hcju/isabelleps huggingface dataset:
f_Exec_Stream_Acc_LocalState__State_Idle_nth: fixes l :: "nat"   and k :: "nat"   and localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and i :: "nat"   and output_fun :: "'d \<Rightarrow> 'g message_af"   and xs :: "'f message_af list"   and c :: "'d" assumes "0 < l"   and "l \<le> k"   and "Exec_Equal_State localState trans_fun"   and "\<And>n. n \<le> i \<Longrightarrow> State_Idle localState output_fun trans_fun (f_Exec_Comp_Stream_Acc_LocalState l localState trans_fun xs c ! n)"   and "i < length xs" shows "f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs c ! i = f_Exec_Comp_Stream_Acc_LocalState l localState trans_fun xs c ! i"
delayed_cond1: assumes "u_max < other.s \<delta>" shows "delayed_safe_distance.no_collision {0..}"
runit'_simps_4: fixes f :: "'a" assumes "C.ide f" shows "C.dom \<r>\<^sub>C\<^sup>-\<^sup>1[f] = f"
prod_dvd_prod_subset: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and B :: "'b set"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.comm_monoid_mult times one"   and "finite B"   and "A \<subseteq> B" shows "dvd.dvd times (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one f B)"
piiq: fixes p :: "'b"   and i :: "'b"   and q :: "'b" assumes "(p, i) \<in> ov \<union> s \<union> m \<union> f\<inverse> \<union> d\<inverse> \<union> e \<union> s\<inverse>"   and "(i, q) \<in> ov\<inverse> \<union> s\<inverse> \<union> m\<inverse> \<union> f \<union> d \<union> e \<union> s" shows "(p, q) \<in> s \<union> s\<inverse> \<union> f \<union> f\<inverse> \<union> d \<union> d\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> m \<union> m\<inverse>"
assoc'_in_hom: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and h :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.ide g"   and "Span.ide h"   and "Span.src f = Span.trg g"   and "Span.src g = Span.trg h" shows "Span.in_hom \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc' (\<odot>) Maps.PRJ\<^sub>0 Maps.PRJ\<^sub>1 f g h, Dom = Cod (f \<circ> g \<circ> h), Cod = Dom ((f \<circ> g) \<circ> h)\<rparr> (f \<circ> g \<circ> h) ((f \<circ> g) \<circ> h)"
simps_3: fixes n :: "nat"   and xs :: "'b list" assumes "0 < n"   and "xs \<noteq> []" shows "chop n xs = take n xs # chop n (drop n xs)"
Inf_lower: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq (Inf A) x"
permute_eq_iff: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "perm"   and x :: "'b"   and y :: "'b" assumes "class.pt permute" shows "(permute p x = permute p y) = (x = y)"
that: fixes s :: "'Q set" assumes "s \<subseteq> Q"   and "q \<in> s"   and "accs \<delta>ss t s" shows "thesis"
lift_sub: fixes f :: "nat \<Rightarrow> real"   and z :: "nat"   and a :: "real mpoly"   and b :: "real mpoly" shows "insertion f (liftPoly 0 z (a - b)) = insertion f (liftPoly 0 z a - liftPoly 0 z b)"
unitfactor_ee: fixes G :: "('c, 'd) monoid_scheme"   and u :: "'c"   and as :: "'c list" assumes "comm_monoid_cancel G"   and "u \<in> Units G"   and "set as \<subseteq> carrier G" shows "essentially_equal G (as[0 := as ! 0 \<otimes>\<^bsub>G\<^esub> u]) as"
mat_of_cols_index: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and ws :: "'a Matrix.vec list" assumes "i < n"   and "j < length ws" shows "mat_of_cols n ws $$ (i, j) = ws ! j $v i"
assms: fixes i :: "nat" assumes "i < length xs" shows "max_Var_floatarith (xs ! i) \<le> F"
set_id: fixes U :: "'b set"   and A :: "'b set" shows "Id\<^bsub>set_cat U\<^esub> A = set_id U A"
homeomorphic_connectedness: fixes s :: "'b set"   and t :: "'c set" assumes "s homeomorphic t" shows "connected s = connected t"
