hcju/isabelleps huggingface dataset:
vp_split_Cons: 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 n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n as n'"   and "as \<noteq> []" obtains a' :: "'a" and as' :: "'a list" where "as = a' # as'" and "n = sourcenode a'" and "valid_edge a'" and "CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges (targetnode a') as' n'"
dist_triangle_eq: fixes a :: "'b"   and c :: "'b"   and b :: "'b" shows "(dist a c = dist a b + dist b c) = (dist a b *\<^sub>R (c - b) + dist b c *\<^sub>R (a - b) = (0::'b))"
sff'_1: shows "f = Polynomial.smult a (\<Prod>(a, i)\<in>set psi. a ^ Suc i)"
on_circline_moebius_circline_I: fixes H :: "circline"   and z :: "complex_homo"   and M :: "moebius" assumes "on_circline H z" shows "on_circline (moebius_circline M H) (moebius_pt M z)"
simps_25: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_aexp_aux (Times e1 e2) (Minus v va) = False"
distrib: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b) \<cdot> c = a \<cdot> c + b \<cdot> c"
comp_arr_dom: fixes f :: "'s"   and a :: "'s" assumes "arr f"   and "local.dom f = a" shows "f \<cdot> a = f"
Int_left_commute: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
less_eq_multiset_empty_left: fixes M :: "'a multiset" shows "{#} \<le> M"
strips_equivalent_to_sas_plus_i_a_I: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops :: "('c, 'd) sas_plus_operator list"   and op' :: "('c \<times> 'd) strips_operator" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"   and "op' \<in> set (map (sasp_op_to_strips \<Psi>) ops)" obtains op :: "('c, 'd) sas_plus_operator" where "op \<in> set ops" and "op' = \<phi>\<^sub>O \<Psi> op"
DiffD1: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "c \<in> A - B" shows "c \<in> A"
get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf: fixes h :: "(_) heap"   and object_ptr :: "(_) object_ptr" assumes "type_wf h" shows "(object_ptr |\<in>| object_ptr_kinds h) = (get object_ptr h \<noteq> None)"
rep_eq: shows "poly_mapping.lookup 1 = (\<lambda>k. (1::'b) when k = (0::'a))"
CollectD: fixes a :: "'e"   and P :: "'e \<Rightarrow> bool" assumes "a \<in> {x. P x}" shows "P a"
F_preserves_objects: shows "F \<^bsub>\<o>\<^esub> \<in> Ob \<rightarrow> Ob\<^bsub>BB\<^esub>"
top_covector_def: shows "top \<equiv> Abs_covector top"
weak_map_of_SomeI: fixes k :: "'a"   and x :: "'b"   and l :: "('a \<times> 'b) list" assumes "(k, x) \<in> set l" shows "\<exists>x. Map.map_of l k = Some x"
imp_imp_compat_eqvR: fixes \<phi> :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla" shows "prv (imp (eqv \<chi>1 \<chi>2) (eqv (imp \<phi> \<chi>1) (imp \<phi> \<chi>2)))"
not_fic_subst1: fixes M :: "trm"   and a :: "coname"   and y :: "name"   and c :: "coname"   and P :: "trm" assumes "\<not> fic M a" shows "\<not> fic (M{y:=<c>.P}) a"
rtranclp_trans: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k"   and z :: "'k" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>*\<^sup>* y z" shows "r\<^sup>*\<^sup>* x z"
lenlex_length: fixes ms :: "'a list"   and ns :: "'a list"   and r :: "('a \<times> 'a) set" assumes "(ms, ns) \<in> lenlex r" shows "length ms \<le> length ns"
inv: shows "A \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) globala P"
