hcju/isabelleps huggingface dataset:
Rep_Abs_Mp: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process" shows "Rep_process (Abs_process ({(tr, ref). tr = [] \<and> ref \<inter> ev ` A = {}} \<union> {(tr, ref). tr \<noteq> [] \<and> hd tr \<in> ev ` A \<and> (\<exists>a. ev a = hd tr \<and> (tl tr, ref) \<in> F (P a))}, {d. d \<noteq> [] \<and> hd d \<in> ev ` A \<and> (\<exists>a. ev a = hd d \<and> tl d \<in> D (P a))})) = ({(tr, ref). tr = [] \<and> ref \<inter> ev ` A = {}} \<union> {(tr, ref). tr \<noteq> [] \<and> hd tr \<in> ev ` A \<and> (\<exists>a. ev a = hd tr \<and> (tl tr, ref) \<in> F (P a))}, {d. d \<noteq> [] \<and> hd d \<in> ev ` A \<and> (\<exists>a. ev a = hd d \<and> tl d \<in> D (P a))})"
nfresh_distinct: fixes S :: "'a set"   and n :: "nat" assumes "finite S" shows "distinct (nfresh S n)"
failingSubFailsI: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and gamma :: "'b" assumes "inherited subs P"   and "\<not> P (tree subs gamma)"   and "\<not> terminal subs gamma"   and "fans subs" shows "\<not> P (tree subs (failingSub subs P gamma))"
eqv_\<phi>: shows "prv (eqv \<phi> (PP (encF N \<langle>\<phi>\<rangle>)))"
AA22: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and N :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b"   and w :: "'a \<Rightarrow> 'c" assumes "|~ P \<and> N \<longrightarrow> \<circle>P \<or> \<circle>Q"   and "|~ P \<and> N \<and> \<langle>A\<rangle>_v \<longrightarrow> \<circle>Q"   and "|~ P \<and> Unchanged w \<longrightarrow> \<circle>P" shows "\<turnstile> \<box>N \<and> \<box>(\<box>P \<longrightarrow> \<diamond>\<langle>A\<rangle>_v) \<longrightarrow> P \<leadsto> Q"
finite'_intersection_of_Int: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "(finite' intersection_of P) S"   and "(finite' intersection_of P) T" shows "(finite' intersection_of P) (S \<inter> T)"
prems_1: shows "Controller_L y (fin_inf_append [Zero] l) l z"
one_plus_ennreal: fixes x :: "real" assumes "0 \<le> x" shows "1 + ennreal x = ennreal (1 + x)"
restrict_current_le: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" shows "(g \<upharpoonleft> \<Gamma> / f) e \<le> g e"
subseq_Cons2_neq: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" assumes "subseq (x # xs) (y # ys)"   and "x \<noteq> y" shows "subseq (x # xs) ys"
filterlim_inverse_at_iff: fixes g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "(LIM x F. inverse (g x) :> at (0::'b)) = filterlim g at_infinity F"
some_sym_eq_trivial: fixes x :: "'a" shows "(SOME y. x = y) = x"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and finite_3 :: "Enum.finite_3" shows "P (case finite_3 of a\<^sub>1 \<Rightarrow> f1 | a\<^sub>2 \<Rightarrow> f2 | a\<^sub>3 \<Rightarrow> f3) = ((finite_3 = a\<^sub>1 \<longrightarrow> P f1) \<and> (finite_3 = a\<^sub>2 \<longrightarrow> P f2) \<and> (finite_3 = a\<^sub>3 \<longrightarrow> P f3))"
permute_list_map: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'b list"   and g :: "'b \<Rightarrow> 'c" assumes "f permutes {..<length xs}" shows "permute_list f (map g xs) = map g (permute_list f xs)"
infsetsum_Times: fixes A :: "'c set"   and B :: "'d set"   and f :: "'c \<times> 'd \<Rightarrow> 'e" assumes "countable A"   and "countable B"   and "f abs_summable_on A \<times> B" shows "infsetsum f (A \<times> B) = (\<Sum>\<^sub>ax\<in>A. \<Sum>\<^sub>ay\<in>B. f (x, y))"
