hcju/isabelleps huggingface dataset:
BlockThrow: fixes V :: "char list"   and T :: "ty"   and a :: "nat"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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" shows "P \<turnstile> \<langle>{V:T; Throw a},s\<rangle> \<rightarrow> \<langle>Throw a,s\<rangle>"
sm: shows "(clinit, Static, [], Void, pns, body) \<in> set ms"
smaller_antisym: fixes a :: "('b, 'c) Combinators"   and b :: "('b, 'c) Combinators"   and l :: "'b set list" assumes "\<not> smaller a b l"   and "in_list a l"   and "in_list b l"   and "singleCombinators [a]"   and "singleCombinators [b]" shows "smaller b a l"
safe_reach_safe_rtrancl: fixes step :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and safe :: "'b \<Rightarrow> bool"   and cfg :: "'b"   and cfg' :: "'b" assumes "safe_reach step safe cfg"   and "step\<^sup>*\<^sup>* cfg cfg'" shows "safe cfg'"
nth_length_takeWhile: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "length (takeWhile P xs) < length xs" shows "\<not> P (xs ! length (takeWhile P xs))"
Int_open: fixes x :: "'a set"   and y :: "'a set" assumes "x open"   and "y open" shows "x \<inter> y open"
case_1: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case Eq of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f1"
proj_arcs_G_simps_2: shows "proj_arcs_G (rev_G uv) = vw"
accI: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "\<And>y. r y x \<Longrightarrow> Wellfounded.accp r y" shows "Wellfounded.accp r x"
rule_\<alpha>_type: fixes N :: "symbol"   and \<alpha> :: "symbol list" assumes "(N, \<alpha>) \<in> \<RR>" shows "is_sentence \<alpha>"
inconsistent_eq_solve_eq_none: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "inconsistent A b = (solve A b = None)"
simps_2: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and L :: "lock_action"   and Ls :: "lock_action list" shows "lock_actions_ok l t (L # Ls) = (lock_action_ok l t L \<and> lock_actions_ok (upd_lock l t L) t Ls)"
cfg_on_not_empty: fixes s :: "'s" shows "cfg_on s \<noteq> {}"
m1: fixes C :: "'a set" assumes "C \<in> \<M>" shows "aff_dim C = - 1"
rexp_DA_def: fixes init :: "'a rexp \<Rightarrow> 'b"   and delta :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and final :: "'b \<Rightarrow> bool"   and L :: "'b \<Rightarrow> 'a list set" shows "rexp_DA init delta final L \<equiv> (\<forall>r. L (init r) = lang r) \<and> (\<forall>a s. L (delta a s) = Deriv a (L s)) \<and> (\<forall>s. final s = ([] \<in> L s))"
simps_5: fixes x :: "'b trm"   and y :: "'b trm"   and "next" :: "indices list"   and u :: "'b trm"   and S :: "'b trm" shows "replace_subterm (x \<cdot> y) (indices.Right # next) u S = (\<exists>S2. replace_subterm y next u S2 \<and> S = x \<cdot> S2)"
shift_operator_ergodic_axioms: shows "shift_operator_ergodic M"
Sup_ereal_close: fixes e :: "ereal"   and S :: "ereal set" assumes "0 < e"   and "\<bar>Sup S\<bar> \<noteq> \<infinity>"   and "S \<noteq> {}" shows "\<exists>x\<in>S. Sup S - e < x"
arc_fst_in_verts: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs G" shows "fst e \<in> pverts G"
inf_assertion: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top"   and "y \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "inf x y \<in> mbt_algebra.assertion one times inf less_eq dual top"
span_eq_dim: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and T :: "'b set" assumes "vector_space scale"   and "module.span scale S = module.span scale T" shows "vector_space.dim scale S = vector_space.dim scale T"