Int_Un_eq_4: fixes T :: "'c set"   and S :: "'c set" shows "T \<union> S \<inter> T = T"
assms_2: assumes "x \<notin>\<^sub>\<circ> A" shows "R"
select_convs_3: fixes TDomain :: "'a"   and TExprSet :: "('a, 'b) Expression set"   and TCodomain :: "'a"   and more :: "'c" shows "TCodomain \<lparr>TDomain = TDomain, TExprSet = TExprSet, TCodomain = TCodomain, \<dots> = more\<rparr> = TCodomain"
add_increasing2: fixes c :: "'e multiset"   and b :: "'e multiset"   and a :: "'e multiset" assumes "{#} \<subseteq># c"   and "b \<subseteq># a" shows "b \<subseteq># a + c"
that: fixes a' :: "'edge"   and as' :: "'edge list" assumes "as = a' # as'"   and "(_High_) -as'\<rightarrow>* n"   and "kind a' = (\<lambda>s. True)\<^sub>\<surd>" shows "thesis"
prv_tms_imp_cong: fixes t1 :: "'trm"   and t1' :: "'trm"   and t2 :: "'trm"   and t2' :: "'trm" assumes "t1 \<in> atrm"   and "t1' \<in> atrm"   and "t2 \<in> atrm"   and "t2' \<in> atrm" shows "prv (imp (eql t1 t1') (imp (eql t2 t2') (eql (tms t1 t2) (tms t1' t2'))))"
results_in_mono_1: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and t1 :: "'c"   and t2 :: "'c"   and s :: "'b" assumes "dataflow_topology summary results_in"   and "t1 \<le> t2" shows "results_in t1 s \<le> results_in t2 s"
ta_tb_wf_4: shows "\<Gamma> ta = \<Gamma> tb"
inverse_subgroupI: fixes H :: "'a set" assumes "subgroup H G (\<cdot>) \<one>" shows "subgroup (inverse ` H) G (\<cdot>) \<one>"
assms_2: shows "\<forall>i. (S i, S (Suc i)) \<in> r"
\<B>: fixes S :: "'a set" assumes "open S" shows "\<exists>U\<subseteq>\<B>. S = \<Union> U"
a1: shows "\<not> (reset_trap_val s = False \<and> get_ET (cpu_reg_val PSR s) = 0)"
assms_1: shows "eoutKnowCorrect C (kKS m)"
\<sigma>_def: shows "\<sigma> = (\<lambda>n. real (divisor_count n))"
not_less_iff_gr_or_eq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" shows "(\<not> less x y) = (less y x \<or> x = y)"
nsumElmAdd: fixes A :: "('c, 'd) aGroup_scheme"   and f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "aGroup A"   and "\<forall>i. f i \<in> carrier A"   and "\<forall>i. g i \<in> carrier A" shows "\<Sigma>\<^sub>e A (\<lambda>i. f i \<plusminus>\<^bsub>A\<^esub> g i) n = \<Sigma>\<^sub>e A f n \<plusminus>\<^bsub>A\<^esub> \<Sigma>\<^sub>e A g n"
divideC_field_simps_1: fixes c :: "complex"   and a :: "'b"   and b :: "'b" assumes "c \<noteq> 0" shows "(a = b /\<^sub>C c) = (c *\<^sub>C a = b)"
balance_in_tree: fixes k :: "'a"   and x :: "'b"   and l :: "('a, 'b) rbt"   and v :: "'a"   and y :: "'b"   and r :: "('a, 'b) rbt" shows "((k, x) \<in> set (entries (balance l v y r))) = ((k, x) \<in> set (entries l) \<or> k = v \<and> x = y \<or> (k, x) \<in> set (entries r))"
bot_set_def: shows "{} = Collect Orderings.bot_class.bot"
sep_conj_assoc: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and R :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj (sep_algebra.sep_conj plus sep_disj P Q) R = sep_algebra.sep_conj plus sep_disj P (sep_algebra.sep_conj plus sep_disj Q R)"
Sup_least: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'d set"   and z :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
False: shows "a \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"