uniform_lim_mult: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'c set"   and l :: "'c \<Rightarrow> 'd"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and m :: "'c \<Rightarrow> 'd" assumes "uniform_limit S f l F"   and "uniform_limit S g m F"   and "bounded (l ` S)"   and "bounded (m ` S)" shows "uniform_limit S (\<lambda>a b. f a b * g a b) (\<lambda>a. l a * m a) F"
OneHAInitState_SAStates: fixes S :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and T :: "'d"   and SA :: "('d, 'e, 'f) seqauto" assumes "S \<in> HAInitStates A"   and "T \<in> HAInitStates A"   and "S \<in> States SA"   and "T \<in> States SA"   and "SA \<in> SAs A" shows "S = T"
geodesic_segment_topology_5: fixes G :: "'b set" assumes "geodesic_segment G" shows "closed G"
Max_wrt_of_weak_ranking: fixes xs :: "'b set list" assumes "is_weak_ranking xs" shows "Max_wrt (of_weak_ranking xs) = (if xs = [] then {} else hd xs)"
is_red_adds_iff: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.is_red F p = (\<exists>f\<in>F. \<exists>v\<in>keys p. f \<noteq> 0 \<and> local.punit.adds_term (punit.lt f) v)"
comp_null_2: fixes f :: "'a2" shows "f \<cdot>\<^sub>2 C2.null = C2.null"
C: fixes x :: "real" assumes "x \<in> {x0..max x0 b}" shows "\<bar>f x\<bar> \<le> C"
neg_zmset_pos: fixes M :: "'b multiset" shows "mset_neg (zmset_of M) = {#}"
intersect_def: fixes M :: "'a multiset"   and S :: "'a set" shows "M \<inter>s S = {#x \<in># M. x \<in> S#}"
Abs_Int_inv_axioms: shows "Abs_Int_inv num' plus' test_num' inv_plus' inv_less' \<gamma>"
utility_ge: fixes outcomes :: "'b set"   and relation :: "('b pmf \<times> 'b pmf) set"   and u :: "'b \<Rightarrow> real"   and p :: "'b pmf"   and q :: "'b pmf" assumes "vNM_utility outcomes relation u"   and "p \<succeq>[relation] q" shows "measure_pmf.expectation q u \<le> measure_pmf.expectation p u"
acyclic_single_step: fixes x :: "'a" assumes "is_acyclic x" shows "x \<le> - 1'"
sup_V_0_left: fixes a :: "V" shows "0 \<squnion> a = a"
whisker_right: fixes f :: "'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.ide f"   and "D.seq \<nu> \<mu>" shows "\<nu> \<cdot>\<^sub>D \<mu> \<star>\<^sub>D f = (\<nu> \<star>\<^sub>D f) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D f)"
expect_1_I: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "P x" shows "1 \<le> \<guillemotleft> P \<guillemotright> x"
Scope: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (c, <<>|s>) \<Rightarrow> s'" shows "\<pi>: (SCOPE c, s) \<Rightarrow> <s|s'>"
fcompose_nzero: fixes p :: "'a poly"   and q2 :: "'a poly"   and q1 :: "'a poly" assumes "p \<noteq> 0"   and "q2 \<noteq> 0"   and "\<forall>c. q1 \<noteq> smult c q2"   and "infinite UNIV" shows "fcompose p q1 q2 \<noteq> 0"
lens_inv_bij: fixes X :: "'d \<Longrightarrow> 'e" assumes "bij_lens X" shows "bij_lens (inv\<^sub>L X)"
XY: shows "entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) = - (LINT x|(S \<Otimes>\<^sub>M T). Pxy x * log b (Pxy x))"
rcosI: fixes G :: "('c, 'd) monoid_scheme"   and h :: "'c"   and H :: "'c set"   and x :: "'c" assumes "Group.group G"   and "h \<in> H"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "h \<otimes>\<^bsub>G\<^esub> x \<in> H #>\<^bsub>G\<^esub> x"
findMin_def: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.findMin bq = (let min = SkewBinomialHeapStruc.getMinTree bq in (SkewBinomialHeapStruc.val min, SkewBinomialHeapStruc.prio min))"
