hcju/isabelleps huggingface dataset:
local_property_eq: fixes P :: "(nat \<Rightarrow> 'f \<Rightarrow> 'g) \<Rightarrow> bool" shows "local_property P = (\<forall>\<sigma> \<tau>. locally_similar \<sigma> \<tau> \<longrightarrow> P \<sigma> = P \<tau>)"
rt_strictly_fresher_fresherD: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt2" shows "the (rt1 dip) \<sqsubseteq> the (rt2 dip)"
sec_id_identified_1: fixes n :: "nat"   and m :: "nat"   and n' :: "nat" shows "{Tok_PriK n, Sec_PriK m, Rev_PriK} \<noteq> {Tok_PriK n', Rev_PriK}"
Healthy_def: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and H :: "('b \<times> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<times> 'b \<Rightarrow> bool" shows "P is H healthy \<equiv> P = H P"
finsum_insert: fixes F :: "'h set"   and a :: "'h"   and f :: "'h \<Rightarrow> 'c" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier M"   and "f a \<in> carrier M" shows "finsum M f (insert a F) = f a \<oplus>\<^bsub>M\<^esub> finsum M f F"
cases: fixes a1 :: "com"   and a2 :: "state" assumes "a1 \<down> a2" obtains f :: "state \<Rightarrow> state set" and s :: "state" where "a1 = Do f__" and "a2 = s__" and "f__ s__ \<noteq> {}"   | c1 :: "com" and s0 :: "state" and c2 :: "com" where "a1 = (c1__; c2__)" and "a2 = s0" and "c1__ \<down> s0" and "\<And>s1. s0 -c1__\<rightarrow> s1 \<Longrightarrow> c2__ \<down> s1"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and c2 :: "com" where "a1 = IF ba__ THEN c1a__ ELSE c2a__" and "a2 = sa__" and "ba__ sa__" and "c1a__ \<down> sa__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c2 :: "com" and c1 :: "com" where "a1 = IF bb__ THEN c1 ELSE c2" and "a2 = sb__" and "\<not> bb__ sb__" and "c2 \<down> sb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = WHILE bc__ DO ca__" and "a2 = sc__" and "\<not> bc__ sc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = WHILE b DO cb__" and "a2 = sd__" and "b sd__" and "cb__ \<down> sd__" and "\<And>t. sd__ -cb__\<rightarrow> t \<Longrightarrow> WHILE b DO cb__ \<down> t"   | p :: "pname" and s :: "state" where "a1 = CALL p" and "a2 = se__" and "body p \<down> se__"   | c :: "com" and f :: "state \<Rightarrow> state" and s :: "state" and g :: "state \<Rightarrow> state \<Rightarrow> state" where "a1 = LOCAL f; c; g" and "a2 = s" and "c \<down> f s"
degree_primitive_part_fract: fixes p :: "'b fract poly" shows "degree (primitive_part_fract p) = degree p"
has_least_fixpoint_def: fixes f :: "'a \<Rightarrow> 'a" shows "has_least_fixpoint f \<equiv> \<exists>x. is_least_fixpoint f x"
chop_assoc1: fixes i :: "nat_int"   and i1 :: "nat_int"   and i2 :: "nat_int"   and i3 :: "nat_int"   and i4 :: "nat_int" shows "N_Chop(i,i1,i2) \<and> N_Chop(i2,i3,i4) \<longrightarrow> N_Chop(i,i1 \<squnion> i3,i4) \<and> N_Chop(i1 \<squnion> i3,i1,i3)"
"4_5": fixes t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (delete\<langle>t,t'\<rangle>) = fv t \<union> fv t'"
prime_divisor_exists_strong: fixes m :: "int" assumes "1 < m"   and "\<not> prime m" shows "\<exists>n k. m = n * k \<and> 1 < n \<and> n < m \<and> 1 < k \<and> k < m"
rhs_def: shows "rhs = \<lfloor>r\<rfloor> div n"
c_dom_def: fixes D :: "'f \<Rightarrow> 'e"   and xp :: "'f \<times> 'g" shows "c_dom D xp \<equiv> D (fst xp)"
