hcju/isabelleps huggingface dataset:
sgn_0_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(sgn a = zero) = (a = zero)"
callee_invariant': fixes y :: "bool list"   and s' :: "'a \<Rightarrow> bool list option"   and s :: "'a \<Rightarrow> bool list option"   and x :: "'a" assumes "(y, s') \<in> set_spmf (PRF.random_oracle s x)"   and "finite (dom s)"   and "x \<in> outs_\<I> \<I>_full" shows "finite (dom s') \<and> y \<in> responses_\<I> \<I>_full x"
assms_3: fixes cc :: "'a multiset" assumes "c' = {#s#} + cc"   and "c = {#r'#} + cc"   and "c' - {#s#} = cc"   and "c - {#r'#} = cc" shows "P"
m_closed: fixes x :: "int vec"   and y :: "int vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "x + y \<in> carrier_vec n"
SXAlloc_def: fixes G :: "prog"   and P :: "'b assn" shows "SXAlloc G P = (\<lambda>Y s Z. \<forall>s'. G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s' \<longrightarrow> P Y s' Z)"
arrI: fixes f :: "'s"   and a :: "'s"   and b :: "'s" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "arr f"
assms_1: shows "x ; 1 \<cdot> x\<^sup>T ; 1 \<cdot> y\<^sup>T ; 1 = 0"
rewrite_negated_src_ports: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'b common_primitive match_expr"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m" shows "matches (\<beta>, \<alpha>) (rewrite_negated_src_ports m) a p = matches (\<beta>, \<alpha>) m a p"
almost_full_on_multisets: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "almost_full_on P A" shows "almost_full_on (mulex_on P A)\<^sup>=\<^sup>= (multisets A)"
commute: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and a :: "'g"   and b :: "'g" assumes "abel_semigroup f" shows "f a b = f b a"
mat_extension_def: fixes m :: "'a mat" shows "mat_extension m = tensor_mat m (1\<^sub>m d2)"
prime_gt_3_impl_p_minus_one_not_prime: fixes p :: "nat" assumes "prime p"   and "3 < p" shows "\<not> prime (p - 1)"
real_linear: fixes a :: "'a"   and b :: "'a" assumes "is_real a"   and "is_real b" shows "a < b \<or> a = b \<or> b < a"
Sup_supdistl_iso: fixes f :: "'a \<Rightarrow> 'c" assumes "Sup \<circ> (`) f \<le> f \<circ> Sup" shows "mono f"
1: fixes g :: "'g"   and P :: "'val set" assumes "redundant_set g P" shows "\<exists>scc\<subseteq>P. redundant_scc g P scc"
anormal_nnf: fixes \<phi> :: "'a fm" assumes "qfree \<phi>"   and "normal \<phi>" shows "\<forall>a\<in>atoms (nnf \<phi>). anormal a"
frameEq_1: fixes A\<^sub>F :: "name list"   and \<Psi> :: "'d"   and \<Psi>' :: "'d" shows "(\<langle>A\<^sub>F, \<Psi>\<rangle> = \<langle>\<epsilon>, \<Psi>'\<rangle>) = (A\<^sub>F = [] \<and> \<Psi> = \<Psi>')"
dijkstra_step_preserves_inv: fixes G :: "graph"   and S :: "state"   and m :: "nat" assumes "Dijkstra.inv G S"   and "is_heap_min m (heap S)" shows "Dijkstra.inv G (dijkstra_step G m S)"
itrm_comp: fixes g :: "'a itrm"   and f :: "'a itrm"   and x :: "'a itrm" shows "Pure \<B> \<diamondop> g \<diamondop> f \<diamondop> x \<simeq> g \<diamondop> (f \<diamondop> x)"
diagram_def: fixes ars :: "('c \<times> 'd \<times> 'c) set"   and d :: "('c \<times> ('d \<times> 'c) list) \<times> ('c \<times> ('d \<times> 'c) list) \<times> ('c \<times> ('d \<times> 'c) list) \<times> 'c \<times> ('d \<times> 'c) list" shows "diagram ars d = (let (\<tau>, \<sigma>, \<sigma>', \<tau>') = d in {\<sigma>, \<tau>, \<sigma>', \<tau>'} \<subseteq> seq ars \<and> fst \<sigma> = fst \<tau> \<and> lst \<sigma> = fst \<tau>' \<and> lst \<tau> = fst \<sigma>' \<and> lst \<sigma>' = lst \<tau>')"
