hcju/isabelleps huggingface dataset:
mem_val_w32_8_mem_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and a :: "word36" assumes "mem_equal s1 s2 a" shows "mem_val_w32 8 a s1 = mem_val_w32 8 a s2"
left_neutral: fixes a :: "'a \<Rightarrow> bool" shows "(\<box> \<and>* a) = a"
sin_arctan: fixes x :: "real" shows "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "int \<Rightarrow> 'c"   and f2 :: "nat \<Rightarrow> mon \<Rightarrow> 'c"   and f3 :: "nat \<Rightarrow> mon \<Rightarrow> 'c"   and mon :: "mon" shows "P (case mon of Mc x \<Rightarrow> f1 x | Minj x xa \<Rightarrow> f2 x xa | MX x xa \<Rightarrow> f3 x xa) = ((\<forall>x1. mon = Mc x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. mon = Minj x21 x22 \<longrightarrow> P (f2 x21 x22)) \<and> (\<forall>x31 x32. mon = MX x31 x32 \<longrightarrow> P (f3 x31 x32)))"
no_outstanding_write_to_read_only_memory_nth_update: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and i :: "nat"   and sb :: "'b memref list"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "no_outstanding_write_to_read_only_memory \<S> ts"   and "i < length ts"   and "no_write_to_read_only_memory \<S> sb" shows "no_outstanding_write_to_read_only_memory \<S> (ts[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
case_cong: fixes list :: "'f list"   and list' :: "'f list"   and f1 :: "'g"   and g1 :: "'g"   and f2 :: "'f \<Rightarrow> 'f list \<Rightarrow> 'g"   and g2 :: "'f \<Rightarrow> 'f list \<Rightarrow> 'g" assumes "list = list'"   and "list' = [] \<Longrightarrow> f1 = g1"   and "\<And>x21 x22. list' = x21 # x22 \<Longrightarrow> f2 x21 x22 = g2 x21 x22" shows "(case list of [] \<Rightarrow> f1 | x21 # x22 \<Rightarrow> f2 x21 x22) = (case list' of [] \<Rightarrow> g1 | x21 # x22 \<Rightarrow> g2 x21 x22)"
CK_nf_pos_set: shows "{f. CK_nf_pos f} = {id, I, K, I \<circ> K, K \<circ> I, I \<circ> K \<circ> I, K \<circ> I \<circ> K}"
omega_sum_refine: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> x \<cdot> (x + y)\<^sup>\<star>" shows "(x + y)\<^sup>\<omega> = x\<^sup>\<omega> + x\<^sup>\<star> \<cdot> y\<^sup>\<omega>"
approx_PInf_emeasure_with_finite: fixes W :: "'a set"   and C :: "real" assumes "W \<in> sets M"   and "emeasure M W = \<infinity>" obtains Z :: "'a set" where "Z \<in> sets M" and "Z \<subseteq> W" and "emeasure M Z < \<infinity>" and "ennreal C < emeasure M Z"
less: shows "cs\<^bsup>path \<sigma>\<^esup> j \<prec> cs\<^bsup>path \<sigma>\<^esup> k"
maximal_def: fixes A :: "'g set"   and P :: "'g set \<Rightarrow> bool" shows "maximal A P = (P A \<and> (\<forall>B. A \<subseteq> B \<and> P B \<longrightarrow> B = A))"
min: fixes y' :: "'a" assumes "y' \<in> B" shows "infdist y A \<le> infdist y' A"
orderE: fixes a :: "'a::semilattice_inf"   and b :: "'a::semilattice_inf" assumes "(a::'a::semilattice_inf) \<le> (b::'a::semilattice_inf)" obtains "(a::'a::semilattice_inf) = a \<sqinter> (b::'a::semilattice_inf)"
distinct_23: shows "OclVoid \<noteq> UnlimitedNatural"
ANR_def: fixes S :: "'c set" shows "ANR S \<equiv> \<forall>U S'. S homeomorphic S' \<and> closedin (top_of_set U) S' \<longrightarrow> (\<exists>T. openin (top_of_set U) T \<and> S' retract_of T)"
dims2_def: shows "st2.dims2 = nths dims (vars1 - {Suc k})"
distinct_tl_verts_imp_distinct: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v"   and "distinct (tl (awalk_verts u p))" shows "distinct p"
