hcju/isabelleps huggingface dataset:
simps: fixes a :: "'k"   and \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" shows "(a \<in> SAT \<Gamma> f) = ((\<exists>x. a = x \<and> x \<in> A \<Gamma>) \<or> (\<exists>x. a = x \<and> weight \<Gamma> x \<le> d_IN f x))"
make_one_new_out_of_two: fixes x :: "'b"   and y :: "'c" assumes "\<exists>n1. new_tv n1 x"   and "\<exists>n2. new_tv n2 y" shows "\<exists>n. new_tv n x \<and> new_tv n y"
Cn_union: fixes Cn :: "'b set \<Rightarrow> 'b set"   and A :: "'b set"   and B :: "'b set" assumes "Tarskian_logic Cn" shows "Cn (Cn A \<union> Cn B) = Cn (A \<union> B)"
no_memb_fold_insert: fixes dxs :: "'c set_dlist"   and x :: "'c"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d" assumes "\<not> member dxs x" shows "DList_Set.fold f (DList_Set.insert x dxs) b = DList_Set.fold f dxs (f x b)"
fmsubset_pred: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap" assumes "fmpred P m"   and "n \<subseteq>\<^sub>f m" shows "fmpred P n"
hom_bind: fixes x :: "'m1"   and f :: "'a \<Rightarrow> 'm1" shows "h (bind1 x f) = bind2 (h x) (h \<circ> f)"
lub_belowI: fixes S :: "'e set"   and x :: "'e" assumes "\<And>y. y \<in> S \<Longrightarrow> y \<sqsubseteq> x" shows "lub S \<sqsubseteq> x"
proj5_invalidate: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat" shows "\<pi>\<^sub>5 (the (invalidate rt dests dip)) = \<pi>\<^sub>5 (the (rt dip))"
rel_bset_def: shows "rel_bset \<equiv> (id ---> set_bset ---> set_bset ---> id) rel_set"
fst_sum_list: fixes xs :: "('a \<times> 'b) list" shows "fst (sum_list xs) = sum_list (map fst xs)"
preserves_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "trg\<^sub>D (G \<mu>) = G.map\<^sub>0 (trg\<^sub>C \<mu>)"
distinct_5: fixes x2 :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and x31 :: "'c"   and x32 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd" shows "Assume x2 \<noteq> Assign x31 x32"
real_power_nat_power: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and c :: "'a" assumes "class.nat_power_normed_field one times plus zero nat_power divide inverse minus uminus dist norm scaleR sgn uniformity open inner real_power"   and "0 < n" shows "real_power (real n) c = nat_power n c"
reachable_state_fragmentsLS: fixes sys :: "('g, 'h, 'i, 'j, 'k, 'l) pre_system_scheme"   and sh :: "('g, 'h, 'i, 'j, 'k) system_state"   and p :: "'i" assumes "reachable_state sys sh" shows "fragmentsLS (GST sh p) \<subseteq> fragments (PGMs sys p) {}"
case_4: fixes f1 :: "'a"   and f2 :: "qtname \<Rightarrow> 'a"   and f3 :: "qtname \<Rightarrow> 'a"   and f4 :: "ty \<Rightarrow> 'a"   and x4 :: "ty" shows "(case ArrayT x4 of NullT \<Rightarrow> f1 | IfaceT x \<Rightarrow> f2 x | ClassT x \<Rightarrow> f3 x | ArrayT x \<Rightarrow> f4 x) = f4 x4"
injects: fixes a :: "'r"   and tree1 :: "'r tree"   and tree2 :: "'r tree"   and a' :: "'r"   and tree1' :: "'r tree"   and tree2' :: "'r tree" assumes "a \<noteq> \<bottom>" shows "(Node\<cdot>a\<cdot>tree1\<cdot>tree2 = Node\<cdot>a'\<cdot>tree1'\<cdot>tree2') = (a = a' \<and> tree1 = tree1' \<and> tree2 = tree2')"
valid_Rcons: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and tr :: "('a, 'b, 'c) trans list"   and trn :: "('a, 'b, 'c) trans" assumes "IO_Automaton.valid step tr"   and "tgtOf (last tr) = srcOf trn"   and "IO_Automaton.validTrans step trn" shows "IO_Automaton.valid step (tr ## trn)"
search_converg_1: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat"   and i :: "nat" assumes "total_cons s"   and "s b \<down>= i"   and "eval (total_cons.r_search s) [b] \<down>" shows "\<phi> i (the (eval (total_cons.r_search s) [b])) \<down>"