split_min_def: fixes t :: "'b tree" shows "Root_Balanced_Tree.split_min t = Time_Monad.val (split_min_tm t)"
simps_985: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f6 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f7 :: "tf \<Rightarrow> b_e list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f8 :: "nat \<Rightarrow> 'a"   and f9 :: "nat \<Rightarrow> 'a"   and f10 :: "nat list \<Rightarrow> nat \<Rightarrow> 'a"   and f11 :: "'a"   and f12 :: "nat \<Rightarrow> 'a"   and f13 :: "nat \<Rightarrow> 'a"   and f14 :: "nat \<Rightarrow> 'a"   and f15 :: "nat \<Rightarrow> 'a"   and f16 :: "nat \<Rightarrow> 'a"   and f17 :: "nat \<Rightarrow> 'a"   and f18 :: "nat \<Rightarrow> 'a"   and f19 :: "t \<Rightarrow> (tp \<times> sx) option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f20 :: "t \<Rightarrow> tp option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f21 :: "'a"   and f22 :: "'a"   and f23 :: "v \<Rightarrow> 'a"   and f24 :: "t \<Rightarrow> unop_i \<Rightarrow> 'a"   and f25 :: "t \<Rightarrow> unop_f \<Rightarrow> 'a"   and f26 :: "t \<Rightarrow> binop_i \<Rightarrow> 'a"   and f27 :: "t \<Rightarrow> binop_f \<Rightarrow> 'a"   and f28 :: "t \<Rightarrow> testop \<Rightarrow> 'a"   and f29 :: "t \<Rightarrow> relop_i \<Rightarrow> 'a"   and f30 :: "t \<Rightarrow> relop_f \<Rightarrow> 'a"   and f31 :: "t \<Rightarrow> cvtop \<Rightarrow> t \<Rightarrow> sx option \<Rightarrow> 'a"   and x311 :: "t"   and x312 :: "cvtop"   and x313 :: "t"   and x314 :: "sx option" shows "(case Cvtop x311 x312 x313 x314 of Unreachable \<Rightarrow> f1 | Nop \<Rightarrow> f2 | Drop \<Rightarrow> f3 | Select \<Rightarrow> f4 | Block x xa \<Rightarrow> f5 x xa | Loop x xa \<Rightarrow> f6 x xa | b_e.If x xa xb \<Rightarrow> f7 x xa xb | Br x \<Rightarrow> f8 x | Br_if x \<Rightarrow> f9 x | Br_table x xa \<Rightarrow> f10 x xa | Return \<Rightarrow> f11 | Call x \<Rightarrow> f12 x | Call_indirect x \<Rightarrow> f13 x | Get_local x \<Rightarrow> f14 x | Set_local x \<Rightarrow> f15 x | Tee_local x \<Rightarrow> f16 x | Get_global x \<Rightarrow> f17 x | Set_global x \<Rightarrow> f18 x | Load x xa xb xc \<Rightarrow> f19 x xa xb xc | Store x xa xb xc \<Rightarrow> f20 x xa xb xc | Current_memory \<Rightarrow> f21 | Grow_memory \<Rightarrow> f22 | C x \<Rightarrow> f23 x | Unop_i x xa \<Rightarrow> f24 x xa | Unop_f x xa \<Rightarrow> f25 x xa | Binop_i x xa \<Rightarrow> f26 x xa | Binop_f x xa \<Rightarrow> f27 x xa | Testop x xa \<Rightarrow> f28 x xa | Relop_i x xa \<Rightarrow> f29 x xa | Relop_f x xa \<Rightarrow> f30 x xa | Cvtop x xa xb xc \<Rightarrow> f31 x xa xb xc) = f31 x311 x312 x313 x314"
matT_vec_multI_def: fixes ze :: "'a"   and pl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and ti :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "'a list list"   and v :: "'a list" shows "matT_vec_multI ze pl ti m v = map (\<lambda>w. scalar_prodI ze pl ti w v) m"
S_parcontractE: fixes r :: "comb" assumes "S \<Rrightarrow>\<^sup>1 r" obtains "r = S"
subst_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and f :: "'d \<Rightarrow> 'e option"   and mem :: "'d \<Rightarrow> 'e" assumes "sifum_security stop eval some_val some_val'" shows "subst f mem = (\<lambda>x. case f x of None \<Rightarrow> mem x | Some v \<Rightarrow> v)"
