hcju/isabelleps huggingface dataset:
vector_space_pair_with_def: fixes plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f"   and scale\<^sub>1 :: "'g \<Rightarrow> 'f \<Rightarrow> 'f"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h"   and scale\<^sub>2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'h" shows "vector_space_pair_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 \<equiv> vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 \<and> vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2"
weak_until_and_left_distrib: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi>\<^sub>1 :: "'b ltln"   and \<phi>\<^sub>2 :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n (\<phi>\<^sub>1 and\<^sub>n \<phi>\<^sub>2) W\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<phi>\<^sub>1 W\<^sub>n \<psi> and\<^sub>n \<phi>\<^sub>2 W\<^sub>n \<psi>"
trans_equiv_less: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "comparator cmp"   and "cmp a b = Equiv"   and "cmp b c = Less" shows "cmp a c = Less"
ennreal_mult_left_cong: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" assumes "a \<noteq> 0 \<Longrightarrow> b = c" shows "a * b = a * c"
defined_offending: fixes G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_graph G"   and "\<not> sinvar G nP" shows "set_offending_flows G nP \<noteq> {}"
I: fixes trn :: "('state, 'act, 'out) trans" assumes "K (srcOf trn)"   and "validTrans trn"   and "\<not> T trn" shows "(\<phi> trn \<longrightarrow> f trn \<noteq> v) \<and> K (tgtOf trn)"
wlp_Loop1: fixes P :: "'b \<Rightarrow> real"   and body :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and G :: "'b \<Rightarrow> bool" assumes "unitary P"   and "nearly_healthy (wlp body)" shows "wlp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P = gfp_exp (\<lambda>Q s. \<guillemotleft> G \<guillemotright> s * wlp body Q s + \<guillemotleft> \<N> G \<guillemotright> s * P s)"
integral_real_bounded: fixes r :: "real"   and M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "0 \<le> r"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M \<le> ennreal r" shows "integral\<^sup>L M f \<le> r"
renamings_apart_renaming: fixes \<rho> :: "'s"   and Cs :: "'a literal multiset list" assumes "\<rho> \<in> set (renamings_apart Cs)" shows "is_renaming \<rho>"
intro: fixes block_collection :: "'b set multiset"   and u_block_size :: "int" assumes "\<And>bl. bl \<in># block_collection \<Longrightarrow> int (card bl) = u_block_size" shows "block_design_axioms block_collection u_block_size"
dom_fmlookup_finite: fixes m :: "'d \<Rightarrow>\<^sub>F 'e" shows "finite (dom (fmlookup m))"
valid_path_def: fixes as :: "'edge list" shows "valid_path as \<equiv> valid_path_aux [] as"
init_fin_\<tau>moves_False: shows "\<tau>multithreaded.init_fin_\<tau>move (\<lambda>_ _ _. False) = (\<lambda>_ _ _. False)"
below_monic_inj: fixes f :: "'a \<Rightarrow> 'b" assumes "below_monic f" shows "inj f"
minor: fixes X :: "V set" assumes "X \<subseteq> elts \<alpha>" shows "ordertype X VWF = \<alpha> \<or> ordertype (elts \<alpha> - X) VWF = \<alpha>"
poincare_line_cvec_cmat_AeqD: fixes z :: "complex \<times> complex"   and w :: "complex \<times> complex"   and A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex" assumes "poincare_line_cvec_cmat z w = (A, B, C, D)" shows "A = D"
axioms_2: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e" assumes "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str" shows "presheaf_of_rings_axioms is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
increasing_measurable_info: fixes f :: "'a \<Rightarrow> 'e"   and s :: "'b"   and N :: "'e measure"   and t :: "'b" assumes "f \<in> F s \<rightarrow>\<^sub>M N"   and "s \<le> t" shows "f \<in> F t \<rightarrow>\<^sub>M N"
simps_1: fixes x :: "'c"   and y :: "'c"   and n :: "nat" shows "univariate_power_list x [(y, n)] = (if x = y then Some n else None)"
simps_15: fixes va :: "'b formula" shows "safe_formula (Neg (Neg va)) = False"
simps_6: fixes x :: "nat"   and a :: "floatarith"   and vs :: "real list" shows "isDERIV x (Arctan a) vs = isDERIV x a vs"
app_3: shows "length \<Gamma>1 = length \<Gamma>2"
mWHILE_unfold: fixes m :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres" shows "m \<bind> WHILE cond step = m \<bind> (\<lambda>x. if cond x then step x \<bind> WHILE cond step else RETURN x)"
