hcju/isabelleps huggingface dataset:
ord_resolve: fixes CAs :: "'a literal multiset list"   and n :: "nat"   and Cs :: "'a literal multiset list"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and D :: "'a literal multiset" assumes "length CAs = n"   and "length Cs = n"   and "length AAs = n"   and "length As = n"   and "n \<noteq> 0"   and "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)"   and "\<forall>i<n. AAs ! i \<noteq> {#}"   and "Some \<sigma> = mgu (set_mset ` set (Map2.map2 add_mset As AAs))"   and "eligible S \<sigma> As (D + negs (mset As))"   and "\<forall>i<n. strictly_maximal_wrt (As ! i \<cdot>a \<sigma>) (Cs ! i \<cdot> \<sigma>)"   and "\<forall>i<n. S (CAs ! i) = {#}" shows "ord_resolve S CAs (D + negs (mset As)) AAs As \<sigma> ((\<Sum>\<^sub># (mset Cs) + D) \<cdot> \<sigma>)"
iso_unit\<^sub>P\<^sub>B\<^sub>U: fixes a :: "'a" assumes "weak_unit a" shows "local.iso (\<i>' a)"
comp1: fixes g :: "'e \<Rightarrow> 'a" assumes "bounded_linear g" shows "bounded_bilinear (\<lambda>x. (**) (g x))"
TailIsPrefixList: fixes tail :: "'b list"   and l1 :: "'b list" assumes "tail \<noteq> []" shows "prefixList l1 (l1 @ tail)"
4: shows "t \<sqinter> v \<sqinter> (u \<sqinter> w) = x"
noVal_def: fixes istate :: "'a"   and step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and T :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and K :: "'a \<Rightarrow> bool"   and v :: "'d" shows "BD_Security.noVal istate step \<phi> f T K v \<equiv> \<forall>s a ou s'. BD_Security.reachNT istate step T s \<and> K s \<and> step s a = (ou, s') \<and> \<phi> (Trans s a ou s') \<longrightarrow> f (Trans s a ou s') \<noteq> v"
scaleC_minus1_left: fixes x :: "'b" shows "- 1 *\<^sub>C x = - x"
lea_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LeA D E F" shows "A \<noteq> B \<and> C \<noteq> B \<and> D \<noteq> E \<and> F \<noteq> E"
simps_2: fixes ac :: "'a list"   and x :: "'a" shows "quicksort_acc ac [x] = x # ac"
closure_of_eq_empty: fixes S :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> topspace X" shows "(X closure_of S = {}) = (S = {})"
seq_simps_3: fixes x :: "'b" assumes "x \<noteq> \<bottom>" shows "seq\<cdot>x = ID"
a_get_tag_name_locs_def: fixes element_ptr :: "(_) element_ptr" shows "CD.a_get_tag_name_locs element_ptr \<equiv> {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name)}"
map_bind_spmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'c spmf"   and g :: "'c \<Rightarrow> 'b spmf" shows "map_spmf f (bind_spmf p g) = bind_spmf p (map_spmf f \<circ> g)"
Inf_pres_iso: fixes f :: "'b \<Rightarrow> 'c" assumes "Inf_pres f" shows "mono f"
lformula_RESTR: fixes \<phi> :: "('a, 'k) aformula" assumes "lformula \<phi>" shows "lformula (RESTR \<phi>)"
4: shows "H * (d \<sqinter> - a)\<^sup>T * (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top) \<le> (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top"
arc_def: fixes g :: "real \<Rightarrow> 'a" shows "arc g = (path g \<and> inj_on g {0..1})"
valid_path_split_1: fixes as :: "'edge list"   and as' :: "'edge list" assumes "valid_path (as @ as')" shows "valid_path as"
addrOf_def: fixes pheap :: "pheap" shows "addrOf pheap = (case pheap of pHeap x1 x2 \<Rightarrow> x2)"
CondLowCompositionality: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and c1 :: "com"   and c2 :: "com"   and b :: "expr" assumes "nonInterference \<Gamma> c1"   and "nonInterference \<Gamma> c2"   and "\<Gamma> \<turnstile> b : Low" shows "nonInterference \<Gamma> (if (b) c1 else c2)"
simps_1: fixes P :: "'b \<Rightarrow> bool" shows "takeWhile P [] = []"