intros_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and b :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P \<inter> b) c\<^sub>1 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P \<inter> - b) c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Cond b c\<^sub>1 c\<^sub>2 Q,A"
start_Subobj: 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 a :: "nat"   and C :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and Cs :: "char list list"   and fs :: "char list \<Rightarrow> val option" assumes "start_heap P a = \<lfloor>(C, S)\<rfloor>"   and "(Cs, fs) \<in> S" shows "Subobjs P C Cs"
fw_upd'_spec: fixes M :: "nat \<times> nat \<Rightarrow> 'b"   and k :: "nat"   and i :: "nat"   and j :: "nat" shows "fw_upd' M k i j \<le> SPEC (\<lambda>M'. M' = uncurry (fw_upd (curry M) k i j))"
preserves_arr: fixes f :: "'d" assumes "D.arr f" shows "D.arr (src\<^sub>D f)"
simps_1: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref" shows "Dag p l r Tip = (p = Null)"
ran_distinct: fixes al :: "('a \<times> 'b) list" assumes "distinct (map fst al)" shows "ran (map_of al) = snd ` set al"
assms_2: shows "\<forall>f T. Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t S \<longrightarrow> T = [] \<or> (\<exists>u\<in>set T. u \<notin> Var ` X)"
loop_orientable_def: fixes x :: "'a" shows "loop_orientable x \<equiv> \<exists>y. loop_orientation x y"
NextPerformOpD_1: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and \<Delta>neg :: "('loc \<times> 't) multiset"   and \<Delta>mint_msg :: "('b \<times> 'loc \<times> 't) multiset"   and \<Delta>mint_self :: "('loc \<times> 't) multiset" assumes "NextPerformOp' c0 c1 p \<Delta>neg \<Delta>mint_msg \<Delta>mint_self" shows "cri.next_performop' (exchange_config c0) (exchange_config c1) p \<Delta>neg \<Delta>mint_msg \<Delta>mint_self"
B: fixes i :: "'m"   and j :: "'n" shows "B $ i $ j \<in> \<rat>"
lasso_of_prpl_def: fixes prpl :: "'c list \<times> 'c list" shows "lasso_of_prpl prpl \<equiv> case prpl of (pr, pl) \<Rightarrow> \<lparr>lasso_reach = pr, lasso_va = hd pl, lasso_cysfx = tl pl\<rparr>"
simps_1: shows "cospan_shape.Dom cospan_shape.AA = cospan_shape.AA"
set_of_appr_nonempty: fixes X :: "'b list" shows "set_of_appr X \<noteq> {}"
lock_actions_ok_append: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and Ls :: "lock_action list"   and Ls' :: "lock_action list" shows "lock_actions_ok l t (Ls @ Ls') = (lock_actions_ok l t Ls \<and> lock_actions_ok (upd_locks l t Ls) t Ls')"
assms_2: shows "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"
inv_ordermap_VWF_mono_le: fixes M :: "V set"   and \<alpha> :: "V"   and \<beta> :: "V" assumes "inv_into M (ordermap M VWF) \<alpha> \<le> inv_into M (ordermap M VWF) \<beta>"   and "M \<subseteq> ON"   and "small M"   and "\<alpha> \<in> elts (ordertype M VWF)"   and "\<beta> \<in> elts (ordertype M VWF)" shows "\<alpha> \<le> \<beta>"
Loop'_def: shows "Loop' \<equiv> {(stk, mrk). distinct stk}"
simps_48: fixes x61 :: "Arraytype"   and x62 :: "nat" shows "nullV \<noteq> arrV x61 x62"