clop_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "cl_op x \<le> cl_op y"
K: shows "Max {i. i < n \<and> offs (enum xs index key n mi ma) 0 ! i \<le> j \<and> 0 < offs_num n xs index key mi ma i} \<in> {i. i < n \<and> offs (enum xs index key n mi ma) 0 ! i \<le> j \<and> 0 < offs_num n xs index key mi ma i}"
measurable_right: fixes M :: "'d measure" shows "right \<in> tree_sigma M \<rightarrow>\<^sub>M tree_sigma M"
closed_segment_line_iff: fixes i :: "real"   and a :: "'b"   and b :: "'b" assumes "0 < i"   and "a \<noteq> b" shows "{a--{a -- b}\<^bsub>i\<^esub>} = {a -- b}\<^bsub>`{0..i}\<^esub>"
Rep_filter_inject: fixes x :: "'c Filters.filter"   and y :: "'c Filters.filter" shows "(Filters.filter.Rep_filter x = Filters.filter.Rep_filter y) = (x = y)"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "a1 >\<^sub>t a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = ta__" and "a2 = sa__" and "wt ta__ >\<^sub>p wt sa__"   | t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = tb__" and "a2 = sb__" and "wt tb__ \<ge>\<^sub>p wt sb__" and "\<not> head tb__ \<le>\<ge>\<^sub>h\<^sub>d head sb__" and "num_args tb__ = 1" and "\<exists>f\<in>local.ground_heads (head tb__). arity_sym f = 1 \<and> wt_sym f = 0" and "arg tb__ \<ge>\<^sub>t sb__"   | t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = tc__" and "a2 = sc__" and "wt tc__ \<ge>\<^sub>p wt sc__" and "head tc__ >\<^sub>h\<^sub>d head sc__"   | t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "wt t \<ge>\<^sub>p wt s" and "head t = head s" and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)"
in_hom: shows "\<guillemotleft>\<omega> : r \<Rightarrow> t\<guillemotright>"
Calling_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('a, 'b) chan"   and p :: "PrIds" shows "Calling ch p \<equiv> cbit<ch!p> \<noteq> rbit<ch!p>"
is_word_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and s :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.is_word \<TT> s = list_all (CFG.is_terminal \<TT>) s"
well_defined_Tensor: fixes M1 :: "'a list list"   and M2 :: "'a list list" assumes "mat (row_length M1) (length M1) M1 \<and> mat (row_length M2) (length M2) M2" shows "mat (row_length M1 * row_length M2) (length M1 * length M2) (M1 \<otimes> M2)"
invariant: fixes r :: "'s"   and s :: "'s"   and st :: "('a list \<times> 's \<times> 's) list \<times> ('s \<times> 's) list \<times> ('s \<times> 's) set" assumes "invariant r s st"   and "test_invariant st" shows "invariant r s (step_invariant st)"
M_7: fixes vs :: "nat \<Rightarrow> nat"   and x :: "nat"   and vs' :: "nat \<Rightarrow> nat" assumes "vs 0 = x"   and "vs 1 = 0"   and "((2, vs), Suc 0, vs') \<in> (step M)\<^sup>*" shows "x \<in> {}"
alphaAbs_qAbs_iff_ex_qAFresh: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and xs :: "'e"   and x :: "'d"   and xs' :: "'e"   and x' :: "'d"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
inf_top_right: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and top :: "'b::type"   and x :: "'b::type" assumes "class.bounded_semilattice_inf_top (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (top::'b::type)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (top::'b::type) = x"
arr_char: fixes f :: "'b \<times> 'b" shows "B.VV.arr f = (B.arr (fst f) \<and> B.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (snd f))"
orthogonal_flow_of_current: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "('a \<times> 'a) \<times> 'a \<times> 'a \<Rightarrow> ennreal"   and S :: "('a \<times> 'a) set" assumes "web_flow (web_of_network \<Delta>) f"   and "orthogonal_current (web_of_network \<Delta>) f S"   and "separating (web_of_network \<Delta>) S"   and "\<And>e. e \<in> \<^bold>E\<^bsub>\<Delta>\<^esub> \<Longrightarrow> 0 < capacity \<Delta> e"   and "source \<Delta> \<noteq> sink \<Delta>"   and "\<And>x. \<not> edge \<Delta> (sink \<Delta>) x"   and "\<not> edge \<Delta> (source \<Delta>) (sink \<Delta>)" shows "orthogonal \<Delta> (flow_of_current \<Delta> f) (cut_of_sep \<Delta> S)"
tannot: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and bv :: "'b state \<times> (v, v) result"   and t0 :: "t" assumes "evaluate ck env s e bv" shows "evaluate ck env s (Tannot e t0) bv"
hn_refine_def: fixes \<Gamma> :: "assn"   and c :: "'a Heap"   and \<Gamma>' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and m :: "'b nres" shows "hn_refine \<Gamma> c \<Gamma>' R m \<equiv> nofail m \<longrightarrow> <\<Gamma>> c <\<lambda>r. \<Gamma>' * (\<exists>\<^sub>Ax. R x r * \<up> (return x \<le> m))>\<^sub>t"
