hcju/isabelleps huggingface dataset:
resSimCases: fixes Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and x :: "name"   and P :: "pi"   and C :: "'a" assumes "eqvt Rel"   and "\<And>Q' a. \<lbrakk>Q \<longmapsto> a[x] \<prec> Q'; a \<noteq> x\<rbrakk> \<Longrightarrow> \<exists>P'. P \<longmapsto> a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"   and "\<And>Q' a y. \<lbrakk>Q \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> Q'; x \<sharp> a; x \<noteq> y; y \<sharp> C\<rbrakk> \<Longrightarrow> \<exists>P'. P \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> P' \<and> derivative P' (<\<nu>x>Q') a y Rel"   and "\<And>Q' \<alpha>. \<lbrakk>Q \<longmapsto> \<alpha> \<prec> Q'; x \<sharp> \<alpha>\<rbrakk> \<Longrightarrow> \<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', <\<nu>x>Q') \<in> Rel" shows "P \<leadsto>[Rel] <\<nu>x>Q"
RuUSdBeqlen: fixes d :: "'d"   and R :: "('com list \<times> 'com list) set"   and V :: "'com list"   and V' :: "'com list" assumes "d_Bisimulation_Up_To_USdB d R"   and "(V, V') \<in> R \<union> (\<approx>\<^bsub>d\<^esub>)" shows "length V = length V'"
neqE: fixes x :: "'a"   and y :: "'a"   and ko :: "'a key_order" assumes "x \<noteq> y" obtains "ko.lt ko x y"   | "ko.lt ko y x"
char_to_digit_simps_9: shows "char_to_digit CHR ''8'' = 8"
max_ideal_iff_max_lideal: fixes I :: "'a set" shows "max_ideal R I (+) (\<cdot>) \<zero> \<one> = max_lideal I R (+) (\<cdot>) \<zero> \<one>"
l_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool" shows "l_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_document get_disconnected_document_locs known_ptr type_wf get_host get_host_locs get_shadow_root get_shadow_root_locs get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_tag_name heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<equiv> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs \<and> l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf \<and> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"
9: shows "\<forall>x y. enum_lex_less_eq x y \<or> enum_lex_less_eq y x"
assms: shows "GEN_ALGO_tag (is_map_to_sorted_list ordR Rk Rv Rm tsl)"
ref_valI: fixes A :: "'a attree"   and A' :: "'a attree" assumes "A \<sqsubseteq> A'"   and "\<turnstile>A'" shows "\<turnstile>\<^sub>V A"
subgraphs_finite: fixes G :: "nat set \<times> nat set set" assumes "finite_graph G" shows "finite (subgraphs G)"
arev_arev_raw: fixes G :: "('e, 'f) pre_digraph"   and arev :: "'f \<Rightarrow> 'f"   and a :: "'f" assumes "bidirected_digraph G arev"   and "a \<in> arcs G" shows "arev (arev a) = a"
llemma: fixes p :: "nat"   and \<C> :: "(nat \<Rightarrow> real) set set" assumes "standard_simplex p \<subseteq> \<Union> \<C>"   and "\<And>U. U \<in> \<C> \<Longrightarrow> openin (powertop_real UNIV) U" obtains d :: "real" where "0 < d" and "\<And>K. \<lbrakk>K \<subseteq> standard_simplex p; \<And>x y i. \<lbrakk>i \<le> p; x \<in> K; y \<in> K\<rbrakk> \<Longrightarrow> \<bar>x i - y i\<bar> \<le> d\<rbrakk> \<Longrightarrow> \<exists>U. U \<in> \<C> \<and> K \<subseteq> U"
derived_set_of_mono: fixes S :: "'c set"   and T :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> T" shows "X derived_set_of S \<subseteq> X derived_set_of T"
imod_infinity_right: fixes n :: "enat" shows "n mod \<infinity> = n"
simps_11: fixes vd :: "'a" shows "(extN.Top \<le> extN.Val vd) = False"
vsv_ex1_app1: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "(b = r\<lparr>a\<rparr>) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r)"
D: shows "distributed M lborel X (\<lambda>x. ennreal (erlang_density k l x))"
mult_less_le_imp_less: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less_eq c d"   and "less_eq zero a"   and "less zero c" shows "less (times a c) (times b d)"
