hcju/isabelleps huggingface dataset:
in_load_tmps_conv: fixes t :: "nat"   and xs :: "instr list" shows "(t \<in> load_tmps xs) = ((\<exists>volatile a. Read volatile a t \<in> set xs) \<or> (\<exists>a sop cond ret A L R W. RMW a t sop cond ret A L R W \<in> set xs))"
zhmset_of_0: shows "zhmset_of 0 = 0"
hyps_3: shows "v \<notin> range_vars \<sigma>1"
mem_Collect_eq: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(a \<in> Set.Collect P) = P a"
fv_regex_alt: fixes m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex" assumes "safe_regex m g r" shows "fv_regex r = \<Union> (fv ` atms r)"
meissel_mertens_def: shows "meissel_mertens = 1 - ln (ln 2) + integral {2..} (\<lambda>t. (\<MM> t - ln t) / (t * (ln t)\<^sup>2))"
1: shows "l' \<subseteq>\<^sub>m [Vs [\<mapsto>] ls', V \<mapsto> v]"
assms_4: assumes "x \<le> 0"   and "y \<le> 0" shows "P (gcd (- x) (- y))"
comp_arr_dom: fixes f :: "'a2"   and a :: "'a2" assumes "A2.arr f"   and "A2.dom f = a" shows "f \<cdot>\<^sub>A\<^sub>2 a = f"
ap_spmf_conv_bind: fixes f :: "('b \<Rightarrow> 'a) spmf"   and x :: "'b spmf" shows "f \<diamondop> x = f \<bind> (\<lambda>f. x \<bind> (\<lambda>x. return_spmf (f x)))"
\<I>_1: shows "subst_domain \<I> = UNIV"
a'_insertion'': fixes x :: "real" shows "insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2) = a'"
star_left_induct: fixes z :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and x :: "'c \<times> 'c \<Rightarrow> 'd" shows "z \<oplus> y \<odot> x \<preceq> x \<longrightarrow> y\<^sup>\<odot> \<odot> z \<preceq> x"
cycle_decomp: fixes u :: "'b list" assumes "u \<noteq> []" shows "cycle u = u @- cycle u"
finite_pdevs_domain: fixes x :: "'b pdevs" shows "finite (pdevs_domain x)"
finfun_comp_update: fixes g :: "'c \<Rightarrow> 'b"   and f :: "('a, 'c) finfun"   and a :: "'a"   and b :: "'c" shows "finfun_comp g (finfun_update f a b) = finfun_update (finfun_comp g f) a (g b)"
T_Sub_5: fixes S' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> U <: S"   and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'"   and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'" shows "P"
nd_fun_plus_assoc: fixes x :: "'a nd_fun"   and y :: "'a nd_fun"   and z :: "'a nd_fun" shows "x + y + z = x + (y + z)"
card_image: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set" assumes "inj_on f A" shows "card (f ` A) = card A"
set_least_squares_approximation_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and b :: "('a, 'c) vec" shows "set_least_squares_approximation A b = {x. \<forall>y. norm (b - A *v x) \<le> norm (b - A *v y)}"
list_emb_Cons2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "P x y"   and "list_emb P xs ys" shows "list_emb P (x # xs) (y # ys)"
add_carrier_vec: fixes v\<^sub>1 :: "'a vec"   and n :: "nat"   and v\<^sub>2 :: "'a vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 + v\<^sub>2 \<in> carrier_vec n"
ifmi'_ifmi: fixes s :: "bdd"   and ni1 :: "nat"   and ni2 :: "nat"   and v :: "nat" assumes "bdd_sane s"   and "bdd_node_valid s ni1"   and "bdd_node_valid s ni2" shows "ifmi' v ni1 ni2 s = Some (ifmi v ni1 ni2 s)"