relH_ref: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap"   and r :: "'b ref" assumes "relH as h h'"   and "addr_of_ref r \<in> as" shows "Ref.get h r = Ref.get h' r"
not_unifD: fixes e :: "('c, 'd) Term.term \<times> ('c, 'd) Term.term" assumes "\<nexists>E'. ({#e#}, E') \<in> unif" shows "(\<exists>x t. (e = (Var x, t) \<or> e = (t, Var x)) \<and> x \<in> vars_term t \<and> is_Fun t) \<or> (\<exists>f g ss ts. e = (Fun f ss, Fun g ts) \<and> (f \<noteq> g \<or> length ss \<noteq> length ts))"
TzQ: fixes C :: "acom"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and n :: "nat"   and s :: "char list \<Rightarrow> int" shows "preTz C e n s = e (postQz C s n)"
get_delayed_write_def: fixes dwp :: "(int \<times> word32 \<times> CPU_register) list" shows "get_delayed_write dwp \<equiv> get_delayed_0 dwp"
status_assn_pure_conv: fixes a :: "'a code_status"   and b :: "'a code_status" shows "status_assn id_assn a b = id_assn a b"
lc_minus_higher: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p" shows "lc (p - higher p v) = lookup p v"
x: shows "x = lincomb_list lc (us[i := us ! i + c \<cdot>\<^sub>v us ! j])"
LID\<^sub>L_def: fixes s :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr" shows "LID\<^sub>L s \<equiv> case s of (\<sigma>, \<tau>, e) \<Rightarrow> LID\<^sub>S \<sigma> \<union> LID\<^sub>S \<tau> \<union> LID\<^sub>E e"
hindered_into_hindered_by: fixes \<Gamma> :: "('a, 'b) web_scheme" assumes "hindered \<Gamma>" shows "\<exists>\<epsilon>>0. hindered_by \<Gamma> \<epsilon>"
1: shows "map (Struct.int intF1 \<xi>) Tl = map (Struct.int intF2 \<xi>) Tl"
casts_casts_eq_result: 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 T :: "ty"   and v :: "val"   and v' :: "val"   and w' :: "val"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and T' :: "ty"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option" assumes "P \<turnstile> T casts v to v' "   and "P \<turnstile> T casts v to w' "   and "is_type P T"   and "P,E \<turnstile> e :: T'"   and "P \<turnstile> T' \<le> T"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>Val v,(h, l)\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "wf_C_prog P" shows "v' = w'"
simps_2: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" shows "fold f (x # xs) = fold f xs \<circ> f x"
\<epsilon>o_def: fixes x :: "'c" shows "\<epsilon>o x = \<psi> x (G x)"
emeasure_spmf_single: fixes p :: "'c spmf"   and x :: "'c" shows "emeasure (measure_spmf p) {x} = ennreal (spmf p x)"
that: fixes X0 :: "'a set" assumes "x0 \<in> X0"   and "open X0"   and "\<forall>x\<in>X0 \<inter> U. \<forall>t\<in>C. dist (fx (x, t)) (fx (x0, t)) \<le> e" shows "thesis"
C_se: shows "C \<subseteq> {(u, p). dist u \<noteq> \<infinity> \<and> awalk u p u \<and> awalk_cost c p < 0}"
strip_while_def: fixes P :: "'c \<Rightarrow> bool" shows "strip_while P = rev \<circ> dropWhile P \<circ> rev"
minsky_reaching_0_def: shows "minsky_reaching_0 = {uu_. \<exists>n M vs vs'. uu_ = n \<and> (M, vs) = decode_minsky_state n \<and> ((Suc 0, vs), 0, vs') \<in> (step M)\<^sup>*}"
h3: shows "insertion f (isolate_variable_sparse p v d) * f v ^ d = g d"
assms: shows "\<phi> (\<sigma>\<^bsub>the_singleton\<^esub>t n)"
assm2: shows "mn \<in> set (normalize_primitive_extract (disc, sel) C f m)"