right_unique_fun: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "right_total A"   and "right_unique B" shows "right_unique (rel_fun A B)"
mult_rel_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> R \<times> (R\<setminus>I)"   and "(r', s') \<in> R \<times> (R\<setminus>I)" shows "cq.mult_rel (cq.frac r s) (cq.frac r' s') = cq.frac (r \<cdot> r') (s \<cdot> s')"
eq_iff: fixes x :: "'a list"   and y :: "'a list" shows "(x = y) = (suffix x y \<and> suffix y x)"
fag_single_elemTr: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "nat" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "\<forall>x. x \<in> add_set f (aug_pm_set z i {a}) n \<longrightarrow> (\<exists>n1. x \<in> add_set f {a} n1) \<or> (\<exists>m1. x \<in> add_set f {\<^sub>i- a} m1) \<or> x = z"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'c"   and bool :: "bool" shows "P (case bool of True \<Rightarrow> f1 | False \<Rightarrow> f2) = ((bool = True \<longrightarrow> P f1) \<and> (bool = False \<longrightarrow> P f2))"
\<pi>''0: shows "\<pi>'' 0 = ipd (\<pi> k)"
ext_wf_bounded_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_wf_bounded ext \<equiv> ext_irrefl_before_trans ext \<and> ext_hd_or_tl ext"
bexE: fixes A :: "'f set"   and P :: "'f \<Rightarrow> bool" assumes "\<exists>x\<in>A. P x" obtains x :: "'f" where "x \<in> A" and "P x"
convergent_limsup_cl: fixes X :: "nat \<Rightarrow> 'a" assumes "convergent X" shows "limsup X = lim X"
simps: fixes V :: "Vars.var set"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" shows "restr_conf V (\<Gamma>, e, S) = (restrictA V \<Gamma>, e, restr_stack V S)"
filtermap_mono: fixes F :: "'b filter"   and F' :: "'b filter"   and f :: "'b \<Rightarrow> 'c" assumes "F \<le> F'" shows "filtermap f F \<le> filtermap f F'"
SCallNonStatic: fixes P :: "(char list list \<times> char list exp) prog"   and ps :: "char list exp list"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and vs :: "val list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> char list exp"   and D :: "char list" assumes "P \<turnstile> \<langle>ps,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>map Val vs,s\<^sub>2\<rangle>"   and "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW IncompatibleClassChangeError,s\<^sub>2\<rangle>"
zero_fpxs_def: shows "0 \<equiv> Abs_fpxs (\<lambda>r. 0::'b)"
epi_hom_induced_relativization: fixes S :: "'b set"   and X :: "'b topology"   and p :: "int" assumes "S = {} \<or> S retract_of_space X" shows "hom_induced p X {} X S id ` carrier (homology_group p X) = carrier (relative_homology_group p X S)"
ctr_transfer: fixes R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R1 :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "(R2 ===> rel_optionT' R1 R2) OptionT OptionT"
vT': shows "P,h \<turnstile> v :\<le> vT'"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "linkage \<Gamma> f" obtains "\<And>x. x \<in> A \<Gamma> \<Longrightarrow> d_OUT f x = weight \<Gamma> x"
eq: shows "g ` S' = g ` {x \<in> S. f (g x) \<noteq> 0} \<inter> g ` {x \<in> S. det (matrix (g' x)) \<noteq> 0}"
tendsto_rabs_zero_iff: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter" shows "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> 0) F"
wf_inst_formula: fixes Q :: "variable \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and f :: "variable \<Rightarrow> object"   and \<phi> :: "PDDL_STRIPS_Semantics.term atom formula" assumes "\<And>x T. is_of_type Q x T \<Longrightarrow> is_of_type objT (f x) T"   and "wf_fmla (ty_term Q constT) \<phi>" shows "wf_fmla objT ((map_formula \<circ> map_atom \<circ> subst_term) f \<phi>)"
emeasure_subprob_space_less_top: fixes M :: "'c pmf"   and A :: "'c set" shows "emeasure (measure_pmf M) A \<noteq> \<top>"