intros_27: fixes vs :: "e list"   and n :: "nat"   and t1s :: "t list"   and t2s :: "t list"   and m :: "nat"   and es :: "b_e list" assumes "const_list vs"   and "length vs = n"   and "length t1s = n"   and "length t2s = m" shows "\<lparr>vs @ [$Block (t1s _> t2s) es]\<rparr> \<leadsto> \<lparr>[Label m [] (vs @ ($* es))]\<rparr>"
troeger_1: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<circ> = x\<^sup>\<circ> * Rf y ((x \<squnion> y)\<^sup>\<circ>)"
inv_equality: fixes G :: "('d, 'e) monoid_scheme"   and y :: "'d"   and x :: "'d" assumes "Group.group G"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "inv\<^bsub>G\<^esub> x = y"
exhaust: fixes y :: "epol" obtains "y = Cext"   | "y = Text"   | "y = Fext"
nonempty_rel: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "K .\<bottom>. \<phi> \<noteq> {}" shows "\<exists>B\<in>K .\<bottom>. \<phi>. \<forall>C\<in>K .\<bottom>. \<phi>. C \<preceq>\<^bsub>K\<^esub> B"
pfa_insert_grow_def: shows "pfa_insert_grow \<equiv> \<lambda>(a, n) i x. pfa_ensure (a, n) (n + 1) \<bind> (\<lambda>a'. pfa_insert a' i x \<bind> return)"
simps_1: fixes x :: "'a" shows "insert_in_list_at_arbitrary_pos x [] = {[x]}"
f_Exec_State_Idle_replicate_NoMsg_gr0_output: fixes localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState c)"   and "0 < n" shows "output_fun (f_Exec_Comp trans_fun NoMsg\<^bsup>n\<^esup> c) = NoMsg"
eval_finalsId: fixes es :: "expr list"   and 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 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)" assumes "finals es" shows "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es,s\<rangle>"
semAssignSC_aux: fixes x :: "'id"   and \<iota> :: "nat"   and e :: "'exp"   and d :: "'d" assumes "e \<equiv>\<^bsub>DA x,htchLoc \<iota>\<^esub> e" shows "NDC d (x:=\<^bsub>\<iota>\<^esub> e) \<or> IDC d (x:=\<^bsub>\<iota>\<^esub> e) (htchLoc (pp (x:=\<^bsub>\<iota>\<^esub> e)))"
descendants_branch: fixes k :: "'b"   and i :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and xs :: "(nat \<times> nat) set"   and extra :: "(('c, 'b) fm list \<times> 'b) list" assumes "descendants k i branch xs" shows "descendants k i (extra @ branch) xs"
Span_eq_combine_set_length_version: fixes Us :: "'a list" assumes "set Us \<subseteq> carrier R" shows "Span K Us = {combine Ks Us |Ks. length Ks = length Us \<and> set Ks \<subseteq> K}"
lookup_raw_eq_valueI: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and k :: "'a"   and v :: "'c" assumes "oalist_inv xs"   and "(k, v) \<in> set (fst xs)" shows "lookup_raw xs k = v"
le_fininfE: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a stream" assumes "traces ind"   and "traces.le_fininf ind w\<^sub>1 w\<^sub>2" obtains v\<^sub>2 :: "'a list" where "v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2" and "traces.le_fin ind w\<^sub>1 v\<^sub>2"
is_aligned_andI2: fixes y :: "'b word"   and n :: "nat"   and x :: "'b word" assumes "is_aligned y n" shows "is_aligned (x AND y) n"
until_or_right_distrib: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n (\<psi>\<^sub>1 or\<^sub>n \<psi>\<^sub>2) = w \<Turnstile>\<^sub>n (\<phi> U\<^sub>n \<psi>\<^sub>1) or\<^sub>n (\<phi> U\<^sub>n \<psi>\<^sub>2)"
moebius_dilatation_def: fixes a :: "real" shows "moebius_dilatation a = moebius_similarity (cor a) 0"
apply_guards_rearrange: fixes x :: "vname gexp"   and G :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" assumes "x \<in> set G" shows "apply_guards G s = apply_guards (x # G) s"
Basis_vec_def: shows "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