induct: fixes x1 :: "char list exp list"   and x2 :: "val list"   and P :: "char list exp list \<Rightarrow> val list \<Rightarrow> bool" assumes "map_val x1 x2"   and "P [] []"   and "\<And>xs ys y. \<lbrakk>map_val xs ys; P xs ys\<rbrakk> \<Longrightarrow> P (Val y # xs) (y # ys)" shows "P x1 x2"
eint_minus_le: fixes b :: "eint"   and c :: "eint" assumes "b < c" shows "0 < c - b"
length_insert_nth: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "length (insert_nth n x xs) = Suc (length xs)"
elims_2: fixes x :: "('c, 'd) trie" assumes "is_empty_trie x" obtains v :: "'d option" and m :: "('c \<times> ('c, 'd) trie) list" where "x = Trie v m" and "v = None \<and> m = []"
nprv_cnjE0: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "cnj \<phi>1 \<phi>2 \<in> F"   and "nprv (insert \<phi>1 (insert \<phi>2 F)) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F \<psi>"
dma\<^sub>C_eq': shows "dma\<^sub>C mem\<^sub>1\<^sub>C [\<parallel>\<^sub>1 A\<^sub>C] = dma\<^sub>C mem\<^sub>2\<^sub>C [\<parallel>\<^sub>2 A\<^sub>C]"
sop_one: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and i :: "'d \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set" assumes "commute_bpp f (aug_pm_set z i (s_set R s A))"   and "assoc_bpp (aug_pm_set z i (s_set R s A)) f"   and "ipp_cond1 (s_set R s A) i"   and "ipp_cond2 z (s_set R s A) i f"   and "ipp_cond3 z i"   and "inv_ipp z i f (s_set R s A)"   and "zeroA z i f (s_set R s A) z"   and "sop_distr2 R s f i (s_set R s A) z"   and "sop_assoc R s (aug_pm_set z i (s_set R s A))"   and "sop_one R s (aug_pm_set z i (s_set R s A))" shows "\<forall>x\<in>addition_set f (aug_pm_set z i (s_set R s A)). 1\<^sub>r \<^sub>s\<cdot> x = x"
local_steps_commute: fixes s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and x :: "('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r' :: "'r"   and y :: "('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr" assumes "s\<^sub>2 = s\<^sub>1(r \<mapsto> x)"   and "s\<^sub>2' = s\<^sub>1(r' \<mapsto> y)"   and "revision_step r' (s\<^sub>1(r \<mapsto> x)) (s\<^sub>1(r \<mapsto> x, r' \<mapsto> y))"   and "revision_step r (s\<^sub>1(r' \<mapsto> y)) (s\<^sub>1(r' \<mapsto> y, r \<mapsto> x))" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
S_M_selects_subseteq: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and M :: "'a literal multiset set"   and C :: "'a literal multiset" assumes "selection S" shows "S_M S M C \<subseteq># C"
is_lub_unique: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "class.po below"   and "po.is_lub below S x"   and "po.is_lub below S y" shows "x = y"
0: shows "alw (\<lambda>js. (shd (stl js), shd js) \<in> Ord \<and> shd (stl js) \<noteq> shd js) ls"
order_refl: fixes ko :: "'a key_order"   and x :: "'a" shows "ko.le ko x x"
set_val_get_tag_name: fixes ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>set_val_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
float_frac_le: fixes a :: "('a, 'b) IEEE.float" shows "fraction a \<le> 2 ^ LENGTH('b) - 1"
is_span: shows "C.span \<mu>.cod.leg0 \<mu>.cod.leg1"
D: fixes r :: "('b \<times> 'b) set"   and \<tau> :: "'b list"   and \<sigma> :: "'b list"   and \<sigma>' :: "'b list"   and \<tau>' :: "'b list" assumes "trans r"   and "irrefl r" shows "D r \<tau> \<sigma> \<sigma>' \<tau>' = ((r|\<tau>'| -s r \<down>l \<sigma>, r|\<tau>|) \<in> mul_eq r \<and> (r|\<sigma>'| -s r \<down>l \<tau>, r|\<sigma>|) \<in> mul_eq r)"