tendsto_at_left_sequentially: fixes b :: "'a"   and a :: "'a"   and X :: "'a \<Rightarrow> 'b"   and L :: "'b" assumes "b < a"   and "\<And>S. \<lbrakk>\<And>n. S n < a; \<And>n. b < S n; incseq S; S \<longlonglongrightarrow> a\<rbrakk> \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L" shows "(X \<longlongrightarrow> L) (at_left a)"
ring_irreducibleE_4: fixes r :: "'a" assumes "r \<in> carrier R"   and "ring_irreducible r" shows "r \<notin> Units R"
eventually_INF_base: fixes B :: "'a set"   and F :: "'a \<Rightarrow> 'b filter"   and P :: "'b \<Rightarrow> bool" assumes "B \<noteq> {}"   and "\<And>a b. \<lbrakk>a \<in> B; b \<in> B\<rbrakk> \<Longrightarrow> \<exists>x\<in>B. F x \<le> inf (F a) (F b)" shows "eventually P (Inf (F ` B)) = (\<exists>b\<in>B. eventually P (F b))"
simps: fixes Rel :: "ccs \<Rightarrow> ccs \<Rightarrow> bool"   and a1 :: "ccs"   and a2 :: "ccs" shows "bangRelp Rel a1 a2 = ((\<exists>P Q. a1 = !P \<and> a2 = !Q \<and> Rel P Q) \<or> (\<exists>R T P Q. a1 = R \<parallel> P \<and> a2 = T \<parallel> Q \<and> Rel R T \<and> bangRelp Rel P Q))"
update_comp: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "State_Monad.update f \<bind> (\<lambda>_. State_Monad.update g) = State_Monad.update (g \<circ> f)"
tr_filter_conc_eq2: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list"   and cs :: "'c set" shows "a @ b = tr_filter (a @ c) cs \<longrightarrow> a = tr_filter a cs"
rcis_cnj: fixes a :: "complex" shows "cnj a = rcis (cmod a) (- arg a)"
leg_simps_3: shows "C.arr assoc'_fgh.dom.leg1"
dg_GRPH_ObjE: fixes \<AA> :: "V"   and \<alpha> :: "V" assumes "\<AA> \<in>\<^sub>\<circ> dg_GRPH \<alpha>\<lparr>Obj\<rparr>" obtains "digraph \<alpha> \<AA>"
fconverse_iff: fixes b :: "V"   and a :: "V"   and r :: "V" shows "([b, a]\<^sub>\<circ> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<bullet>) = ([a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r)"
ex_cf_sol: fixes p :: "'a" assumes "p \<in> carrier R" shows "\<exists>c. cf_sol R S X p c"
ge: shows "Suc (degree f - 1) \<le> degree (pderiv f)"
edges_def: fixes w :: "'c \<times> 'c \<Rightarrow> enat" shows "WGraph.edges w \<equiv> {(u, v). w (u, v) \<noteq> \<infinity>}"
y: shows "\<forall>y'\<in>set ys. y < y'"
leg0_simps_3: shows "trg rs.p\<^sub>0 = src ((tab\<^sub>0 r)\<^sup>* \<star> tab\<^sub>1 s)"
adj_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplexEndomorphism X f"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<sim> D" shows "f ` C \<sim> f ` D"
p_geI: fixes p' :: "real" assumes "1 \<le> (\<Sum>i<k. as ! i * bs ! i powr p')" shows "p' \<le> p"
Weierstrass_m_test'_ev: fixes A :: "'b set"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>A. norm (f n x) \<le> M n"   and "summable M" shows "uniformly_convergent_on A (\<lambda>n x. \<Sum>i<n. f i x)"
Object_widen: fixes P :: "'a prog"   and T :: "ty" assumes "P \<turnstile> Class Object \<le> T" shows "T = Class Object"