congruent_triangle_def: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" shows "congruent_triangle a1 b1 c1 a2 b2 c2 \<equiv> (dist a1 b1 = dist a2 b2 \<and> dist a1 c1 = dist a2 c2 \<and> dist b1 c1 = dist b2 c2) \<and> angle b1 a1 c1 = angle b2 a2 c2 \<and> angle a1 b1 c1 = angle a2 b2 c2 \<and> angle a1 c1 b1 = angle a2 c2 b2"
conjugate_conjl2: fixes y :: "'a" shows "conjugate ((\<lhd>) y) = (\<lambda>x. x \<rhd> y)"
lens_comp_quotient: fixes Y :: "'b \<Longrightarrow> 'c"   and X :: "'d \<Longrightarrow> 'b" assumes "weak_lens Y" shows "(X ;\<^sub>L Y) /\<^sub>L Y = X"
Basic: fixes f :: "'c \<Rightarrow> 'c"   and s :: "'c" shows "(Some (Basic f), s) -c\<rightarrow> (None, f s)"
fps_divide_by_mult': fixes h :: "'a fps"   and f :: "'a fps"   and g :: "'a fps" assumes "subdegree h \<le> subdegree f" shows "f / (g * h) = f / h / g"
Un: fixes S :: "'a set"   and T :: "'a set" assumes "P S"   and "P T"   and "interior S \<inter> interior T = {}" shows "P (S \<union> T)"
DAimplieMR: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "DenyAll \<in> set p" shows "applied_rule_rev C x p \<noteq> \<bottom>"
inf_bot_left: fixes x :: "'a multiset" shows "{#} \<inter># x = {#}"
simps: fixes n :: "nat"   and l :: "real" shows "\<Phi> (n, l) = (if f2' * l \<le> real n then ai * (real n - f2' * l) else if real n \<le> f1' * l \<and> l0 \<le> l / c then ad * (f1' * l - real n) else 0)"
simps_4: fixes f1 :: "'b"   and f2 :: "'d \<Rightarrow> 'b" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
AE_Ball_mp: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<forall>x\<in>space M. P x"   and "AE x in M. P x \<longrightarrow> Q x" shows "almost_everywhere M Q"
qGoodAbs_iff_goodAbs_asAbs: fixes qA :: "('a, 'b, 'varSort, 'var, 'c) qAbs" shows "goodAbs (asAbs qA) = qGoodAbs qA"
widen_Object: fixes G :: "prog"   and T :: "ref_ty" assumes "isrtype G T"   and "ws_prog G" shows "G\<turnstile>RefT T\<preceq>Class Object"
absorb_iff2: fixes b :: "'k multiset"   and a :: "'k multiset" shows "(b \<subseteq># a) = (a \<inter># b = b)"
S: shows "S \<in> carrier_mat CARD('nr) CARD('nc)"
wf_measure: fixes f :: "'g \<Rightarrow> nat" shows "wf (measure f)"
can_sync_def: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and LT :: "('l + 't + 't) set" shows "t \<turnstile> \<langle>x, m\<rangle> LT \<wrong> \<equiv> \<exists>ta x' m'. t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle> \<and> LT = collect_waits ta"
zero_reg_lan_def: shows "0 \<equiv> Abs_reg_lan 0"
img_fst: fixes a :: "'c"   and b :: "'d"   and S :: "('c \<times> 'd) set" assumes "(a, b) \<in> S" shows "a \<in> fst ` S"
wca_on_total_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "wca_on A f"   and "f_range_on A f"   and "decisive_on A f" shows "total_on A (rwp_on A f)"
Disjoint_graph_aux_def: shows "Disjoint_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>Disjoint_aux z t u. x1 = (t, u) \<and> x2 = All2 z t (Neg (Var z IN u)) \<and> atom z \<sharp> (t, u))"
measure_Union: fixes M :: "'b measure"   and A :: "'b set"   and B :: "'b set" assumes "emeasure M A \<noteq> \<infinity>"   and "emeasure M B \<noteq> \<infinity>"   and "A \<in> sets M"   and "B \<in> sets M"   and "A \<inter> B = {}" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M B"
dec_add_to_counter: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_dec ck mn env s d res" shows "\<forall>s' r extra. res = (s', r) \<and> ck = True \<and> r \<noteq> Rerr (Rabort Rtimeout_error) \<longrightarrow> evaluate_dec True mn env (update_clock (\<lambda>_. clock s + extra) s) d (update_clock (\<lambda>_. clock s' + extra) s', r)"
