hcju/isabelleps huggingface dataset:
mod_by_1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a one = zero"
greater_iff_sym_less: fixes cmp :: "'b comparator"   and b :: "'b"   and a :: "'b" shows "(compare cmp b a = Greater) = (compare cmp a b = Less)"
prems_3: shows "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"
cobounded1: fixes a :: "'d"   and b :: "'d" shows "a \<le> max a b"
sum_list_inc_gt0: fixes u :: "nat list"   and y :: "nat list"   and i :: "nat" assumes "0 < sum_list u"   and "length y = length u" shows "0 < sum_list (inc y i u)"
assms_1: shows "f \<in> carrier (function_ring (carrier Zp) Zp)"
next_sendupd_vacant: fixes s :: "('d, 'c, 'e) configuration_scheme stream"   and t :: "'c" assumes "NrecVacantUpto (shd s) t"   and "next_sendupd s" shows "NrecVacantUpto (shd (stl s)) t"
e2: shows "e2 \<in> UNIV \<rightarrow> {- 1..1}"
omit_branch_mem_diff_opening: fixes i :: "'c"   and branch :: "(('d, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set"   and ps :: "('d, 'c) fm list"   and a :: "'c" assumes "only_touches i branch xs"   and "(ps, a) \<in>. branch"   and "i \<noteq> a" shows "(ps, a) \<in>. omit_branch xs branch"
one_smult: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and m :: "'b" assumes "RModule R smult M"   and "m \<in> M" shows "smult (1::'a) m = m"
bind_lifted_simps_1: fixes f :: "'b \<rightarrow> 'a\<cdot>lifted" shows "\<bottom> \<bind> f = \<bottom>"
almost_additive_converges_2: fixes u :: "nat \<Rightarrow> real"   and C :: "real"   and k :: "nat" assumes "\<And>m n. \<bar>u (m + n) - u m - u n\<bar> \<le> C" shows "\<bar>u k - real k * lim (\<lambda>n. u n / real n)\<bar> \<le> C"
embed_underS2: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set"   and g :: "'d \<Rightarrow> 'e"   and a :: "'d" assumes "Well_order r"   and "embed r s g"   and "a \<in> Field r" shows "g ` underS r a = underS s (g a)"
less_le_trans: fixes x :: "'loc \<times> 't"   and y :: "'loc \<times> 't"   and z :: "'loc \<times> 't" assumes "x<\<^sub>py"   and "y\<le>\<^sub>pz" shows "x<\<^sub>pz"
\<alpha>: shows "\<alpha> \<in> elts (ordertype M r)"
exhaust: fixes y :: "('a, 'b, 'c) new_thread_action" obtains x11 :: "'a" and x12 :: "'b" and x13 :: "'c" where "y = NewThread x11 x12 x13"   | x21 :: "'a" and x22 :: "bool" where "y = ThreadExists x21 x22"
b: shows "r \<down>l (\<tau> @ \<sigma>') = r \<down>m (r|\<sigma>'| + r|\<tau>|)"
smult_minus_right: fixes a :: "'b"   and p :: "'b poly" shows "smult a (- p) = - smult a p"
nprv_clear4_3: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>2 :: "'c"   and \<phi>4 :: "'c"   and \<psi> :: "'c"   and \<phi>3 :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2, \<phi>4} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "\<psi> \<in> fmla" shows "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2, \<phi>3, \<phi>4} \<psi>"
4644: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<sqinter> - y) \<sqinter> (x \<sqinter> - (y \<squnion> z)) = bot"
fds_converges_add: fixes f :: "'a fds"   and s :: "'a"   and g :: "'a fds" assumes "fds_converges f s"   and "fds_converges g s" shows "fds_converges (f + g) s"
has_derivative_bounded_linear: fixes f :: "'d \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e"   and F :: "'d filter" assumes "(f has_derivative f') F" shows "bounded_linear f'"
