hcju/isabelleps huggingface dataset:
exec1_split: fixes P :: "instr list"   and c :: "instr list"   and P' :: "instr list"   and i :: "int"   and s :: "(char list \<Rightarrow> int) \<times> int list"   and j :: "int"   and s' :: "(char list \<Rightarrow> int) \<times> int list" assumes "P @ c @ P' \<turnstile> (size P + i, s) \<rightarrow> (j, s')"   and "0 \<le> i"   and "i < size c" shows "c \<turnstile> (i, s) \<rightarrow> (j - size P, s')"
st_3: shows "\<forall>x\<in>i - S. f differentiable at x within i"
is_int_t_def: fixes t :: "t" shows "is_int_t t = (case t of T_f32 \<Rightarrow> False | T_f64 \<Rightarrow> False | _ \<Rightarrow> True)"
comp_fun_idem_semilattice_set_apply: fixes f :: "'b semilattice_set" shows "comp_fun_idem (semilattice_set_apply f)"
d_loc_ax: fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> d y) = d (x \<cdot> y)"
intro: fixes p :: "nat" assumes "2 < p"   and "prime (2 ^ p - 1)" shows "mersenne_prime p"
ln_over_ln_ln_primorial'_asymp_equiv: shows "(\<lambda>k. ln (real (primorial' k)) / ln (ln (real (primorial' k)))) \<sim>[sequentially] real"
inject_4: fixes x4 :: "'c set"   and y4 :: "'c set" shows "(Abs x4 = Abs y4) = (x4 = y4)"
base: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option" shows "accepts_trace e s r []"
l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs \<equiv> type_wf = ShadowRootClass.type_wf \<and> set_shadow_root = set_shadow_root \<and> set_shadow_root_locs = set_shadow_root_locs"
le20: shows "tensor_mat (1\<^sub>m d1) m2 \<le>\<^sub>L tensor_mat (1\<^sub>m d1) (1\<^sub>m d2)"
inv_mult_units: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "Group.monoid G"   and "x \<in> Units G"   and "y \<in> Units G" shows "inv\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> y) = inv\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x"
recursively_enumerable_def: fixes X :: "nat set" shows "recursively_enumerable X \<equiv> X = {} \<or> (\<exists>f. recfn 1 f \<and> Partial_Recursive.total f \<and> X = {the (eval f [x]) |x. x \<in> UNIV})"
swap_sym: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set" shows "X #[x \<and> y]_zs = X #[y \<and> x]_zs"
freshRes: fixes a :: "name"   and P :: "pi" assumes "a \<sharp> P" shows "<\<nu>a>P \<sim>\<^sub>e P"
prj_simps_1: shows "C.arr trg_f.prj\<^sub>1"
cycle_free_antimono: fixes g :: "'b ugraph"   and g' :: "'b ugraph" assumes "edges g \<subseteq> edges g'"   and "cycle_free g'" shows "cycle_free g"
split_min': fixes P :: "'b \<Rightarrow> bool"   and i :: "'b"   and j :: "'b" shows "P (min i j) = ((min i j = i \<longrightarrow> P i) \<and> (min i j = j \<longrightarrow> P j))"
Int_Un_eq_1: fixes S :: "'e set"   and T :: "'e set" shows "S \<inter> T \<union> S = S"
nsetTr2: fixes j :: "nat"   and a :: "nat"   and b :: "nat" assumes "j \<in> nset (Suc a) (Suc b)" shows "j - Suc 0 \<in> nset a b"
LNil_less_LCons: fixes a :: "'b"   and t :: "'b llist" shows "LNil < a ## t"
extensional_set: fixes a :: "'t"   and b :: "'t" assumes "S'.ide a"   and "S'.ide b"   and "S'.set a = S'.set b" shows "a = b"
opnet_connect: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d net_state"   and i :: "nat"   and i' :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set"   and t :: "'d net_state"   and t' :: "'d net_state"   and T :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, s), connect(i, i'), \<sigma>', s') \<in> S"   and "((\<sigma>, t), connect(i, i'), \<sigma>', t') \<in> T" shows "((\<sigma>, SubnetS s t), connect(i, i'), \<sigma>', SubnetS s' t') \<in> opnet_sos S T"
