> 'b"   and stack_elem :: "stack_elem" shows "P (case stack_elem of Alts x xa \<Rightarrow> f1 x xa | Arg x \<Rightarrow> f2 x | Upd x \<Rightarrow> f3 x | Dummy x \<Rightarrow> f4 x) = ((\<forall>x11 x12. stack_elem = Alts x11 x12 \<longrightarrow> P (f1 x11 x12)) \<and> (\<forall>x2. stack_elem = Arg x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x3. stack_elem = Upd x3 \<longrightarrow> P (f3 x3)) \<and> (\<forall>x4. stack_elem = Dummy x4 \<longrightarrow> P (f4 x4)))"
SeqStTermP_graph_aux_def: shows "SeqStTermP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SeqStTermP_aux l s k v i sl sl' m n sm sm' sn sn' t u. x1 = (v, i, t, u, s, k) \<and> x2 = VarP v AND LstSeqP s k (HPair t u) AND All2 l (SUCC k) (SyntaxN.Ex sl (SyntaxN.Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND ((Var sl EQ v AND Var sl' EQ i OR (IndP (Var sl) OR Var sl NEQ v) AND Var sl' EQ Var sl) OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND Var sl EQ Q_Eats (Var sm) (Var sn) AND Var sl' EQ Q_Eats (Var sm') (Var sn'))))))))))) \<and> atom l \<sharp> (s, k, v, i, sl, sl', m, n, sm, sm', sn, sn') \<and> atom sl \<sharp> (s, v, i, sl', m, n, sm, sm', sn, sn') \<and> atom sl' \<sharp> (s, v, i, m, n, sm, sm', sn, sn') \<and> atom m \<sharp> (s, n, sm, sm', sn, sn') \<and> atom n \<sharp> (s, sm, sm', sn, sn') \<and> atom sm \<sharp> (s, sm', sn, sn') \<and> atom sm' \<sharp> (s, sn, sn') \<and> atom sn \<sharp> (s, sn') \<and> atom sn' \<sharp> s)"
inj_on_restrict_eq: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "inj_on (restrict f A) A = inj_on f A"
is_SDG_path_Append: 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 Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n'' :: "'b SDG_node"   and ns' :: "'b SDG_node list"   and n' :: "'b SDG_node"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' ns' n'"   and "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n''" shows "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (ns @ ns') n'"
assms: shows "ys \<noteq> zs \<or> y \<notin> {z1, z2}"
coinduct: fixes X :: "('test, 'atom, 'choice) cmd \<Rightarrow> bool"   and x :: "('test, 'atom, 'choice) cmd" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>c. x = c \<and> (\<forall>x xa. xa < brn c \<longrightarrow> x \<approx> eff c x xa \<and> (X (local.cont c x xa) \<or> discr (local.cont c x xa)))" shows "discr x"
approx_form_aform: fixes p :: "nat"   and f :: "form"   and VS :: "(real \<times> real pdevs) list"   and vs :: "real list" assumes "approx_form_aform p f VS"   and "vs \<in> Joints VS" shows "interpret_form f vs"
fk'_1: fixes t :: "real" assumes "0 < t" shows "(\<lambda>x. nth_derivative k f x 1) differentiable at t"
remove_cycles_subs: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list" shows "set (remove_cycles xs x ys) \<subseteq> set xs \<union> set ys"
intros_1: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xs :: "'c stream" assumes "\<phi> xs" shows "ev \<phi> xs"
mem_Collect_eq: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(a \<in> p2s P) = P a"
inv_prod_rel_eq: fixes P :: "('f \<times> 'd) set"   and Q :: "('g \<times> 'e) set" shows "(P \<times>\<^sub>r Q)\<inverse> = P\<inverse> \<times>\<^sub>r Q\<inverse>"
path_2: fixes G :: "('e, 'f) Graph_scheme"   and v :: "'e"   and w :: "'e" assumes "Graph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w"   and "v \<noteq> w" shows "Graph.walk G [v, w] \<and> distinct [v, w]"
assms_1: assumes "c = \<bottom>" shows "e = \<bottom>"
v: shows "f `` {v} \<subseteq> vertices (S (max i j))"