Parallel_Strong_Soundness_aux_aux: fixes c :: "'b ann_com"   and b :: "'b"   and co :: "'b ann_com option"   and t :: "'b"   and i :: "nat"   and Ts :: "('b ann_com option \<times> 'b set) list"   and j :: "nat" assumes "(Some c, b) -1\<rightarrow> (co, t)"   and "i < length Ts"   and "OG_Hoare.com (Ts ! i) = Some c"   and "\<And>i. i < length Ts \<Longrightarrow> if OG_Hoare.com (Ts ! i) = None then b \<in> OG_Hoare.post (Ts ! i) else b \<in> pre (the (OG_Hoare.com (Ts ! i))) \<and> \<turnstile> the (OG_Hoare.com (Ts ! i))  OG_Hoare.post (Ts ! i)"   and "interfree Ts"   and "j < length Ts \<and> i \<noteq> j" shows "if OG_Hoare.com (Ts ! j) = None then t \<in> OG_Hoare.post (Ts ! j) else t \<in> pre (the (OG_Hoare.com (Ts ! j))) \<and> \<turnstile> the (OG_Hoare.com (Ts ! j))  OG_Hoare.post (Ts ! j)"
interp: fixes x :: "complex"   and y :: "complex" assumes "(x, y) \<in> set (zip (map (unity_root k) (map int [0..<k])) ws)" shows "poly (finite_fourier_poly ws) x = y"
no_await_no_await: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and b :: "'BExp"   and c' :: "('Var, 'AExp, 'BExp) Stmt" assumes "no_await c" shows "c \<noteq> Await b c'"
sg_subset_elem: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and h :: "'e" assumes "Group G"   and "G \<guillemotright> H "   and "h \<in> H" shows "h \<in> carrier G"
ltl_to_generalized_rabin_af\<^sub>\<UU>_wellformed: fixes \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set" assumes "finite \<Sigma>"   and "range w \<subseteq> \<Sigma>" shows "ltl_to_rabin_af_unf \<Sigma> w"
sharing_consistent_all_acquired: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "sharing_consistent \<S> \<O> sb" shows "all_acquired sb \<subseteq> dom \<S> \<union> \<O>"
ESem_subst_same: fixes \<rho> :: "Vars.var \<Rightarrow> 'Value"   and x :: "Vars.var"   and y :: "Vars.var"   and e :: "exp" assumes "\<rho> x = \<rho> y" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e[x::=y] \<rbrakk>\<^bsub>\<rho>\<^esub>"
simps_2: fixes u :: "nat"   and x :: "nat"   and y :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "isPath u ((x, y) # p) v = (u = x \<and> (x, y) \<in> E \<and> isPath y p v)"
diff_diff_eq2: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.group_add minus plus zero uminus" shows "minus a (minus b c) = minus (plus a c) b"
elims: fixes x :: "nterm"   and y :: "name option" assumes "unconst x = y" obtains name :: "name" where "x = Nconst name" and "y = Some name"   | v :: "name" where "x = Nvar v__" and "y = None"   | v :: "name" and va :: "nterm" where "x = (\<Lambda>\<^sub>n va__. vaa__)" and "y = None"   | v :: "nterm" and va :: "nterm" where "x = v $\<^sub>n va" and "y = None"
indep: shows "P.indep_vars (\<lambda>i. borel) (\<lambda>i x. x i) UNIV"
Hide_is_CSP: fixes cs :: "'c set"   and P :: "('c, 'd) action" shows "is_CSP_process (R (\<lambda>(S, S'). \<exists>s. diff_tr S' S = tr_filter (s - tr S) cs \<and> action.relation_of P (S, S'\<lparr>tr := s, ref := ref S' \<union> cs\<rparr>)) ;; action.relation_of Skip)"
fact_ge_self: fixes n :: "nat" shows "n \<le> fact n"
simps_4: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and a :: "'b"   and as :: "'b list"   and b :: "'b"   and bs :: "'b list" shows "lex_ext_unbounded f (a # as) (b # bs) = (let (stri, nstri) = f a b in if stri then (True, True) else if nstri then lex_ext_unbounded f as bs else (False, False))"
build_rel_SPEC_conv: fixes \<alpha> :: "'b \<Rightarrow> 'c"   and I :: "'b \<Rightarrow> bool"   and \<Phi> :: "'c \<Rightarrow> bool" shows "\<Down> (br \<alpha> I) (SPEC \<Phi>) = SPEC (\<lambda>x. I x \<and> \<Phi> (\<alpha> x))"