is_real_inversion: fixes x :: "complex" assumes "is_real x"   and "x \<noteq> 0" shows "is_real (to_complex (inversion (of_complex x)))"
ord_less_eq_trans: fixes a :: "'a Word.word"   and b :: "'a Word.word"   and c :: "'a Word.word" assumes "a <s b"   and "b = c" shows "a <s c"
obj_sumE: fixes s :: "'e + 'f"   and P :: "bool" assumes "\<forall>x. s = Inl x \<longrightarrow> P"   and "\<forall>x. s = Inr x \<longrightarrow> P" shows "P"
must_sync_def: fixes t :: "'t"   and x :: "'x1"   and m :: "'m1" shows "t \<turnstile> \<langle>x, m\<rangle> \<wrong>1 = (\<exists>taa x' m' s. r1.r_syntax t x m taa x' m' \<and> shr s = m \<and> r1.actions_ok s t taa)"
sec_id_pubk_more_rel: fixes m :: "nat"   and m' :: "nat"   and n :: "nat" assumes "{m, m'} \<subseteq> bad_sec_prik"   and "(n, m) \<notin> bad_id"   and "(n, m') \<in> bad_id" shows "s\<^sub>0 \<Turnstile> sec_id_pubk_more n m"
fin_R_L: shows "finite (vertices (snd R) - vertices (fst R))"
chine_composite: shows "\<nu>\<pi>.chine = \<langle>\<nu>.chine \<cdot> \<p>\<^sub>1[\<nu>.dom.leg0, \<pi>.dom.leg1] \<lbrakk>\<nu>.cod.leg0, \<pi>.cod.leg1\<rbrakk> \<pi>.chine \<cdot> \<p>\<^sub>0[\<nu>.dom.leg0, \<pi>.dom.leg1]\<rangle>"
weak_barbed_coupled_simulation_versus_simulation_and_contrasimulation: fixes Rel :: "('a \<times> 'a) set"   and CWB :: "('a, 'b) calculusWithBarbs" shows "weak_barbed_coupled_simulation Rel CWB = (weak_barbed_simulation Rel CWB \<and> weak_barbed_contrasimulation Rel CWB)"
face_im: fixes X :: "'b set set"   and w :: "'b set"   and y :: "'c set"   and f :: "'b \<Rightarrow> 'c" assumes "SimplicialComplex X"   and "w \<in> X"   and "order.greater_eq (f ` w) y" shows "y \<in> f \<turnstile> X"
execute_bind_1: fixes f :: "'a Heap"   and h :: "heap"   and x :: "'a"   and h' :: "heap"   and g :: "'a \<Rightarrow> 'b Heap" assumes "execute f h = Some (x, h')" shows "execute (f \<bind> g) h = execute (g x) h'"
assms_3: shows "(i, Store e1 mo i) \<in> set (node_deliver_messages xs)"
equivalence_map_preserved_by_iso: fixes f :: "'a"   and f' :: "'a" assumes "B.equivalence_map f"   and "f \<cong>\<^sub>B f'" shows "B.equivalence_map f'"
simps_5: fixes v :: "'a"   and l :: "value list"   and a :: "'a aexp" shows "gexp_constrains (In v l) a = aexp_constrains (V v) a"
Cons_1: assumes "(f, g) \<in> set ps" shows "monom_mult ((1::'b) / lc (fst f)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst f)) (fst f) \<in> set (pdata_pairs_to_list ps)"
poisson_density'_def: fixes rate :: "real"   and k :: "int" shows "poisson_density' rate k = ennreal (pmf (poisson_pmf rate) (nat k) * indicat_real ({0<..} \<times> {0..}) (rate, k))"
s_mono: fixes u :: "real"   and t :: "real" assumes "u \<le> t"   and "0 \<le> u" shows "ego2.s u \<le> ego2.s t"
inj_circline_set: fixes H :: "circline"   and H' :: "circline" assumes "circline_set H = circline_set H'"   and "circline_set H \<noteq> {}" shows "H = H'"
Suc_le_length_iff: fixes n :: "nat"   and xs :: "'g list" shows "(Suc n \<le> length xs) = (\<exists>x ys. xs = x # ys \<and> n \<le> length ys)"
less_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less" shows "less x y = (less_eq x y \<and> x \<noteq> y)"