mkVarChannelProc_gState_progress_rel: fixes prog :: "program"   and g :: "'a gState_scheme"   and v :: "procVarDecl"   and p :: "pState" assumes "gState_inv prog g" shows "(g, fst (mkVarChannelProc v g p)) \<in> gState_progress_rel prog"
IH: shows "finite {uu_. \<exists>\<delta> T'. uu_ = ((S @ send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t # S', \<theta>), S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>, \<theta> \<circ>\<^sub>s \<delta>) \<and> simple S \<and> Fun f T' \<in> M \<and> Some \<delta> = mgu (Fun f T) (Fun f T')}"
Const_imp_Term: fixes x :: "hf" assumes "Const x" shows "Term x"
K33_contractedI: fixes G :: "'b pair_pre_digraph"   and H :: "'b pair_pre_digraph" assumes "subdivision_pair G H"   and "K\<^bsub>3,3\<^esub> (with_proj G)" shows "K\<^bsub>3,3\<^esub> (with_proj (gen_contr_graph (with_proj H) (verts3 (with_proj H))))"
hyps_1: shows "P \<turnstile> C \<preceq>\<^sup>* C'"
ext_inject: fixes p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'b word"   and p_dst :: "'b word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'c"   and p_iiface' :: "char list"   and p_oiface' :: "char list"   and p_src' :: "'b word"   and p_dst' :: "'b word"   and p_proto' :: "8 word"   and p_sport' :: "16 word"   and p_dport' :: "16 word"   and p_tcp_flags' :: "tcp_flag set"   and p_payload' :: "char list"   and more' :: "'c" shows "(\<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr> = \<lparr>p_iiface = p_iiface', p_oiface = p_oiface', p_src = p_src', p_dst = p_dst', p_proto = p_proto', p_sport = p_sport', p_dport = p_dport', p_tcp_flags = p_tcp_flags', p_payload = p_payload', \<dots> = more'\<rparr>) = (p_iiface = p_iiface' \<and> p_oiface = p_oiface' \<and> p_src = p_src' \<and> p_dst = p_dst' \<and> p_proto = p_proto' \<and> p_sport = p_sport' \<and> p_dport = p_dport' \<and> p_tcp_flags = p_tcp_flags' \<and> p_payload = p_payload' \<and> more = more')"
assms_1: shows "f \<le> w \<squnion> w\<^sup>T"
matrix_def: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" shows "matrix f = (\<chi>i j. f (axis j (1::'a)) $ i)"
roots: shows "rat_roots2 p = map ((+) (- b2a)) (sqrt_rat bac)"
b_all: shows "basis_completion v = v # concat (map (\<lambda>i. if i \<noteq> k then [unit_vec n i] else []) [0..<n])"
weak_supplementation: fixes y :: "i"   and x :: "i" assumes "PP y x" shows "\<exists>z. P z x \<and> \<not> O z y"
returns_result_select_result: fixes h :: "'m"   and f :: "('m, 'n, 'o) prog" assumes "h \<turnstile> ok f" shows "h \<turnstile> f \<rightarrow>\<^sub>r |h \<turnstile> f|\<^sub>r"
lt_ge_tt: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "tt p \<preceq>\<^sub>t lt p"
and_abs_def: shows "and_abs \<equiv> map_fun Rep (map_fun Rep Abs) LTLAnd"
connected_with_outside: fixes S :: "'a set" assumes "closed S"   and "connected S" shows "connected (S \<union> outside S)"
below_DEFL_udom: fixes A :: "udom defl" shows "A \<sqsubseteq> DEFL(udom)"
imageE: fixes b :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "s2p (\<P> f A) b" obtains x :: "'b" where "b = f x" and "s2p A x"
OstrowskiTr1: fixes v :: "'a \<Rightarrow> ant"   and s :: "'a"   and t :: "'a" assumes "valuation K v"   and "s \<in> carrier K"   and "t \<in> carrier K"   and "0 \<le> v s"   and "v t < 0" shows "s \<plusminus> t \<noteq> \<zero>"
assms_5: shows "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])"