induct: fixes x1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and x2 :: "('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list"   and x3 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and P :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set \<Rightarrow> ('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list \<Rightarrow> ('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set \<Rightarrow> bool" assumes "T.\<tau>rtrancl3p x1 x2 x3"   and "\<And>s. P s [] s"   and "\<And>s s' tls s'' tl. \<lbrakk>redT s tl s'; \<not> \<tau>move s tl s'; T.\<tau>rtrancl3p s' tls s''; P s' tls s''\<rbrakk> \<Longrightarrow> P s (tl # tls) s''"   and "\<And>s s' tls s'' tl. \<lbrakk>redT s tl s'; \<tau>move s tl s'; T.\<tau>rtrancl3p s' tls s''; P s' tls s''\<rbrakk> \<Longrightarrow> P s tls s''" shows "P x1 x2 x3"
measure_uniform_measure: fixes M :: "'a measure"   and A :: "'a set"   and B :: "'a set" assumes "emeasure M A \<noteq> 0"   and "emeasure M A \<noteq> \<infinity>"   and "B \<in> sets M" shows "Sigma_Algebra.measure (uniform_measure M A) B = Sigma_Algebra.measure M (A \<inter> B) / Sigma_Algebra.measure M A"
closed_sphere: fixes a :: "'d"   and r :: "real" shows "closed (sphere a r)"
all: shows "\<forall>a''. valid_edge a'' \<and> sourcenode a'' = sourcenode a' \<and> intra_kind (kind a'') \<longrightarrow> a'' = a"
subgroup_is_comm_group: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := J\<rparr>)" shows "comm_group (G\<lparr>carrier := J, carrier := H\<rparr>)"
step_5: shows "\<forall>p\<in>set (fst r). \<exists>na\<le>n. (p, na) \<in> derivable R"
Ana_keys_wf: fixes t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and R :: "('fun, 'var) Term.term list"   and k :: "('fun, 'var) Term.term"   and f :: "'fun"   and T :: "('fun, 'var) Term.term list" assumes "Ana t = (K, R)"   and "\<And>g S. Fun g S \<sqsubseteq> t \<Longrightarrow> length S = arity g"   and "k \<in> set K"   and "Fun f T \<sqsubseteq> k" shows "length T = arity f"
Remove_def: fixes ES :: "('b list set \<times> 'b trm set) set"   and X :: "'b list set"   and xrhs :: "'b trm set" shows "Remove ES X xrhs \<equiv> Subst_all (ES - {(X, xrhs)}) X (Arden X xrhs)"
repcs_unbounded_AE_non_loop_end_strong: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and st :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> R_G.cfg_on (abss st)"   and "st \<in> S"   and "alternating cfg" shows "AE \<omega> in MDP.MC.T (repcs st cfg). \<forall>u. (\<forall>c\<in>\<X>. real (k c) < u c) \<longrightarrow> \<not> ev (alw (\<lambda>xs. shd xs = u)) (smap (snd \<circ> state) \<omega>)"
de_morgans_2: fixes a :: "trilean"   and b :: "trilean" shows "\<not>? a \<and>? b = (\<not>? a) \<or>? (\<not>? b)"
m_s2: fixes m :: "'a \<Rightarrow> nat"   and h :: "nat"   and X :: "char list set"   and S1 :: "'a st"   and S2 :: "'a st" assumes "Measure m h"   and "finite X"   and "fun S1 = fun S2 on - X"   and "S1 < S2" shows "Measure1.m_s m S2 X < Measure1.m_s m S1 X"
degeneralize_nodes_finite: fixes A :: "('b, 'a) ngbta" shows "finite (NBTA.nodes (degeneralize A)) = finite (NGBTA.nodes A)"
ssum_map_sinr': fixes g :: "'f \<rightarrow> 'e"   and f :: "'h \<rightarrow> 'g"   and x :: "'f" assumes "g\<cdot>\<bottom> = \<bottom>" shows "ssum_map\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>x) = sinr\<cdot>(g\<cdot>x)"
ideal_explicit2: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "finite S" shows "ring.ideal_generated plus zero uminus times S = {y. \<exists>f. comm_monoid_add.sum plus zero (\<lambda>i. times (f i) i) S = y}"