perm_is_sort_respecting: fixes f :: "atom \<Rightarrow> atom"   and a :: "atom" assumes "f \<in> perm" shows "sort_of (f a) = sort_of a"
ins_neq_Leaf: fixes k :: "'b"   and p :: "'c"   and t :: "('b \<times> 'c) tree" shows "ins k p t \<noteq> \<langle>\<rangle>"
rep_eq: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and xa :: "'a" shows "mapping_of (monom x xa) = Poly_Mapping.single x xa"
order_greater_zero: fixes f :: "'a perm"   and a :: "'a" shows "0 < order f a"
RID\<^sub>SE: fixes r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option" assumes "r \<in> RID\<^sub>S \<sigma>" obtains l :: "'b" and v :: "('a, 'b, 'c) val" where "\<sigma> l = Some v" and "r \<in> RID\<^sub>V v"
simps_32: shows "ShiftRightSigned \<noteq> Eq"
vars\<^sub>s\<^sub>t_is_fv\<^sub>s\<^sub>t_bvars\<^sub>s\<^sub>t: fixes S :: "('b, 'a) strand_step list" shows "vars\<^sub>s\<^sub>t S = fv\<^sub>s\<^sub>t S \<union> bvars\<^sub>s\<^sub>t S"
clock_repSim_def: shows "clock_repSim \<equiv> snd"
elims_2: fixes x :: "('b, 'c) Combinators list" assumes "noDenyAll1 x" obtains xs :: "('b, 'c) Combinators list" where "x = DenyAll # xs" and "noDenyAll xs"   | "x = []" and "noDenyAll []"   | vb :: "'b" and vc :: "'b" and va :: "('b, 'c) Combinators list" where "x = DenyAllFromTo vb__ vc__ # va__" and "noDenyAll (DenyAllFromTo vb__ vc__ # va__)"   | vb :: "'b" and vc :: "'b" and vd :: "'c" and va :: "('b, 'c) Combinators list" where "x = AllowPortFromTo vba__ vca__ vd # vaa__" and "noDenyAll (AllowPortFromTo vba__ vca__ vd # vaa__)"   | vb :: "('b, 'c) Combinators" and vc :: "('b, 'c) Combinators" and va :: "('b, 'c) Combinators list" where "x = vb \<oplus> vc # va" and "noDenyAll (vb \<oplus> vc # va)"
eval_wlp_Embed: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "wlp (Embed t) = t"
alw_mp: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xs :: "'c stream"   and \<psi> :: "'c stream \<Rightarrow> bool" assumes "alw \<phi> xs"   and "alw (\<lambda>xs. \<phi> xs \<longrightarrow> \<psi> xs) xs" shows "alw \<psi> xs"
vector_add_component: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec"   and i :: "'b" shows "(x + y) $ i = x $ i + y $ i"
simps_2: fixes a :: "'d"   and H :: "'d list"   and iostep :: "'d \<Rightarrow> 'c \<Rightarrow> ('b \<times> 'c) option"   and \<sigma> :: "'c" shows "mbind (a # H) iostep \<sigma> = (case iostep a \<sigma> of None \<Rightarrow> Some ([], \<sigma>) | Some (out, \<sigma>') \<Rightarrow> case mbind H iostep \<sigma>' of None \<Rightarrow> Some ([out], \<sigma>') | Some (outs, \<sigma>'') \<Rightarrow> Some (out # outs, \<sigma>''))"
nlists_empty: fixes n :: "nat" shows "nlists {} n = (if n = 0 then {[]} else {})"
invertibleI: fixes u :: "'a"   and v :: "'a" assumes "u \<cdot> v = \<one>"   and "v \<cdot> u = \<one>"   and "u \<in> N"   and "v \<in> N" shows "sub.invertible u"
inject_8: fixes x91 :: "'f"   and x92 :: "'d set"   and x93 :: "('d, 'e, 'f) com"   and y91 :: "'f"   and y92 :: "'d set"   and y93 :: "('d, 'e, 'f) com" shows "(Guard x91 x92 x93 = Guard y91 y92 y93) = (x91 = y91 \<and> x92 = y92 \<and> x93 = y93)"
finpref_mono2: fixes s :: "'b llist"   and t :: "'b llist"   and A :: "'b set" assumes "s \<le> t" shows "finpref A s \<subseteq> finpref A t"
tuple_eqI: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f0 :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f1 :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "seq (pr0 b c) f"   and "seq (pr1 b c) f"   and "local.comp (pr0 b c) f = f0"   and "local.comp (pr1 b c) f = f1" shows "f = HF'.tuple f1 f0"