prog_sifum_secure_cont_def: fixes cmds :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list" shows "prog_sifum_secure_cont cmds = (\<forall>mem\<^sub>1 mem\<^sub>2. INIT mem\<^sub>1 \<and> INIT mem\<^sub>2 \<and> mem\<^sub>1 =\<^sup>l mem\<^sub>2 \<longrightarrow> (\<forall>sched cms\<^sub>1' mem\<^sub>1'. (cmds, mem\<^sub>1) \<rightarrow>\<^bsub>sched\<^esub> (cms\<^sub>1', mem\<^sub>1') \<longrightarrow> (\<exists>cms\<^sub>2' mem\<^sub>2'. (cmds, mem\<^sub>2) \<rightarrow>\<^bsub>sched\<^esub> (cms\<^sub>2', mem\<^sub>2') \<and> map snd cms\<^sub>1' = map snd cms\<^sub>2' \<and> length cms\<^sub>2' = length cms\<^sub>1' \<and> (\<forall>x. dma mem\<^sub>1' x = Low \<and> (x \<in> \<C> \<or> (\<forall>i<length cms\<^sub>1'. x \<notin> snd (cms\<^sub>1' ! i) AsmNoReadOrWrite)) \<longrightarrow> mem\<^sub>1' x = mem\<^sub>2' x))))"
assms_2: fixes x :: "'b" assumes "x \<in> X" shows "h x = i x"
invert_side_of_triangle_1: fixes h :: "'d"   and f :: "'d"   and g :: "'d" assumes "D.arr h"   and "f \<cdot>\<^sub>D g = h"   and "D.iso f" shows "D.seq (D.inv f) h \<and> g = D.inv f \<cdot>\<^sub>D h"
empty_subsetI: fixes A :: "'d set" shows "{} \<subseteq> A"
ddcc_imp_e: shows "\<forall>I. I \<Turnstile>m DD + CC \<longrightarrow> I \<Turnstile> E"
is_path_undir_simps_2: fixes G :: "('c, 'd) graph"   and v :: "'c"   and w :: "'d"   and v' :: "'c" assumes "valid_graph G" shows "is_path_undir G v [(v, w, v')] v' = ((v, w, v') \<in> edges G \<or> (v', w, v) \<in> edges G)"
d1: fixes k :: "nat" assumes "k < n" shows "M k * A * adjoint (M k) \<in> carrier_mat d d"
lookup_map_keys: fixes f :: "'c \<Rightarrow> 'b"   and m :: "('c, 'a) mapping"   and x :: "'b" shows "Mapping.lookup (map_keys f m) x = (if f -` {x} \<noteq> {} then Mapping.lookup m (THE k. f -` {x} = {k}) else None)"
language_ltlc_def: fixes \<phi> :: "'b ltlc" shows "language_ltlc \<phi> \<equiv> {\<xi>. \<xi> \<Turnstile>\<^sub>c \<phi>}"
simps_1: fixes t1 :: "('c \<times> nat) tree"   and a :: "'c"   and lva :: "nat"   and t2 :: "('c \<times> nat) tree"   and b :: "'c"   and lvb :: "nat"   and t3 :: "('c \<times> nat) tree"   and c :: "'c"   and lvc :: "nat"   and t4 :: "('c \<times> nat) tree" shows "split \<langle>t1, (a, lva), \<langle>t2, (b, lvb), \<langle>t3, (c, lvc), t4\<rangle>\<rangle>\<rangle> = (if lva = lvb \<and> lvb = lvc then \<langle>\<langle>t1, (a, lva), t2\<rangle>, (b, lva + 1), \<langle>t3, (c, lva), t4\<rangle>\<rangle> else \<langle>t1, (a, lva), \<langle>t2, (b, lvb), \<langle>t3, (c, lvc), t4\<rangle>\<rangle>\<rangle>)"
surjective: fixes r :: "('a, 'b, 'c) SES_rec_scheme" shows "r = \<lparr>S_SES = S_SES r, s0_SES = s0_SES r, E_SES = E_SES r, I_SES = I_SES r, O_SES = O_SES r, T_SES = T_SES r, \<dots> = SES_rec.more r\<rparr>"
toplevel_inters_finite: fixes r :: "'c rexp" shows "finite (toplevel_inters r)"
reachableI_app_iso: fixes u :: "'c"   and v :: "'c"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "u \<rightarrow>\<^sup>*\<^bsub>app_iso hom G\<^esub> v"   and "H.digraph_isomorphism hom" shows "iso_verts hom u \<rightarrow>\<^sup>*\<^bsub>app_iso hom (app_iso hom G)\<^esub> iso_verts hom v"
next_eq: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<circle>(F = G) = (\<circle>F = \<circle>G)"
filter_inf_filter: fixes F :: "'a set"   and x :: "'a" assumes "ord_class.filter F" shows "ord_class.filter (\<Up>{y. \<exists>z\<in>F. x \<sqinter> z = y})"