comp_inj: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and g :: "'f \<Rightarrow> 'g"   and C :: "'g set" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "g \<in> B \<rightarrow> C"   and "inj_on g B" shows "inj_on (compose A g f) A"
id_moebius_cmat_def: shows "id_moebius_cmat = eye"
hom_carrier: fixes h :: "'a \<Rightarrow> 'b"   and G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme" assumes "h \<in> hom G H" shows "h ` carrier G \<subseteq> carrier H"
simps_2: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "dfa_of_pf' n (Le ks l) = ineq_dfa n ks l"
ssubseqs_Cons: fixes x :: "'b"   and xs :: "'b list" shows "ssubseqs (x # xs) = (#) x ` ssubseqs xs \<union> ssubseqs xs"
Integer_ge_less_than_def: fixes d :: "int" shows "Integer_ge_less_than d = {(MkI\<cdot>z', MkI\<cdot>z) |z z'. d \<le> z' \<and> z' < z}"
assms_2: shows "\<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) (the_enat \<langle>cid #\<^bsub>enat n\<^esub>inf_llist t\<rangle>)"
nhds_countable: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.first_countable_topology open" obtains X :: "nat \<Rightarrow> 'a set" where "decseq X" and "\<And>n. open (X n)" and "\<And>n. x \<in> X n" and "topological_space.nhds open x = (INF n. principal (X n))"
bexpI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and A :: "('a \<times> 'b) set" assumes "P x y"   and "(x, y) \<in> A" shows "\<exists>(x, y)\<in>A. P x y"
shunt1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<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 x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less_eq (inf x y) z = less_eq x (sup (uminus y) z)"
assms_11: shows "short_ch X \<longrightarrow> Q\<^sub>x \<in> X \<and> Q\<^sub>z \<in> X \<and> (\<forall>Q\<^sub>y\<in>\<E>. [[Q\<^sub>x Q\<^sub>y Q\<^sub>z]] \<longrightarrow> Q\<^sub>y \<in> \<emptyset> Q b)"
l5_12_a: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C" shows "Tarski_neutral_dimensionless.Le Bet Cong A B A C \<and> Tarski_neutral_dimensionless.Le Bet Cong B C A C"
prime_elem_dvd_power_iff: fixes p :: "'a"   and n :: "nat"   and x :: "'a" assumes "prime_elem p"   and "0 < n" shows "(p dvd x ^ n) = (p dvd x)"
num_s_is_min: fixes v :: "'a" assumes "v \<in> verts G"   and "v \<noteq> s"   and "dist v \<noteq> \<infinity>" shows "0 < num v"
mult_unit_dvd_iff: 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 b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times (times a b) c = dvd.dvd times a c"
fps_const_compose: fixes a :: "'b"   and b :: "'b fps" shows "fps_const a oo b = fps_const a"
preserves_dom: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.dom \<a>\<^sub>B[fst f, fst (snd f), snd (snd f)] = B.HoHV (B.VVV.dom f)"
Diagonalize_Inv: fixes t :: "'c local.term" assumes "Can t" shows "\<^bold>\<lfloor>Inv t\<^bold>\<rfloor> = Inv \<^bold>\<lfloor>t\<^bold>\<rfloor>"
lcp_ext_left: fixes z :: "'b list"   and u :: "'b list"   and v :: "'b list" shows "z \<cdot> u \<and>\<^sub>p z \<cdot> v = z \<cdot> (u \<and>\<^sub>p v)"
pmaxsimpchainD_distinct: fixes xs :: "'a set list" assumes "pgallery xs" shows "distinct xs"
elims_3: fixes x :: "'a edge_kind"   and xa :: "'a" assumes "\<not> pred x xa" obtains P :: "'a \<Rightarrow> bool" and s :: "'a" where "x = (P)\<^sub>\<surd>" and "xa = s" and "\<not> P s"
