hcju/isabelleps huggingface dataset:
minus_add_minus_mat: fixes u :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'b mat"   and w :: "'b mat" assumes "u \<in> carrier_mat nr nc"   and "v \<in> carrier_mat nr nc"   and "w \<in> carrier_mat nr nc" shows "u - (v + w) = u - v - w"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and s :: "'a stream" shows "(x # xs) @- s = x ## xs @- s"
times_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "times_hom hom \<equiv> \<forall>x y. hom (x * y) = hom x * hom y"
SOME_Basis: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(SOME i. i \<in> Basis) \<in> Basis"
select_convs_9: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getWatch2 \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getWatch2"
two_lists_cases_snoc_Cons': fixes as :: "'b list"   and bs :: "'c list" obtains "as = []" and "bs = []"   | y :: "'c" and ys :: "'c list" where "as = []" and "bs = y__ # ys__"   | xs :: "'b list" and x :: "'b" where "as = xs__ @ [x__]" and "bs = []"   | xs :: "'b list" and x :: "'b" and y :: "'c" and ys :: "'c list" where "as = xs @ [x]" and "bs = y # ys"
sum_squares_ge_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "less_eq zero (plus (times x x) (times y y))"
merge_empty: fixes f :: "'a \<Rightarrow> 'm" shows "merge cempty f = return cempty"
unat_split_asm: fixes P :: "nat \<Rightarrow> bool"   and x :: "'b word" shows "P (unat x) = (\<nexists>n. word_of_nat n = x \<and> n < 2 ^ LENGTH('b) \<and> \<not> P n)"
AllN: shows "\<forall>var L. freeIn var \<phi> \<longrightarrow> (\<exists>x. eval \<phi> (L[var := x])) = (\<forall>x. eval \<phi> (L[var := x]))"
set_sel_2: fixes a :: "'c llist"   and x :: "'c" assumes "\<not> lnull a"   and "x \<in> lset (ltl a)" shows "x \<in> lset a"
select_convs_2: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" shows "\<one>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
card_of_ordLeq_infinite: fixes A :: "'f set"   and B :: "'g set" assumes "|A| \<le>o |B|"   and "infinite A" shows "infinite B"
norm_Pair: fixes a :: "'b"   and b :: "'c" shows "norm (a, b) = sqrt ((norm a)\<^sup>2 + (norm b)\<^sup>2)"
rt_fresherI2: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> kD rt1"   and "dip \<in> kD rt2"   and "nsqn rt1 dip = nsqn rt2 dip"   and "the (dhops rt2 dip) \<le> the (dhops rt1 dip)" shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"
L\<^sub>1_p: shows "L\<^sub>1 \<subseteq> C\<^sub>1 \<and> L\<^sub>1 \<cdot>\<^sub>l\<^sub>s \<eta> = L\<^sub>1' \<and> C\<^sub>1 - L\<^sub>1 \<cdot>\<^sub>l\<^sub>s \<eta> = C\<^sub>1' - L\<^sub>1'"
finite_Fvars_tm: fixes t :: "tm" shows "finite (Fvars t)"
bind_returns_heap_E4: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and h'' :: "'a"   and h' :: "'a" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h''"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'" obtains x :: "'d" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h' \<turnstile> g x \<rightarrow>\<^sub>h h''"
wagner_2_var: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (y \<cdot> x)\<^sup>\<omega> \<le> (x \<cdot> y)\<^sup>\<omega>"
simps_5: fixes P :: "'a prog"   and C :: "String.literal" shows "is_type P (Class C) = is_class P C"
l: fixes sh :: "('a, 'b, 'c, 'd, 'e) system_state"   and p :: "'c"   and ls' :: "('a, 'b, 'd, 'e) com list \<times> 'b option \<times> 'e" assumes "reachable_state sys sh"   and "P sh"   and "GST sh p \<rightarrow>\<^bsub>\<tau>\<^esub> ls'" shows "P \<lparr>GST = (GST sh)(p := ls'), HST = HST sh\<rparr>"
\<omega>consistent_def: shows "\<omega>consistent \<equiv> \<forall>\<phi>\<in>fmla. \<forall>x\<in>var. Fvars \<phi> = {x} \<longrightarrow> (\<forall>n\<in>num. prv (neg (subst \<phi> n x))) \<longrightarrow> \<not> prv (neg (neg (exi x \<phi>)))"