inter_guards_Catch: fixes a1 :: "('d, 'e, 'f) com"   and a2 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" shows "((Catch a1 a2 \<inter>\<^sub>g c2) = Some c) = (\<exists>b1 b2 d1 d2. c2 = Catch b1 b2 \<and> (a1 \<inter>\<^sub>g b1) = Some d1 \<and> (a2 \<inter>\<^sub>g b2) = Some d2 \<and> c = Catch d1 d2)"
smooth_on_subset: fixes k :: "enat"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "k-smooth_on T f"   and "S \<subseteq> T" shows "k-smooth_on S f"
inject_2: fixes x31 :: "nat"   and x32 :: "'b list"   and x33 :: "'b list"   and y31 :: "nat"   and y32 :: "'b list"   and y33 :: "'b list" shows "(App x31 x32 x33 = App y31 y32 y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
cl_subset: fixes X :: "'a set" assumes "X \<subseteq> carrier" shows "X \<subseteq> cl X"
replace_1: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and j :: "nat"   and a :: "nat \<Rightarrow> nat"   and x :: "nat \<Rightarrow> nat" assumes "kuhn_simplex p n base upd s"   and "j < n"   and "a \<in> s"   and "\<forall>x\<in>s - {a}. x j = p"   and "x \<in> s" shows "a \<le> x"
nrename_aredu: fixes M :: "trm"   and x :: "name"   and y :: "name"   and M' :: "trm" assumes "M[x\<turnstile>n>y] \<longrightarrow>\<^sub>a M'"   and "x \<noteq> y" shows "\<exists>M0. M0[x\<turnstile>n>y] = M' \<and> M \<longrightarrow>\<^sub>a M0"
rel_refl_strong: fixes x :: "('b, 'd, 'e, 'f, 'g, 'h, 'i) G"   and R1a :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R2a :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> set1_G x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> set2_G x \<Longrightarrow> R2a z2 z2" shows "rell_G R1a R2a x x"
CYK_Lang: fixes S :: "'c"   and G :: "('c \<times> ('c, 'd) RHS) list"   and w :: "'d list" shows "(S \<in> CYK G w 0 (length w)) = (w \<in> Lang G S)"
foldr_append: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and xs :: "'h list"   and ys :: "'h list"   and a :: "'g" shows "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
doubleRestriction: fixes P :: "('a \<times> 'b) set"   and X :: "'a set"   and Y :: "'a set" shows "(P || X) || Y = P || (X \<inter> Y)"
push_bit_of_1: fixes n :: "nat" shows "push_bit n (1::'a) = (2::'a) ^ n"
find_closest_pair_code_eq: fixes \<delta> :: "real"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and \<delta>' :: "int"   and ps :: "(int \<times> int) list"   and C\<^sub>0 :: "int \<times> int"   and C\<^sub>1 :: "int \<times> int"   and \<Delta>' :: "int"   and C\<^sub>0' :: "int \<times> int"   and C\<^sub>1' :: "int \<times> int" assumes "\<delta> = dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))"   and "\<delta>' = dist_code c\<^sub>0 c\<^sub>1"   and "sorted_snd ps"   and "(C\<^sub>0, C\<^sub>1) = find_closest_pair (c\<^sub>0, c\<^sub>1) ps"   and "(\<Delta>', C\<^sub>0', C\<^sub>1') = find_closest_pair_code (\<delta>', c\<^sub>0, c\<^sub>1) ps" shows "C\<^sub>0 = C\<^sub>0' \<and> C\<^sub>1 = C\<^sub>1'"
simps_2: fixes n :: "nat"   and R :: "('a \<times> 'a) set" shows "R ^^ Suc n = R ^^ n ; R"
countable_Fpow: fixes S :: "'c set" assumes "countable S" shows "countable (Fpow S)"
proj_poly_syz_lift_poly_syz: fixes i :: "nat"   and n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "i < n" shows "proj_poly_syz n (lift_poly_syz n p i) = p"
Rep_trm_inject: fixes x :: "trm"   and y :: "trm" shows "(Rep_trm x = Rep_trm y) = (x = y)"
simps: fixes name :: "char list"   and preconds :: "(nat \<times> nat) list"   and effects :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and cost :: "nat" shows "rem_implicit_pres (name, preconds, effects, cost) = (name, implicit_pres effects @ preconds, map rem_effect_implicit_pres effects, cost)"
sim_actions_action_obsD: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat" assumes "E [\<approx>] E'" shows "action_obs E a \<approx> action_obs E' a"
n_kat_2_opp: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y \<cdot> !z = !x \<cdot> y \<cdot> !z) = \<lparr>y\<rparr>!z\<lparr>!x \<cdot> y\<rparr>"