acomp_size: fixes a :: "aexp" shows "1 \<le> size (acomp a)"
power_similiar_sq_mtx_diag_eq: fixes P :: "'b sq_mtx"   and A :: "'b sq_mtx"   and f :: "'b \<Rightarrow> real"   and n :: "nat" assumes "mtx_invertible P"   and "A = P\<^sup>-\<^sup>1 * sq_mtx_diag f * P" shows "A ^ n = P\<^sup>-\<^sup>1 * (\<d>\<i>\<a>\<g> i. f i ^ n) * P"
sorted_sublist_wrt_refl: fixes i :: "nat"   and xs :: "'c list"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "i < length xs" shows "sorted_sublist_wrt R xs i i"
prime: assumes "prime n" shows "fermat_test n = return_pmf True"
simps: fixes n :: "nat" shows "the_enat (enat n) = n"
Card_order_cexp: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('c \<times> 'c) set" shows "Card_order (r1 ^c r2)"
xmin_1: shows "0 < xmin \<and> \<not> (s (e_snoc (local.adverse z i j \<triangleright> xmin) 0) \<noteq> s (local.adverse z i j \<triangleright> xmin) \<or> s (e_snoc (local.adverse z i j \<triangleright> xmin) 1) \<noteq> s (local.adverse z i j \<triangleright> xmin))"
step: shows "trsys.inf_step mthr.silent_move2 s ttas"
summable: fixes u :: "complex" assumes "cmod u < 1" shows "summable (\<lambda>n. g n * u ^ n)"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Even of Even \<Rightarrow> f1 | Odd \<Rightarrow> f2) = f1"
isomorphic_implies_ide_1: fixes f :: "'d"   and g :: "'d" assumes "D.isomorphic f g" shows "D.ide f"
function_smult_closed: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and f :: "'c \<Rightarrow> 'a"   and S :: "'c set" assumes "ring_functions R"   and "a \<in> carrier R"   and "f \<in> carrier (function_ring S R)" shows "a \<odot>\<^bsub>function_ring S R\<^esub> f \<in> carrier (function_ring S R)"
nth_mem: fixes n :: "nat"   and xs :: "'a list" assumes "n < |xs|" shows "xs ! n \<in> set xs"
minim_inField: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "local.minim B \<in> Field r"
vquot_dbfm_empty: fixes A :: "dbfm" shows "vquot_dbfm {} A = quot_dbfm A"
ipresIGWls_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and MOD :: "('h, 'i, 'j, 'e, 'k, 'l, 'g, 'm, 'n) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "ipresIGWls h (FixSyn.termMOD Delta) MOD = FixSyn.ipresWls Delta h MOD"
measurable_restrict: fixes I :: "'d set"   and X :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and N :: "'e measure"   and M :: "'d \<Rightarrow> 'f measure" assumes "\<And>i. i \<in> I \<Longrightarrow> X i \<in> N \<rightarrow>\<^sub>M M i" shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M I M"
state_to_strips_state_dom_is: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>" shows "dom (\<phi>\<^sub>S \<Psi> s) = (\<Union>v\<in>{v |v. v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+) \<and> s v \<noteq> None}. {(v, a) |a. a \<in> \<R>\<^sub>+ \<Psi> v})"