pre_pure_rule': fixes b :: "bool"   and f :: "'b Heap"   and Q :: "'b \<Rightarrow> assn" shows "(\<not> <\<up> b> f <Q>) = (\<not> <emp> f <Q> \<and> b)"
dist_fst_le: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "dist (fst x) (fst y) \<le> dist x y"
from_nat_eq_imp_eq: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and x :: "nat"   and xa :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.from_nat Abs x = mod_type.from_nat Abs xa"   and "x < CARD('d)"   and "xa < CARD('d)" shows "x = xa"
catch_gpv_def: fixes gpv :: "('f, 'g, 'h) gpv" shows "catch_gpv gpv = TRY map_gpv Some id gpv ELSE Generative_Probabilistic_Value.Done None"
IH1: assumes "Vagree \<nu> \<nu>' (FVDiff t1)" shows "frechet I t1 (fst \<nu>) (snd \<nu>) = frechet I t1 (fst \<nu>') (snd \<nu>')"
axioms_2: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "H \<lhd> G" shows "Group.group G"
sym_diff: fixes A :: "('b \<times> 'b) set"   and B :: "('b \<times> 'b) set" assumes "sym A"   and "sym B" shows "sym (A - B)"
fold_congs_3: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "l r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "l_update f r = l_update f' r'"
reach_def: fixes \<Sigma> :: "'b set"   and \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a" shows "reach \<Sigma> \<delta> q\<^sub>0 = {run \<delta> q\<^sub>0 w n |w n. range w \<subseteq> \<Sigma>}"
Zero_Mem_sf: fixes i :: "name" shows "Sigma_fm (Zero IN Var i)"
bisim_invD2: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and ta2 :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and s2' :: "'x2 \<times> 'm2" assumes "bisim_inv"   and "t \<turnstile> s1 \<approx> s2"   and "t \<turnstile> s2 -2-ta2\<rightarrow> s2'" shows "\<exists>s1'. t \<turnstile> s1' \<approx> s2'"
ListAassign: fixes upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and x1 :: "char list"   and x2 :: "aexp"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "preList upds (x1 ::= x2) l s = postList upds l (s[x2/x1])"
intro: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma> {c} \<Gamma>'"   and "mds_consistent mds \<Gamma>"   and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2" shows "\<langle>c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>'\<^esub> \<langle>c, mds, mem\<^sub>2\<rangle>"
distinct_zipI1: fixes xs :: "'a list"   and ys :: "'b list" assumes "distinct xs" shows "distinct (zip xs ys)"
set_pmf_of_multiset: fixes M :: "'b multiset" assumes "M \<noteq> {#}" shows "set_pmf (pmf_of_multiset M) = set_mset M"
simps_12: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v2 :: "'a" shows "local.less_aexp_aux (Plus e1 e2) (V v2) = False"
ifmi_saneI: fixes s :: "bdd"   and v :: "nat"   and ni1 :: "nat"   and ni2 :: "nat"   and ni :: "nat"   and s' :: "bdd" assumes "bdd_sane s"   and "ifmi v ni1 ni2 s = (ni, s')" shows "bdd_sane s'"
trans: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b"   and "b \<subseteq># c" shows "a \<subseteq># c"
partial_function_definitions_def: fixes leq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lub :: "'f set \<Rightarrow> 'f" shows "partial_function_definitions leq lub \<equiv> ((\<forall>x. leq x x) \<and> (\<forall>x y z. leq x y \<longrightarrow> leq y z \<longrightarrow> leq x z)) \<and> (\<forall>x y. leq x y \<longrightarrow> leq y x \<longrightarrow> x = y) \<and> (\<forall>A x. Complete_Partial_Order.chain leq A \<longrightarrow> x \<in> A \<longrightarrow> leq x (lub A)) \<and> (\<forall>A z. Complete_Partial_Order.chain leq A \<longrightarrow> (\<forall>x. x \<in> A \<longrightarrow> leq x z) \<longrightarrow> leq (lub A) z)"
Cod_norm: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "monoidal_language.Cod C (free_monoidal_category.norm C t) = monoidal_language.Cod C t"