rev_filter: fixes P :: "'k \<Rightarrow> bool"   and xs :: "'k list" shows "rev (filter P xs) = filter P (rev xs)"
less_synlr_def: fixes x :: "('a, 'b) synlr"   and y :: "('a, 'b) synlr" shows "x < y \<equiv> unsynlr x \<subset> unsynlr y"
AuthResUns: fixes evsAResU :: "event list"   and C :: "agent"   and k :: "nat"   and M :: "agent"   and i :: "nat"   and KP :: "nat"   and KC1 :: "nat"   and KM :: "nat"   and PIHead :: "msg"   and LID_M :: "nat"   and XID :: "nat"   and HOD :: "msg"   and PurchAmt :: "nat"   and P_I :: "msg"   and EKj :: "nat"   and HOIData :: "msg"   and P :: "agent" assumes "evsAResU \<in> set_pur"   and "C = Cardholder k"   and "M = Merchant i"   and "Key KP \<notin> used evsAResU"   and "KP \<in> symKeys"   and "CardSecret k = 0"   and "KC1 \<in> symKeys"   and "KM \<in> symKeys"   and "PIHead = \<lbrace>Number LID_M, Number XID, HOD, Number PurchAmt, Agent M\<rbrace>"   and "P_I = EXHcrypt KC1 EKj \<lbrace>PIHead, HOIData\<rbrace> (Pan (pan C))"   and "Gets P (EncB (priSK M) KM (pubEK P) \<lbrace>Number LID_M, Number XID, HOIData, HOD\<rbrace> P_I) \<in> set evsAResU" shows "Says P M (EncB (priSK P) KP (pubEK M) \<lbrace>Number LID_M, Number XID, Number PurchAmt\<rbrace> authCode) # evsAResU \<in> set_pur"
inject_4: fixes x4 :: "real mpoly"   and y4 :: "real mpoly" shows "(Neq x4 = Neq y4) = (x4 = y4)"
measurable_pair: fixes f :: "'c \<Rightarrow> 'd \<times> 'e"   and M :: "'c measure"   and M1 :: "'d measure"   and M2 :: "'e measure" assumes "fst \<circ> f \<in> M \<rightarrow>\<^sub>M M1"   and "snd \<circ> f \<in> M \<rightarrow>\<^sub>M M2" shows "f \<in> M \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"
consistency_onI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<And>B C. \<lbrakk>B \<subseteq> A; C \<subseteq> B; f B \<subseteq> C\<rbrakk> \<Longrightarrow> f B = f C" shows "consistency_on A f"
homotopic_constant_maps: fixes X :: "'a topology"   and X' :: "'b topology"   and a :: "'b"   and b :: "'b" shows "homotopic_with (\<lambda>x. True) X X' (\<lambda>x. a) (\<lambda>x. b) = (topspace X = {} \<or> path_component_of X' a b)"
Ab_def: shows "Ab \<equiv> quot_type.abs eq Abs_msg"
distinct_1: fixes a :: "'a"   and b' :: "'b" shows "Inl a \<noteq> Inr b'"
path_append: fixes g :: "'b ugraph"   and u :: "'b"   and p1 :: "('b \<times> 'b) list"   and p2 :: "('b \<times> 'b) list"   and w :: "'b" shows "path g u (p1 @ p2) w = (\<exists>v. path g u p1 v \<and> path g v p2 w)"
int_and_assoc: fixes x :: "int"   and y :: "int"   and z :: "int" shows "(x AND y) AND z = x AND y AND z"
naturality: fixes f :: "'a" assumes "arr f" shows "\<t>[cod f] \<cdot> local.map f = \<Omega>.map f \<cdot> \<t>[local.dom f]"
vc_rec_var: fixes S :: "'v set"   and V :: "'v set"   and v :: "'v"   and it :: "'v set"   and V' :: "'v set" assumes "rpre S (V, v)"   and "fe_inv S V v it (V', False)" shows "(V', V) \<in> finite_psupset local.reachable"
trailIsDecreasedByDeciedUnitPropagateAndBackjump: fixes stateA :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set"   and stateB :: "State" assumes "invariantsHoldInState stateA F0 decisionVars"   and "appliedDecide stateA stateB decisionVars \<or> appliedUnitPropagate stateA stateB F0 decisionVars \<or> appliedBackjump stateA stateB" shows "(getM stateB, getM stateA) \<in> lexLessRestricted (vars F0 \<union> decisionVars)"
subgroup_indicator_le: fixes a :: "'a"   and n :: "nat"   and H :: "'a set" assumes "a [^] n \<in> H"   and "0 < n"   and "a \<in> carrier G" shows "subgroup_indicator G H a \<le> n"