a: shows "\<forall>y>0. \<exists>a\<in>{T \<in> existence_ivl0 x. 0 < T \<and> flow0 x T = x}. a < y"
new_spairs_alt_spp: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "map (app_pair spp_of) (new_spairs bs p) = new_spairs_spp (map spp_of bs) (spp_of p)"
bisimWeakCongruence: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<sim> Q" shows "P \<cong> Q"
cbox_top_un: fixes y\<^sub>0 :: "real"   and y\<^sub>1 :: "real"   and y\<^sub>2 :: "real"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real" assumes "y\<^sub>0 \<le> y\<^sub>1"   and "y\<^sub>1 \<le> y\<^sub>2" shows "cbox (x\<^sub>0, y\<^sub>0) (x\<^sub>1, y\<^sub>1) \<union> cbox (x\<^sub>0, y\<^sub>1) (x\<^sub>1, y\<^sub>2) = cbox (x\<^sub>0, y\<^sub>0) (x\<^sub>1, y\<^sub>2)"
Span_mem_imp_non_trivial_combine: fixes Us :: "'a list"   and a :: "'a" assumes "set Us \<subseteq> carrier R"   and "a \<in> Span K Us" obtains k :: "'a" and Ks :: "'a list" where "k \<in> K - {\<zero>}" and "set Ks \<subseteq> K" and "length Ks = length Us" and "combine (k # Ks) (a # Us) = \<zero>"
Healthy_intro: fixes H :: "'a upred \<Rightarrow> 'a upred"   and P :: "'a upred" assumes "H P = P" shows "P is H"
component_of_not_node: fixes r :: "'b"   and g :: "'b ugraph" assumes "r \<notin> nodes g" shows "component_of g r = graph {r} {}"
ge_trans: fixes y :: "'c"   and x :: "'c"   and z :: "'c" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
desired_trace_always_exists: fixes t :: "('a, 'b, 'c) event list" assumes "trace init t final" shows "\<exists>t'. t' <~~> t \<and> (\<exists>i. (\<forall>j<i. \<not> postrecording_event t' j) \<and> (\<forall>j\<ge>i. \<not> prerecording_event t' j) \<and> i \<le> length t' \<and> trace init t' final)"
lim_cnj: fixes f :: "'a \<Rightarrow> complex"   and l :: "complex"   and F :: "'a filter" shows "((\<lambda>x. cnj (f x)) \<longlongrightarrow> cnj l) F = (f \<longlongrightarrow> l) F"
DiffE: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A - B" obtains "c \<in> A" and "c \<notin> B"
rel_interior_eq_between: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "convex T" shows "(rel_interior S = rel_interior T) = (rel_interior S \<subseteq> T \<and> T \<subseteq> closure S)"
dom_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "S.arr (S.mkArr A B F)" shows "S.dom (S.mkArr A B F) = S.mkIde A"
rep_eq: fixes x :: "'b word \<Rightarrow> 'c word" shows "is_up x = (LENGTH('b) \<le> LENGTH('c))"
last_plus: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m \<le> n" shows "sum g {m..n} = g n + sum g {m..<n}"
UP_one_closed: shows "\<one>\<^bsub>UP R\<^esub> \<in> carrier (UP R)"
small: shows "small ((\<lambda>x. x\<lparr>0\<^sub>\<nat>\<rparr>) ` elts r)"
consistentE: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "CardinalityArityTransformation Aexp Aheap"   and "CardinalityArityTransformation.consistent Aexp (ae, a, as) (\<Gamma>, e, S)" obtains "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, as) (\<Gamma>, e, S)" and "\<And>x. x \<in> thunks \<Gamma> \<Longrightarrow> ae x = up\<cdot>0"
ih: shows "X = A ^^ Suc n @@ X \<union> (\<Union>m\<le>n. A ^^ m @@ B)"
2: shows "(X, X #[x' \<and> x]_xs) \<in> swapped"
free2: shows "dfree ($f fid1 (\<lambda>i. if i = vid1 then Var vid1 else Const 0))"
inner_product_space_axioms: shows "inner_product_space (*\<^sub>R) (\<bullet>)"
Un_empty_left: fixes B :: "'e set" shows "{} \<union> B = B"
simps_2: fixes uu :: "('b \<times> nat) tree"   and uv :: "'b"   and n :: "nat"   and uw :: "('b \<times> nat) tree" shows "mht \<langle>uu, (uv, n), uw\<rangle> = n"