InitStatus_def: fixes A :: "('d, 'e, 'f) hierauto" shows "InitStatus A \<equiv> Abs_status (A, InitConf A, {}, HAInitValue A)"
eqvt: fixes x1 :: "trm"   and x2 :: "trm"   and pi :: "(name \<times> name) list" assumes "x1 \<Down> x2" shows "pi \<bullet> x1 \<Down> pi \<bullet> x2"
indRelLTPO_to_TRel_4: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q" shows "\<forall>TP TQ. TP \<in>T P \<and> TQ \<in>T Q \<longrightarrow> (TP, TQ) \<in> TRel\<^sup>+"
cas_true: shows "I \<Turnstile>m mset CAs \<cdot>cm \<sigma> \<cdot>cm \<eta>"
some_sol: shows "(sol has_derivative (\<lambda>v'. v' *\<^sub>R ODE_sem I ODE (sol s))) (at s within {0..t})"
big_step_determ: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int"   and u :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (c, s) \<Rightarrow> t"   and "\<pi>: (c, s) \<Rightarrow> u" shows "u = t"
right_neutral: fixes a :: "'b::bounded_semilattice_inf_top" shows "(a::'b::bounded_semilattice_inf_top) \<sqinter> \<top> = a"
i_shrink_const: fixes k :: "nat"   and m :: "'b message_af" assumes "0 < k" shows "(\<lambda>x. m) \<div> k = (\<lambda>x. m)"
less_aexp_aux_antisym: fixes x :: "'a aexp"   and y :: "'a aexp" shows "local.less_aexp_aux x y = (\<not> local.less_aexp_aux y x \<and> x \<noteq> y)"
exec1s_impl_execs: fixes cs :: "com list"   and s :: "state"   and t :: "state" assumes "(cs, s) \<rightarrow>\<^sup>* ([], t)" shows "s =cs\<Rightarrow> t"
III: shows "- \<rho> = - (- (\<rho> \<squnion> - \<rho> \<squnion> - \<rho>) \<squnion> \<rho>)"
left_neutral: fixes a :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "Var \<circ>\<^sub>s a = a"
disj_zero_right: fixes x :: "'c" shows "x OR (0::'c) = x"
dominates_unsnoc: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and g :: "'a"   and n :: "'b"   and m :: "'b"   and m' :: "'b" assumes "graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry"   and "invar g"   and "graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g n m"   and "m' \<in> set (graph_path_base.predecessors inEdges' g m)"   and "n \<noteq> m" shows "graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g n m'"
square_free_imp_resultant_non_zero: fixes f :: "int poly" assumes "square_free f" shows "resultant f (pderiv f) \<noteq> 0"
infsetsum'_Diff: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and A :: "'c set" assumes "infsetsum'_converges f B"   and "infsetsum'_converges f A"   and "A \<subseteq> B" shows "infsetsum' f (B - A) = infsetsum' f B - infsetsum' f A"
op_norm_matrix_matrix_mult_le: fixes A :: "(('b, 'e) vec, 'd) vec"   and B :: "(('b, 'c) vec, 'e) vec" shows "\<parallel>A ** B\<parallel>\<^sub>o\<^sub>p \<le> (\<parallel>A\<parallel>\<^sub>o\<^sub>p) * (\<parallel>B\<parallel>\<^sub>o\<^sub>p)"
g2: shows "lookup (monom_mult c s p) ((s + t) \<oplus> lt f) \<noteq> (0::'b)"
indicator_scaleR_eq_if: fixes S :: "'b set"   and x :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "indicat_real S x *\<^sub>R f x = (if x \<in> S then f x else (0::'a))"
assms_1: fixes k :: "real" shows "(f k has_integral y k) s"
arr_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "arr \<a>[a, b, c]"