oneElementSetCharacterization: fixes l :: "'a list"   and a :: "'a" shows "(set l = {a}) = (remdups l = [a])"
mset_s_mul_ext: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool" shows "((mset xs, mset ys) \<in> s_mul_ext {(x, y). snd (f x y)} {(x, y). fst (f x y)}) = fst (mul_ext f xs ys)"
dims_output_deep_model: fixes rs :: "nat list"   and j :: "nat"   and m :: "real mat convnet" assumes "2 \<le> length rs"   and "\<And>r. r \<in> set rs \<Longrightarrow> 0 < r"   and "j < rs ! 0"   and "remove_weights m = deep_model_l rs" shows "dims (tensors_from_net m $ j) = replicate (2 ^ (length rs - 2)) (last rs)"
PO_l2_inv6: shows "reach l2 \<subseteq> l2_inv6"
new_element_shadow_root_opt: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr" shows "h' \<turnstile> get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t new_element_ptr shadow_root_opt \<rightarrow>\<^sub>r None"
closed_connected_component: fixes S :: "'c set"   and x :: "'c" assumes "closed S" shows "closed (connected_component_set S x)"
new_Addr_def: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" shows "new_Addr h \<equiv> if \<exists>a. h a = None then \<lfloor>SOME a. h a = None\<rfloor> else None"
rel_option_iff: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d option"   and y :: "'e option" shows "rel_option R x y = (case (x, y) of (None, None) \<Rightarrow> True | (None, Some a) \<Rightarrow> False | (Some x, None) \<Rightarrow> False | (Some x, Some y) \<Rightarrow> R x y)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.map GF.map \<eta>"
eucl_truncate_down_def: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and 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"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Basis_list :: "'a list"   and eucl_down :: "int \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_down :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_up :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and q :: "nat"   and b :: "'a" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "eucl_truncate_down q b = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (truncate_down q (inner b i)) i) Basis"
inf_top_right: fixes x :: "'a" shows "x \<sqinter> \<top> = x"
obj_simps_2: fixes a :: "'a" assumes "S.obj a" shows "S.src a = a"
hRe_minus: fixes z :: "complex star" shows "hRe (- z) = - hRe z"
P''''Chain: shows "P'''' \<Longrightarrow>\<^sub>\<tau> P'"
Exit_no_return_target: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "targetnode a = Exit" shows "False"
Class_widen_Class: fixes 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"   and 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 D :: "char list" assumes "wf_prog wf_md P"   and "is_class P C" shows "P \<turnstile> Class C \<le> Class D = (P \<turnstile> Path C to D unique)"
h2'_def: shows "h2' \<equiv> \<lambda>z. a * (POWSER g (a * z) * POWSER' f (a * z) - POWSER f (a * z) * POWSER' g (a * z)) / (POWSER g (a * z))\<^sup>2 + Polygamma 1 (1 + z) + Polygamma 1 (1 - z)"
simps_3: fixes \<phi> :: "'b ltl" shows "\<^bold>G (F \<phi>) = \<^bold>G \<phi>"
theSubjectFresh: fixes \<alpha> :: "'a action"   and x :: "name" assumes "\<alpha> \<noteq> \<tau>" shows "x \<sharp> the (subject \<alpha>) = x \<sharp> subject \<alpha>"
dualA_iff: fixes cl :: "'b potype" assumes "PO cl" shows "pset (dual cl) = pset cl"
simps_1: fixes cs :: "'edge list" shows "upd_rev_cs cs [] = cs"
NF_stepI: fixes s :: "'a"   and M :: "('a \<times> ('a, 'b) Op) set"   and vs :: "'b \<Rightarrow> nat" assumes "s \<notin> fst ` M" shows "(s, vs) \<in> NF (step M)"
allDefs_disjoint: fixes n :: "'b"   and g :: "'a"   and m :: "'b" assumes "n \<in> set (\<alpha>n g)"   and "m \<in> set (\<alpha>n g)"   and "n \<noteq> m" shows "allDefs g n \<inter> allDefs g m = {}"
monom_Suc: fixes a :: "'a"   and n :: "nat" shows "Polynomial.monom a (Suc n) = pCons (0::'a) (Polynomial.monom a n)"
merge_monoms_merge_monoms: shows "(merge_monoms, merge_monoms) \<in> monom_rel \<rightarrow> monom_rel \<rightarrow> monom_rel"
finite_ta_RegionsD: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)" assumes "finite_ta A" obtains k :: "'e \<Rightarrow> nat" and v :: "'e \<Rightarrow> nat" and n :: "nat" and x :: "'e" where "Regions' (clk_set A) v n x" and "valid_abstraction A (clk_set A) (\<lambda>x. real (k x))" and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"