generate_field_incl: fixes H :: "'a set" assumes "H \<subseteq> carrier R" shows "generate_field R H \<subseteq> carrier R"
is_sublist_in: fixes a :: "'a"   and as :: "'a list"   and vs :: "'a list" assumes "is_sublist (a # as) vs" shows "a \<in> set vs"
in_mktop: fixes x :: "alt"   and y :: "alt"   and L :: "(alt \<times> alt) set"   and z :: "alt" shows "(x <\<^bsub>mktop L z\<^esub> y) = (x \<noteq> z \<and> (if y = z then x \<noteq> y else x <\<^bsub>L\<^esub> y))"
zero_closed: fixes G :: "'a set" assumes "Group G" shows "(0::'a) \<in> G"
div_mult_swap: 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 c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c b" shows "times a (divide b c) = divide (times a b) c"
heap_del_min: fixes insert :: "'b \<Rightarrow> 'b tree \<Rightarrow> 'b tree"   and del_min :: "'b tree \<Rightarrow> 'b tree"   and q :: "'b tree" assumes "Heap insert del_min"   and "heap q" shows "heap (del_min q)"
assms: shows "R[a\<turnstile>c>b] = OrR2 <c>.M e"
P1_a: fixes x :: "'a" shows "x l\<rightarrow> x = (1::'a)"
surj_Sup_pres_bij_on: fixes f :: "'d \<Rightarrow> 'c" assumes "top_pres ((`) f)"   and "Sup_pres f" shows "bij_betw f (range (radj f \<circ> f)) UNIV"
EncodeProposition_def: fixes x :: "\<kappa>"   and p :: "\<o>" shows "x \<^bold>\<Sigma> p \<equiv> \<lparr>A!,x\<rparr> \<^bold>& \<lbrace>x,\<^bold>\<lambda>x. p\<rbrace>"
assms_1: shows "S \<subseteq> \<Union> \<C>"
isomorphic_implies_hpar_4: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and f' :: "'b" assumes "horizontal_composition V H src trg"   and "category.isomorphic V f f'" shows "trg f = trg f'"
invfun_surj: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B" shows "surj_to (f\<inverse>\<^bsub>B,A\<^esub>) B A"
nodes_target: fixes p :: "'a"   and execute :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and enabled :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and initial :: "'a \<Rightarrow> bool"   and r :: "'b list" assumes "p \<in> transition_system_initial.nodes execute enabled initial"   and "transition_system.path execute enabled r p" shows "fold execute r p \<in> transition_system_initial.nodes execute enabled initial"
select_convs_2: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'a" shows "cMassF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cMassF"
analz_Key_image_insert_eq: fixes K :: "nat"   and evs :: "event list"   and KK :: "nat set" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> set_mr" shows "analz (Key ` insert K KK \<union> knows Spy evs) = insert (Key K) (analz (Key ` KK \<union> knows Spy evs))"
unat_mult_lem: fixes x :: "'b word"   and y :: "'b word" shows "(unat x * unat y < 2 ^ LENGTH('b)) = (unat (x * y) = unat x * unat y)"
Units_eq: fixes G :: "('b, 'c) monoid_scheme" assumes "Group.group G" shows "Units G = carrier G"
exact_decompD_zero: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and h' :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U' :: "'x set" assumes "exact_decomp 0 ps"   and "(h, U) \<in> set (ps\<^sub>+)"   and "(h', U') \<in> set (ps\<^sub>+)"   and "poly_deg h = poly_deg h'" shows "(h, U) = (h', U')"
sup_apply: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(f \<squnion> g) x = f x \<squnion> g x"
inj_onI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. \<lbrakk>s2p A x; s2p A y; f x = f y\<rbrakk> \<Longrightarrow> x = y" shows "inj_on f A"