successI: fixes f :: "'b Heap"   and h :: "heap" assumes "execute f h \<noteq> None" shows "success f h"
bot_annir': fixes x :: "'a" shows "x \<cdot> \<bottom> = \<bottom>"
funcsetI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f \<in> A \<rightarrow> B"
sel_3: fixes R :: "('f, 'g, 'h) Resumption.resumption set" assumes "\<not> (\<forall>r\<in>R. Resumption.resumption.is_Done r)" shows "Resumption.resumption.resume (resumption_lub R) = (\<lambda>inp. resumption_lub ((\<lambda>c. c inp) ` Resumption.resumption.resume ` (R \<inter> {r. \<not> Resumption.resumption.is_Done r})))"
rep_eq: fixes x :: "('a \<times> 'b) list" shows "impl_of (bulkload x) = rbt_bulkload x"
valid_refs_inv_dequeue_Mutate: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state"   and m' :: "'e"   and r :: "'h"   and f :: "'f"   and opt_r' :: "'h option"   and ws :: "('f, 'g, 'h) mem_store_action list" assumes "valid_refs_inv s"   and "sys_mem_store_buffers (mutator m') s = mw_Mutate r f opt_r' # ws" shows "valid_refs_inv (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_fields := (obj_fields obj)(f := opt_r')\<rparr>) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(mutator m' := ws)\<rparr>))"
config_rand_distinct: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and init :: "'c list"   and qs :: "'c list" shows "\<forall>x\<in>set_pmf (config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). distinct (fst x) = distinct init"
uedge_commute: fixes a :: "'b"   and b :: "'b" shows "uedge (a, b) = uedge (b, a)"
succ_def: fixes G :: "'a graph"   and v :: "'a" shows "succ G v \<equiv> {v'. (v, v') \<in> edges G}"
HEndPhase1_HInv4a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4a s q"   and "Inv2a s" shows "HInv4a s' q"
ind_one_str_def: fixes V :: "'b set" shows "ind_one_str V \<equiv> one_str (X \<inter> V)"
hyps_1: shows "subdivision (with_proj G, swap_in (parcs G)) (I, rev_I)"
trans_def: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state" shows "local.trans cf = map_pmf (\<lambda>i. if proper (fst cf) then cont_eff cf i else cf) (wt_pmf cf)"
presAtm_def: fixes atm :: "'atom" shows "presAtm atm \<equiv> \<forall>s. s \<approx> aval atm s"
Call: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and n :: "nat"   and a' :: "val"   and s1 :: "state"   and args :: "expr list"   and vs :: "val list"   and s2 :: "state"   and D :: "qtname"   and mode :: "inv_mode"   and statT :: "ref_ty"   and mn :: "mname"   and pTs :: "ty list"   and s3 :: "state"   and s3' :: "state"   and accC :: "qtname"   and v :: "val"   and s4 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s2"   and "D = invocation_declclass G mode (snd s2) a' statT \<lparr>name = mn, parTs = pTs\<rparr>"   and "s3 = init_lvars G D \<lparr>name = mn, parTs = pTs\<rparr> mode a' vs s2"   and "s3' = check_method_access G accC statT mode \<lparr>name = mn, parTs = pTs\<rparr> a' s3"   and "G\<turnstile>s3' \<midarrow>Methd D \<lparr>name = mn, parTs = pTs\<rparr>-\<succ>v\<midarrow>n\<rightarrow> s4" shows "G\<turnstile>Norm s0 \<midarrow>{accC,statT,mode}e\<cdot>mn( {pTs}args)-\<succ>v\<midarrow>n\<rightarrow> set_lvars (locals (snd s2)) s4"
rep_eq: fixes x :: "Vars.var"   and xa :: "Vars.var"   and xb :: "CoCalls" shows "x--xa\<in>xb = ((x, xa) \<in> Rep_CoCall xb)"