scale_zero_right: fixes a :: "rat" shows "a *R (0::'a) = (0::'a)"
dim_kernel_zero_jordan_block_pow: fixes n :: "nat"   and k :: "nat" shows "vectorspace.dim class_ring (module_vec TYPE('a) n\<lparr>carrier := mat_kernel (jordan_block n (0::'a) ^\<^sub>m k)\<rparr>) = min k n"
space_eq_univ: fixes M :: "'b measure" assumes "finite_borel_measure' M" shows "space M = UNIV"
ccSUP_upper: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and i :: "'c"   and f :: "'c \<Rightarrow> 'b" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "i \<in> A" shows "less_eq (f i) (Sup (f ` A))"
lipschitz_on_mono: fixes M :: "real"   and E :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and D :: "'b set"   and L :: "real" assumes "M-lipschitz_on E f"   and "D \<subseteq> E"   and "M \<le> L" shows "L-lipschitz_on D f"
map_of_eq_Some_iff: fixes xys :: "('a \<times> 'b) list"   and x :: "'a"   and y :: "'b" assumes "distinct (map fst xys)" shows "(Map.map_of xys x = Some y) = ((x, y) \<in> set xys)"
measure_subprob: fixes M :: "'b pmf" shows "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
set_tag_name_get_tag_name: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and tag :: "char list"   and h' :: "(_) heap" assumes "h \<turnstile> set_tag_name element_ptr tag \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_tag_name element_ptr \<rightarrow>\<^sub>r tag"
schroeder_4_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> - z) = (z \<sqinter> id y \<le> - x)"
ext_inject: fixes Leg0 :: "'b"   and Leg1 :: "'b"   and more :: "'c"   and Leg0' :: "'b"   and Leg1' :: "'b"   and more' :: "'c" shows "(\<lparr>Leg0 = Leg0, Leg1 = Leg1, \<dots> = more\<rparr> = \<lparr>Leg0 = Leg0', Leg1 = Leg1', \<dots> = more'\<rparr>) = (Leg0 = Leg0' \<and> Leg1 = Leg1' \<and> more = more')"
path_fg: shows "path_image \<gamma> \<subseteq> s - zeros_fg"
simps_2: fixes kts :: "(('c \<times> 'd) \<times> ('c, 'd) rbt) list"   and c :: "color"   and l :: "('c, 'd) rbt"   and k :: "'c"   and v :: "'d"   and r :: "('c, 'd) rbt" shows "rbt_entries_next (kts, Branch c l k v r) = rbt_entries_next (((k, v), r) # kts, l)"
prems_2: shows "n1 < length (al @ [a]) \<and> j1 < f ((al @ [a]) ! n1)"
sum_component: fixes f :: "'c \<Rightarrow> ('a, 'b) vec"   and S :: "'c set"   and i :: "'b" shows "sum f S $ i = (\<Sum>x\<in>S. f x $ i)"
map_simp_ide: fixes a :: "'a" assumes "ide a" shows "\<l>.map a = pr0 (cod \<iota>) a"
inverse_unique: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.inverse_arrows f g" shows "VV.inv f = g"
Suc_1: fixes j :: "nat"   and r :: "mregex" assumes "j \<le> i" shows "RPDi j (MStar r) \<subseteq> {MStar r} \<union> MTimesL (MStar r) (\<Union>j\<le>i. RPDi j r)"
agents_eq: shows "agents = {i. prefs_from_table xs i \<noteq> (\<lambda>_ _. False)}"
marker_present: shows "Marker \<in> set (msgs (S t (i + 1)) cid)"
simps_2: fixes n :: "nat"   and s :: "'a" shows "fuse.iter (Suc n) s = (if guard s then iter (N s) s \<bind> fuse.iter n else return_spmf s)"
wqo_on_imp_qo_on: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "wqo_on P A" shows "qo_on P A"