axioms_1: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "Dynkin_system \<Omega> M" shows "subset_class \<Omega> M"
assms_2: fixes x :: "'a" assumes "x \<in> T - S" shows "g x = f x"
hf_pres_fst: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and k :: "bool" shows "fst (hf_pres R k) = R"
fofu_partial_correct: shows "fofu \<le> SPEC isMaxFlow"
inv_def: shows "inv = (\<lambda>\<delta>p (Q', W', \<delta>d'). hs.invar Q' \<and> ls.invar \<delta>d' \<and> (\<exists>Wn. distinct Wn \<and> set Wn = f_succ \<delta>p `` {(q1, q2)} - hs.\<alpha> Q \<and> W' = Wn @ Wtl \<and> hs.\<alpha> Q' = hs.\<alpha> Q \<union> f_succ \<delta>p `` {(q1, q2)}) \<and> ls.\<alpha> \<delta>d' = ls.\<alpha> \<delta>d \<union> {r \<in> \<delta>p. lhs r = (q1, q2)})"
N_of_state_state_of_wstate_wN_of_wstate: fixes C :: "'a literal multiset"   and St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat" assumes "C \<in> N_of_wstate St" shows "\<exists>i. (C, i) \<in># wN_of_wstate St"
get_mode_is_l_get_mode: shows "l_get_mode ShadowRootClass.type_wf get_mode get_mode_locs"
nb\<^sub>2: shows "v \<in> {v \<in> set (\<Pi>\<^sub>\<V>). (\<Pi>\<^sub>I) v \<noteq> None}"
le_supI2: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and b :: "'b::type"   and a :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (b::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) ((sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (b::'b::type))"
ord_equiv_box: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and F :: "('g, 'h) Order_scheme" assumes "Order D"   and "Order E"   and "Order F"   and "ord_equiv D E"   and "ord_equiv D F" shows "ord_equiv E F"
aristo_conversion1: fixes A :: "'a set"   and B :: "'a set" assumes "A E B" shows "B E A"
word_diff_less: fixes n :: "'b word"   and m :: "'b word" assumes "0 < n"   and "0 < m"   and "n \<le> m" shows "m - n < m"
1: fixes a :: "alt"   and b :: "alt"   and a' :: "alt"   and b' :: "alt"   and P :: "indi \<Rightarrow> (alt \<times> alt) set"   and P' :: "indi \<Rightarrow> (alt \<times> alt) set" assumes "a \<noteq> b"   and "a' \<noteq> b'"   and "a \<noteq> b'"   and "b \<noteq> a'"   and "P \<in> Prof"   and "P' \<in> Prof"   and "\<forall>i. (a <\<^bsub>P i\<^esub> b) = (a' <\<^bsub>P' i\<^esub> b')"   and "a <\<^bsub>F P\<^esub> b" shows "a' <\<^bsub>F P'\<^esub> b'"
dis: shows "{0..1} \<inter> {u. g u \<in> closure (- S)} \<noteq> {}"
map_cong0: fixes x :: "'e pmf"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "\<And>z. z \<in> set_pmf x \<Longrightarrow> f z = g z" shows "map_pmf f x = map_pmf g x"
edges_of_walk_middle: fixes v :: "'a"   and w :: "'a"   and xs :: "'a list"   and xs' :: "'a list" shows "(v, w) \<in> H.edges_of_walk (xs @ v # w # xs')"
lfp_trans_sound: fixes P :: "'a \<Rightarrow> real"   and T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and v :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "sound P"   and "le_trans (T v) v"   and "\<And>P. sound P \<Longrightarrow> sound (v P)" shows "sound (lfp_trans T P)"
mono_plus': fixes a1 :: "'a"   and b1 :: "'a"   and a2 :: "'a"   and b2 :: "'a" assumes "a1 \<le> b1"   and "a2 \<le> b2" shows "plus' a1 a2 \<le> plus' b1 b2"
sub_bot_least: fixes x :: "'a" shows "bot \<le> - x"
homeomorphic_space_def: fixes X :: "'c topology"   and Y :: "'d topology" shows "X homeomorphic_space Y \<equiv> \<exists>f g. homeomorphic_maps X Y f g"