B_greater_zero: fixes p :: "nat"   and a :: "int"   and x :: "int" assumes "GAUSS p a"   and "x \<in> GAUSS.B p a" shows "0 < x"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and f3 :: "'b val \<Rightarrow> 'a"   and f4 :: "String.literal \<Rightarrow> 'a"   and f5 :: "ty \<Rightarrow> 'a"   and f6 :: "'a"   and f7 :: "'a"   and f8 :: "'a"   and f9 :: "String.literal \<Rightarrow> String.literal \<Rightarrow> 'a"   and f10 :: "String.literal \<Rightarrow> String.literal \<Rightarrow> 'a"   and f11 :: "String.literal \<Rightarrow> String.literal \<Rightarrow> 'a"   and f12 :: "ty \<Rightarrow> 'a"   and f13 :: "ty \<Rightarrow> 'a"   and f14 :: "String.literal \<Rightarrow> nat \<Rightarrow> 'a"   and f15 :: "'a"   and f16 :: "'a"   and f17 :: "'a"   and f18 :: "'a"   and f19 :: "bop \<Rightarrow> 'a"   and f20 :: "int \<Rightarrow> 'a"   and f21 :: "int \<Rightarrow> 'a"   and f22 :: "'a"   and f23 :: "'a"   and f24 :: "'a"   and instr :: "'b instr" shows "P (case instr of Load x \<Rightarrow> f1 x | Store x \<Rightarrow> f2 x | Push x \<Rightarrow> f3 x | New x \<Rightarrow> f4 x | NewArray x \<Rightarrow> f5 x | ALoad \<Rightarrow> f6 | AStore \<Rightarrow> f7 | ALength \<Rightarrow> f8 | Getfield x xa \<Rightarrow> f9 x xa | Putfield x xa \<Rightarrow> f10 x xa | CAS x xa \<Rightarrow> f11 x xa | Checkcast x \<Rightarrow> f12 x | Instanceof x \<Rightarrow> f13 x | Invoke x xa \<Rightarrow> f14 x xa | Return \<Rightarrow> f15 | Pop \<Rightarrow> f16 | Dup \<Rightarrow> f17 | Swap \<Rightarrow> f18 | BinOpInstr x \<Rightarrow> f19 x | Goto x \<Rightarrow> f20 x | IfFalse x \<Rightarrow> f21 x | ThrowExc \<Rightarrow> f22 | MEnter \<Rightarrow> f23 | MExit \<Rightarrow> f24) = ((\<forall>x1. instr = Load x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. instr = Store x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x3. instr = Push x3 \<longrightarrow> P (f3 x3)) \<and> (\<forall>x4. instr = New x4 \<longrightarrow> P (f4 x4)) \<and> (\<forall>x5. instr = NewArray x5 \<longrightarrow> P (f5 x5)) \<and> (instr = ALoad \<longrightarrow> P f6) \<and> (instr = AStore \<longrightarrow> P f7) \<and> (instr = ALength \<longrightarrow> P f8) \<and> (\<forall>x91 x92. instr = Getfield x91 x92 \<longrightarrow> P (f9 x91 x92)) \<and> (\<forall>x101 x102. instr = Putfield x101 x102 \<longrightarrow> P (f10 x101 x102)) \<and> (\<forall>x111 x112. instr = CAS x111 x112 \<longrightarrow> P (f11 x111 x112)) \<and> (\<forall>x12. instr = Checkcast x12 \<longrightarrow> P (f12 x12)) \<and> (\<forall>x13. instr = Instanceof x13 \<longrightarrow> P (f13 x13)) \<and> (\<forall>x141 x142. instr = Invoke x141 x142 \<longrightarrow> P (f14 x141 x142)) \<and> (instr = Return \<longrightarrow> P f15) \<and> (instr = Pop \<longrightarrow> P f16) \<and> (instr = Dup \<longrightarrow> P f17) \<and> (instr = Swap \<longrightarrow> P f18) \<and> (\<forall>x19. instr = BinOpInstr x19 \<longrightarrow> P (f19 x19)) \<and> (\<forall>x20. instr = Goto x20 \<longrightarrow> P (f20 x20)) \<and> (\<forall>x21. instr = IfFalse x21 \<longrightarrow> P (f21 x21)) \<and> (instr = ThrowExc \<longrightarrow> P f22) \<and> (instr = MEnter \<longrightarrow> P f23) \<and> (instr = MExit \<longrightarrow> P f24))"
is_basis: fixes p :: "'b set \<Rightarrow> bool"   and B :: "'b set set" assumes "countable_basis p B" shows "topological_space.topological_basis p B"
hyps: fixes m' :: "nat \<Rightarrow> nat"   and m :: "nat \<Rightarrow> nat" assumes "a \<in> (\<Union>a\<in>set ls. case a of (uu_, uua_, uub_, sb, uuc_, uud_, uue_) \<Rightarrow> outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb))" shows "flush_all_until_volatile_write ls m' a = flush_all_until_volatile_write ls m a"
Q: fixes x :: "'a"   and y :: "'b" assumes "P x y" shows "Q1 x = Q2 y"
preserves_dom: fixes f :: "'a" assumes "S.arr f" shows "S.dom (S.src f) = S.src (S.dom f)"