strong_induct: fixes P :: "'a \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a"   and lam :: "lam" assumes "\<And>name z. P z (VAR name)"   and "\<And>lam1 lam2 z. \<lbrakk>\<And>z. P z lam1; \<And>z. P z lam2\<rbrakk> \<Longrightarrow> P z (APP lam1 lam2)"   and "\<And>name lam z. \<lbrakk>name \<sharp> z; \<And>z. P z lam\<rbrakk> \<Longrightarrow> P z LAM [name].lam"   and "\<And>nat z. P z (NUM nat)"   and "\<And>lam1 lam2 z. \<lbrakk>\<And>z. P z lam1; \<And>z. P z lam2\<rbrakk> \<Longrightarrow> P z lam1 -- lam2"   and "\<And>lam1 lam2 z. \<lbrakk>\<And>z. P z lam1; \<And>z. P z lam2\<rbrakk> \<Longrightarrow> P z lam1 ++ lam2"   and "\<And>z. P z TRUE"   and "\<And>z. P z FALSE"   and "\<And>lam1 lam2 lam3 z. \<lbrakk>\<And>z. P z lam1; \<And>z. P z lam2; \<And>z. P z lam3\<rbrakk> \<Longrightarrow> P z (IF lam1 lam2 lam3)"   and "\<And>name lam z. \<lbrakk>name \<sharp> z; \<And>z. P z lam\<rbrakk> \<Longrightarrow> P z FIX [name].lam"   and "\<And>lam z. (\<And>z. P z lam) \<Longrightarrow> P z (ZET lam)"   and "\<And>lam1 lam2 z. \<lbrakk>\<And>z. P z lam1; \<And>z. P z lam2\<rbrakk> \<Longrightarrow> P z (EQI lam1 lam2)" shows "P z lam"
simps_1: shows "invh \<langle>\<rangle> = True"
elims_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq x" obtains "x = []"   | xa :: "'a" and ys :: "'a list" where "x = xa # ys" and "Ball (set ys) (less_eq xa) \<and> linorder.sorted less_eq ys"
weakening: fixes \<C> :: "t_context"   and es :: "b_e list"   and t1s :: "t list"   and t2s :: "t list"   and ts :: "t list" assumes "\<C> \<turnstile> es : t1s _> t2s" shows "\<C> \<turnstile> es : ts @ t1s _> ts @ t2s"
ord_pp_strict_punit_alt: shows "ord_pp_strict_punit = lt_of_nat_term_order"
non_speculative_nthI: fixes obs :: "('a, 'b) obs_event action llist"   and P :: "'c prog"   and vs :: "'a \<times> addr_loc \<Rightarrow> 'a val set" assumes "\<And>i ad al v. \<lbrakk>enat i < llength obs; lnth obs i = NormalAction (ReadMem ad al v); non_speculative P vs (ltake (enat i) obs)\<rbrakk> \<Longrightarrow> v \<in> w_values P vs (list_of (ltake (enat i) obs)) (ad, al)" shows "non_speculative P vs obs"
assms: shows "h \<turnstile> ok get_root_node_si ptr"
points_index_one_not_unique_block: fixes B :: "'a set multiset"   and ps :: "'a set"   and bl :: "'a set"   and bl' :: "'a set" assumes "B index ps = 1"   and "ps \<subseteq> bl"   and "bl \<in># B"   and "bl' \<in># remove1_mset bl B" shows "\<not> ps \<subseteq> bl'"
set_eq: shows "set (verticesFrom f v') = \<V> f"
SetPC_def: fixes f :: "'d \<Rightarrow> bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and p :: "'c \<Rightarrow> 'd \<Rightarrow> real" shows "SetPC f p \<equiv> \<lambda>ab P s. \<Sum>a\<in>supp (p s). p s a * f a ab P s"
with_take_cols_from_matrix: shows "vec_space.rank (length (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) (take_cols_from_matrix (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))\<^sup>T (map snd (pivot_positions (gauss_jordan_single (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))\<^sup>T)))) = dim_col (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))"
I_in_set_ar_cos: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I" shows "I \<in> set_ar_cos R I"
ij_1: shows "{i, j} \<subseteq> gen P I"
assoc_def: shows "assoc \<equiv> \<lambda>\<mu> \<nu> \<pi>. transformation_by_components.map VVV.comp (\<bullet>) HoHV (\<lambda>fgh. \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 (fst fgh) (fst (snd fgh)) (snd (snd fgh)), Dom = Dom ((fst fgh \<star> fst (snd fgh)) \<star> snd (snd fgh)), Cod = Cod (fst fgh \<star> fst (snd fgh) \<star> snd (snd fgh))\<rparr>) (\<mu>, \<nu>, \<pi>)"
null_sets_outer: fixes S :: "'c set"   and M :: "'c measure" shows "(S \<in> null_sets (completion M)) = (\<forall>e>0. \<exists>T\<in>fmeasurable (completion M). S \<subseteq> T \<and> Sigma_Algebra.measure (completion M) T < e)"
Inv_preserves_Can_2: fixes t :: "'c local.term" assumes "Can t" shows "Dom (Inv t) = Cod t"
fmapU_errorT_def: shows "fmapU \<equiv> errorT_map\<cdot>ID"