mutual_information_eq_entropy_conditional_entropy_distr: fixes S :: "'d measure"   and T :: "'e measure"   and X :: "'a \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'e"   and Py :: "'e \<Rightarrow> real"   and Pxy :: "'d \<times> 'e \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M S X (\<lambda>x. ennreal (Px x))"   and "\<And>x. x \<in> space S \<Longrightarrow> 0 \<le> Px x"   and "distributed M T Y (\<lambda>x. ennreal (Py x))"   and "\<And>x. x \<in> space T \<Longrightarrow> 0 \<le> Py x"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (Pxy x))"   and "\<And>x y. \<lbrakk>x \<in> space S; y \<in> space T\<rbrakk> \<Longrightarrow> 0 \<le> Pxy (x, y)"   and "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))"   and "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"   and "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))" shows "mutual_information b S T X Y = entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
cases: fixes G :: "prog"   and a1 :: "qtname \<times> memberdecl"   and a2 :: "qtname" assumes "G \<turnstile> a1 member_of a2" obtains m :: "qtname \<times> memberdecl" and C :: "qtname" where "a1 = m__" and "a2 = Cb__" and "G\<turnstile> mbr m__ declared_in Cb__" and "declclass m__ = Cb__"   | m :: "qtname \<times> memberdecl" and C :: "qtname" and S :: "qtname" where "a1 = m" and "a2 = C" and "G \<turnstile> m inheritable_in pid C" and "G\<turnstile> memberid m undeclared_in C" and "G\<turnstile>C\<prec>\<^sub>C1S" and "G \<turnstile> Class S accessible_in pid C" and "G \<turnstile> m member_of S"
basic_subtype_irrefl: fixes \<tau> :: "'a basic_type" assumes "\<tau> < \<tau>" shows "False"
nn_integral_has_integral_lebesgue: fixes f :: "'c \<Rightarrow> real"   and I :: "real"   and \<Omega> :: "'c set" assumes "\<And>x. 0 \<le> f x"   and "(f has_integral I) \<Omega>" shows "\<integral>\<^sup>+ x. ennreal (indicat_real \<Omega> x * f x) \<partial>lborel = ennreal I"
symclpI2: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "r y x" shows "symclp r x y"
size_eq_Suc_imp_eq_union: fixes M :: "'k multiset"   and n :: "nat" assumes "size M = Suc n" shows "\<exists>a N. M = add_mset a N"
degree_higher_pderiv: fixes n :: "nat"   and p :: "'a poly" shows "degree ((pderiv ^^ n) p) = degree p - n"
dghm_is_dghm_if_ge_Limit: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<beta>\<^esub> \<BB>"
path_first_loc: fixes loc :: "'loc"   and loc' :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list" assumes "path loc loc' xs"   and "xs \<noteq> []" shows "(case xs ! 0 of (s, l, t) \<Rightarrow> s) = loc"
sup_bot_right: fixes x :: "('a \<times> ('b \<times> 'a) list) set" shows "x \<union> t_zero = x"
nn_integral_le_const: fixes c :: "ennreal"   and f :: "'a \<Rightarrow> ennreal"   and M :: "'a pmf" assumes "0 \<le> c"   and "AE x in measure_pmf M. f x \<le> c" shows "integral\<^sup>N (measure_pmf M) f \<le> c"
ind_hyp_l1: fixes l2 :: "'a list" assumes "distinct l2 \<and> sorted l2" shows "distinct (diff_sorted l1 l2) \<and> sorted (diff_sorted l1 l2) \<and> set (diff_sorted l1 l2) = set l1 - set l2"
keys_map_entry: fixes k :: "'c"   and f :: "'d \<Rightarrow> 'd"   and m :: "('c, 'd) mapping" shows "Mapping.keys (Mapping.map_entry k f m) = Mapping.keys m"
assms_3: fixes i :: "nat" assumes "i < length xs" shows "xs' ! i = xs ! f i"
simps_7: fixes k' :: "WS1S_Prelim.order"   and \<phi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula"   and k :: "WS1S_Prelim.order" shows "FV (FAll k' \<phi>) k = (if k' = k then (\<lambda>x. x - 1) ` (FV \<phi> k - {0}) else FV \<phi> k)"
simps_7: fixes p :: "pvar"   and \<phi> :: "'a dfmla" shows "FV (Exi p \<phi>) = FV \<phi> - {p}"