si_empty2: fixes X :: "'c event set"   and y :: "'c event"   and t :: "'c event list" shows "setinterleaving ([], X, y # t) = (if y \<in> X then {} else {y # u |u. u setinterleaves (([], t), X)})"
list_all2_fusion_def: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and g1 :: "('e, 'g) generator"   and g2 :: "('f, 'h) generator"   and s1 :: "'g"   and s2 :: "'h" shows "list_all2_fusion P g1 g2 s1 s2 = list_all2 P (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
brI: fixes a :: "'d"   and \<alpha> :: "'e \<Rightarrow> 'd"   and c :: "'e"   and I :: "'e \<Rightarrow> bool" assumes "a = \<alpha> c"   and "I c" shows "(c, a) \<in> br \<alpha> I"
assms: shows "flowpipe0 (fst ` X0) x1 x1 aba bba"
sup_impl_inf: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<squnion> b) l\<rightarrow> c = (a l\<rightarrow> c) \<sqinter> (b l\<rightarrow> c)"
pp_of_map_component: fixes f :: "'k \<Rightarrow> 'k"   and v :: "'t" shows "pp_of_term (map_component f v) = pp_of_term v"
seteqI: fixes A :: "'b set"   and B :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B"   and "\<And>x. x \<in> B \<Longrightarrow> x \<in> A" shows "A = B"
BinOp1Reds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and bop :: "bop"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e \<guillemotleft>bop\<guillemotright> e\<^sub>2,s,b\<rangle> \<rightarrow>* \<langle>e' \<guillemotleft>bop\<guillemotright> e\<^sub>2,s',b'\<rangle>"
log_mult_ext3: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "0 \<le> x"   and "0 < y"   and "0 < z" shows "x * log b (x * y * z) = x * log b (x * y) + x * log b z"
eval_Lunit_Ide: fixes t :: "'a local.term" assumes "Ide t" shows "\<lbrace>\<^bold>\<l>\<^bold>[t\<^bold>]\<rbrace> = \<l>[\<lbrace>t\<rbrace>]"
VectorSpaceHom: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "FGModuleHom G smult V smult' T" shows "VectorSpaceHom (aezfun_scalar_mult.fsmult smult) V (aezfun_scalar_mult.fsmult smult') T"
simps_1: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "pre SKIP Q = (\<lambda>s. eSuc (Q s))"
simps: fixes a :: "object atom formula list"   and d :: "object atom formula list"   and s :: "object atom formula set" shows "apply_effect (Effect a d) s = s - set d \<union> set a"
field_differentiable_compose: fixes f :: "'a \<Rightarrow> 'a"   and z :: "'a"   and g :: "'a \<Rightarrow> 'a" assumes "f field_differentiable at z"   and "g field_differentiable at (f z)" shows "g \<circ> f field_differentiable at z"
cond_imp_distr: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and R :: "'b \<times> 'b \<Rightarrow> bool"   and S :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<longrightarrow> Q \<triangleleft> b \<triangleright>  R \<longrightarrow> S) = ((P \<triangleleft> b \<triangleright>  R) \<longrightarrow> (Q \<triangleleft> b \<triangleright>  S))"
II1: fixes \<sigma> :: "'b" assumes "I {} \<sigma>" shows "P \<sigma>"
card_noteq_1_has_more: fixes ES :: "('b \<times> 'c) set"   and X :: "'b"   and xrhs :: "'c" assumes "Cond ES"   and "(X, xrhs) \<in> ES"   and "finite ES" shows "\<exists>(Y, yrhs)\<in>ES. (X, xrhs) \<noteq> (Y, yrhs)"
col_line_eq_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points" assumes "A \<noteq> B"   and "B \<noteq> C"   and "col A B C" shows "line A B = line B C"