arg_min_on_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "ord.arg_min_on less f S = ord.arg_min less f (\<lambda>x. x \<in> S)"
card_decreases: fixes V :: "'a set"   and y :: "'a"   and D :: "'a \<Rightarrow> bool"   and W :: "'a set"   and brk :: "bool" assumes "finite V"   and "y \<notin> V"   and "local.dfs_reachable_invar D V (insert y W) brk" shows "card ((E_of_succ succ)\<^sup>* `` set w\<^sub>0 - insert y V) < card ((E_of_succ succ)\<^sup>* `` set w\<^sub>0 - V)"
p2: shows "{OrdP (Var k)} \<turnstile> (SyntaxN.All si (SyntaxN.All li (SyntaxN.All j (SyntaxN.All j' (SyntaxN.All sj (SyntaxN.All lj (SeqQuoteP (Var i) (Var i') (Var si) (Var li) IMP SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP HaddP (Var li) (Var lj) (Var k) IMP (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j'))))))))))(i'::=Var i')"
df_1: shows "degree (F (i - 1)) \<le> degree (Polynomial.smult (\<alpha> i) (F (i - 2)))"
simple_restrict_some: fixes i :: "nat"   and X :: "'c option list"   and A :: "nat set" assumes "i < length X"   and "i \<in> A" shows "restrict A X ! i = X ! i"
Beta_strong_induct: fixes x1 :: "Lambda_Z.term"   and x2 :: "Lambda_Z.term"   and P :: "'a \<Rightarrow> Lambda_Z.term \<Rightarrow> Lambda_Z.term \<Rightarrow> bool"   and c :: "'a" assumes "x1 \<rightarrow>\<^sub>\<beta> x2"   and "\<And>x t s c. \<lbrakk>{atom x} \<sharp>* c; atom x \<sharp> t\<rbrakk> \<Longrightarrow> P c (App (Abs x s) t) (subst x t s)"   and "\<And>s t u c. \<lbrakk>s \<rightarrow>\<^sub>\<beta> t; \<And>c. P c s t\<rbrakk> \<Longrightarrow> P c (App s u) (App t u)"   and "\<And>s t u c. \<lbrakk>s \<rightarrow>\<^sub>\<beta> t; \<And>c. P c s t\<rbrakk> \<Longrightarrow> P c (App u s) (App u t)"   and "\<And>s t x c. \<lbrakk>{atom x} \<sharp>* c; s \<rightarrow>\<^sub>\<beta> t; \<And>c. P c s t\<rbrakk> \<Longrightarrow> P c (Abs x s) (Abs x t)" shows "P c x1 x2"
induct: fixes x1 :: "ty"   and x2 :: "ty"   and P :: "ty \<Rightarrow> ty \<Rightarrow> bool" assumes "erase_graph x1 x2"   and "\<And>erase_sum. P One One"   and "\<And>erase_sum \<tau>\<^sub>1 \<tau>\<^sub>2. \<lbrakk>erase_graph \<tau>\<^sub>1 (erase_sum \<tau>\<^sub>1); P \<tau>\<^sub>1 (erase_sum \<tau>\<^sub>1); erase_graph \<tau>\<^sub>2 (erase_sum \<tau>\<^sub>2); P \<tau>\<^sub>2 (erase_sum \<tau>\<^sub>2)\<rbrakk> \<Longrightarrow> P (Fun \<tau>\<^sub>1 \<tau>\<^sub>2) (Fun (erase_sum \<tau>\<^sub>1) (erase_sum \<tau>\<^sub>2))"   and "\<And>erase_sum \<tau>\<^sub>1 \<tau>\<^sub>2. \<lbrakk>erase_graph \<tau>\<^sub>1 (erase_sum \<tau>\<^sub>1); P \<tau>\<^sub>1 (erase_sum \<tau>\<^sub>1); erase_graph \<tau>\<^sub>2 (erase_sum \<tau>\<^sub>2); P \<tau>\<^sub>2 (erase_sum \<tau>\<^sub>2)\<rbrakk> \<Longrightarrow> P (Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2) (Syntax.Sum (erase_sum \<tau>\<^sub>1) (erase_sum \<tau>\<^sub>2))"   and "\<And>erase_sum \<tau>\<^sub>1 \<tau>\<^sub>2. \<lbrakk>erase_graph \<tau>\<^sub>1 (erase_sum \<tau>\<^sub>1); P \<tau>\<^sub>1 (erase_sum \<tau>\<^sub>1); erase_graph \<tau>\<^sub>2 (erase_sum \<tau>\<^sub>2); P \<tau>\<^sub>2 (erase_sum \<tau>\<^sub>2)\<rbrakk> \<Longrightarrow> P (Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2) (Syntax.Prod (erase_sum \<tau>\<^sub>1) (erase_sum \<tau>\<^sub>2))"   and "\<And>erase_sum \<alpha> \<tau>. \<lbrakk>erase_graph \<tau> (erase_sum \<tau>); P \<tau> (erase_sum \<tau>)\<rbrakk> \<Longrightarrow> P (Mu \<alpha> \<tau>) (Mu \<alpha> (erase_sum \<tau>))"   and "\<And>erase_sum \<alpha>. P (Alpha \<alpha>) (Alpha \<alpha>)"   and "\<And>erase_sum \<tau>. \<lbrakk>erase_graph \<tau> (erase_sum \<tau>); P \<tau> (erase_sum \<tau>)\<rbrakk> \<Longrightarrow> P (AuthT \<tau>) (erase_sum \<tau>)" shows "P x1 x2"