red_single_lt_distinct_lt: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and g :: "'t \<Rightarrow>\<^sub>0 'c"   and h :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "red_single f g h t"   and "g \<noteq> 0"   and "lt g \<noteq> lt f" shows "lt f = t \<oplus> lt h"
drop_bit_Suc: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> '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"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit (Suc n) a = drop_bit n (divide a (numeral.numeral one plus (num.Bit0 num.One)))"
wrapB_unwrapB_body: fixes f :: "nat discr Box \<rightarrow> nat discr Box" assumes "f\<cdot>\<bottom> = \<bottom>" shows "(wrapB oo unwrapB)\<cdot>f = f"
of_complex_def: fixes c :: "complex" shows "of_complex c = c *\<^sub>C (1::'a)"
m: shows "max_Var_floatariths fas \<le> 2 * DIM('a)"
ord_nat: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>" shows "int (nat (ord_Zp x)) = ord_Zp x"
Rs_def: shows "Rs \<equiv> \<lambda>n. {R. linorder_on {..<n} R}"
sum_upto_sum_divisors: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and x :: "real" shows "sum_upto (\<lambda>n. \<Sum>d | d dvd n. f n d) x = sum_upto (\<lambda>k. sum_upto (\<lambda>d. f (d * k) k) (x / real k)) x"
l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent get_parent_locs get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_ancestors adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_owner_document insert_before insert_before_locs DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs"
const: fixes name :: "name"   and rhs :: "value"   and rs :: "(name \<times> value) list"   and \<Gamma> :: "(name, value) fmap" assumes "(name, rhs) \<in> set rs" shows "rs, \<Gamma> \<turnstile>\<^sub>v Sconst name \<down> rhs"
large_m_def: fixes u :: "int poly"   and vs :: "'a poly list" shows "large_m u vs = (\<forall>v n. v dvd u \<longrightarrow> degree v \<le> degree_bound vs \<longrightarrow> 2 * \<bar>coeff v n\<bar> < m)"
mod_minus_right: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo a (uminus b) = uminus (modulo (uminus a) b)"
g_f: fixes x :: "'a" assumes "x \<in> A" shows "g (f x) = x"
rerr_invalid_def: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and m :: "msg" shows "rerr_invalid crt m \<equiv> case m of Rerr destsc x \<Rightarrow> \<forall>ripc\<in>dom destsc. ripc \<in> iD crt \<and> the (destsc ripc) = sqn crt ripc | _ \<Rightarrow> True"
assms_3: shows "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
add_diff_inverse: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># b" shows "a + (b - a) = b"
strand_sem_subst_subst_idem: fixes \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and M :: "('fun, 'var) Term.term set"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"   and "\<lbrakk>M; S \<cdot>\<^sub>s\<^sub>t \<delta>\<rbrakk>\<^sub>c (\<delta> \<circ>\<^sub>s \<theta>)"   and "subst_idem \<delta>" shows "\<lbrakk>M; S\<rbrakk>\<^sub>c (\<delta> \<circ>\<^sub>s \<theta>)"
h1: shows "fst ` rel_interior S = rel_interior {y. f y \<noteq> {}}"
cinner_ge_zero: fixes x :: "'a" shows "0 \<le> cinner x x"
compress_normalize_primitive_None: fixes m :: "'c match_expr"   and disc :: "'c \<Rightarrow> bool"   and sel :: "'c \<Rightarrow> 'd"   and C :: "'d \<Rightarrow> 'c"   and f :: "'d negation_type list \<Rightarrow> ('d list \<times> 'd list) option"   and \<gamma> :: "('c \<Rightarrow> 'e \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'e \<Rightarrow> bool)"   and a :: "action"   and p :: "'e" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "compress_normalize_primitive (disc, sel) C f m = None"   and "\<And>as. f as = None \<Longrightarrow> \<not> matches \<gamma> (alist_and (NegPos_map C as)) a p" shows "\<not> matches \<gamma> m a p"