HInv2_def: fixes s :: "state" shows "HInv2 s = (Inv2a s \<and> Inv2b s \<and> Inv2c s)"
Un_tables_def: fixes ts :: "('c \<Rightarrow> 'd set) set" shows "Un_tables ts = (\<lambda>k. \<Union>t\<in>ts. t k)"
summable_arctan_series: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "summable (\<lambda>k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))"
pxs_simp: shows "pos xs = (\<forall>e. e \<in> set xs \<longrightarrow> 0 < e)"
nn_integral_integrable_on: fixes f :: "'c \<Rightarrow> real" assumes "f \<in> borel_measurable borel"   and "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel < \<infinity>" shows "f integrable_on UNIV"
DAnotTL: fixes xs :: "('a, 'b) Combinators list" assumes "xs \<noteq> []"   and "wellformed_policy1 (xs @ [DenyAll])" shows "False"
poly_of_vec_add: fixes a :: "'b vec"   and b :: "'b vec" assumes "dim_vec a = dim_vec b" shows "poly_of_vec (a + b) = poly_of_vec a + poly_of_vec b"
size_eq_find_index_conv: fixes xs :: "'b list"   and n :: "nat"   and P :: "'b \<Rightarrow> bool" assumes "length xs = n" shows "(n = find_index P xs) = (\<forall>x\<in>set xs. \<not> P x)"
lockstep_to_plus_forward_simulation: fixes match :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and step1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s1' :: "'a"   and step2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s1 :: "'a"   and s2 :: "'b" assumes "\<And>s1 s2 s2'. \<lbrakk>match s1 s2; step1 s1 s1'\<rbrakk> \<Longrightarrow> \<exists>s2'. step2 s2 s2' \<and> match s1' s2'"   and "match s1 s2"   and "step1 s1 s1'" shows "\<exists>s2'. step2\<^sup>+\<^sup>+ s2 s2' \<and> match s1' s2'"
le_less_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<le> y"   and "y < z" shows "x < z"
succsr_is_node: fixes x :: "nat"   and y :: "nat" assumes "(x, y) \<in> (succsr (rquot_succs A n))\<^sup>*"   and "dfa_is_node A x" shows "dfa_is_node A y"
abs_INV_trans_transfer: fixes R :: "('a \<times> 'b) set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme"   and I :: "'a set"   and J :: "'a set" assumes "PO_refines R Ta Tc"   and "{I} TS.trans Ta {> J}" shows "{R `` I} TS.trans Tc {> R `` J}"
triangular_imp_diagonal_1x2: fixes A :: "'b mat" assumes "A \<in> carrier_mat 1 2"   and "admits_triangular_reduction A" shows "admits_diagonal_reduction A"
rep_mult_rep_non_zero: fixes p :: "proj2"   and A :: "cltn2" shows "proj2_rep p v* cltn2_rep A \<noteq> 0"
H_def: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "H p x q = (tp \<cdot> x \<le> x \<cdot> tq)"
\<tau>move2_iff: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" shows "\<tau>move2 P h stk e pc xcp = (pc < length (compE2 e) \<and> (xcp = None \<longrightarrow> \<tau>instr P h stk (compE2 e ! pc)))"
Grp_apply: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'd"   and x :: "'b"   and y :: "'d" shows "Grp A f x y = (y = f x \<and> x \<in> A)"
\<chi>'_def: shows "\<chi>' = H.lift_character (\<chi>, z)"
linear_conv_bounded_linear: fixes f :: "'c \<Rightarrow> 'd" shows "linear f = bounded_linear f"
square_free_factorization_order_root_no_mem: fixes p :: "'c poly"   and c :: "'c"   and bs :: "('c poly \<times> nat) list"   and x :: "'c" assumes "square_free_factorization p (c, bs)"   and "p \<noteq> 0"   and "\<And>a i. (a, i) \<in> set bs \<Longrightarrow> poly a x \<noteq> (0::'c)" shows "order x p = 0"
converseD: fixes a :: "'d"   and b :: "'e"   and r :: "('e \<times> 'd) set" assumes "(a, b) \<in> r\<inverse>" shows "(b, a) \<in> r"
decompose_prod_root_unity_def: fixes p :: "'a poly" shows "decompose_prod_root_unity p = decompose_prod_root_unity_main p (degree p)"