assms: shows "((\<forall>i t. i \<le> n \<and> SEQ i s1 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {}) \<and> (\<forall>i t. i \<le> n \<and> SEQ i s2 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {}) \<longrightarrow> (\<exists>t1. Some t1 = SEQ n s1 \<and> (\<exists>t2. Some t2 = SEQ n s2 \<and> get_S (cpu_reg_val PSR t1) = 0 \<and> get_S (cpu_reg_val PSR t2) = 0 \<and> low_equal t1 t2))) \<and> (\<forall>i t. i \<le> Suc n \<and> SEQ i s1 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {}) \<and> (\<forall>i t. i \<le> Suc n \<and> SEQ i s2 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {})"
all_mix_pmf_indiff_indiff_best_worst: fixes l :: "'a pmf"   and b :: "'a pmf"   and w :: "'a pmf" assumes "l \<in> local.\<P>"   and "b \<in> local.best"   and "w \<in> local.worst"   and "b \<approx>[\<R>] w" shows "\<forall>\<alpha>\<in>{0..1}. l \<approx>[\<R>] mix_pmf \<alpha> b w"
pdt_def: shows "pdt = {(x, y). x \<noteq> y \<and> y pd\<rightarrow> x}"
pmf_neq_exists_less: fixes M :: "'d pmf"   and N :: "'d pmf" assumes "M \<noteq> N" shows "\<exists>x. pmf M x < pmf N x"
terminal_def: fixes b :: "'a" shows "terminal b \<equiv> ide b \<and> (\<forall>a. ide a \<longrightarrow> (\<exists>!f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright>))"
C: assumes "a n \<noteq> - 1" shows "- 1 \<le> a (Suc n)"
from_Gromov_completion_continuous_2: shows "continuous_on (range to_Gromov_completion) from_Gromov_completion"
drop_suffix_hd_css_step: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "('e, 'd, 'f) com"   and ps :: "('e, 'd, 'f) com list"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and pnorm :: "('e, 'd, 'f) com list"   and pabr :: "('e, 'd, 'f) com list"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (p # ps @ cs, css, s) \<rightarrow> (cs', [(pnorm @ ps @ cs, pabr @ ps @ cs)] @ css, t)" shows "\<Gamma>\<turnstile> (p # ps, css, s) \<rightarrow> (cs', [(pnorm @ ps, pabr @ ps)] @ css, t)"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>c e. P (Eq (CN 0 c e))"   and "\<And>c e. P (NEq (CN 0 c e))"   and "\<And>c e. P (Lt (CN 0 c e))"   and "\<And>c e. P (Le (CN 0 c e))"   and "P T"   and "P F"   and "\<And>va. P (Lt (C va))"   and "\<And>va. P (Lt (Bound va))"   and "\<And>vd vb vc. P (Lt (CN (Suc vd) vb vc))"   and "\<And>va. P (Lt (Neg va))"   and "\<And>va vb. P (Lt (Add va vb))"   and "\<And>va vb. P (Lt (Sub va vb))"   and "\<And>va vb. P (Lt (Mul va vb))"   and "\<And>va. P (Lt (Floor va))"   and "\<And>va vb vc. P (Lt (CF va vb vc))"   and "\<And>va. P (Le (C va))"   and "\<And>va. P (Le (Bound va))"   and "\<And>vd vb vc. P (Le (CN (Suc vd) vb vc))"   and "\<And>va. P (Le (Neg va))"   and "\<And>va vb. P (Le (Add va vb))"   and "\<And>va vb. P (Le (Sub va vb))"   and "\<And>va vb. P (Le (Mul va vb))"   and "\<And>va. P (Le (Floor va))"   and "\<And>va vb vc. P (Le (CF va vb vc))"   and "\<And>v. P (Gt v)"   and "\<And>v. P (Ge v)"   and "\<And>va. P (Eq (C va))"   and "\<And>va. P (Eq (Bound va))"   and "\<And>vd vb vc. P (Eq (CN (Suc vd) vb vc))"   and "\<And>va. P (Eq (Neg va))"   and "\<And>va vb. P (Eq (Add va vb))"   and "\<And>va vb. P (Eq (Sub va vb))"   and "\<And>va vb. P (Eq (Mul va vb))"   and "\<And>va. P (Eq (Floor va))"   and "\<And>va vb vc. P (Eq (CF va vb vc))"   and "\<And>va. P (NEq (C va))"   and "\<And>va. P (NEq (Bound va))"   and "\<And>vd vb vc. P (NEq (CN (Suc vd) vb vc))"   and "\<And>va. P (NEq (Neg va))"   and "\<And>va vb. P (NEq (Add va vb))"   and "\<And>va vb. P (NEq (Sub va vb))"   and "\<And>va vb. P (NEq (Mul va vb))"   and "\<And>va. P (NEq (Floor va))"   and "\<And>va vb vc. P (NEq (CF va vb vc))"   and "\<And>v va. P (Dvd v va)"   and "\<And>v va. P (NDvd v va)"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