L_flat_Prf2: fixes s :: "'b list"   and r :: "'b rexp" assumes "s \<in> lang r" shows "\<exists>v. \<turnstile> v : r \<and> flat v = s"
tiTable_SampleT_def: fixes req :: "nat \<Rightarrow> reqType list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st_in :: "nat \<Rightarrow> GatewayStatus"   and buffer_in :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and st_out :: "nat \<Rightarrow> GatewayStatus" shows "tiTable_SampleT req a1 stop lose st_in buffer_in ack i1 vc st_out \<equiv> \<forall>t r x y z. (st_in t = init_state \<and> req t = [init] \<longrightarrow> ack t = [call] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = call) \<and> (st_in t = init_state \<and> req t \<noteq> [init] \<longrightarrow> ack t = [init_state] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = init_state) \<and> ((st_in t = call \<or> st_in t = connection_ok \<and> r \<noteq> [send]) \<and> req t = r \<and> lose t = [False] \<longrightarrow> ack t = [connection_ok] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = connection_ok) \<and> ((st_in t = call \<or> st_in t = connection_ok \<or> st_in t = sending_data) \<and> lose t = [True] \<longrightarrow> ack t = [init_state] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = init_state) \<and> (st_in t = connection_ok \<and> req t = [send] \<and> lose t = [False] \<longrightarrow> ack t = [sending_data] \<and> i1 t = buffer_in t \<and> vc t = [] \<and> st_out t = sending_data) \<and> (st_in t = sending_data \<and> a1 t = [] \<and> lose t = [False] \<longrightarrow> ack t = [sending_data] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = sending_data) \<and> (st_in t = sending_data \<and> a1 t = [sc_ack] \<and> lose t = [False] \<longrightarrow> ack t = [voice_com] \<and> i1 t = [] \<and> vc t = [vc_com] \<and> st_out t = voice_com) \<and> (st_in t = voice_com \<and> stop t = [] \<and> lose t = [False] \<longrightarrow> ack t = [voice_com] \<and> i1 t = [] \<and> vc t = [vc_com] \<and> st_out t = voice_com) \<and> (st_in t = voice_com \<and> stop t = [] \<and> lose t = [True] \<longrightarrow> ack t = [voice_com] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = voice_com) \<and> (st_in t = voice_com \<and> stop t = [stop_vc] \<longrightarrow> ack t = [init_state] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = init_state)"
3: shows "\<exists>x<Suc (n + m). \<phi> i x \<noteq> f x"
in_outs'_restrict_gpvD: fixes x :: "'a"   and \<I> :: "('a, 'c) \<I>"   and gpv :: "('b, 'a, 'c) gpv" assumes "x \<in> outs'_gpv (restrict_gpv \<I> gpv)" shows "x \<in> outs_\<I> \<I>"
ta_bisim_mono: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and ta1 :: "('f, 'a, 'b, 'c, 'g, 'h) thread_action"   and ta2 :: "('f, 'a, 'd, 'e, 'g, 'h) thread_action"   and bisim' :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool" assumes "ta_bisim bisim ta1 ta2"   and "\<And>t s1 s2. bisim t s1 s2 \<Longrightarrow> bisim' t s1 s2" shows "ta_bisim bisim' ta1 ta2"
admits_diagonal_imp_admits_triangular: assumes "Ball (carrier_mat 1 2) admits_diagonal_reduction" shows "All admits_triangular_reduction"
size_3: fixes x3 :: "fmla" shows "size (neg x3) = size x3 + Suc 0"
distinct_imp_distinct_take: fixes l :: "('a \<times> 'b) list list"   and x :: "nat" assumes "distinct (map fst (concat l))" shows "distinct (map fst (concat (take x l)))"
length_getTvars: fixes \<sigma>l :: "'tp list" shows "length (getTvars \<sigma>l) = length \<sigma>l"