intro: fixes n :: "nat"   and fs :: "int vec list" assumes "cof_vec_space.lin_indpt_list n (map of_int_hom.vec_hom fs)" shows "fs_int_indpt n fs"
euclidean_size_mod_ring_def: fixes x :: "'a mod_ring" shows "euclidean_size x = (if x = 0 then 0 else 1)"
order_iff: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (a = a \<sqinter> b)"
finite_imageI: fixes F :: "'c set"   and h :: "'c \<Rightarrow> 'd" assumes "finite F" shows "finite (h ` F)"
Int_emptyI: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> False" shows "A \<inter> B = {}"
A: fixes s :: "'n list" assumes "s \<in># csp" shows "\<exists>p. s = [entry fg p] \<and> initialproc fg p"
fun_mono: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and D :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "C \<le> A"   and "B \<le> D" shows "A ===> B \<le> C ===> D"
wf_induct: fixes r :: "('r \<times> 'r) set"   and P :: "'r \<Rightarrow> bool"   and a :: "'r" assumes "wf r"   and "\<And>x. \<forall>y. (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
equal_pdevs_def: fixes a :: "'a pdevs"   and b :: "'a pdevs" shows "equal_class.equal a b = (a = b)"
rep_eq: fixes x :: "'a dense"   and xa :: "'a dense" shows "Rep_dense (x \<squnion> xa) = Rep_dense x \<squnion> Rep_dense xa"
odd_real_root_power_cancel: fixes n :: "nat"   and x :: "real" assumes "odd n" shows "root n (x ^ n) = x"
rw_monom: shows "monom (g (Suc b)) (Suc b) mod f = monom (g (Suc b)) (Suc b)"
finite_field_ops32: fixes pp :: "uint32" assumes "p = int_of_uint32 pp"   and "p \<le> 65535" shows "field_ops (finite_field_ops32 pp) mod_ring_rel32"
obj_is_equivalence_map: fixes a :: "'b" assumes "D.obj a" shows "D.equivalence_map a"
tendsto_sandwich: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and net :: "'d filter"   and h :: "'d \<Rightarrow> 'c"   and c :: "'c" assumes "class.order_topology less_eq less open"   and "\<forall>\<^sub>F n in net. less_eq (f n) (g n)"   and "\<forall>\<^sub>F n in net. less_eq (g n) (h n)"   and "filterlim f (topological_space.nhds open c) net"   and "filterlim h (topological_space.nhds open c) net" shows "filterlim g (topological_space.nhds open c) net"
finite_big_d': fixes y :: "nat list"   and i :: "nat" shows "finite (set (big_d' a b y i))"
ps2: fixes b :: "real"   and a :: "real" assumes "\<not> 4 * b \<le> 3"   and "b \<le> 1"   and "0 \<le> a"   and "a \<le> 1" shows "p ((a - 1) * (4 * b - 3) + 1) \<in> s"
degree_aform_err_arctan_aform_err: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "arctan_aform_err p X = Some Y"   and "degree_aform_err X \<le> n" shows "degree_aform_err Y \<le> n"
abs_e4: shows "\<bar>e4\<bar> \<le> snd (trunc_bound_pdevs p (add_pdevs (fst (trunc_bound_pdevs p (scaleR_pdevs (fst (fst Y)) (snd (fst X))))) (fst (trunc_bound_pdevs p (scaleR_pdevs (fst (fst X)) (snd (fst Y)))))))"
Seq: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and R :: "'e set"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and Q :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c\<^sub>1 R,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Seq c\<^sub>1 c\<^sub>2 Q,A"
prj_simps_3: shows "C.arr doms.Prj\<^sub>0"
spmf_Domainp_rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "Domainp (rel_spmf R) = pred_spmf (Domainp R)"
disc_2: fixes n :: "enat"   and xs :: "'c llist" assumes "n \<noteq> 0"   and "\<not> lnull xs" shows "\<not> lnull (ltake n xs)"