exec_gpv_map_gpv': fixes callee :: "'i \<Rightarrow> 'j \<Rightarrow> ('k \<times> 'i) spmf"   and f :: "'l \<Rightarrow> 'h"   and g :: "'m \<Rightarrow> 'j"   and h :: "'k \<Rightarrow> 'n"   and gpv :: "('l, 'm, 'n) gpv"   and s :: "'i" shows "exec_gpv callee (map_gpv' f g h gpv) s = map_spmf (map_prod f id) (exec_gpv (map_fun id (map_fun g (map_spmf (map_prod h id))) callee) gpv s)"
dg_prod_Dom_vdomain: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V" shows "\<D>\<^sub>\<circ> (dg_prod I \<AA>\<lparr>Dom\<rparr>) = dg_prod I \<AA>\<lparr>Arr\<rparr>"
le_divide_eq_1_pos: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less_eq one (divide b a) = less_eq a b"
CAS1Red1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr1"   and e3 :: "'addr expr1" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>e\<bullet>compareAndSwap(D\<bullet>F, e2, e3),s\<rangle> -ta\<rightarrow> \<langle>e'\<bullet>compareAndSwap(D\<bullet>F, e2, e3),s'\<rangle>"
Re_poly_of_real: fixes p :: "complex poly"   and x :: "real" shows "Re (poly p (complex_of_real x)) = poly (map_poly Re p) x"
ipcidr_union_set_uncurry: fixes ips :: "('a word \<times> nat) set" shows "ipcidr_union_set ips = \<Union> (uncurry ipset_from_cidr ` ips)"
ivl_disj_un_singleton_2: fixes u :: "'c" shows "{..<u} \<union> {u} = {..u}"
sylvester_mat_def: fixes p :: "'c poly"   and q :: "'c poly" shows "sylvester_mat p q \<equiv> sylvester_mat_sub (degree p) (degree q) p q"
countable_relpow: fixes X :: "('b \<times> 'b) set"   and Y :: "'b set"   and i :: "nat" assumes "\<And>Y. countable Y \<Longrightarrow> countable (X `` Y)"   and "countable Y" shows "countable ((X ^^ i) `` Y)"
create_character_data_known_ptr: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list"   and new_character_data_ptr :: "(_) character_data_ptr" assumes "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>r new_character_data_ptr" shows "known_ptr (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr)"
assms_3: fixes p :: "'a poly"   and a :: "'a" assumes "Q p" shows "Q ([:- a, 1::'a:] * p)"
flow0_defined: fixes xa :: "real"   and x0 :: "'a" assumes "xa \<in> rev.existence_ivl0 x0" shows "rev.flow0 x0 xa \<in> X"
image_iff: fixes z :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(z \<in> f ` A) = (\<exists>x\<in>A. z = f x)"
gen_ramanujan_sum_distrib_left: fixes a :: "nat"   and k :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> complex"   and g :: "nat \<Rightarrow> complex" assumes "0 < a"   and "0 < k"   and "0 < m"   and "coprime a k"   and "coprime k m"   and "multiplicative_function f"   and "multiplicative_function g" shows "s f g (m * k) a = s f g m a * g k"
fps_divide_shift_numer: fixes n :: "nat"   and f :: "'a fps"   and g :: "'a fps" assumes "n \<le> subdegree f" shows "fps_shift n f / g = fps_shift n (f / g)"
component_of_def: fixes g :: "'b ugraph"   and r :: "'b" shows "component_of g r \<equiv> ins_node r (restrict_nodes g (reachable_nodes g r))"
h9: shows "(\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f \<noteq> 0)) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
IH: assumes "a \<sqsubseteq> a'"   and "eq a' v1 v2" shows "eq a v1 v2"
wls_fresh_swap_exchange2: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x2 :: "'var"   and x1 :: "'var" assumes "wls s X" shows "fresh xs x2 (X #[x2 \<and> x1]_xs) = fresh xs x1 X"
simps_1: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signals :: "signal \<Rightarrow> nat"   and chan :: "chan set"   and bad :: "agent set"   and more :: "'a" shows "chan \<lparr>ik = ik, secret = secret, progress = progress, signals = signals, chan = chan, bad = bad, \<dots> = more\<rparr> = chan"