induct: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and b :: "'b"   and x :: "('a, 'b, 'c, 'd) converter"   and P :: "('a, 'b, 'c, 'd) converter \<Rightarrow> bool" assumes "resultsp_converter \<I> \<I>' b x"   and "\<And>conv' conv a. \<lbrakk>(b, conv') \<in> results_gpv \<I>' (run_converter conv a); a \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> P conv"   and "\<And>b' conv' conv a. \<lbrakk>(b', conv') \<in> results_gpv \<I>' (run_converter conv a); resultsp_converter \<I> \<I>' b conv'; P conv'; a \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> P conv" shows "P x"
G_face_cycle_set_a: shows "G.face_cycle_set a = {a}"
assms_5: shows "p \<succeq>[\<R>] r \<and> (r, p) \<notin> \<R>"
\<R>\<^sub>3_mem_eq: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" shows "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2"
\<alpha>_def: shows "\<alpha> \<equiv> fun_of_rel R"
BIT_c_2: fixes x :: "'a"   and y :: "'a"   and init :: "'a list"   and v :: "'a list" assumes "x \<noteq> y"   and "init \<in> {[x, y], [y, x]}"   and "v \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom x])" shows "Partial_Cost_Model.config'_rand BIT (type0 init x y) v = type0 init x y"
vars_add_monom: fixes p2 :: "'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a"   and p1 :: "'a mpoly" assumes "p2 = MPoly_Type.monom m a"   and "m \<notin> keys (mapping_of p1)" shows "vars (p1 + p2) = vars p1 \<union> vars p2"
croot_def: fixes n :: "nat"   and x :: "complex" shows "croot n x = rcis (root n (cmod x)) (arg x / real n)"
assms_1: shows "w1 \<equiv>\<^sub>U y"
cases: fixes a1 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a2 :: "nat \<Rightarrow> nat"   and a3 :: "pr_operation'"   and a4 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a5 :: "nat \<Rightarrow> nat" assumes "((a1, a2), a3, a4, a5) \<in> pr_algo_lts'" obtains f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = fa__" and "a2 = la__" and "a3 = NONSAT_PUSH'" and "a4 = push_effect fa__ e__" and "a5 = la__" and "Height_Bounded_Labeling c s t fa__ la__" and "nonsat_push_precond fa__ la__ e__"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = fb__" and "a2 = lb__" and "a3 = SAT_PUSH' e" and "a4 = push_effect fb__ e" and "a5 = lb__" and "Height_Bounded_Labeling c s t fb__ lb__" and "sat_push_precond fb__ lb__ e"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and u :: "nat" where "a1 = f" and "a2 = l" and "a3 = RELABEL'" and "a4 = f" and "a5 = relabel_effect f l u" and "Height_Bounded_Labeling c s t f l" and "relabel_precond f l u"
rbt_union_is_rbt: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lt :: "('a, 'b) RBT_Impl.rbt"   and rt :: "('a, 'b) RBT_Impl.rbt" assumes "class.linorder less_eq less"   and "ord.is_rbt less lt"   and "ord.is_rbt less rt" shows "ord.is_rbt less (ord.rbt_union less lt rt)"
distinct_adj_remdups_adj: fixes xs :: "'g list" shows "distinct_adj (remdups_adj xs)"
assms: shows "h \<turnstile> f \<rightarrow>\<^sub>r x"
ipassmt_iface_constrain_srcip_mexpr_def: fixes ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and ifce :: "iface" shows "ipassmt_iface_constrain_srcip_mexpr ipassmt ifce = (case ipassmt ifce of None \<Rightarrow> Match (IIface ifce) | Some ips \<Rightarrow> MatchAnd (Match (IIface ifce)) (match_list_to_match_expr (map (Match \<circ> Src) (map (uncurry IpAddrNetmask) ips))))"
vsv_vlrestriction: fixes A :: "V" shows "vsv (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
odd_of_distinct_def: fixes p :: "nat \<Rightarrow> nat" shows "odd_of_distinct p = (\<lambda>i. if odd i then \<Sum>j | p (2 ^ j * i) = 1. 2 ^ j else 0)"