ProblemIkTpartG_GEModel_axioms: shows "ProblemIkTpartG_GEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl eintT eintF eintP"
starfun_lambda_cancel: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and x' :: "'d star" shows "(*f* (\<lambda>h. f (x + h))) x' = (*f* f) (star_of x + x')"
sqrt_approx_main_impl: fixes x :: "'a" assumes "(0::'a) < x" shows "sqrt_approx_main_impl \<epsilon> \<bar>x\<bar> x = sqrt_approx_main x"
lub_sum: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'a" assumes "\<forall>k. f k \<le> x" shows "comm_monoid_add.sum (\<squnion>) bot f {k. True} \<le> x"
infinite_Bpow: fixes r :: "('a \<times> 'a) set"   and A :: "'b set" assumes "Card_order r"   and "Field r \<noteq> {}"   and "infinite A" shows "infinite (Bpow r A)"
finite_tvs: fixes t :: "Core.term" shows "finite (tvs t)"
1: fixes x :: "bool \<times> bool list list"   and xa :: "bool"   and y :: "bool list list" assumes "x = iter M T"   and "(xa, y) = x"   and "xa" shows "P M y"
zero_index: fixes i :: "'c" shows "0 $ i = (0::'b)"
disc_11: fixes x3 :: "nat" shows "\<not> is_REL (HOLD x3)"
subset_closed_def: fixes C :: "'c set set" shows "subset_closed C = (\<forall>S'\<in>C. \<forall>S\<subseteq>S'. S \<in> C)"
ldrop_0: fixes xs :: "'c llist" shows "ldrop 0 xs = xs"
power_commuting_commutes: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and n :: "nat" assumes "class.monoid_mult one times"   and "times x y = times y x" shows "times (power.power one times x n) y = times y (power.power one times x n)"
cSup_eq_Max: fixes X :: "'a set" assumes "finite X"   and "X \<noteq> {}" shows "Sup X = Max X"
real_inprod_cvec: fixes z :: "complex \<times> complex" shows "is_real \<langle>z,z\<rangle>"
word_less_1: fixes x :: "'a Word.word" shows "(x < 1) = (x = 0)"
fx: fixes x :: "'a"   and t :: "real" assumes "x \<in> U"   and "t \<in> {a..b}" shows "((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
Radon_Nikodym_finite_measure: fixes N :: "'a measure" assumes "finite_measure N"   and "sets N = sets M"   and "absolutely_continuous M N" shows "\<exists>f\<in>borel_measurable M. density M f = N"
rep_eq: fixes x :: "('d \<times> 'e) fset" shows "is_fmap x = is_map (fset x)"
symmetric: fixes x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>int\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>int\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(x \<doteq> y) = (y \<doteq> x)"
hoare_tripleD_3: fixes P :: "assn"   and c :: "'d Heap"   and Q :: "'d \<Rightarrow> assn"   and h :: "heap"   and as :: "nat set"   and \<sigma> :: "heap option"   and r :: "'d" assumes "<P> c <Q>"   and "(h, as) \<Turnstile> P"   and "run c (Some h) \<sigma> r" shows "relH {a. a < lim h \<and> a \<notin> as} h (the_state \<sigma>)"
h01: shows "h ` S \<subseteq> ball 0 1"
Joints_ConsD: fixes x :: "'b"   and xs :: "'b list"   and X :: "'b \<times> 'b pdevs"   and XS :: "('b \<times> 'b pdevs) list" assumes "x # xs \<in> Joints (X # XS)" shows "xs \<in> Joints XS"
cospan: shows "trg (tab\<^sub>1 g) = trg (tab\<^sub>0 f)"
target: fixes T1 :: "'a"   and T2 :: "'a"   and TRel :: "('a \<times> 'a) set"   and Enc :: "'b \<Rightarrow> 'a" assumes "(T1, T2) \<in> TRel" shows "(TargetTerm T1, TargetTerm T2) \<in> encoding.indRelLT Enc TRel"
pairs_sup_closed: fixes x :: "'c regular"   and y :: "'c dense Filters.filter"   and z :: "'c regular"   and w :: "'c dense Filters.filter" assumes "(x, y) \<in> stone_phi.pairs"   and "(z, w) \<in> stone_phi.pairs" shows "stone_phi.pairs_sup (x, y) (z, w) \<in> stone_phi.pairs"
prefs_wf'_1: fixes i :: "'a" assumes "i \<in> agents" shows "total_preorder_on alts (R i)"