IH: shows "\<exists>\<xi>. \<sigma> i = \<xi> \<and> (\<xi>, p) \<in> reachable A I"
sel_1: fixes eff :: "'b \<Rightarrow> 'a \<Rightarrow> 'a fset \<Rightarrow> bool"   and rs :: "'b stream"   and s :: "'a" shows "tree.root (RuleSystem_Defs.mkTree eff rs s) = (s, shd (RuleSystem_Defs.trim eff rs s))"
flush_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (flush_instr instr s1)) \<and> t2 = snd (fst (flush_instr instr s2))" shows "low_equal t1 t2"
simps_4: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Minus f) env = map_option tm_neg (approx_tm prec ord I a f env)"
independent_insert: fixes a :: "'d"   and S :: "'d set" shows "cindependent (insert a S) = (if a \<in> S then cindependent S else cindependent S \<and> a \<notin> cspan S)"
comp_null_2: fixes f :: "('o, 'a) arr" shows "COMP f null = null"
gen_iapath_def: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "sdG.gen_iapath V u p v \<equiv> u \<in> V \<and> v \<in> V \<and> sdG.apath u p v \<and> set (sdG.inner_verts p) \<inter> V = {} \<and> p \<noteq> []"
winding_number_uminus_comp: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "valid_path \<gamma>"   and "- z \<notin> path_image \<gamma>" shows "winding_number (uminus \<circ> \<gamma>) z = winding_number \<gamma> (- z)"
simps_8: fixes f1 :: "T1 \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "rec_T2 f1 f2 (T2.D x2) = f2 x2"
TSL: shows "SIDE_GEN_ALGO (is_map_to_sorted_list R Rk Rv Rs tsl)"
of_nat_power: fixes m :: "nat"   and n :: "nat" shows "of_nat (m ^ n) = of_nat m ^ n"
surj_fn: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "surj f" shows "surj (f ^^ n)"
project_Constrains: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set" shows "(F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> A Co B) = (extend h F \<squnion> G \<in> extend_set h A Co extend_set h B)"
Events_def: shows "Events \<equiv> fst \<circ> snd \<circ> snd \<circ> Rep_status"
llist_of: fixes ys :: "'a list" assumes "xs = llist_of ys" shows "thesis"
mHom_compos: fixes L :: "('i, 'b, 'j) Module_scheme"   and N :: "('k, 'b, 'l) Module_scheme"   and f :: "'i \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'k" assumes "R module L"   and "R module N"   and "f \<in> mHom R L M"   and "g \<in> mHom R M N" shows "compos L g f \<in> mHom R L N"
sublmm: fixes \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "((s, t) \<cdot>\<^sub>p \<delta> \<circ>\<^sub>s \<I> \<notin> set D \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>) = list_all (\<lambda>d. Ineq' \<delta> [(pair (s, t), pair d)]) D"
prefix_of_replace_prefix: fixes f :: "'c set \<Rightarrow> 'c set"   and \<pi> :: "'c prefix"   and \<sigma> :: "'c trace" assumes "prefix_of (pmap_\<Gamma> f \<pi>) \<sigma>" shows "prefix_of \<pi> (replace_prefix \<pi> \<sigma>)"
sub: shows "f ` ({0..<M} \<inter> v0) \<subseteq> {0..<card ({0..<M} \<inter> v0)}"
rtranclp_imp_relpowp: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "P\<^sup>*\<^sup>* x y" shows "\<exists>n. (P ^^ n) x y"
oinvariant_def: fixes A :: "('e \<times> 'f, 'g) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<times> 'f \<Rightarrow> bool" shows "(A \<Turnstile> (S, U \<rightarrow>) P) = (\<forall>s\<in>oreachable A S U. P s)"