OA_sub: fixes ODE :: "('sf, 'sz) ODE"   and B :: "('sz + 'sz) set"   and A :: "('sz + 'sz) set" assumes "Oadmit \<sigma> ODE B"   and "A \<subseteq> B" shows "Oadmit \<sigma> ODE A"
GroupHomSetI: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and H :: "'b set" assumes "GroupHom G T"   and "T ` G \<subseteq> H" shows "T \<in> GroupHomSet G H"
select_convs_3: fixes bar1 :: "'b"   and bar2 :: "'c"   and bar3 :: "'a"   and bar21 :: "'c \<times> 'b"   and bar32 :: "'a \<times> 'c"   and bar31 :: "'a \<times> 'b"   and more :: "'d" shows "bar3 \<lparr>bar1 = bar1, bar2 = bar2, bar3 = bar3, bar21 = bar21, bar32 = bar32, bar31 = bar31, \<dots> = more\<rparr> = bar3"
mult_mod_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "times (modulo a b) c = modulo (times a c) (times b c)"
cring_simprules_15: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "cring R" shows "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
simps_5: fixes f1 :: "'h"   and f2 :: "'i \<Rightarrow> 'h"   and x2 :: "'i" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
varnum_plus: fixes X :: "'a set"   and s :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a \<Rightarrow>\<^sub>0 'c" shows "varnum X (s + t) = max (varnum X s) (varnum X t)"
in'_1: shows "real_of_rat l \<le> real_of_rat y"
Int_Un_eq_2: fixes S :: "'d set"   and T :: "'d set" shows "S \<inter> T \<union> T = T"
Ints_mat_elements_mat: fixes A :: "'b mat" shows "(A \<in> \<int>\<^sub>m) = (elements_mat A \<subseteq> \<int>)"
pp_dist_pp_sup: fixes x :: "'a"   and y :: "'a" shows "- - (- - x \<squnion> - - y) = - - (x \<squnion> y)"
Call: fixes \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and p :: "'b"   and as :: "('a, 'b, 'c) ann list"   and n :: "nat"   and P :: "('a, 'b, 'c) ann"   and r :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and b :: "('a, 'b, 'c) com"   and F :: "'c set"   and Q :: "'a set"   and A :: "'a set" assumes "\<Theta> p = Some as"   and "as ! n = P"   and "r \<subseteq> pre P"   and "\<Gamma> p = Some b"   and "n < length as"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P b Q, A" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnCall r n) (Call p) Q, A"
assoc'_naturality: fixes \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "D.arr \<mu>"   and "D.arr \<nu>"   and "D.arr \<tau>"   and "src\<^sub>D \<mu> = trg\<^sub>D \<nu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<tau>" shows "\<a>\<^sub>D\<^sup>-\<^sup>1[D.cod \<mu>, D.cod \<nu>, D.cod \<tau>] \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>) = ((\<mu> \<star>\<^sub>D \<nu>) \<star>\<^sub>D \<tau>) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[D.dom \<mu>, D.dom \<nu>, D.dom \<tau>]"
closed_Collect_neg: fixes "open" :: "'b set \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "class.topological_space open"   and "open {x. P x}" shows "topological_space.closed open {x. \<not> P x}"
smooth_on_inner: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x \<bullet> g x)"
R: shows "R \<subseteq> {..n} \<times> {..n}"
PartDom_def: fixes d :: "'a dataspace"   and n :: "nat" shows "d !D! n = Rep_dataspace d ! n"
FG_constituents_irr: assumes "of_nat (card G) \<noteq> (0::'a)" shows "\<forall>U\<in>set FG_constituents. IrrFinGroupRepresentation G (*) U"