valid_nodes: fixes n :: "'node"   and S :: "'node set" assumes "n \<in> backward_slice S" shows "valid_node n"
assms_4: shows "prv (dsj \<phi>1 \<phi>2)"
map_Inf_subset: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "set (map Inf (F m)) \<subseteq> set (map Inf (F n))"
widens_trans: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and Ss :: "ty list"   and Ts :: "ty list"   and Us :: "ty list" assumes "P \<turnstile> Ss [\<le>] Ts"   and "P \<turnstile> Ts [\<le>] Us" shows "P \<turnstile> Ss [\<le>] Us"
True: shows "poly p a = 0 \<and> p \<noteq> 0"
simps: fixes z :: "'e list"   and W :: "'e list set"   and p :: "'e list" shows "mcp z W p = (prefix p z \<and> p \<in> W \<and> (\<forall>p'. prefix p' z \<and> p' \<in> W \<longrightarrow> length p' \<le> length p))"
k_isNode_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'g"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'h"   and jviewInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'g"   and jviewIncr :: "'b \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'g"   and simf :: "'e Trace \<Rightarrow> 'i"   and simRels :: "'b \<Rightarrow> ('i \<times> 'i) set"   and simVal :: "'i \<Rightarrow> 'c \<Rightarrow> bool"   and simAbs :: "'j \<Rightarrow> 'i set"   and simObs :: "'b \<Rightarrow> 'j \<Rightarrow> 'h"   and simInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'j"   and simTrans :: "'b \<Rightarrow> 'j \<Rightarrow> 'j list"   and simAction :: "'b \<Rightarrow> 'j \<Rightarrow> 'd list"   and aOps :: "('k, 'j, 'd list) MapOps"   and tOps :: "('l, 'j \<times> 'h, 'j) MapOps"   and a :: "'b"   and ec :: "'j" assumes "AlgorithmForAgent jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction aOps tOps" shows "AlgorithmForAgent.k_isNode jkbp envInit envAction envTrans envVal jview simf simAbs a ec \<equiv> simAbs ec \<in> (\<lambda>t. simf ` {t' \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview. jview a t' = jview a t}) ` PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview"
zero: fixes a :: "real" shows "a *\<^sub>R (0::'b) = (0::'b)"
zero_right: fixes a :: "'c \<Rightarrow>\<^sub>L 'b" shows "a *\<^sub>v (0::'c) = (0::'b)"
r_length: fixes e :: "nat" shows "eval r_length [e] \<down>= e_length e"
lemma_3_10_i_iii: fixes H :: "'a set" assumes "H \<in> normalfilters" shows "cong_r H = cong_l H"
wt: shows "Env,empty_dt\<Turnstile>es\<Colon>\<doteq>T"
mds_consistent_def: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and dma :: "'b \<Rightarrow> Sec"   and mds :: "Mode \<Rightarrow> 'b set"   and \<Gamma> :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "sifum_types.mds_consistent dma mds \<Gamma> \<equiv> dom \<Gamma> = {x. dma x = Low \<and> x \<in> mds AsmNoRead \<or> dma x = High \<and> x \<in> mds AsmNoWrite}"
intros: fixes uu :: "nat" shows "Regions.isIntv (Regions.intv.Intv uu)"
pd_node1: fixes y :: "'n"   and x :: "'n" assumes "y pd\<rightarrow> x" shows "y \<in> local.nodes"
type_definition_Domainp: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and T :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "type_definition Rep Abs A"   and "T \<equiv> \<lambda>x y. x = Rep y" shows "Domainp T = (\<lambda>x. x \<in> A)"
must_sync_def: fixes t :: "'t"   and x :: "'x"   and m :: "'m" shows "t \<turnstile> \<langle>x, m\<rangle> \<wrong> = (\<exists>ta x' m' s. t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle> \<and> shr s = m \<and> actions_ok s t ta)"
assms: shows "FL_valid_Tree\<^sub>\<alpha> P t"