coincide_hp'_def: fixes \<alpha> :: "('d, 'e, 'f) hp" shows "coincide_hp' \<alpha> = (\<forall>I J. coincide_hp \<alpha> I J \<and> ode_sem_equiv \<alpha> I)"
word_and_full_mask_simp: shows "x && mask LENGTH(64) = x"
r_one: fixes x :: "cltn2" assumes "x \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "x \<otimes>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> \<one>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> = x"
sepref_relpropI: fixes P :: "'d \<Rightarrow> 'b"   and R :: "'d" shows "P R = CONSTRAINT P R"
prune_def: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" shows "prune A w V \<equiv> {v \<in> V. Bex (greachable A w V v) (gaccepting A)}"
IH: shows "suffixes (xs @ ys) = suffixes ys @ map (\<lambda>xs'. xs' @ ys) (tl (suffixes xs))"
bounded_by_def: fixes b :: "real"   and P :: "'b \<Rightarrow> real" shows "bounded_by b P \<equiv> \<forall>x. P x \<le> b"
2: shows "PROP P (smap fst x ||| smap snd x)"
skl1_defs_1: shows "skl1 \<equiv> \<lparr>init = skl1_init, trans = skl1_trans, obs = id\<rparr>"
STCal_step_1: fixes S :: "'procS"   and P' :: "('procS, 'procT) Proc" shows "SourceTerm S \<longmapsto>STCal Source Target P' = (\<exists>S'. S' \<in>S P' \<and> S \<longmapsto>Source S')"
rec: fixes f :: "('e \<Rightarrow> 'f) \<Rightarrow> 'd"   and x :: "'e \<Rightarrow> 'f" shows "rec_envT f (EnvT x) = f x"
wfLearnLessState: fixes learnL :: "(Literal list list \<times> Literal list list) set" assumes "wf learnL" shows "wf (learnLessState learnL)"
get_parent_pure: fixes ptr :: "(_) node_ptr"   and h :: "(_) heap" shows "pure (get_parent ptr) h"
power_Suc0_right: fixes a :: "'a" shows "a ^ Suc zero_class.zero = a"
strand_sem_Send_map_2: fixes T :: "('fun, 'var) Term.term list"   and M :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<And>t. t \<in> set T \<Longrightarrow> \<lbrakk>M; [send\<langle>t\<rangle>\<^sub>s\<^sub>t]\<rbrakk>\<^sub>d \<I>" shows "\<lbrakk>M; map Send T\<rbrakk>\<^sub>d \<I>"
simps_2: fixes xs :: "'g list" shows "shuffles xs [] = {xs}"
finfun_default_update_const: fixes f :: "('b, 'a) finfun"   and a :: "'b"   and b :: "'a" shows "finfun_default (finfun_update f a b) = finfun_default f"
le_imp_inv: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "b = a + (b - a)"
smodule_ideal_coeff_def: fixes R :: "('i, 'j) Ring_scheme"   and M :: "('h, 'i, 'k) Module_scheme"   and A :: "'i set" shows "A \<odot>\<^bsub>R\<^esub> M = linear_span R M A (carrier M)"
pbij_vwb_is_bij_lens: fixes X :: "'a \<Longrightarrow> 'b" assumes "pbij_lens X"   and "vwb_lens X" shows "bij_lens X"
src_trg: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.src (Span.trg \<mu>) = Span.trg \<mu>"
elims_2: fixes x :: "'a list"   and xa :: "'a list" assumes "local.less_eq_list x xa" obtains ys :: "'a list" where "xa__ = []" and "xa = ys__"   | x :: "'a" and xs :: "'a list" and y :: "'a" and ys :: "'a list" where "xa__ = x # xs" and "xa = y # ys" and "x < y \<or> x = y \<and> local.less_eq_list xs ys"
mult_isotone: fixes w :: "'a"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "w \<le> y"   and "x \<le> z" shows "w * x \<le> y * z"
step: shows "S t i \<turnstile> t ! i \<mapsto> S t (Suc i)"
fls_inverse_X_inv': assumes "inverse (1::'a) = (1::'a)" shows "inverse fls_X_inv = fls_X"
induct: fixes x :: "'b set"   and B :: "'b set set"   and P :: "'b set \<Rightarrow> bool" assumes "x \<in> topo B"   and "\<And>x. x \<in> B \<Longrightarrow> P x"   and "\<And>x y. \<lbrakk>x \<in> topo B; P x; y \<in> topo B; P y\<rbrakk> \<Longrightarrow> P (x \<inter> y)"   and "\<And>M. \<lbrakk>\<And>x. x \<in> M \<Longrightarrow> x \<in> topo B; \<And>x. x \<in> M \<Longrightarrow> P x\<rbrakk> \<Longrightarrow> P (\<Union> M)" shows "P x"