cnf_of_encode_initial_state_set_1: fixes \<Pi> :: "'b strips_problem"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "v \<in> dom (\<Pi>\<^sub>I)" shows "(\<Pi>\<^sub>I) v = Some True \<longrightarrow> (\<exists>!C. C \<in> cnf (\<Phi>\<^sub>I \<Pi>) \<and> C = {(State 0 (index (\<Pi>\<^sub>\<V>) v))\<^sup>+})"
1: fixes i :: "nat" shows "set_integrable lborel {l i..u i} f"
cut_def: fixes f :: "'b \<Rightarrow> 'c"   and R :: "('b \<times> 'b) set"   and x :: "'b" shows "cut f R x = (\<lambda>y. if (y, x) \<in> R then f y else undefined)"
trans: fixes a :: "'loc \<times> 't"   and b :: "'loc \<times> 't"   and c :: "'loc \<times> 't" assumes "a\<le>\<^sub>pb"   and "b\<le>\<^sub>pc" shows "a\<le>\<^sub>pc"
steps_add: fixes P :: "'a"   and Cal :: "'a processCalculus"   and Q :: "'a"   and R :: "'a" assumes "P \<longmapsto>Cal* Q"   and "Q \<longmapsto>Cal* R" shows "P \<longmapsto>Cal* R"
distinct_15: shows "cospan_shape.AA \<noteq> cospan_shape.AT"
confluence_to_local_confluence: fixes y :: "'a"   and x :: "'a" assumes "y\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star> \<cdot> y\<^sup>\<star>" shows "y \<cdot> x \<le> x\<^sup>\<star> \<cdot> y\<^sup>\<star>"
rem_UnPath_odd: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "finite E"   and "finite V"   and "is_trail v ps v'"   and "odd (degree v' G)" shows "int (num_of_odd_nodes (rem_unPath ps G)) = int (num_of_odd_nodes G) + (if odd (degree v G) \<and> v \<noteq> v' then - 2 else 0)"
intros_1: fixes A :: "'b set" assumes "finite A" shows "R A 0"
tab_simps_5: shows "D.cod (D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho>) = F r \<star>\<^sub>D F f"
same_messages_if_no_occurrence_trace: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and j :: "nat"   and p :: "'a"   and q :: "'a"   and cid :: "nat" assumes "trace init t final"   and "i \<le> j"   and "j \<le> length t"   and "\<forall>k. i \<le> k \<and> k < j \<longrightarrow> occurs_on (t ! k) \<noteq> p \<and> occurs_on (t ! k) \<noteq> q"   and "channel cid = Some (p, q)" shows "msgs (S t i) cid = msgs (S t j) cid \<and> cs (S t i) cid = cs (S t j) cid"
nvl_length: fixes e :: "nat list"   and v :: "nat"   and f :: "face" assumes "hd e = 0" shows "|natToVertexList v f e| = |e|"
mult_cancel_left1: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and c :: "'c"   and b :: "'c" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(c = times c b) = (c = zero \<or> b = one)"
bv_to_nat_def: shows "bv_to_nat = foldl (\<lambda>bn b. 2 * bn + bitval b) 0"
normFaces_def: fixes fl :: "face list" shows "normFaces fl \<equiv> map normFace fl"
RPCbusy: fixes rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and p :: "PrIds"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan" shows "\<turnstile> $Calling rcv p \<and> $(rst!p) = #rpcB \<longrightarrow> \<not> RPCNext send rcv rst p"
is_solution_imp_solve_system: fixes x :: "('b, 'c) vec"   and A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "is_solution x A b" shows "is_solution x (fst (solve_system A b)) (snd (solve_system A b))"
c'_def: shows "c' \<equiv> \<lambda>i. if i \<le> k then c i else z"
simps_1: fixes lo :: "'b wordinterval" shows "routing_port_ranges [] lo = (if wordinterval_empty lo then [] else [(routing_oiface undefined, lo)])"