norm_nth_eucl: fixes x :: "'b" shows "norm x = L2_set (($\<^sub>e) x) {..<DIM('b)}"
finite_refines_card_le: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "finite (A // R)"   and "R \<subseteq> S"   and "equiv A R"   and "equiv A S" shows "card (A // S) \<le> card (A // R)"
lin_independent_obtain_unique_scalars: fixes ms :: "'v list"   and n :: "'v" assumes "set ms \<subseteq> V"   and "R_lin_independent ms"   and "n \<in> RSpan ms" shows "\<exists>!rs. set rs \<subseteq> UNIV \<and> length rs = length ms \<and> n = rs \<bullet>\<cdot> ms"
fmap_add_ltr_def: fixes m1 :: "('a, 'b) fmap"   and m2 :: "('a, 'b) fmap" shows "m1 ++ m2 \<equiv> m2 ++\<^sub>f m1"
closureE: fixes x :: "'a"   and a :: "'a set"   and c :: "'a set" assumes "x \<in> closure a" obtains "\<not> c closed"   | "\<not> a \<subseteq> c"   | "x \<in> c"
bconf_Throw: fixes P :: "'a prog"   and sh :: "sheap"   and e :: "'b exp"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (throw e,b) \<surd>) = (P,sh \<turnstile>\<^sub>b (e,b) \<surd>)"
induct: fixes P :: "latin_type \<Rightarrow> bool"   and latin_type :: "latin_type" assumes "P Row"   and "P Col"   and "P Num" shows "P latin_type"
foldl_correct: fixes f :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'a"   and \<sigma> :: "'a"   and t :: "('b, 'c) FingerTreeStruc" shows "FingerTreeStruc.foldl f \<sigma> t = foldl f \<sigma> (FingerTreeStruc.toList t)"
F'_def: shows "F' = fds_shift (- complex_of_real (\<sigma> - 1)) F"
take_Suc: shows "take k (replicate (Suc k) (2::'a)) = replicate k (2::'a)"
exhaust: fixes y :: "('b, 'c, 'd) Tree" obtains x1 :: "('b, 'c, 'd) Tree set['b]" where "y = tConj x1"   | x2 :: "('b, 'c, 'd) Tree" where "y = tNot x2"   | x3 :: "'c" where "y = tPred x3"   | x41 :: "'d" and x42 :: "('b, 'c, 'd) Tree" where "y = tAct x41 x42"
sup_bind: fixes P :: "'b pred"   and Q :: "'b pred"   and R :: "'b \<Rightarrow> 'a pred" shows "bind (P \<squnion> Q) R = bind P R \<squnion> bind Q R"
backward_finite_path_connected_aux: fixes r :: "'a"   and x :: "'a" assumes "backward_finite_path_root r x" shows "x\<^sup>T ; r ; x\<^sup>T \<le> x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>"
simps_1: fixes x :: "'d" shows "x\<^bsup>0\<^esup> = []"
identity\<^sub>E_def: shows "basic_identity\<^sub>E \<equiv> \<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>O!,y\<^sup>P\<rparr> \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>))"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_mono: fixes M :: "(('a, 'b) Term.term \<times> ('c, 'b) Term.term) list"   and N :: "(('a, 'b) Term.term \<times> ('c, 'b) Term.term) list" assumes "set M \<subseteq> set N" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s M \<subseteq> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s N"
nsum_zeroTr: fixes A :: "('c, 'd) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "aGroup A" shows "(\<forall>i\<le>n. f i = \<zero>\<^bsub>A\<^esub>) \<longrightarrow> \<Sigma>\<^sub>e A f n = \<zero>\<^bsub>A\<^esub>"
word_uint_eqI: fixes a :: "'a word"   and b :: "'a word" assumes "uint a = uint b" shows "a = b"
\<epsilon>_def: shows "\<phi>\<psi>.\<epsilon> \<equiv> transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) (F \<circ> G) \<phi>\<psi>.\<epsilon>o"
insert_image: fixes x :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "x \<in> A" shows "insert (f x) (f ` A) = f ` A"