pred_inc: fixes n :: "Arity" shows "pred\<cdot>(inc\<cdot>n) = n"
o_inv_distrib: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b" assumes "bij f"   and "bij g" shows "inv (f \<circ> g) = inv g \<circ> inv f"
x: fixes y :: "'b" assumes "x \<le> y" shows "P y"
memi_agree_def: fixes sm :: "'a list"   and j :: "nat option"   and m :: "'a option" shows "memi_agree sm j m = ((\<exists>j' m'. j = Some j' \<and> j' < length sm \<and> m = Some m' \<and> sm ! j' = m') \<or> j = None \<and> m = None)"
useful_tautologies_7: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<not>\<phi> \<^bold>\<rightarrow> \<psi> \<^bold>\<rightarrow> (\<^bold>\<not>\<psi> \<^bold>\<rightarrow> \<phi>) in v]"
map_le_cong: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option"   and s :: "'a \<Rightarrow> 'b option" assumes "\<And>x. m1 x = m2 x" shows "(m1 \<subseteq>\<^sub>m s) = (m2 \<subseteq>\<^sub>m s)"
not_NoMsg_imp_last_message: fixes i :: "nat"   and xs :: "'b message_af list" assumes "i < length xs"   and "xs ! i \<noteq> NoMsg" shows "last_message xs \<noteq> NoMsg"
nofail_def: fixes S :: "'a nres" shows "nofail S \<equiv> S \<noteq> FAIL"
us: shows "us = getFrN (map snd txs) (map fst txs) [eqv \<phi>1 \<phi>2] (length txs)"
assms: shows "x \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>"
adjust_fetch_norm: fixes st :: "nat"   and tp :: "'a list"   and ap :: "(action \<times> nat) list"   and b :: "cell"   and aa :: "action"   and ns :: "nat" assumes "0 < st"   and "st \<le> length tp div 2"   and "fetch ap st b = (aa, ns)"   and "ns \<noteq> 0" shows "fetch (adjust0 ap) st b = (aa, ns)"
map_of_zip_map: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" shows "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
ide_dom: fixes f :: "'d" assumes "Fop.Aop.arr f" shows "Fop.Aop.ide (Fop.Aop.dom f)"
ct_list_eq_common_tsome_list: fixes xs :: "ct list"   and ys :: "ct list" shows "ct_list_eq xs ys = (\<exists>zs. ct_list_eq xs (to_ct_list zs) \<and> ct_list_eq (to_ct_list zs) ys)"
WriteNonVolatile: fixes a :: "nat"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O>s :: "nat set list"   and i :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool" assumes "a \<in> \<O>"   and "a \<notin> dom \<S>" shows "\<O>s,i\<turnstile> (Write False a (D, f) A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
Y: shows "igWls (asIMOD SEM) (asSort ys) Y"
induct: fixes P :: "'a tree23s \<Rightarrow> bool"   and a0 :: "'a tree23s" assumes "\<And>t. P (T t)"   and "\<And>v va vb. P (join_adj (TTs v va vb)) \<Longrightarrow> P (TTs v va vb)" shows "P a0"
ta_upd_proj_simps_2: fixes ta :: "('d, 'a, 'b, 'c, 'e, 'f) thread_action"   and obs :: "'f" shows "\<lbrace>ta_update_obs ta obs\<rbrace>\<^bsub>t\<^esub> = \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"
star_decom: fixes x :: "'b list"   and A :: "'b list set" assumes "x \<in> star A"   and "x \<noteq> []" shows "\<exists>a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> A \<and> b \<in> star A"
integral_cong_AE: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L M g"
simps_2: shows "cospan_shape.Cod cospan_shape.BB = cospan_shape.BB"
divide_divide_eq_left': fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (divide a b) c = divide a (times c b)"
cblinfun_id_cblinfun_apply: shows "cblinfun_apply id_cblinfun = (\<lambda>x. x)"
errMOD_igSwapIGAbsSTR: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igAbsIPresIGWls Delta MOD"   and "igWlsDisj MOD"   and "igSwapIPresIGWls MOD"   and "FixSyn.igSwapIGAbs Delta MOD" shows "igSwapIGAbsSTR (FixSyn.errMOD Delta MOD)"