\<phi>_eq: shows "d\<^sub>0e\<^sub>0.\<phi> = (\<lambda>\<mu>. if \<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright> then ((e\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[d\<^sub>1, cod \<mu> \<star> e\<^sub>0, d\<^sub>0]) \<cdot> \<a>[e\<^sub>1, d\<^sub>1, (cod \<mu> \<star> e\<^sub>0) \<star> d\<^sub>0] \<cdot> ((e\<^sub>1 \<star> d\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[cod \<mu>, e\<^sub>0, d\<^sub>0]) \<cdot> (local.inv \<epsilon>\<^sub>1 \<star> cod \<mu> \<star> local.inv \<epsilon>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[cod \<mu> \<star> src d\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[cod \<mu>]) \<cdot> \<mu> else null)"
nzero: shows "\<forall>x. d \<le> x \<and> x < c \<longrightarrow> poly p x \<noteq> 0"
weak_coinduct: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>[X] Q \<and> (Q, P) \<in> X" shows "P \<sim> Q"
GPicard1: fixes S :: "complex set"   and w :: "complex"   and r :: "real"   and Y :: "(complex \<Rightarrow> complex) set"   and X :: "(complex \<Rightarrow> complex) set" assumes "open S"   and "connected S"   and "w \<in> S"   and "0 < r"   and "Y \<subseteq> X"   and "\<And>h. h \<in> X \<Longrightarrow> h holomorphic_on S"   and "\<And>h z. \<lbrakk>h \<in> X; z \<in> S\<rbrakk> \<Longrightarrow> h z \<noteq> 0 \<and> h z \<noteq> 1"   and "\<And>h. h \<in> Y \<Longrightarrow> cmod (h w) \<le> r" obtains B :: "real" and Z :: "complex set" where "0 < B" and "open Z" and "w \<in> Z" and "Z \<subseteq> S" and "\<And>h z. \<lbrakk>h \<in> Y; z \<in> Z\<rbrakk> \<Longrightarrow> cmod (h z) \<le> B"
coincidence_hp_fml: fixes \<alpha> :: "('d, 'e, 'f) hp"   and \<phi> :: "('d, 'e, 'f) formula" shows "(hpsafe \<alpha> \<longrightarrow> coincide_hp' \<alpha>) \<and> (fsafe \<phi> \<longrightarrow> coincide_fml \<phi>)"
assms_1: shows "coincide_on \<sigma> \<sigma>' (vars_of_cl (cl_ecl C))"
appendPath2: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and Ds :: "char list list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C Cs"   and "Subobjs P (last Cs) Ds"   and "last Cs = hd Ds"   and "wf_prog wf_md P" shows "Subobjs P C (Cs @ tl Ds)"
intro: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "EdomArityAnalysis Aexp"   and "SubstArityAnalysis_axioms Aexp" shows "SubstArityAnalysis Aexp"
isInIDo: fixes p :: "('d, 'e) Combinators list"   and s :: "('d, 'e) Combinators" assumes "noDenyAll p"   and "s \<in> set (insertDenies p)" shows "\<exists>!a. s = DenyAllFromTo (first_srcNet a) (first_destNet a) \<oplus> DenyAllFromTo (first_destNet a) (first_srcNet a) \<oplus> a \<and> a \<in> set p"
hom: shows "homotopic_with_canon (\<lambda>x. True) S (sphere (0::'a) 1) (\<lambda>x. (x - a) /\<^sub>R norm (x - a)) (\<lambda>x. (x - b) /\<^sub>R norm (x - b))"
exhaust_4: fixes x :: "4" shows "x = 1 \<or> x = 2 \<or> x = 3 \<or> x = 4"
foreach_def: fixes u :: "'b"   and adjs :: "('v \<times> nat) list"   and Q\<pi> :: "('v \<Rightarrow> enat) \<times> ('v \<Rightarrow> 'b option)" shows "foreach u adjs Q\<pi> = foldr (foreach_body u) adjs Q\<pi>"
rm_vars_subst_apply_ident: fixes t :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and vs :: "'b set" assumes "t \<cdot> \<theta> = t" shows "t \<cdot> rm_vars vs \<theta> = t"
RModuleEnd_axioms_def: fixes M :: "'a set"   and T :: "'a \<Rightarrow> 'a" shows "RModuleEnd_axioms M T \<equiv> T ` M \<subseteq> M"
MM1: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "\<turnstile> F = G" shows "\<turnstile> \<box>F = \<box>G"
not_cong_4321: 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 D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Cong A B C D" shows "\<not> Cong D C B A"
in_ap_setE: fixes x :: "'c"   and f :: "('d \<Rightarrow> 'c) set"   and y :: "'d set" assumes "x \<in> f \<diamondop> y" obtains f' :: "'d \<Rightarrow> 'c" and y' :: "'d" where "x = f' y'" and "f' \<in> f" and "y' \<in> y"