it_to_sorted_list_def: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and s :: "'f set" shows "it_to_sorted_list R s \<equiv> SPEC (\<lambda>l. distinct l \<and> s = set l \<and> sorted_wrt R l)"
that: fixes B :: "'a cset" assumes "A = cinsert x B"   and "\<not> cin x B" shows "thesis"
nat_0_iff: fixes i :: "int" shows "(nat i = 0) = (i \<le> 0)"
select_convs_4: fixes statename :: "nat option"   and datastate :: "nat \<Rightarrow>f value option"   and action :: "String.literal \<times> value list"   and "output" :: "value option list"   and more :: "'a" shows "output \<lparr>statename = statename, datastate = datastate, action = action, output = output, \<dots> = more\<rparr> = output"
DirProd_group: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme" assumes "Group.group G"   and "Group.group H" shows "Group.group (G \<times>\<times> H)"
CAS_\<tau>execI3: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e'' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1"   and D :: "String.literal"   and F :: "String.literal"   and e' :: "'addr expr1"   and v :: "'addr val"   and v' :: "'addr val" assumes "\<tau>exec_move ci P t e'' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (e\<bullet>compareAndSwap(D\<bullet>F, e', e'')) h (stk @ [v, v'], loc, length (compE2 e) + length (compE2 e') + pc, xcp) (stk' @ [v, v'], loc', length (compE2 e) + length (compE2 e') + pc', xcp')"
RESUMPTION_corec_type: fixes f :: "'j \<Rightarrow> ('h, 'g, 'f, 'i, 'j) resumption_corec"   and e :: "'j" shows "RESUMPTION_corec f e \<in> resumption"
upper_triangular_mult_diagonal: fixes A :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'b) vec"   and i :: "'b" assumes "upper_triangular A"   and "upper_triangular B" shows "(A ** B) $ i $ i = A $ i $ i * B $ i $ i"
exhaust: fixes y :: "('e, 'f) lit" obtains x1 :: "('e, 'f) atm" where "y = Pos x1"   | x2 :: "('e, 'f) atm" where "y = Neg x2"
outside_subset_convex: fixes T :: "'a set"   and S :: "'a set" assumes "convex T"   and "S \<subseteq> T" shows "- T \<subseteq> outside S"
insert_I_eq: shows "insert a I = {i. i < dim_row A \<and> i \<in> insert a I}"
S2MemUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [RNext rmCh mm ires p]_(m p) \<and> $S2 rmhist p \<longrightarrow> unchanged m p"
has_vector_derivative_add_const: fixes g :: "real \<Rightarrow> 'a"   and z :: "'a"   and f' :: "'a"   and net :: "real filter" shows "((\<lambda>t. g t + z) has_vector_derivative f') net = (g has_vector_derivative f') net"
is_lasso_prpl_conv: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and m :: "'a"   and prpl :: "('Q \<times> nat) list \<times> ('Q \<times> nat) list" shows "gb_graph.is_lasso_prpl (b_graph.to_gbg_ext (degeneralize_ext ma) m) prpl = (snd prpl \<noteq> [] \<and> gb_graph.is_lasso (b_graph.to_gbg_ext (degeneralize_ext ma) m) (lasso_of_prpl prpl))"
take_not_empty_conv: fixes n :: "nat"   and xs :: "'b list" shows "(xs \<down> n \<noteq> []) = (0 < n \<and> xs \<noteq> [])"
prepend_prepend: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b llist" shows "prepend xs (prepend ys zs) = prepend (xs @ ys) zs"
Glb_in_L: fixes A :: "'a set" assumes "A \<subseteq> L" shows "Glb A \<in> L"
map_simp_ide: fixes a :: "'d" assumes "D.ide a" shows "F\<eta>'o\<epsilon>'F.map a = (F \<circ> \<eta>'.map) a \<cdot>\<^sub>C (\<epsilon>'.map \<circ> F) a"
minus_one_cong_solve_weak: fixes n :: "nat"   and x :: "nat"   and y :: "nat" assumes "1 < n"   and "x \<in> totatives n"   and "y \<in> totatives n"   and "[x = n - 1] (mod n)"   and "[x * y = 1] (mod n)" shows "y = n - 1"