nargs_\<rho>s: shows "enat (num_args (subst \<rho> s)) < arity_hd (head (subst \<rho> s))"
hyps_3: assumes "X \<subseteq> Sup_llist Xs" shows "\<exists>k. X \<subseteq> Sup_upto_llist Xs (enat k)"
bind_duplicate: fixes m :: "'m"   and f :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'm" shows "bind m (\<lambda>x. bind m (f x)) = bind m (\<lambda>x. f x x)"
CESem_simps_3: fixes x :: "Vars.var"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "\<N>\<lbrakk> Var x \<rbrakk>\<^bsub>\<rho>\<^esub> = (\<Lambda> (C\<cdot>r). \<rho> x\<cdot>r)"
assms_1: shows "a \<leftrightarrow>\<^sup>* b"
conj_idem: fixes conj :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and chaos :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice" assumes "conjunction (conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (chaos::'b::refinement_lattice)" shows "(conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (a::'b::refinement_lattice) a = a"
varsSubsetValuation: fixes valuation1 :: "Literal list"   and valuation2 :: "Literal list" assumes "set valuation1 \<subseteq> set valuation2" shows "vars valuation1 \<subseteq> vars valuation2"
dvd_antisym: fixes m :: "nat"   and n :: "nat" assumes "m dvd n"   and "n dvd m" shows "m = n"
ffilter_eq_fempty_iff_2: fixes P :: "'b \<Rightarrow> bool"   and X :: "'b fset" shows "({||} = ffilter P X) = (\<forall>x. x |\<in>| X \<longrightarrow> \<not> P x)"
span_li_not_depend_1: fixes S :: "'c set"   and N :: "'c set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule K N V" shows "LinearCombinations.module.span K (vs N) S = local.module.span S"
cobounded1: fixes a :: "'b"   and b :: "'b" shows "a \<le> a + b"
Strong_d_Bisimulation_def: fixes d :: "'d"   and R :: "('com list \<times> 'com list) set" shows "Strong_d_Bisimulation d R \<equiv> sym R \<and> (\<forall>(V, V')\<in>R. length V = length V') \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. \<forall>m1 m1' m2 W. \<langle>V ! i,m1\<rangle> \<rightarrow> \<langle>W,m2\<rangle> \<and> m1 =\<^bsub>d\<^esub> m1' \<longrightarrow> (\<exists>W' m2'. \<langle>V' ! i,m1'\<rangle> \<rightarrow> \<langle>W',m2'\<rangle> \<and> (W, W') \<in> R \<and> m2 =\<^bsub>d\<^esub> m2'))"
nonneg_cexpr_subst_val: fixes V :: "nat set"   and t :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and v :: "val" assumes "nonneg_cexpr (shift_var_set V) (case_nat t \<Gamma>) e"   and "val_type v = t" shows "nonneg_cexpr V \<Gamma> (cexpr_subst_val e v)"
mult_ac_2: fixes a :: "'a"   and b :: "'a" shows "a \<cdot> b = b \<cdot> a"
S\<^sub>e_E2: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and s :: "'a" assumes "Reachability_Problem K S S1 S2"   and "s \<in> S1"   and "\<And>t. (s, t) \<in> Markov_Decision_Process.E K \<Longrightarrow> t \<in> Reachability_Problem.S\<^sub>e K S S1 S2" shows "s \<in> Reachability_Problem.S\<^sub>e K S S1 S2"
hyps: assumes "list_all (\<lambda>A. A \<noteq> {}) XS" shows "finite (listset XS) = list_all finite XS"
valid_cfg_state_in_S: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and cfg :: "'a cfg" assumes "Finite_Markov_Decision_Process K S"   and "cfg \<in> Finite_Markov_Decision_Process.valid_cfg K S" shows "state cfg \<in> S"
10: shows "F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot"
intra_path_vp: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit.intra_path n as n'" shows "CFGExit.valid_path' n as n'"
bool_sat_test_def: shows "bool_sat_test \<equiv> sat_test ifex_of"
sconcat_sfoldl_aux: fixes z :: "[:'a:]"   and xs :: "[:[:'a:]:]" shows "sfoldl\<cdot>sappend\<cdot>z\<cdot>xs = z :@ sconcat\<cdot>xs"