ras_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>ras_rel \<equiv> br ras_raw_\<alpha> ras_raw_invar O \<langle>R\<rangle>list_rel"
f: shows "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>"
assms_3: shows "\<not> (l = 0 \<and> \<bar>m\<bar> = \<infinity> \<or> m = 0 \<and> \<bar>l\<bar> = \<infinity>)"
norm_le: fixes x :: "'a"   and y :: "'b" shows "(norm x \<le> norm y) = (inner x x \<le> inner y y)"
WeaklyContingent_def: shows "WeaklyContingent \<equiv> \<lambda>F. Contingent F \<^bold>& (\<^bold>\<forall>x. \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<^bold>\<box>\<lparr>F,x\<^sup>P\<rparr>)"
fixed_selector_induced_subgraph: fixes H :: "nat set \<times> nat set set"   and f :: "nat set \<Rightarrow> nat set \<times> nat set set"   and V :: "nat set"   and G :: "nat set \<times> nat set set" assumes "is_fixed_selector H f"   and "card (uverts H) = card V"   and "finite V"   and "subgraph (f V) (induced_subgraph V G)"   and "V \<subseteq> uverts G"   and "uwellformed G" shows "H \<sqsubseteq> G"
elims_3: fixes x :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" assumes "\<not> invpst x" obtains l :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" and xa :: "'c \<times> 'd" and uu_ :: "'e" and mkp :: "'c \<times> 'd" and r :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" where "x = \<langle>l, (xa, uu_, mkp), r\<rangle>" and "\<not> (invpst l \<and> invpst r \<and> is_min2 mkp (set (Tree2.inorder l @ xa # Tree2.inorder r)))"
ipresIGWlsAll_def: fixes h :: "'c \<Rightarrow> 'd"   and hA :: "'e \<Rightarrow> 'f"   and MOD :: "('g, 'h, 'i, 'j, 'k, 'l, 'c, 'e, 'm) model_scheme"   and MOD' :: "('n, 'o, 'i, 'j, 'p, 'q, 'd, 'f, 'r) model_scheme" shows "ipresIGWlsAll h hA MOD MOD' \<equiv> ipresIGWls h MOD MOD' \<and> ipresIGWlsAbs hA MOD MOD'"
fresh_atm_3: fixes a :: "W.tvar"   and b :: "W.var" shows "a \<sharp> b"
properfactor_trans2: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.monoid G"   and "properfactor G a b"   and "b divides\<^bsub>G\<^esub> c"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "properfactor G a c"
Sup_pres_map_dual_var: fixes f :: "'b \<Rightarrow> 'c" shows "Sup_pres f = Inf_pres (\<partial>\<^sub>F f)"
mult_neg_neg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less a zero"   and "less b zero" shows "less zero (times a b)"
vector_conv_covector: fixes v :: "'a" shows "vector v = covector (v\<^sup>T)"
OTNSEp: fixes p :: "('b, 'c) Combinators list" assumes "noDenyAll1 p"   and "OnlyTwoNets p" shows "OnlyTwoNets (separate p)"
type_less_eq_x_Sequence_intro: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> = Sequence \<rho>"   and "\<rho> \<le> \<sigma>" shows "\<tau> \<le> Sequence \<sigma>"
simps_12: fixes v :: "'b acom"   and va :: "'b acom"   and vb :: "bexp"   and vc :: "'b acom"   and vd :: "'b acom"   and ve :: "'b" shows "(v;; va \<sqsubseteq> IF vb THEN vc ELSE vd {ve}) = False"
loc_imps_fw_M_in_implications': fixes c :: "('loc, 't) configuration"   and loc1 :: "'loc"   and loc2 :: "'loc"   and M :: "'t zmultiset"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and t :: "'t" assumes "loc_imps_fw c loc1 loc2 M xs"   and "inv_imps_work_sum c"   and "inv_implications_nonneg c"   and "worklists_vacant_to c t"   and "0 < zcount M t" shows "\<exists>s\<le>t. s \<in>\<^sub>A frontier (c_imp c loc2)"