set_alternating_list: fixes n :: "nat"   and s :: "'a"   and t :: "'a" shows "order.greater_eq {s, t} (set (alternating_list n s t))"
fwd: shows "collinear u v (v \<leadsto> d) \<longrightarrow> (\<exists>\<beta>. from u to v = (- \<beta>) ** d)"
lunit'_simps_4: fixes f :: "'a" assumes "C.ide f" shows "C.dom \<l>\<^sub>C\<^sup>-\<^sup>1[f] = f"
Rep_mod_ring: fixes x :: "'b mod_ring" shows "Rep_mod_ring x \<in> {0..<int CARD('b)}"
sorted_wrt_upper_sublist_still_sorted: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and hi' :: "nat"   and hi :: "nat"   and lo :: "'d"   and lo' :: "'d"   and xs' :: "'c list" assumes "sorted_sublist_wrt R xs (hi' + 1) hi"   and "lo \<le> lo'"   and "hi < length xs"   and "\<forall>j. hi' < j \<and> j \<le> hi \<longrightarrow> xs' ! j = xs ! j"   and "length xs' = length xs" shows "sorted_sublist_wrt R xs' (hi' + 1) hi"
update_convs_1: fixes p_iiface' :: "char list \<Rightarrow> char list"   and p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a 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 :: "'b" shows "p_iiface_update p_iiface' \<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_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>"
of_int_floor_le: fixes x :: "'a" shows "of_int \<lfloor>x\<rfloor> \<le> x"
ae: shows "AE x in M. (\<forall>n\<in>I. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n. n \<in> I \<and> P n x))"
add_left_cancel: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b = a + c) = (b = c)"
rbtreeify_f_rec_aux_lemma: fixes k :: "nat"   and n :: "nat"   and k' :: "nat" assumes "k - n div 2 = Suc k'"   and "n \<le> k"   and "n mod 2 = Suc 0" shows "k' - n div 2 = k - n"
eqp: fixes x :: "'a \<times> nat" assumes "x \<in> set (zip xs [0..<length xs])" shows "(snd x \<in> - B) = (snd x \<in> A - B)"
collinear_between_cases: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "collinear {a, b, c} = (between (b, c) a \<or> between (c, a) b \<or> between (a, b) c)"
Icc_subset_Ici_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and h :: "'a"   and l' :: "'a" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq l h \<subseteq> ord.atLeast less_eq l') = (\<not> less_eq l h \<or> less_eq l' l)"
fds_nth_fds_pos: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "0 < n" shows "fds_nth (fds f) n = f n"
inf_distrib_def: fixes f :: "'c \<Rightarrow> 'd" shows "inf_distrib f \<equiv> strict f \<and> cont f"
aff_dim_linear_image_le: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "linear f" shows "aff_dim (f ` S) \<le> aff_dim S"
NOTLEFT_eqvt_coname: fixes pi :: "(coname \<times> coname) list"   and B :: "ty"   and X :: "ctrm set" shows "pi \<bullet> NOTLEFT (NOT B) X = NOTLEFT (NOT B) (pi \<bullet> X)"
one_neq_zero: fixes one :: "'d"   and zero :: "'d" assumes "class.zero_neq_one one zero" shows "one \<noteq> zero"
supp_Upd: fixes v :: "Vars.var" shows "supp (Upd v) = supp v"
valid_add_new_ts_mmsaux': fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and auxlist :: "(nat \<times> 'a option list set) list"   and nt :: "nat" assumes "valid_mmsaux args cur aux auxlist"   and "cur \<le> nt" shows "valid_mmsaux args nt (add_new_ts_mmsaux' args nt aux) auxlist"
ide_cod: fixes f :: "'a" assumes "Left.arr f" shows "Left.ide (Left.cod f)"
widen_is_class_type_of: fixes T :: "ty"   and C :: "String.literal"   and P :: "'c prog"   and T' :: "ty" assumes "class_type_of' T = \<lfloor>C\<rfloor>"   and "P \<turnstile> T' \<le> T"   and "T' \<noteq> NT" obtains C' :: "String.literal" where "class_type_of' T' = \<lfloor>C'\<rfloor>" and "P \<turnstile> C' \<preceq>\<^sup>* C"