assms_2: shows "integral\<^sup>N (count_space B) g \<noteq> \<top>"
env_restr_below_itself: fixes S :: "'d set"   and m :: "'d \<Rightarrow> 'e" shows "m f|` S \<sqsubseteq> m"
is_singleton_altdef: fixes A :: "'d set" shows "is_singleton A = (card A = 1)"
fake_beta_act: fixes \<psi> :: "\<kappa> \<Rightarrow> \<o>"   and x :: "\<nu>" assumes "\<And>G \<phi>. IsProperInX (\<lambda>x. \<lparr>G,\<^bold>\<iota>y. \<phi> y x\<rparr>)" shows "[\<lparr>backdoor \<psi>,x\<^sup>P\<rparr> \<^bold>\<equiv> \<psi> (x\<^sup>P) in dw]"
mat_kernelD_1: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> mat_kernel A" shows "v \<in> carrier_vec nc"
div_cfg_def: fixes cfg :: "('a \<times> ('c \<Rightarrow> real)) cfg" shows "div_cfg cfg \<equiv> AE \<omega> in MDP.MC.T cfg. divergent (smap (snd \<circ> state) \<omega>)"
assms: shows "xs <\<^sub>l\<^sub>e\<^sub>x ys"
isin_sorted: fixes xs :: "'b list"   and a :: "'b"   and ys :: "'b list"   and x :: "'b" assumes "sorted_less (xs @ a # ys)" shows "(x \<in> set (xs @ a # ys)) = (if x < a then x \<in> set xs else x \<in> set (a # ys))"
aeq: shows "a = (\<Sum>s\<in>S. b s *\<^sub>R s)"
assms: shows "g \<turnstile> n-ns @ n' # ns'\<rightarrow>m"
assms_2: shows "PTA.collect_clkvt A \<subseteq> X"
lub_def: fixes S :: "'b set"   and po :: "'b potype" shows "lub S po \<equiv> least (\<lambda>x. \<forall>y\<in>S. (y, x) \<in> order po) po"
Spy_analz_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> bankerberos" shows "(Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
rtrancl_idemp: fixes r :: "('c \<times> 'c) set" shows "(r\<^sup>*)\<^sup>* = r\<^sup>*"
Y: shows "next_dim.cone Y = next_dim.polyhedral_cone M"
negligible_subset: fixes s :: "'a set"   and t :: "'a set" assumes "negligible s"   and "t \<subseteq> s" shows "negligible t"
linorder_cases: fixes y :: "'a"   and x :: "'a" obtains "y < x"   | "x = y"   | "x < y"
tuple_in_hom: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "\<guillemotleft>hfsetcat.tuple f g : local.dom f \<rightarrow> hfsetcat.prod (cod f) (cod g)\<guillemotright>"
fall_def: fixes F :: "('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream list \<Rightarrow> bool" shows "fall F \<equiv> neg (exi (\<lambda>\<pi>. neg (F \<pi>)))"
BRANCH_def: fixes c :: "'f"   and rs :: "'i \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set" shows "BRANCH c rs = In1 (In1 (Scons (Old_Datatype.Leaf (Inl c)) (Old_Datatype.Lim rs)))"
partition_def: fixes A :: "'d set"   and B :: "'d set set" shows "Congruence.partition A B \<equiv> (\<forall>a. a \<in> A \<longrightarrow> (\<exists>!b. b \<in> B \<and> a \<in> b)) \<and> (\<forall>b. b \<in> B \<longrightarrow> b \<subseteq> A)"