lm017: fixes P :: "('b \<times> 'a) set"   and X :: "'b set" shows "P `` (X \<inter> Domain P) = P `` X"
gbinomial_0_left: fixes k :: "nat" shows "(0::'a) gchoose k = (if k = 0 then 1::'a else (0::'a))"
Card_ord_of_nat: fixes n :: "nat" shows "Card (n\<^sub>\<nat>)"
key_order_of_nat_term_order_def: fixes to :: "'b nat_term_order" shows "key_order_of_nat_term_order to = Abs_key_order (nat_term_compare to)"
hom_mult: fixes A :: "'a mat"   and B :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "mat\<^sub>h (A * B) = mat\<^sub>h A * mat\<^sub>h B"
is_abelian_group: fixes R :: "('a, 'b) ring_scheme" assumes "ring R" shows "abelian_group R"
l_eq: shows "(set (kv # l) = map_to_set m) = (set l = map_to_set (m(k := None)) \<and> m k = Some v)"
valid_add_new_auxlist: shows "valid_muaux args nt (add_new_muaux args rel1 rel2 nt aux) auxlist'"
const_name_simps_1: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.const_name unconst (const name) = name"
2: shows "wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t (\<A> @ [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)]) = wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t \<A> \<union> fv t"
Cons_3: shows "\<forall>rs. set rs \<subseteq> R \<and> length rs = length (m # ms) \<and> rs \<bullet>\<cdot> (m # ms) = (0::'m) \<longrightarrow> set rs \<subseteq> 0"
suffixes_conv_prefixes: fixes xs :: "'a list" shows "suffixes xs = map rev (prefixes (rev xs))"
derived_set_of_topspace: fixes X :: "'c topology" shows "X derived_set_of topspace X = {x \<in> topspace X. \<not> openin X {x}}"
cl_def: fixes X :: "'a set" shows "\<E>.cl X \<equiv> {x \<in> carrier. \<E>.rank_of (insert x X) = \<E>.rank_of X}"
measure_notin_sets: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<notin> sets M" shows "Sigma_Algebra.measure M A = 0"
bytes_takefill_def: shows "bytes_takefill \<equiv> map_fun Rep_uint8 (map_fun id (map_fun Rep_bytes Abs_bytes)) (\<lambda>a. takefill (Abs_uint8 a))"
natUnion_elem: fixes x :: "'a"   and f :: "nat \<Rightarrow> 'a set"   and n :: "nat" assumes "x \<in> f n" shows "x \<in> natUnion f"
Rcm: shows "take_cols (gauss_jordan_single A) (map snd (pivot_positions (gauss_jordan_single A))) \<in> carrier_mat n (length (map snd (pivot_positions (gauss_jordan_single A))))"
L: shows "fill xs (ns[index key x (length ns) mi ma := Suc (ns ! index key x (length ns) mi ma)]) index key ub mi ma = take (ns ! index key x (length ns) mi ma) (fill xs (ns[index key x (length ns) mi ma := Suc (ns ! index key x (length ns) mi ma)]) index key ub mi ma) @ None # drop (Suc (ns ! index key x (length ns) mi ma)) (fill xs (ns[index key x (length ns) mi ma := Suc (ns ! index key x (length ns) mi ma)]) index key ub mi ma)"
invariant3p_correct_jvm_state_mexecT: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and hconf :: "'c \<Rightarrow> bool"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and spurious_wakeups :: "bool" assumes "JVM_conf_read addr2thread_id thread_id2addr empty_heap allocate typeof_addr heap_read heap_write hconf P"   and "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" shows "invariant3p (multithreaded_base.redT JVM_final (\<lambda>t ((xcp, frstls), h) ta ((xcp', frstls'), h'). JVM_heap_base.exec_1 addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t (xcp, h, frstls) ta (xcp', h', frstls')) convert_RA) (JVM_heap_conf_base.correct_jvm_state thread_id2addr empty_heap allocate typeof_addr hconf P \<Phi>)"