lenlex_length: fixes ms :: "'b list"   and ns :: "'b list"   and r :: "('b \<times> 'b) set" assumes "(ms, ns) \<in> lenlex r" shows "length ms \<le> length ns"
coeff_map_poly_hom: fixes p :: "rat poly"   and i :: "nat" shows "coeff (map_poly of_rat p) i = of_rat (coeff p i)"
inside_subset: fixes U :: "'b set"   and T :: "'b set"   and S :: "'b set" assumes "connected U"   and "\<not> bounded U"   and "T \<union> U = - S" shows "inside S \<subseteq> T"
le_list_def: fixes s :: "'b list"   and t :: "'b list" shows "(s \<le> t) = (\<exists>r. s @ r = t)"
seq_comp_secure_aux_2_case_1: fixes I :: "('c \<times> 'c) set"   and D :: "'d option \<Rightarrow> 'c"   and P :: "'d option process"   and xs :: "'d option list"   and zs :: "'d option list"   and X :: "'d option set"   and y :: "'d option"   and Q :: "'d option process" assumes "secure_termination I D"   and "sequential P"   and "secure P I D"   and "xs @ zs \<notin> sentences P"   and "(xs @ zs, X) \<in> failures P"   and "None \<notin> set xs"   and "None \<notin> set zs"   and "(xs @ [y], {}) \<in> seq_comp_failures P Q" shows "(xs @ y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs X) \<in> seq_comp_failures P Q"
split: fixes P :: "'h \<Rightarrow> bool"   and f1 :: "'h"   and f2 :: "'i \<Rightarrow> 'h"   and option :: "'i option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
scale_scale: fixes a :: "'a"   and b :: "'a"   and x :: "'b" shows "a *s b *s x = (a * b) *s x"
backtrack_nsc: fixes init_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'c"   and assert_nsc :: "'a \<Rightarrow> 'c \<Rightarrow> 'a list + 'c"   and check_nsc :: "'c \<Rightarrow> 'a list + 'c"   and solution_nsc :: "'c \<Rightarrow> (nat, 'b) mapping"   and checkpoint_nsc :: "'c \<Rightarrow> 'd"   and backtrack_nsc :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and invariant_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and checked_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and nsc :: "('a \<times> 'b ns_constraint) list"   and J :: "'a set"   and s :: "'c"   and c :: "'d"   and K :: "'a set"   and s' :: "'c"   and s'' :: "'c" assumes "Incremental_NS_Constraint_Ops init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc"   and "checked_nsc nsc J s"   and "checkpoint_nsc s = c"   and "invariant_nsc nsc K s'"   and "backtrack_nsc c s' = s''"   and "J \<subseteq> K" shows "invariant_nsc nsc J s''"
assms_1: shows "possible_steps e s r l i = {|(s', t)|}"
shows_list_aux_def: fixes s :: "'c \<Rightarrow> char list \<Rightarrow> char list"   and xs :: "'c list" shows "shows_list_aux s xs = shows_list_gen s ''[]'' ''['' '', '' '']'' xs"
union_inter: fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "finite B" shows "prod g (A \<union> B) * prod g (A \<inter> B) = prod g A * prod g B"
elims: fixes x :: "checker_type"   and xa :: "ct list"   and y :: "checker_type" assumes "consume x xa = y" obtains ts :: "t list" and cons :: "ct list" where "x = Type ts" and "xa = cons__" and "y = (if ct_suffix cons__ (to_ct_list ts) then Type (take (length ts - length cons__) ts) else Bot)"   | cts :: "ct list" and cons :: "ct list" where "x = TopType cts" and "xa = cons" and "y = (if ct_suffix cons cts then TopType (take (length cts - length cons) cts) else if ct_suffix cts cons then TopType [] else Bot)"   | uv_ :: "ct list" where "x = Bot" and "xa = uv_" and "y = Bot"
set_union: fixes xs :: "'e list"   and ys :: "'e list" shows "set (List.union xs ys) = set xs \<union> set ys"