copy_first_matrix_dim_1: fixes nr :: "nat"   and nc :: "nat" shows "dim_row (copy_first_matrix nr nc) = nr"
dsr4: shows "ad (ad (zero_class.zero::'a)) = (zero_class.zero::'a)"
set_add_comm: fixes R :: "('c, 'd) ring_scheme"   and I :: "'c set"   and J :: "'c set" assumes "ring R"   and "I \<subseteq> carrier R"   and "J \<subseteq> carrier R" shows "I <+>\<^bsub>R\<^esub> J = J <+>\<^bsub>R\<^esub> I"
union_of_empty: fixes P :: "'b set set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> bool" assumes "P {}" shows "(P union_of Q) {}"
simps_4: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Trg (t \<^bold>\<cdot> u) = Trg t"
Group_FGModuleHomSet: fixes smult' :: "('f, 'g) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and W :: "'a set" assumes "FGModule G smult' W" shows "Group (FGModuleHomSet G (\<cdot>) V smult' W)"
comm_semigroup_pair_def: fixes f\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "comm_semigroup_pair f\<^sub>a f\<^sub>b \<equiv> comm_semigroup f\<^sub>a \<and> comm_semigroup f\<^sub>b"
kab_props: shows "kab \<noteq> (0::'a) \<and> dirA = kab ** dirB"
spr_simAction: fixes t :: "'s Trace"   and ec :: "'s odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRsingle.sim_equiv_class a t" shows "set (local.spr_simAction a ec) = set (jAction SPR.MC t a)"
g'_def: shows "g' = (\<lambda>z. - x * y / (1 - (1 - x * y) * z)\<^sup>2)"
remove_child_locs_impl: shows "remove_child_locs = a_remove_child_locs"
same_awalk_by_same_end: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and q :: "('a \<times> 'a) list"   and w :: "'a" assumes "verts3 (with_proj G) \<subseteq> V"   and "V \<subseteq> pverts G"   and "awalk u p v"   and "awalk u q w"   and "hd p = hd q"   and "p \<noteq> []"   and "q \<noteq> []"   and "progressing p"   and "progressing q"   and "v \<in> V"   and "w \<in> V"   and "set (inner_verts p) \<inter> V = {}"   and "set (inner_verts q) \<inter> V = {}" shows "p = q"
AB_2: shows "(cst, cst') \<in> code_status_status_rel"
left_idem: fixes a :: "'c"   and b :: "'c" shows "a \<squnion> (a \<squnion> b) = a \<squnion> b"
imp_sim: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F \<^bold>\<rightarrow> G" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> F \<^bold>\<or> G"
compact_space_alt: fixes X :: "'b topology" shows "compact_space X = (\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> topspace X \<subseteq> \<Union> \<U> \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> topspace X \<subseteq> \<Union> \<F>))"
columnvector_def: fixes v :: "('a, 'c) Finite_Cartesian_Product.vec" shows "columnvector v = (\<chi>i j. v $h i)"
diagonal_step_JNF_dim: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and d :: "'b"   and v :: "'b" assumes "A \<in> carrier_mat m n" shows "diagonal_step_JNF A i j d v \<in> carrier_mat m n"
lemma_3_1: fixes i :: "int"   and j :: "int" assumes "i \<le> j" shows "K i j \<le> K i (j + 1)"
valofd_less_2p: fixes x :: "('a, 'b) IEEE.float" shows "valofd x < 2 / 2 ^ bias TYPE(('a, 'b) IEEE.float)"
boundary_chain_diff: fixes one_chain :: "('a \<times> 'b) set"   and s :: "('a \<times> 'b) set" assumes "boundary_chain one_chain" shows "boundary_chain (one_chain - s)"
uniq_opp: fixes a :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point" assumes "ell_field_class.add a p\<^sub>1 p\<^sub>2 = Infinity" shows "p\<^sub>2 = opp p\<^sub>1"
abs_rangeI: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and s :: "(nat \<times> nat) list"   and v :: "nat"   and a :: "nat" assumes "abs_ast_prob problem"   and "ast_problem.wf_partial_state problem s"   and "(v, a) \<in> set s" shows "ast_problem.abs_range_map problem v \<noteq> None"