vbrelation_Limit_vsubset_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<AA>\<lparr>Arr\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) \<subseteq>\<^sub>\<circ> Vset \<alpha>" shows "\<FF>\<lparr>ArrMap\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
part_equiv_def: fixes R :: "('a \<times> 'a) set" shows "part_equiv R \<equiv> sym R \<and> trans R"
init_vmsaux_def: shows "init_vmsaux = (\<lambda>_. (0, []))"
le_infI: fixes x :: "'b \<times> 'c"   and a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" assumes "less_eq_prod x a"   and "less_eq_prod x b" shows "less_eq_prod x (inf_prod a b)"
ConsThrow: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e # es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>throw e' # es,s\<^sub>1\<rangle>"
in_set_conv_iff: fixes x :: "'a"   and n :: "nat"   and xs :: "'a list" shows "(x \<in> set (take n xs)) = (\<exists>i<n. i < length xs \<and> xs ! i = x)"
Max_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.Max less_eq \<equiv> semilattice_set.F (ord.max less_eq)"
rquot_DFS_def: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat" shows "rquot_DFS A n \<equiv> wf_dfa A n"
Cauchy_convergent: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'd" assumes "class.complete_space dist uniformity open"   and "uniform_space.Cauchy uniformity X" shows "topological_space.convergent open X"
sep_conj_impl1: fixes P :: "'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "\<And>h. P h \<Longrightarrow> I h"   and "(P \<and>* R) h" shows "(I \<and>* R) h"
induct: fixes P :: "(int poly \<times> nat) list \<Rightarrow> bool"   and a0 :: "(int poly \<times> nat) list" assumes "\<And>xs. \<lbrakk>\<And>x y. \<lbrakk>\<not> length xs \<le> 1; (x, y) = partition_factors xs\<rbrakk> \<Longrightarrow> P x; \<And>x y. \<lbrakk>\<not> length xs \<le> 1; (x, y) = partition_factors xs\<rbrakk> \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P xs" shows "P a0"
one_le_prod: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and I :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "\<And>i. i \<in> I \<Longrightarrow> less_eq one (f i)" shows "less_eq one (comm_monoid_mult.prod times one f I)"
add_zero_right: fixes x :: "'a" assumes "x \<in> V" shows "x + (0::'a) = x"
sum_arb: fixes A :: "'a set"   and x :: "'a"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "finite A"   and "x \<in> A"   and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> g y = h y" shows "sum g A = sum h (A - {x}) + g x"
space_empty_eq_bot: fixes a :: "'c measure" shows "(space a = {}) = (a = bot)"
simps: fixes P :: "real \<Rightarrow> bool"   and a1 :: "real"   and a2 :: "real" shows "finite_Psegments P a1 a2 = ((\<exists>a b. a1 = a \<and> a2 = b \<and> b \<le> a) \<or> (\<exists>s a b. a1 = a \<and> a2 = b \<and> s \<in> {a..<b} \<and> (s = a \<or> P s) \<and> (\<forall>t\<in>{s<..<b}. P t) \<and> finite_Psegments P a s) \<or> (\<exists>s a b. a1 = a \<and> a2 = b \<and> s \<in> {a..<b} \<and> (s = a \<or> P s) \<and> (\<forall>t\<in>{s<..<b}. \<not> P t) \<and> finite_Psegments P a s))"
asymp_equiv_symI: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<sim>[F] g" shows "g \<sim>[F] f"
indis_sym: fixes s :: "Concrete.var \<Rightarrow> nat"   and s' :: "Concrete.var \<Rightarrow> nat" assumes "Example_PL.indisAbbrev s s'" shows "Example_PL.indisAbbrev s' s"
Linear_order_in_diff_Id: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "Linear_order r"   and "a \<in> Field r"   and "b \<in> Field r" shows "((a, b) \<in> r) = ((b, a) \<notin> r - Id)"
\<tau>diverge_trancl_coinduct: fixes X :: "'s2 \<Rightarrow> bool"   and s :: "'s2" assumes "X s"   and "\<And>s. X s \<Longrightarrow> \<exists>s'. s -\<tau>2\<rightarrow>+ s' \<and> (X s' \<or> s' -\<tau>2\<rightarrow> \<infinity>)" shows "s -\<tau>2\<rightarrow> \<infinity>"