pre_2: shows "Apre \<union> Bpre = \<omega>_limit_set x"
integrable_of_real: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "integrable M (\<lambda>x. of_real (f x))"
real_interval_limpt: fixes a :: "real"   and b :: "real" assumes "a < b" shows "b islimpt {a..<b}"
ctr_transfer: fixes R3 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R1 :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'i \<Rightarrow> bool" shows "(R3 ===> rel_writerT R1 R2 R3) WriterT WriterT"
comp_L_eq_comp_R: shows "\<lbrakk>\<lbrakk>t\<^sub>0\<rbrakk>\<rbrakk> \<odot> \<lbrakk>\<lbrakk>\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>u\<^sub>1\<rbrakk>\<rbrakk> \<odot> \<lbrakk>\<lbrakk>\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>"
intros_9: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c2 :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b"   and c1 :: "('a, 'c) com" assumes "PL.transT tval aval (c2, s) s'" shows "PL.transC tval aval (c1 | c2, s) (c1, s')"
mod_frame_fwd: fixes Ps :: "assn"   and h :: "heap \<times> nat set"   and P :: "assn"   and R :: "assn"   and F :: "assn" assumes "h \<Turnstile> Ps"   and "P \<Longrightarrow>\<^sub>A R"   and "Ps \<Longrightarrow>\<^sub>A P * F" shows "h \<Turnstile> R * F"
sum_subtractf_nat: fixes A :: "'a set"   and g :: "'a \<Rightarrow> nat"   and f :: "'a \<Rightarrow> nat" assumes "\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x" shows "(\<Sum>x\<in>A. f x - g x) = sum f A - sum g A"
assms_5: shows "l \<circ> g = h \<circ> l"
one_less_Fract_iff: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "(1 < Fract a b) = (b < a)"
set_iterator_linord_def: fixes it :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and S0 :: "'a set" shows "set_iterator_linord it S0 \<equiv> set_iterator_genord it S0 (\<le>)"
simps: fixes p :: "'d pmf"   and a :: "'d cset" shows "cr_pmf_cset p a = (a = cset_pmf p)"
tendsto_Im4: fixes g :: "'a \<Rightarrow> octo"   and a :: "octo"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Im4 (g x)) \<longlongrightarrow> Im4 a) F"
supp_Act\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>"   and f :: "'d"   and \<alpha> :: "'c" assumes "finite (supp t\<^sub>\<alpha>)" shows "supp (Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha>) = supp f \<union> (supp \<alpha> \<union> supp t\<^sub>\<alpha> - bn \<alpha>)"
bp_sol_finite: fixes P :: "'a set set" assumes "bp P" shows "finite P"
DynCom: fixes P :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "'d \<Rightarrow> ('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c s Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P DynCom c Q,A"
h0: shows "\<forall>x. h (0, x) = p x"
supportE_twicepreT: fixes C1 :: "acom"   and C2 :: "acom"   and e :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> nat" shows "supportE (\<lambda>l. preT C1 (preT C2 (e l))) \<subseteq> supportE e"
observable_behaviour_refinement: fixes R :: "('a \<times> 'b) set"   and pi :: "'c \<Rightarrow> 'd"   and Sa :: "('a, 'd) spec"   and Sc :: "('b, 'c) spec"   and bc :: "'c list" assumes "refines R pi Sa Sc"   and "bc \<in> obeh Sc" shows "map pi bc \<in> obeh Sa"
assms: shows "\<forall>i\<in>fvi (Suc b) \<phi>. i < n"