wf_imp_SN: fixes A :: "('a \<times> 'a) set" assumes "wf (A\<inverse>)" shows "SN A"
of_int_round_gt: fixes x :: "'a" shows "x - (1::'a) / (2::'a) < of_int (round x)"
\<T>o_e_ide_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and e :: "'b"   and b :: "'a" assumes "yoneda_lemma C S \<phi> F a"   and "e \<in> set_category_data.set S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (F a)"   and "partial_magma.ide C b" shows "partial_magma.in_hom S (yoneda_lemma.\<T>o C S \<phi> F a e b) (yoneda_functor.Y C S \<phi> a b) (F b)"
is_valid_problem_sas_plus_then_1: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). \<R>\<^sub>+ \<Psi> v \<noteq> {}"
stateful_strand_step_subst_inI_4: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and A :: "('a, 'b) stateful_strand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "insert\<langle>t,s\<rangle> \<in> set A" shows "insert\<langle>t \<cdot> \<theta>,s \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
induct: fixes x1 :: "('fun, 'var) Term.term set"   and x2 :: "('fun, 'var) Term.term"   and P :: "('fun, 'var) Term.term set \<Rightarrow> ('fun, 'var) Term.term \<Rightarrow> bool" assumes "x1 \<turnstile>\<^sub>c x2"   and "\<And>t M. t \<in> M \<Longrightarrow> P M t"   and "\<And>T f M. \<lbrakk>length T = arity f; public f; \<And>t. t \<in> set T \<Longrightarrow> M \<turnstile>\<^sub>c t; \<And>t. t \<in> set T \<Longrightarrow> P M t\<rbrakk> \<Longrightarrow> P M (Fun f T)" shows "P x1 x2"
word_of_int_Ex: fixes x :: "'b word" shows "\<exists>y. x = word_of_int y"
mult_cancel_left2: fixes 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 c :: "'a"   and a :: "'a" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(times c a = c) = (c = zero \<or> a = one)"
OR_def: fixes x2 :: "ty"   and x1 :: "ty" shows "x2 OR x1 \<equiv> Abs_ty (OR_Rep (Rep_ty x2) (Rep_ty x1))"
int_minus: fixes n :: "nat"   and m :: "nat" shows "int (n - m) = int (nat (int n - int m))"
arc_regular: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_stone_relation_algebra.arc x" shows "matrix_p_algebra.regular x"
that: fixes a :: "'a" assumes "a \<in> A"   and "ys = [a]" shows "thesis"
dinvar_def: fixes \<sigma> :: "'V set \<times> ('V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option)" shows "dinvar \<sigma> \<equiv> let (wl, res) = \<sigma> in wl \<subseteq> V \<and> (\<forall>v\<in>V. \<forall>p. res v = Some p \<longrightarrow> is_path v0 p v \<and> int_vertices p \<subseteq> V - wl) \<and> (\<forall>v\<in>V - wl. \<forall>p. is_path v0 p v \<longrightarrow> path_weight' (res v) \<le> path_weight' (Some p)) \<and> (\<forall>v\<in>wl. \<forall>p. is_path v0 p v \<and> int_vertices p \<subseteq> V - wl \<longrightarrow> path_weight' (res v) \<le> path_weight' (Some p))"
IS_LEFT_UNIQUE_def: fixes R :: "('d \<times> 'e) set" shows "IS_LEFT_UNIQUE R \<equiv> single_valued (R\<inverse>)"
split_int_ge_real': fixes a :: "int"   and b :: "real" shows "(0 \<le> real_of_int a + b) = (0 \<le> real_of_int a + real_of_int \<lfloor>b\<rfloor> \<or> real_of_int a + real_of_int \<lfloor>b\<rfloor> = 0 \<and> real_of_int \<lfloor>b\<rfloor> - b < 0)"
exhaust: fixes y :: "('c, 'd) step" obtains "y = Done"   | x2 :: "'d" where "y = Skip x2"   | x31 :: "'c" and x32 :: "'d" where "y = Yield x31 x32"
b_e_cases: obtains "b_e = Unop_i t iop"   | "b_e = Unop_f t fop"   | "b_e = Testop t testop"
normalize_def: fixes x :: "'a" shows "Projections.normalize x = (1 / norm x) *\<^sub>R x"
IH: shows "(\<forall>x\<in>F. \<exists>y. P y x) \<longrightarrow> (\<exists>f. \<forall>x\<in>F. P (f x) x)"