elims: fixes x :: "'c acom"   and y :: "com" assumes "strip x = y" obtains P :: "'c" where "x = SKIP {P__}" and "y = com.SKIP"   | xa :: "char list" and e :: "aexp" and P :: "'c" where "x = xa ::= e {Pa__}" and "y = xa ::= e"   | c1 :: "'c acom" and c2 :: "'c acom" where "x = c1__;; c2__" and "y = strip c1__;; strip c2__"   | b :: "bexp" and c1 :: "'c acom" and c2 :: "'c acom" and P :: "'c" where "x = IF b__ THEN c1 ELSE c2 {Pb__}" and "y = IF b__ THEN strip c1 ELSE strip c2"   | Inv :: "'c" and b :: "bexp" and c :: "'c acom" and P :: "'c" where "x = {Inv} WHILE b DO c {P}" and "y = WHILE b DO strip c"
lc_def: shows "lc = fst (rat_to_normalized_int_poly p)"
Abs_fps_inverse: fixes y :: "nat \<Rightarrow> 'a" assumes "y \<in> {f. True}" shows "fps_nth (Abs_fps y) = y"
holomorphic_on_def: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set" shows "f holomorphic_on s \<equiv> \<forall>x\<in>s. f field_differentiable at x within s"
no_msgs_if_no_channel: shows "\<forall>i. channel i = None \<longrightarrow> msgs init i = []"
llist_all2_ldropnI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist"   and ys :: "'b llist"   and n :: "nat" assumes "llist_all2 P xs ys" shows "llist_all2 P (ldropn n xs) (ldropn n ys)"
f: fixes x :: "'b" assumes "x \<in> vertices G" shows "(f \<circ> f) x = f x"
H2: fixes x :: "nat \<Rightarrow> state" assumes "v (first x) = c" shows "Q"
w_simps_2: shows "arr TTfgh_TfTgh.chine"
countable_INT: fixes i :: "'b set"   and I :: "'b set set" assumes "i \<in> I"   and "countable (id i)" shows "countable (\<Inter> (id ` I))"
src_def: fixes \<mu> :: "'a arrow_of_spans_data" shows "src \<mu> \<equiv> if arr \<mu> then mkObj (C.cod (Leg0 (Dom \<mu>))) else null"
subst_sound: fixes F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool"   and C :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "eval\<^sub>c F G C" shows "eval\<^sub>c F G (C \<cdot>\<^sub>l\<^sub>s \<sigma>)"
singleton: fixes x :: "'a"   and X :: "'a set" shows "(x \<in> linear_cf X) = (linear_cf X = {x})"
set_pmf_binomial_eq: fixes p :: "real"   and n :: "nat" assumes "0 \<le> p"   and "p \<le> 1" shows "set_pmf (binomial_pmf n p) = (if p = 0 then {0} else if p = 1 then {n} else {..n})"
elims_2: fixes x :: "fm" assumes "qfree x" obtains p :: "fm" where "x = NOT p__" and "qfree p__"   | p :: "fm" and q :: "fm" where "x = And pa__ q__" and "qfree pa__ \<and> qfree q__"   | p :: "fm" and q :: "fm" where "x = Or pb__ qa__" and "qfree pb__ \<and> qfree qa__"   | p :: "fm" and q :: "fm" where "x = Imp pc__ qb__" and "qfree pc__ \<and> qfree qb__"   | p :: "fm" and q :: "fm" where "x = Iff p q" and "qfree p \<and> qfree q"   | "x = T"   | "x = F"   | v :: "Ferrack.num" where "x = Lt v__"   | v :: "Ferrack.num" where "x = Le va__"   | v :: "Ferrack.num" where "x = Gt vb__"   | v :: "Ferrack.num" where "x = Ge vc__"   | v :: "Ferrack.num" where "x = Eq vd__"   | v :: "Ferrack.num" where "x = NEq v"
word_of_integer_def: shows "word_of_integer = word_of_int \<circ> int_of_integer"
top_add_clock: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and tp :: "top0"   and s' :: "'b state"   and r :: "(v sem_env, v) result" assumes "evaluate_top ck env s tp (s', r)"   and "\<not> ck" shows "\<exists>c. evaluate_top True env (update_clock (\<lambda>_. c) s) tp (update_clock (\<lambda>_. 0) s', r)"
simps_20: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x41 :: "'e list"   and x42 :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list" shows "(case \<forall>x41\<langle>\<or>\<noteq>: x42\<rangle>\<^sub>s\<^sub>t of send\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f1 x | receive\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f3 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f4 x xa) = f4 x41 x42"
skip: fixes \<iota> :: "nat"   and m :: "'id \<Rightarrow> 'val" shows "\<langle>skip\<^bsub>\<iota>\<^esub>,m\<rangle> \<rightarrow>\<lhd>[]\<rhd> \<langle>None,m\<rangle>"
transfer: shows "rel_fun pcr_float (rel_fun pcr_float pcr_float) (*) (*)"