cases: fixes x :: "'b \<times> 'b tree" obtains a :: "'b" and t :: "'b tree" where "x = (a, t)"
map_simp: fixes f :: "'a" assumes "B.arr f" shows "B.map f = f"
ekey_oracle_conv: fixes \<eta> :: "nat"   and \<sigma> :: "('ekey \<times> 'dkey \<times> 'cipher list) option"   and x :: "unit" shows "ekey_oracle \<eta> \<sigma> x = (case \<sigma> of None \<Rightarrow> map_spmf (\<lambda>(ekey, dkey). (ekey, Some (ekey, dkey, []))) (key_gen \<eta>) | Some (ekey, rest) \<Rightarrow> return_spmf (ekey, Some (ekey, rest)))"
vunion_vsingleton: fixes a :: "V"   and A :: "V" shows "set {a} \<union>\<^sub>\<circ> A = vinsert a A"
28270: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- - (x \<sqinter> y) \<squnion> - (- x \<sqinter> z) = - (- x \<sqinter> z)"
piecewise_continuous_on_cong: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c"   and I :: "'c set"   and J :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "a = c"   and "b = d"   and "I = J"   and "\<And>x. \<lbrakk>c \<le> x; x \<le> d; x \<notin> J\<rbrakk> \<Longrightarrow> f x = g x" shows "piecewise_continuous_on a b I f = piecewise_continuous_on c d J g"
trans: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a \<preceq> b"   and "b \<preceq> c" shows "a \<preceq> c"
locate: fixes al :: "'a list"   and f :: "'a \<Rightarrow> nat"   and i :: "nat" assumes "\<And>a. a \<in> set al \<Longrightarrow> 0 < f a"   and "i < lsum f al" shows "locate1 f al i < length al \<and> locate2 f al i < f (al ! locate1 f al i) \<and> i = lsum f (take (locate1 f al i) al) + locate2 f al i"
generate: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "principalideal I R" shows "\<exists>i\<in>carrier R. I = Idl\<^bsub>R\<^esub> {i}"
executions_def: fixes ioa :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set" shows "executions ioa \<equiv> {e. snd e 0 \<in> starts_of ioa \<and> is_execution_fragment ioa e}"
zero_power2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero 2 = zero"
sup_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "sup (sup x y) z = sup x (sup y z)"
of_real_1: shows "of_real 1 = (1::'c)"
Cons: shows "\<forall>a b. l \<in> a \<otimes> b \<longrightarrow> l @ [x] \<in> a @ [x] \<otimes> b"
f: fixes F :: "nat \<Rightarrow> 'b poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>" shows "f i = lead_coeff (F i)"
\<R>\<^sub>3_elim': fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" obtains \<Gamma> :: "'Var \<Rightarrow> Sec option" and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" and c :: "('Var, 'AExp, 'BExp) Stmt" where "c\<^sub>1'__ = c\<^sub>1''__ ;; c__" and "\<langle>c\<^sub>1''__, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>''__\<^esub> \<langle>c\<^sub>2''__, mds, mem\<^sub>2\<rangle>" and "\<turnstile> \<Gamma>''__ {c__} \<Gamma>'__" and "c\<^sub>2'__ = c\<^sub>2''__ ;; c__" and "mds\<^sub>2 = mds"   | \<Gamma> :: "'Var \<Rightarrow> Sec option" and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" and c :: "('Var, 'AExp, 'BExp) Stmt" where "c\<^sub>1'__ = c\<^sub>1'''__ ;; ca__" and "\<langle>c\<^sub>1'''__, mds, mem\<^sub>1\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>'''__\<^esub> \<langle>c\<^sub>2'''__, mds, mem\<^sub>2\<rangle>" and "\<turnstile> \<Gamma>'''__ {ca__} \<Gamma>'__" and "c\<^sub>2'__ = c\<^sub>2'''__ ;; ca__" and "mds\<^sub>2 = mds"   | \<Gamma> :: "'Var \<Rightarrow> Sec option" and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" and c :: "('Var, 'AExp, 'BExp) Stmt" where "c\<^sub>1'__ = c\<^sub>1 ;; c" and "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" and "\<turnstile> \<Gamma> {c} \<Gamma>'__" and "c\<^sub>2'__ = c\<^sub>2 ;; c" and "mds\<^sub>2 = mds"