same_carrier: fixes C :: "'c set"   and A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "finite C"   and "A \<subseteq> C"   and "B \<subseteq> C"   and "\<And>a. a \<in> C - A \<Longrightarrow> g a = \<^bold>1"   and "\<And>b. b \<in> C - B \<Longrightarrow> h b = \<^bold>1" shows "(F g A = F h B) = (F g C = F h C)"
next_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<circle>P = (\<lambda>\<sigma>. P (\<sigma> |\<^sub>s 1))"
t_1: shows "t \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"
ordered_map_def: fixes S :: "('c \<times> 'd) set" shows "ordered_map S = map (get_map S) (sorted_list_of_set (fst ` S))"
project_set_I: fixes z :: "'c"   and C :: "'c set" assumes "z \<in> C" shows "f z \<in> project_set h C"
acyclic_3a_spannable_implies_6: fixes x :: "'a" assumes "acyclic_3a x"   and "spannable x"   and "symmetric x" shows "acyclic_6 x"
wf_sig_imp_exe_wf_sig': fixes \<Sigma> :: "exesignature" assumes "exe_sig_conds \<Sigma>"   and "wf_sig (translate_signature \<Sigma>)" shows "exe_wf_sig \<Sigma>"
cr_envT_stateT_simps: fixes s :: "'f"   and M :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" shows "cr_envT_stateT s M (EnvT f) (StateT g) = M (f s) (g s)"
prems_3: shows "sequence_result (Rval y # ys) = Rval vs"
rel_filter_neg_distr_cond'_def: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C' :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_filter_neg_distr_cond' C C' = (left_total C \<and> right_unique C \<or> right_total C' \<and> left_unique C')"
delta_interpretation: fixes \<delta> :: "'c"   and def :: "'c" assumes "(0::'c) < \<delta>"   and "\<delta> \<le> def" shows "SN_strict_mono_ordered_semiring_1 def (delta_gt \<delta>) delta_mono"
Domain_mono: fixes r :: "('b \<times> 'c) set"   and s :: "('b \<times> 'c) set" assumes "r \<subseteq> s" shows "Domain r \<subseteq> Domain s"
ell2norm_xl: fixes \<epsilon> :: "real"   and n :: "nat" assumes "I \<epsilon> \<le> n"   and "0 < \<epsilon>" shows "ell2_norm (\<lambda>a. x n a - l a) \<le> \<epsilon>"
simps_19: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and f4 :: "('c, 'd) Combinators \<Rightarrow> ('c, 'd) Combinators \<Rightarrow> 'b"   and x41 :: "('c, 'd) Combinators"   and x42 :: "('c, 'd) Combinators" shows "(case x41 \<oplus> x42 of DenyAll \<Rightarrow> f1 | DenyAllFromTo x xa \<Rightarrow> f2 x xa | AllowPortFromTo x xa xb \<Rightarrow> f3 x xa xb | x \<oplus> xa \<Rightarrow> f4 x xa) = f4 x41 x42"
hyps: fixes \<Pi> :: "(interface, 'a) chain"   and p1 :: "assertion"   and p2 :: "assertion" assumes "length cs = chainlen \<Pi>"   and "p1 = asn (pre \<Pi>)"   and "p2 = asn (post \<Pi>)"   and "\<And>i. i < chainlen \<Pi> \<Longrightarrow> prov_triple (asn (fst3 (nthtriple \<Pi> i)), cs ! i, asn (thd3 (nthtriple \<Pi> i)))" shows "prov_triple (p1, foldr (;;) cs Skip, p2)"
disc_1: fixes x1 :: "'a" shows "is_dyn_operand (OpDyn x1)"
gSubstInp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'b, 'i, 'j) Recursion.model_scheme"   and ys :: "'e"   and Y' :: "('c, 'd, 'e, 'h, 'g) qTerm set"   and Y :: "'b"   and y :: "'h"   and inp' :: "'a \<Rightarrow> ('c, 'd, 'e, 'h, 'g) qTerm set option"   and inp :: "'a \<Rightarrow> 'b option" shows "gSubstInp MOD ys Y' Y y inp' inp \<equiv> lift2 (gSubst MOD ys Y' Y y) inp' inp"
crel_vs1_executeD: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'g"   and b :: "'h Heap"   and heap :: "heap" assumes "crel_vs1 R a b"   and "P heap"   and "state_dp_consistency.cmem heap" obtains x :: "'h" and heap' :: "heap" where "execute b heap = Some (x, heap')" and "P heap'" and "state_dp_consistency.cmem heap'" and "R a x"