cases: fixes x :: "('d, 'e) ioa \<times> 'd \<times> ('e \<times> 'd) list" obtains A :: "('d, 'e) ioa" and s :: "'d" and ps :: "('e \<times> 'd) list" and p' :: "'e \<times> 'd" and p :: "'e \<times> 'd" where "x = (Aa__, s__, Cons pa__ (Cons p' ps))"   | A :: "('d, 'e) ioa" and s :: "'d" and p :: "'e \<times> 'd" where "x = (Ab__, sa__, [p])"   | A :: "('d, 'e) ioa" and s :: "'d" where "x = (A, s, [])"
7: shows "coreflexive (e * (f \<sqinter> - q)\<^sup>T)"
assms_7: shows "set us \<inter> FvarsT r = {}"
subst_lsst_prefix: fixes B :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<theta> :: "'f \<Rightarrow> ('e, 'f) Term.term" assumes "prefix B (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)" shows "\<exists>C. C \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta> = B \<and> prefix C A"
differentiable_on_eqI: fixes g :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "g differentiable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x"   and "open S" shows "f differentiable_on S"
tensor_mat_adjoint: fixes m1 :: "'a mat"   and dims :: "nat list"   and vars :: "nat set"   and m2 :: "'a mat" assumes "m1 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m2 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)" shows "adjoint (tensor_mat dims vars m1 m2) = tensor_mat dims vars (adjoint m1) (adjoint m2)"
ContractsWithSubstitutes_axioms: shows "ContractsWithSubstitutes Xd Xh Pd Ch"
list_to_fun_def: fixes ts :: "'b list"   and cs :: "'a list"   and t :: "'b" shows "list_to_fun ts cs t = (case map_of (zip ts cs) t of None \<Rightarrow> 0::'a | Some c \<Rightarrow> c)"
lderivs_One: fixes xs :: "'a list" shows "lderivs xs One \<in> {Zero, One}"
finite_measure_Diff': fixes A :: "bool stream set"   and B :: "bool stream set" assumes "A \<in> events"   and "B \<in> events" shows "prob (A - B) = prob A - prob (A \<inter> B)"
k: shows "f k = (\<^bold>\<not> \<^bold>\<forall> p)"
BlockRedSome: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and V :: "char list"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and v :: "val"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l(V := None), sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "l' V = \<lfloor>v\<rfloor>"   and "\<not> assigned V e" shows "P \<turnstile> \<langle>{V:T; e},(h, l, sh),b\<rangle> \<rightarrow> \<langle>{V:T; V:=Val v;; e'},(h', l'(V := l V), sh'),b'\<rangle>"
image_mset_of_subset: fixes A :: "'b multiset"   and \<eta> :: "'c \<Rightarrow> 'b"   and C' :: "'c multiset" assumes "A \<subseteq># image_mset \<eta> C'" shows "\<exists>A'. image_mset \<eta> A' = A \<and> A' \<subseteq># C'"
rivest_perfect_hiding: fixes \<eta> :: "nat"   and \<A> :: "(nat \<times> nat \<Rightarrow> ((nat \<times> nat) \<times> 'b) spmf) \<times> (nat \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "rivest_commit.perfect_hiding_ind_cpa \<eta> \<A>"
reads_def: fixes S :: "('d \<Rightarrow> 'd \<Rightarrow> bool) set"   and getter :: "('d, 'e, 'f) prog"   and h :: "'d"   and h' :: "'d" shows "reads S getter h h' = ((\<forall>P\<in>S. reflp P \<and> transp P) \<and> ((\<forall>P\<in>S. P h h') \<longrightarrow> preserved getter h h'))"
SPRAY_path: fixes P :: "'a set"   and x :: "'a" assumes "P \<in> SPRAY x" shows "P \<in> \<P>"
fs1_def: shows "fs1 = rev (remdups (rev (removeAll 0 fs)))"
rep_eq: fixes x :: "Vars.var ttree" shows "ccApprox x = lub (ccFromList ` Rep_ttree x)"