set_zip_ACS: shows "set (zip (filter (\<lambda>m. \<not> implc_isIFS m) (get_impl M)) (filter (\<lambda>m. \<not> c_isIFS m) (get_spec M))) \<subseteq> set M"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "dec_after_clear (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> ml + mr = Suc m \<and> ml = Suc m \<and> r \<noteq> [] \<and> r \<noteq> [] \<and> (if lm1 = [] then l = Oc \<up> ml @ Bk # Bk # ires else l = Oc \<up> ml @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> (tl r = Bk # <lm2> @ Bk \<up> rn \<or> tl r = [] \<and> lm2 = []))"
gt_trans: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and u :: "('b, 'a) tm"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "rpo ground_heads_var gt_sym extf arity_sym arity_var"   and "rpo.gt ground_heads_var gt_sym extf u t"   and "rpo.gt ground_heads_var gt_sym extf t s" shows "rpo.gt ground_heads_var gt_sym extf u s"
length_rev: fixes xs :: "'b list" shows "length (rev xs) = length xs"
float_plus_up_float: fixes x :: "real"   and y :: "real"   and p :: "nat" assumes "x \<in> float"   and "y \<in> float" shows "plus_up p x y \<in> float"
defs_uses_disjoint_inv: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)"   and "redundant g" shows "defs g n \<inter> uses' g n = {}"
adopt_node_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> adopt_node_locs parent owner_document document_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "DocumentClass.type_wf h = DocumentClass.type_wf h'"
set_filter: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "set (filter P xs) = {x \<in> set xs. P x}"
poly_mapping_nat_ge_0: fixes m :: "nat \<Rightarrow>\<^sub>0 nat" shows "0 \<le> m"
cases: fixes x :: "'a cstate \<times> auth_query" obtains m :: "'a" where "x = (Store m__, Forward)"   | m :: "'a" where "x = (Store ma__, Drop)"   | m :: "'a" where "x = (Store m, Look)"   | uu_ :: "auth_query" where "x = (Void, uu___)"   | uu_ :: "auth_query" where "x = (cstate.Fail, uua___)"   | v :: "'a" and uu_ :: "auth_query" where "x = (cstate.Collect v__, uu_)"   | "x = (Void, Look)"   | "x = (cstate.Fail, Look)"   | v :: "'a" where "x = (cstate.Collect va__, Look)"   | "x = (Void, Drop)"   | "x = (cstate.Fail, Drop)"   | v :: "'a" where "x = (cstate.Collect v, Drop)"
conj_commute: fixes x :: "'a"   and y :: "'a" shows "x \<^bold>\<squnion> y = y \<^bold>\<squnion> x"
restrictD: fixes G :: "('d, 'e) labeled_graph" assumes "graph G" shows "edges G \<subseteq> {(l, v1, v2). v1 \<in> vertices G \<and> v2 \<in> vertices G}"
vlesspoll_def: fixes A :: "V"   and B :: "V" shows "(A \<prec>\<^sub>\<circ> B) = (A \<lesssim>\<^sub>\<circ> B \<and> \<not> A \<approx>\<^sub>\<circ> B)"
measure_sums_UNION: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and R :: "nat set"   and B :: "nat \<Rightarrow> 'b set" assumes "measure_space M"   and "\<forall>j1\<in>R. \<forall>j2\<in>R. j1 \<noteq> j2 \<longrightarrow> B j1 \<inter> B j2 = {}"   and "\<forall>j\<in>R. B j \<in> measurable_sets M" shows "(\<lambda>n. Measure.measure M (if n \<in> R then B n else {})) sums Measure.measure M (\<Union> (B ` R))"
sint_up_scast: fixes w :: "'b word" assumes "is_up scast" shows "sint (scast w) = sint w"
Min_le_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq (linorder.Min less_eq A) x = (\<exists>a\<in>A. less_eq a x)"
REC_def: fixes B :: "('d \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'c"   and x :: "'d" shows "REC B x \<equiv> if trimono B then lfp B x else top"