C_DenyAll: fixes xs :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" shows "C (list2FWpolicy (xs @ [DenyAll])) x = \<lfloor>deny ()\<rfloor>"
OrdP_IndP_Q_Ind: fixes x :: "tm" shows "{OrdP x} \<turnstile> IndP (Q_Ind x)"
X1_def: shows "X1 = X \<inter> {x. norm x = 1}"
preserves_reflects_arr: fixes f :: "('a S.E.term, 'a) S.arr" shows "S.arr (S.src f) = S.arr f"
inv_solve_right': fixes G :: "('d, 'e) monoid_scheme"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "Group.group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(b \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> c = a) = (b = a \<otimes>\<^bsub>G\<^esub> c)"
simps_6: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type" shows "cexpr_type \<Gamma> (\<integral>\<^sub>c e \<partial>t) = (if cexpr_type (case_nat t \<Gamma>) e = Some REAL then Some REAL else None)"
fst_conv: fixes x1 :: "'h"   and x2 :: "'i" shows "fst (x1, x2) = x1"
read_actions_change_prefix: fixes r :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and n :: "enat"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "r \<in> read_actions E"   and "ltake n E [\<approx>] ltake n E'"   and "enat r < n" shows "r \<in> read_actions E'"
Iio_eq_empty_iff: fixes n :: "'a" shows "({..<n} = {}) = (n = \<bottom>)"
coeff_Var2: fixes x :: "nat"   and y :: "nat" assumes "x \<noteq> y" shows "coeff (Var x) y = 0"
prems_1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set (x__ # xs)"   and "y \<in> set (x__ # xs)" shows "f x \<circ> f y = f y \<circ> f x"
simps_2: shows "sqrt_approx_main x = (if (0::'a) < x then if x * x - n < \<epsilon> then x else sqrt_approx_main ((n / x + x) / (2::'a)) else (0::'a))"
assoc: fixes a :: "'g multiset"   and b :: "'g multiset"   and c :: "'g multiset" shows "a \<union># b \<union># c = a \<union># (b \<union># c)"
abs_eq: fixes x :: "'c" assumes "eq_onp in_p_image x x" shows "- Abs_regular x = Abs_regular (- x)"
listset_nthD: fixes vs :: "'a list"   and VS :: "'a set list"   and i :: "nat" assumes "vs \<in> listset VS"   and "i < length vs" shows "vs ! i \<in> VS ! i"
f2: shows "mix_pmf \<alpha> u v = mix_pmf (1 - \<alpha>) v u"
lookup_update': fixes k :: "'b"   and v :: "'a"   and m :: "('b, 'a) mapping"   and k' :: "'b" shows "Mapping.lookup (Mapping.update k v m) k' = (if k = k' then Some v else Mapping.lookup m k')"
assms_3: shows "op' \<in> set (map (sasp_op_to_strips \<Psi>) ops)"
simps_1: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'c" shows "ord.rbt_lookup less rbt.Empty k = None"
unique2: 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 C :: "char list"   and Cs :: "char list list"   and C' :: "char list"   and Cs' :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C (Cs @ C' # Cs')"   and "wf_prog wf_md P" shows "C' \<notin> set Cs"
rel_star_contl: fixes X :: "('a \<times> 'b) set"   and Y :: "('b \<times> 'b) set" shows "X O Y\<^sup>* = (\<Union>n. X O rel_dioid.power Y n)"
x_is_inc: shows "x = inc_dot (length (item_\<alpha> x)) (init_item (item_nonterminal x, item_rhs x) 0)"