hyps_4: shows "insert k K = dom \<iota>\<^sub>m"
class_of_eq: shows "stk1.class_of = stk2.class_of"
rsp: shows "eq_onp (\<lambda>f. bij f \<and> finite {a. f a \<noteq> a}) id id"
lift_self_disjoint: fixes x :: "V"   and u :: "V" shows "x \<sqinter> lift x u = 0"
preserves_ide: fixes a :: "'b" assumes "D.ide a" shows "D.ide (D.R a)"
rev_append: fixes xs :: "'b list"   and ys :: "'b list" shows "rev (xs \<cdot> ys) = rev ys \<cdot> rev xs"
conuc_Sup_qclosed: fixes f :: "'b \<Rightarrow> 'b" assumes "conucleus f" shows "Sup_closed_set (Fix f) \<and> comp_closed_set (Fix f)"
quotient_def: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "A // r = (\<Union>x\<in>A. {r `` {x}})"
exec_step_ind_Getfield_Static: fixes v :: "val"   and stk :: "val list"   and D :: "char list"   and fs :: "Objects.fields"   and h :: "heap"   and P :: "jvm_prog"   and F :: "char list"   and t :: "ty"   and C :: "char list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "v = hd stk"   and "(D, fs) = the (h (the_Addr v))"   and "v \<noteq> Null"   and "P \<turnstile> D has F,Static:t in C" shows "exec_step_ind (StepI (Getfield F C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt IncompatibleClassChangeError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
weight_gt_0: fixes w :: "'c \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'c \<times> nat \<Rightarrow> 'c \<times> nat \<Rightarrow> bool"   and pr_weak :: "'c \<times> nat \<Rightarrow> 'c \<times> nat \<Rightarrow> bool"   and least :: "'c \<Rightarrow> bool"   and scf :: "'c \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and t :: "('c, 'd) Term.term" assumes "admissible_kbo w w0 pr_strict pr_weak least scf" shows "0 < weight_fun.weight w w0 scf t"
formulaTrueSubset: fixes formula :: "Literal list list"   and valuation :: "Literal list"   and formula' :: "Literal list list" assumes "formulaTrue formula valuation"   and "\<forall>clause. clause el formula' \<longrightarrow> clause el formula" shows "formulaTrue formula' valuation"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and acc :: "'b list" shows "remove_cycles (x # xs) y acc = (if x = y then remove_cycles xs y [x] else remove_cycles xs y (x # acc))"
termi_merge_c1: shows "\<Gamma>\<turnstile>mark_guards f c1 \<down> Normal s"
a_closed: fixes G :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "abelian_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> y \<in> carrier G"
subst_transform: fixes a :: "Arity"   and e :: "exp"   and x :: "Vars.var"   and y :: "Vars.var" shows "(ccTransform a e)[x::=y] = ccTransform a e[x::=y]"
adm_eq: fixes u :: "'g \<Rightarrow> 'h"   and v :: "'g \<Rightarrow> 'h" assumes "cont u"   and "cont v" shows "adm (\<lambda>x. u x = v x)"
is_ground_atms_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and AA :: "'b set" shows "substitution_ops.is_ground_atms subst_atm AA = (\<forall>A\<in>AA. substitution_ops.is_ground_atm subst_atm A)"
in_filtermap_iff: fixes f' :: "'b \<Rightarrow> 'c"   and h' :: "'d \<Rightarrow> 'b"   and F' :: "'d filter"   and g' :: "'b \<Rightarrow> 'c" shows "(f' \<in> \<Omega>[filtermap h' F'](g')) = ((\<lambda>x. f' (h' x)) \<in> \<Omega>[F'](\<lambda>x. g' (h' x)))"
mod_diff1_eq2: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a \<le> b"   and "b mod m < a mod m" shows "(b - a) mod m = m + b mod m - a mod m"
diff_empty: fixes M :: "'k multiset" shows "M - {#} = M \<and> {#} - M = {#}"