cong_exp_iff_simps_6: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and q :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus num.One) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus (num.Bit0 n)) (numeral.numeral one plus (num.Bit0 q))) = False"
residue_simple_pole_deriv: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and g :: "complex \<Rightarrow> complex"   and z :: "complex"   and g' :: "complex" assumes "f holomorphic_on s"   and "g holomorphic_on s"   and "open s"   and "connected s"   and "z \<in> s"   and "(g has_field_derivative g') (at z)"   and "f z \<noteq> 0"   and "g z = 0"   and "g' \<noteq> 0" shows "residue (\<lambda>w. f w / g w) z = f z / g'"
assms_3: shows "src f \<cong> g \<star> f"
rel_refl_strong: fixes x :: "('d, 'e, 'f) gpv"   and R1a :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> results'_gpv x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> outs'_gpv x \<Longrightarrow> R2a z2 z2" shows "rel_gpv R1a R2a x x"
sigma_finite_pair_measure: fixes A :: "'a measure"   and B :: "'b measure" assumes "sigma_finite_measure A"   and "sigma_finite_measure B" shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)"
tauActTauChain: fixes P :: "pi"   and P' :: "pi" assumes "P \<longmapsto> \<tau> \<prec> P'" shows "P \<Longrightarrow>\<^sub>\<tau> P'"
s_UnauthP: fixes v :: "Syntax.term"   and \<pi> :: "Syntax.term list"   and h :: "hash" assumes "value v" shows "\<lless>\<pi>, Unauth (Hashed h v)\<ggreater> P\<rightarrow> \<lless>\<pi> @ [\<lparr>v\<rparr>], v\<ggreater>"
float_lt_trans: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float"   and c :: "('a, 'b) IEEE.float" assumes "is_finite a"   and "is_finite b"   and "is_finite c"   and "a < b"   and "b < c" shows "a < c"
bi_unique_rel_set_lemma: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'a set"   and Y :: "'b set" assumes "bi_unique R"   and "rel_set R X Y" obtains f :: "'a \<Rightarrow> 'b" where "Y = f ` X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
lspasl_d_der: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "heap_sep_algebra.tern_rel plus sep_disj h1 h1 h2" shows "heap_sep_algebra.tern_rel plus sep_disj zero zero zero \<and> h1 = zero \<and> h2 = zero"
assms_1: shows "v \<notin> set (add_effects_of op)"
normalize_quot_eq_iff: fixes b :: "'a"   and d :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<noteq> (0::'a)"   and "d \<noteq> (0::'a)" shows "(normalize_quot (a, b) = normalize_quot (c, d)) = (a * d = b * c)"
simps: fixes x :: "'b"   and S :: "('c \<times> 'd \<times> 'c) set"   and S' :: "('c \<times> 'd \<times> 'c) set" shows "\<^bold>\<upharpoonleft>\<^sub>x (S, S') = (\<Union> (\<upharpoonleft>\<^sub>x ` S), \<Union> (\<upharpoonleft>\<^sub>x ` S'))"
mono_infI: fixes f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" assumes "mono f"   and "mono g" shows "mono (inf f g)"
attractor_set_VVp: fixes v :: "'a"   and p :: "Player"   and w :: "'a"   and W :: "'a set" assumes "v \<in> VV p"   and "v \<rightarrow> w"   and "w \<in> attractor p W" shows "v \<in> attractor p W"
antisym_action_order: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist" shows "antisymp (action_order E)"