eval\<^sub>c\<^sub>s_bij: fixes b_of_a :: "'a \<Rightarrow> 'b"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool"   and Cs :: "fterm literal set set" assumes "bij b_of_a" shows "eval\<^sub>c\<^sub>s (F_conv b_of_a F) (G_conv b_of_a G) Cs = eval\<^sub>c\<^sub>s F G Cs"
p': shows "positive (matrix_sum d (\<lambda>k. adjoint (M k) * M k - adjoint (M k) * P k * M k) n)"
outer_pasch_R1: fixes P :: "'p"   and Q :: "'p"   and C :: "'p"   and A :: "'p"   and B :: "'p" assumes "Col P Q C"   and "Bet A C P"   and "Bet B Q C" shows "\<exists>X. Bet A X B \<and> Bet P Q X"
set_object_value_def: fixes obj_id :: "obj_id_t"   and val :: "obj_t"   and s :: "state_t" shows "set_object_value obj_id val s = s\<lparr>obj := (obj s)(obj_id := val)\<rparr>"
abs_ast_operator_section_def: shows "abs_ast_operator_section \<equiv> map abs_ast_operator ast\<delta>"
minus_add_minus_mat: fixes u :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'c mat"   and w :: "'c mat" assumes "u \<in> carrier_mat nr nc"   and "v \<in> carrier_mat nr nc"   and "w \<in> carrier_mat nr nc" shows "u - (v + w) = u - v - w"
simps_23: fixes x51 :: "Core.term"   and x52 :: "Core.term"   and x3 :: "nat" shows "x51 $ x52 \<noteq> Bv x3"
assms: shows "ev (Not \<circ> HLD {y}) (y ## xs)"
split_correct: fixes x :: "real"   and I :: "float interval" assumes "x \<in>\<^sub>r I" shows "list_ex (in_real_interval x) (split I)"
insert_def: fixes x :: "'b"   and ts :: "'b tree list" shows "insert x ts = ins_tree (Node 0 x []) ts"
\<sigma>_def: fixes P :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" shows "M.\<sigma> P = map M.indexed_const P"
finite_lprefix_def: shows "Coinductive_List.finite_lprefix = (\<sqsubseteq>)"
ring_1_axioms: shows "class.ring_1 (1::'a) (*) (+) (0::'a) (-) uminus"
Busemann_function_quasi_morphism: fixes xi :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b"   and z :: "'b" shows "\<bar>Busemann_function_at xi x y + Busemann_function_at xi y z - Busemann_function_at xi x z\<bar> \<le> 2 * deltaG TYPE('b)"
n_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "!x \<cdot> (!y \<oplus> !z) = !x \<cdot> !y \<oplus> !x \<cdot> !z"
subtuple_acyclic'': fixes ym :: "'e \<rightharpoonup>\<^sub>f 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xm :: "'e \<rightharpoonup>\<^sub>f 'd" assumes "acyclicP_on (fmran' ym) R"   and "subtuple R\<^sup>*\<^sup>* xm ym"   and "subtuple R ym xm" shows "xm = ym"
rm_vars_dom: fixes V :: "'a set"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "subst_domain (rm_vars V s) = subst_domain s - V"
resCasesB': fixes x' :: "name"   and P :: "pi"   and a :: "subject"   and y' :: "name"   and P' :: "pi" assumes "<\<nu>x'>P \<longmapsto> a\<guillemotleft>y'\<guillemotright> \<prec> P'" obtains P :: "pi" and aa :: "name" and b :: "name" and P'a :: "pi" where "<\<nu>x'>Pb__ = <\<nu>b>Pc__" and "a\<guillemotleft>y'\<guillemotright> \<prec> P' = aa__<\<nu>b> \<prec> P'a__" and "Pc__ \<longmapsto> aa__[b] \<prec> P'a__" and "aa__ \<noteq> b"   | P :: "pi" and aa :: "subject" and x :: "name" and P'a :: "pi" and y :: "name" where "<\<nu>x'>Pb__ = <\<nu>y>P" and "a\<guillemotleft>y'\<guillemotright> \<prec> P' = aa\<guillemotleft>x\<guillemotright> \<prec> <\<nu>y>P'a" and "P \<longmapsto> aa\<guillemotleft>x\<guillemotright> \<prec> P'a" and "y \<sharp> aa" and "y \<noteq> x" and "x \<sharp> P" and "x \<sharp> aa"
f1: shows "rk {A, B, C, A', B', C', P} = 3"
max_input_Plus: fixes a1 :: "vname aexp"   and a2 :: "vname aexp" shows "max_input (Plus a1 a2) = max (max_input a1) (max_input a2)"
Sup_atMost: fixes y :: "'a" shows "\<Sqinter> ord.atMost (\<lambda>x y. y \<le> x) y = y"