initialproc_def: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and p :: "'b" shows "initialproc fg p \<equiv> p = main fg \<or> (\<exists>u v. (u, Spawn p, v) \<in> edges fg)"
parallelE: fixes xs :: "'c list"   and ys :: "'c list" assumes "xs \<parallel> ys" obtains "\<not> prefix xs ys \<and> \<not> prefix ys xs"
Cons_9: shows "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) M"
semiattractive_axioms: shows "semiattractive A (\<lambda>x y. y \<sqsubseteq> x)"
List_Ref: fixes h :: "'c \<Rightarrow> 'c ref"   and a :: "'c"   and as :: "'c list" shows "List h (Ref a) as = (\<exists>bs. as = a # bs \<and> List h (h a) bs)"
interior_injective_linear_image: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "linear f"   and "inj f" shows "interior (f ` S) = f ` interior S"
group_ow_def: fixes U :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b" shows "group_ow U f z inverse \<equiv> semigroup_ow U f \<and> group_ow_axioms U f z inverse"
map_invar_impl_def: fixes R :: "'a \<Rightarrow> ('b \<times> 'c ifex) set"   and m :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b option"   and s :: "'a" shows "bdd_impl_cmp_pre.map_invar_impl R m s = (\<forall>ii ti ei ri. m (ii, ti, ei) = Some ri \<longrightarrow> (\<exists>i t e. (ri, ifex_ite_opt i t e) \<in> R s \<and> (ii, i) \<in> R s \<and> (ti, t) \<in> R s \<and> (ei, e) \<in> R s))"
ivl_disj_un_6: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "{l..<u} \<union> {u} = {l..u}"
closed_halfspace_Re_le: fixes b :: "real" shows "closed {z. Re z \<le> b}"
mset_factors_m_def: fixes F :: "int poly multiset"   and p :: "int poly" shows "mset_factors_m F p \<equiv> F \<noteq> {#} \<and> (\<forall>f. f \<in># F \<longrightarrow> irreducible_m f) \<and> p =m \<Prod>\<^sub># F"
mertens_second_theorem_2: shows "(\<lambda>x. prime_sum_upto (\<lambda>p. 1 / real p) x - ln (ln x) - meissel_mertens) \<in> O(\<lambda>x. 1 / ln x)"
right_gpv_map: fixes f :: "'f \<Rightarrow> 'a"   and g :: "'g \<Rightarrow> 'c"   and gpv :: "('f, 'g, 'e) gpv"   and h :: "'h \<Rightarrow> 'b" shows "right_gpv (map_gpv f g gpv) = map_gpv f (map_sum h g) (right_gpv gpv)"
Max_ge: fixes A :: "'a DBMEntry set"   and x :: "'a DBMEntry" assumes "finite A"   and "x \<in> A" shows "x \<preceq> linordered_monoid.Max A"
atU_union: fixes U :: "'a set"   and c1 :: "'a list multiset"   and c2 :: "'a list multiset" shows "atU U (c1 + c2) = (atU U c1 \<or> atU U c2)"
hom_components_homogeneous: fixes q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "q \<in> hom_components p" shows "homogeneous q"
hypnat_diff_add_inverse2: fixes m :: "nat star"   and n :: "nat star" shows "m + n - n = m"
split_stream_first: fixes A :: "'b set"   and xs :: "'b stream" assumes "A \<inter> sset xs \<noteq> {}" obtains ys :: "'b list" and a :: "'b" and zs :: "'b stream" where "xs = ys @- a ## zs" and "A \<inter> set ys = {}" and "a \<in> A"
carrier_is_subring: shows "subring (carrier (K [X])) (K [X])"
psi_bounds_induct_2: fixes n :: "nat" shows "psi n - psi (n div 2) \<le> real n * ln 2 + (4 * ln (real (if n = 0 then 1 else n)) + 3)"
top_evaluate_not_timeout: fixes env :: "v sem_env"   and s :: "'b state"   and tp :: "top0"   and s' :: "'b state"   and r :: "(v sem_env, v) result" assumes "evaluate_top False env s tp (s', r)" shows "r \<noteq> Rerr (Rabort Rtimeout_error)"
domStart: fixes t :: "'b"   and p1 :: "'b \<Rightarrow> 'c option"   and p2 :: "'b \<Rightarrow> 'c option" assumes "t \<in> dom p1" shows "(p1 \<Oplus> p2) t = p1 t"