assms_5: fixes xs :: "'b set list" assumes "xs \<in> set (map snd xss)" shows "\<Union> (set xs) = alts"
psubset_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<subset> B"   and "B \<subset> C" shows "A \<subset> C"
addNonSyncEmpty: fixes sa :: "'c event list"   and A :: "'c event set"   and u :: "'c event list"   and y1 :: "'c event" shows "sa setinterleaves (([], u), A) \<and> y1 \<notin> A \<longrightarrow> sa @ [y1] setinterleaves (([y1], u), A) \<and> sa @ [y1] setinterleaves (([], u @ [y1]), A)"
nonzero_mult_divide_mult_cancel_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'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"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times a c) (times b c) = divide a b"
mulex_on_mono: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b multiset"   and N :: "'b multiset" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "mulex_on P A M N" shows "mulex_on Q A M N"
assms_3: shows "A' \<in> set (tr\<^sub>p\<^sub>c A [])"
cfs_smult: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "a \<in> carrier R"   and "p \<in> carrier P" shows "(a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> p n"
find_max_works: fixes x :: "nat"   and xs :: "nat list" assumes "x \<in> set xs" shows "x \<le> find_max xs"
try_gpv_parametric: fixes A :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_fun (rel_gpv A C) (rel_fun (rel_gpv A C) (rel_gpv A C)) try_gpv try_gpv"
nhds_core: fixes U :: "'a set"   and x :: "'a" assumes "U \<in> nhds x" shows "x \<in> core U"
unat_of_nat_len: fixes x :: "nat" assumes "x < 2 ^ LENGTH('a)" shows "unat (word_of_nat x) = x"
monoseq_imp_convergent_iff_Bseq: fixes f :: "nat \<Rightarrow> real" assumes "monoseq f" shows "convergent f = Bseq f"
assms: shows "evaluate_list ck env s [e] (s', Rerr err)"
countable_def: fixes S :: "'c set" shows "countable S = (\<exists>f. inj_on f S)"
axioms_2: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set" assumes "open_cover_of_subset X is_open U I C" shows "cover_of_subset X U I C"
bbox_def: fixes x :: "'a"   and y :: "'a" shows "antirange_semiring.bbox (\<cdot>) ar x y = ar (ar y \<cdot> x)"
span_add: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x \<in> More_Modules.ideal S"   and "y \<in> More_Modules.ideal S" shows "x + y \<in> More_Modules.ideal S"
abs_eq_iff: fixes x :: "conc_interface"   and y :: "conc_interface" shows "(abs_interface x = abs_interface y) = (x \<simeq> y)"
path_2: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a" assumes "Graph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w"   and "v \<noteq> w" shows "Graph.walk G [v, w] \<and> distinct [v, w]"
valid_line_cases: fixes l :: "color list" assumes "valid_line l" shows "l = [] \<or> l ! 0 = B \<and> valid_line (tl l) \<or> 3 \<le> length l \<and> (\<forall>i<length l. l ! i = R) \<or> (\<exists>j<length l. 3 \<le> j \<and> (\<forall>i<j. l ! i = R) \<and> l ! j = B \<and> valid_line (drop (j + 1) l))"
dual_upset: shows "ord.upset (\<lambda>x y. y \<le> x) = \<down>"
arg_rcis_cis: fixes n :: "real"   and x :: "real" assumes "0 < n" shows "arg (rcis n x) = arg (cis x)"
vector_diff_chain_within: fixes f :: "real \<Rightarrow> real"   and f' :: "real"   and x :: "real"   and s :: "real set"   and g :: "real \<Rightarrow> 'b"   and g' :: "'b" assumes "(f has_vector_derivative f') (at x within s)"   and "(g has_vector_derivative g') (at (f x) within f ` s)" shows "(g \<circ> f has_vector_derivative f' *\<^sub>R g') (at x within s)"