insert_ident: fixes x :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "x \<notin> A"   and "x \<notin> B" shows "(insert x A = insert x B) = (A = B)"
insert_seq_hd: fixes start :: "'a option"   and xs :: "('a \<times> 'a option) list" assumes "insert_seq start xs" shows "\<exists>oid. hd xs = (oid, start)"
rep_eq: fixes x :: "('d, 'c) sq_matrix"   and xa :: "'c"   and xb :: "('d, 'c) vec" shows "to_fun (upd_row x xa xb) = (\<lambda>i j. if i = xa then xb $ j else to_fun x i j)"
null_is_valid: fixes bot :: "'d"   and null :: "'d" assumes "class.null bot null" shows "null \<noteq> bot"
simps: fixes callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and c :: "('b, 'e, 'f) gpv"   and s :: "'d" shows "exec_gpv callee c s = the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). exec_gpv callee (c x) y))"
cset_mapM_Some_image: fixes A :: "'c cset" shows "cset_mapM (Some `\<^sub>c A) = Some A"
vec_to_iarray_basis_null_space: fixes A :: "(('d, 'e) vec, 'f) vec" shows "vec_to_iarray ` basis_null_space A = basis_null_space_iarrays (matrix_to_iarray A)"
take_bit_Suc_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "take_bit (Suc n) one = one"
set_eq_subset: fixes A :: "'d set"   and B :: "'d set" shows "(A = B) = (A \<subseteq> B \<and> B \<subseteq> A)"
ground_heads_def: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and ground_heads_var :: "'b \<Rightarrow> 'a set" shows "ground_heads gt_sym arity_sym arity_var ground_heads_var \<equiv> gt_sym gt_sym \<and> ground_heads_axioms arity_sym arity_var ground_heads_var"
quadratic_hensel_binary_def: fixes p :: "int"   and n :: "nat"   and C :: "int poly"   and D :: "int poly"   and H :: "int poly" shows "quadratic_hensel_binary p n C D H = (case euclid_ext_poly_dynamic p D H of (S, T) \<Rightarrow> quadratic_hensel_main C p S T D H n)"
kcomp_klift: fixes f :: "'c \<Rightarrow> 'e set"   and g :: "'e \<Rightarrow> 'd set" shows "f \<circ>\<^sub>K g = bd\<^sub>\<F> g \<circ> f"
sum_eq_0_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "finite F" shows "(comm_monoid_add.sum plus zero f F = zero) = (\<forall>a\<in>F. f a = zero)"
uminus_perm_def: fixes p :: "perm" shows "- p = Abs_perm (inv (Rep_perm p))"
zero_le_divide_1_iff: fixes a :: "'b" shows "((0::'b) \<le> (1::'b) / a) = ((0::'b) \<le> a)"
im_has_bottom: assumes "domain.has_bottom P" shows "codomain.has_bottom (f ` P)"
Units_l_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> Units (add_monoid R)"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus> y = x \<oplus> z) = (y = z)"
mayDiverge_coind: fixes phi :: "('test, 'atom) com \<Rightarrow> 'state \<Rightarrow> bool"   and c :: "('test, 'atom) com"   and s :: "'state" assumes "phi c s"   and "\<And>c s. phi c s \<Longrightarrow> \<exists>c' s'. (c, s) \<rightarrow>c (c', s') \<and> (phi c' s' \<or> mayDiverge c' s')" shows "mayDiverge c s"
IS: fixes m :: "'a" assumes "m \<le> lfp f"   and "m \<le> P" shows "f m \<le> P"
periodic_imp_\<omega>_limit_set: fixes K :: "'a set"   and x :: "'a"   and y :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K"   and "periodic_orbit y"   and "range (flow0 y) \<subseteq> \<omega>_limit_set x" shows "range (flow0 y) = \<omega>_limit_set x"
simps_2: fixes ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" shows "inorder (Node ts t) = inorder_list ts @ inorder t"
sel_1: fixes x :: "complex"   and y :: "complex" shows "Re (x + y) = Re x + Re y"
3: fixes x :: "'a" assumes "x \<in> (\<lambda>x. x /\<^sub>R norm x) ` B" shows "norm x = 1"
retract_of_space_section_map: fixes S :: "'b set"   and X :: "'b topology" shows "(S retract_of_space X) = (S \<subseteq> topspace X \<and> section_map (subtopology X S) X id)"
path_strategies_monotone_Suc: fixes P :: "'a llist"   and n :: "nat" assumes "lset P \<subseteq> S"   and "valid_path P"   and "path_conforms_with_strategy p P well_ordered_strategy"   and "enat (Suc n) < llength P" shows "(path_strategies P $ Suc n, path_strategies P $ n) \<in> r"
None_eq_map_option_iff: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e option" shows "(None = map_option f x) = (x = None)"
successively_Cons: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "successively P (x # xs) = (xs = [] \<or> P x (lead_coeff xs) \<and> successively P xs)"