emp_shift: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and I :: "nat set" assumes "PL_Indis cval indis" shows "(PL_Indis.shift cl n I = {}) = (I = {})"
invoke_empty: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and args :: "'d" shows "\<not> h \<turnstile> ok invoke [] ptr args"
dis_rews_1: shows "null\<cdot>\<bottom> = \<bottom>"
emax_pos_1: fixes x :: "('a, 'b) IEEE.float itself" shows "0 < emax x"
mkVarChannel_gState_inv: fixes prog :: "program"   and g :: "'a gState_scheme"   and p :: "pState"   and v :: "varDecl"   and upd :: "((String.literal, variable) assoc_list \<Rightarrow> (String.literal, variable) assoc_list) \<Rightarrow> 'a gState_scheme \<times> pState \<Rightarrow> 'a gState_scheme \<times> pState" assumes "gState_inv prog g"   and "\<And>k v' cs. toVariable g p v = (k, v', cs) \<Longrightarrow> gState_inv prog (fst (upd (lm.update k v') (g, p)))" shows "gState_inv prog (fst (mkVarChannel v upd g p))"
ocomplete_no_change: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d net_state"   and a :: "'e node_action"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d net_state"   and T :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set"   and j :: "nat" assumes "((\<sigma>, s), a, \<sigma>', s') \<in> ocnet_sos T"   and "j \<notin> net_ips s" shows "\<sigma>' j = \<sigma> j"
permutation_inverse: fixes p :: "'b \<Rightarrow> 'b" assumes "permutation p" shows "permutation (inv p)"
lex_two_compat': fixes ns1 :: "('b \<times> 'b) set"   and s1 :: "('b \<times> 'b) set"   and s2 :: "('c \<times> 'c) set"   and ns2 :: "('c \<times> 'c) set"   and ab1 :: "'b \<times> 'c"   and ab2 :: "'b \<times> 'c"   and ab3 :: "'b \<times> 'c" assumes "ns1 O s1 \<subseteq> s1"   and "s1 O ns1 \<subseteq> s1"   and "s1 O s1 \<subseteq> s1"   and "ns1 O ns1 \<subseteq> ns1"   and "s2 O ns2 \<subseteq> s2"   and "(ab1, ab2) \<in> lex_two s1 ns1 s2"   and "(ab2, ab3) \<in> lex_two s1 ns1 ns2" shows "(ab1, ab3) \<in> lex_two s1 ns1 s2"
refine: fixes mrel :: "('a \<times> 'b) set"   and node_rel :: "('c \<times> 'd) set"   and node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'a) gen_g_impl_scheme"   and G :: "('d, 'b) graph_rec_scheme"   and g :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'e) gen_g_impl_scheme" assumes "fr_graph_impl_loc mrel node_rel node_eq_impl node_hash_impl node_def_hash_size G_impl G" shows "RETURN (compute_SCC_tr node_eq_impl node_hash_impl node_def_hash_size g) \<le> compute_SCC_code node_eq_impl node_hash_impl node_def_hash_size g"
prems_2: shows "ifex_sat_list (IF v t e) = Some u"
Impl_body_eq: fixes Z :: "state"   and M :: "cname \<times> mname" shows "(\<lambda>t. \<exists>n. Z -Impl M-n\<rightarrow> t) = (\<lambda>t. \<exists>n. Z -body M-n\<rightarrow> t)"
comp_same: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b" assumes "comparator comp" shows "comp x x = Eq"
rtranclp_into_tranclp2: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
quadratic_sub_neq: fixes var :: "nat"   and L :: "real list"   and Dv :: "real"   and Cv :: "real"   and c :: "real mpoly"   and a :: "real mpoly"   and Av :: "real"   and b :: "real mpoly"   and Bv :: "real"   and d :: "real mpoly"   and p :: "real mpoly" assumes "var < length L"   and "Dv \<noteq> 0"   and "0 \<le> Cv"   and "var \<notin> vars c"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) a = Av"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) b = Bv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) c = Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) d = Dv" shows "aEval (Neq p) (L[var := (Av + Bv * sqrt Cv) / Dv]) = eval (quadratic_sub var a b c d (Neq p)) (L[var := sqrt Cv])"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) (\<lambda>f. f \<otimes>\<^sub>C C.cod \<iota>\<^sub>C) (\<lambda>f. f \<otimes>\<^sub>C C.cod \<iota>\<^sub>C) (\<lambda>f. f \<otimes>\<^sub>C C.cod \<iota>\<^sub>C)"