is_AddrE: fixes v :: "'b val" assumes "is_Addr v" obtains a :: "'b" where "v = Addr a"
assms_6: shows "\<^bold>\<lfloor>v\<^bold>\<rfloor> = \<^bold>\<lfloor>w\<^bold>\<rfloor>"
tendsto_const: fixes "open" :: "'b set \<Rightarrow> bool"   and k :: "'b"   and F :: "'d filter" assumes "class.topological_space open" shows "LIM x F. k :> topological_space.nhds open k"
le_supI: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<le> x"   and "b \<le> x" shows "max a b \<le> x"
cdiv: fixes c :: "'a"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "O[F](\<lambda>x. f x / c) = O[F](f)"
almost_full_on_finite_subsetE: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set" assumes "reflp P"   and "almost_full_on P S" obtains T :: "'c set" where "finite T" and "T \<subseteq> S" and "\<And>s. s \<in> S \<Longrightarrow> \<exists>t\<in>T. P t s"
partition_filter_conv: fixes f :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "partition f xs = (filter f xs, filter (Not \<circ> f) xs)"
fds_of_real_norm: shows "fds_of_real (fds_norm f) = f"
distinct_7: shows "hp_Idle \<noteq> hp_IdleMarkSweep"
assms: shows "v \<in> \<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S))"
Rep_matrix_inject: fixes x :: "'b matrix"   and y :: "'b matrix" shows "(Rep_matrix x = Rep_matrix y) = (x = y)"
rad_of_strict_monoI: fixes x :: "real"   and y :: "real" assumes "x < y" shows "rad_of x < rad_of y"
Vagree_stateinterpol: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and S :: "variable set" shows "Vagree (stateinterpol \<nu> \<omega> S) \<nu> S"
inf_top_right: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and top :: "'d"   and x :: "'d" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "inf x top = x"
a_get_disconnected_nodes_def: fixes document_ptr :: "(_) document_ptr" shows "local.a_get_disconnected_nodes document_ptr = get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes"
not_zero_j_A: shows "\<not> is_zero_row j A"
paths_injs_simps: fixes t :: "'b ttree"   and t' :: "'b ttree" shows "(paths t = paths t') = (t = t')"
assms_1: shows "wf\<^sub>s\<^sub>t X A"
set_rev_mp: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
subsets_are_rows: fixes subsets :: "nat list list"   and signs :: "rat list list" shows "\<forall>i<length subsets. row (alt_matrix_A signs subsets) i = vec (length signs) (\<lambda>j. z (subsets ! i) (signs ! j))"
dvdm_def: fixes f :: "int poly"   and g :: "int poly" shows "pll.dvdm f g = (\<exists>h. pll.eq_m g (f * h))"
eq_Max_iff: fixes A :: "'t set"   and m :: "'t" assumes "finite A"   and "A \<noteq> {}" shows "(m = ord_term_lin.Max A) = (m \<in> A \<and> (\<forall>a\<in>A. a \<preceq>\<^sub>t m))"
Diagonalize_preserves_Can: fixes t :: "'a local.term" assumes "Can t" shows "Can \<^bold>\<lfloor>t\<^bold>\<rfloor>"
funcset_Un_left: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d set"   and C :: "'e set" shows "(f \<in> A \<union> B \<rightarrow> C) = (f \<in> A \<rightarrow> C \<and> f \<in> B \<rightarrow> C)"
vsv_vrrestriction: fixes Aa :: "V"   and A :: "V" shows "vsv ((r \<restriction>\<^sup>l\<^sub>\<circ> Aa) \<restriction>\<^sup>r\<^sub>\<circ> A)"
l8_4: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong B C C'" shows "Tarski_neutral_dimensionless.Per Bet Cong A B C'"