boundedE: fixes b :: "'c \<times> 'd"   and c :: "'c \<times> 'd"   and a :: "'c \<times> 'd" assumes "less_eq_prod (sup_prod b c) a" obtains "less_eq_prod b a" and "less_eq_prod c a"
cop_F_CH_CD_on_disjoint: shows "Xd ` CH fp \<inter> Xd ` (CD_on ds (- Rf CH fp) - fp) = {}"
c': shows "\<not> multSubst \<Gamma>' \<and> \<not> multSubst \<Delta>'"
hyps_2: shows "r # rs' = butlast (butlast (butlast rs))"
intros_7: fixes v\<^sub>1 :: "Syntax.term"   and v\<^sub>2 :: "Syntax.term" assumes "value v\<^sub>1"   and "value v\<^sub>2" shows "value (Syntax.Pair v\<^sub>1 v\<^sub>2)"
op2_rep2_left2_right2_strict_1: fixes pat :: "[:'r:]" shows "op2\<cdot>pat\<cdot>\<bottom> = \<bottom>"
nb: shows "set (List.product [0..<t] (\<Pi>\<^sub>\<O>)) = {0..<t} \<times> set (\<Pi>\<^sub>\<O>)"
same_upto_refl: fixes j :: "nat"   and A :: "'a mat" shows "local.same_upto j A A"
rAlpha: fixes xvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and C :: "'d"   and p :: "(name \<times> name) list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* C"   and "xvec \<sharp>* (p \<bullet> xvec)"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "(p \<bullet> xvec) \<sharp>* N"   and "(p \<bullet> xvec) \<sharp>* P'"   and "Prop C \<Psi> P M xvec N P'" shows "Prop C \<Psi> P M (p \<bullet> xvec) (p \<bullet> N) (p \<bullet> P')"
fds_abs_converges_truncate: fixes n :: "nat"   and f :: "'a fds"   and s :: "'a" shows "fds_abs_converges (fds_truncate n f) s"
Spy_see_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> kerbV" shows "(Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
affine_dependent_explicit: fixes p :: "'c set" shows "affine_dependent p = (\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = (0::'c))"
Body_triple_valid_0: fixes P :: "'b assn"   and pn :: "pname"   and Q :: "'b assn" shows "|=0:{P}. BODY pn .{Q}"
simps_15: fixes v :: "'a local.term" shows "Diag \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[v\<^bold>] = False"
ring_hom_mult: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
injI: fixes f :: "'c \<Rightarrow> 'd" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "inj f"
right_connected: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and \<mu>' :: "'b"   and \<nu> :: "'b" assumes "weak_composition V H"   and "partial_magma.arr V (V \<mu> \<mu>')" shows "(H \<nu> \<mu> \<noteq> partial_magma.null V) = (H \<nu> \<mu>' \<noteq> partial_magma.null V)"
eq: fixes c :: "'a"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "c \<noteq> (0::'a)" shows "poly_of_pm x (monomial c t * q) = poly_of_pm x (monomial c t) * poly_of_pm x q"
inverse_minus_eq: fixes a :: "'a" shows "inverse (- a) = - inverse a"
take_decr_non_empty: fixes xs :: "int list" assumes "xs \<noteq> []" shows "take_decr xs \<noteq> []"
classical_P_prv: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "let PP = \<lambda>t. subst P t xx in prv (neg (neg (PP \<langle>\<phi>\<rangle>)))" shows "prv (PP \<langle>\<phi>\<rangle>)"