hcju/isabelleps huggingface dataset:
CACH_NEW: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and c :: "Class"   and Mspec :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and Minv :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and n :: "nat"   and k :: "nat"   and G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" assumes "ins_is C m l (new c)"   and "MST\<down>(C, m) = Some (Mspec, Minv, Anno)"   and "Anno\<down>l = None"   and "n = k + 1"   and "derivAssum G C m (l + 1) (Cachera k)" shows "deriv G C m l (Cachera n)"
returns_to_return_time_pos: fixes P :: "'a set"   and x :: "'a" assumes "closed P" shows "returns_to P x = (0 < return_time P x)"
transpose_columnvector: fixes v :: "('a, 'b) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.transpose (columnvector v) = rowvector v"
conj_subst: fixes P :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "P = R" shows "(P \<and> Q) = (R \<and> Q)"
remove_child_locs_def: shows "remove_child_locs \<equiv> l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child_locs set_child_nodes_locs set_disconnected_nodes_locs"
iterates_lmap: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "iterates f x = LCons x (lmap f (iterates f x))"
path_component_mem_1: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "path_component S x y" shows "x \<in> S"
homeomorphic_empty_space: fixes X :: "'a topology"   and Y :: "'b topology" assumes "X homeomorphic_space Y" shows "(topspace X = {}) = (topspace Y = {})"
nat_pow_Suc2: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp" shows "x [^] Suc n = x \<otimes> x [^] n"
simps: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "median3 x y z = (if x \<le> y \<and> y \<le> z then y else if x \<le> z \<and> z \<le> y then z else if y \<le> x \<and> x \<le> z then x else if y \<le> z \<and> z \<le> x then z else if z \<le> x \<and> x \<le> y then x else y)"
simps_2: fixes n :: "nat"   and c :: "'b list" shows "mk_cycles_path (Suc n) c = c @ mk_cycles_path n c"
elims_2: fixes x :: "('a \<times> 'c) list \<times> 'o"   and xa :: "('a \<times> 'c) list \<times> 'o" assumes "oalist_eq_raw x xa" obtains xs :: "('a \<times> 'c) list" and ox :: "'o" and ys :: "('a \<times> 'c) list \<times> 'o" where "x = (xs, ox)" and "xa = ys" and "xs = sort_oalist_aux ox ys"
x_y_def: shows "x_y \<equiv> [:[:0::'a, 1::'a:], - 1:]"
irreducible_def: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a" shows "Divisibility.irreducible G a = (a \<notin> Units G \<and> (\<forall>b\<in>carrier G. Divisibility.properfactor G b a \<longrightarrow> b \<in> Units G))"
leg0_simps_3: shows "trg (tab\<^sub>0 r) = src r"
sel_2: fixes x2 :: "'b" shows "projr (Inr x2) = x2"
def_scheme': fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and x :: "'b" shows "f self x \<equiv> \<lambda>\<tau>. SOME res. let res = \<lambda>_. res in if \<tau> \<Turnstile> \<delta> self \<and> f_\<upsilon> x \<tau> then \<tau> \<Turnstile> PRE self x \<and> \<tau> \<Turnstile> POST self x res else \<tau> \<Turnstile> res \<triangleq> invalid"
chine_simps_3: shows "C.cod \<mu>\<nu>_\<pi>.chine = \<mu>\<nu>_\<pi>.cod.apex"
compat_add_opp: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point" assumes "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "ell_field_class.add a p\<^sub>1 p\<^sub>2 = ell_field_class.add a p\<^sub>1 (opp p\<^sub>2)"   and "p\<^sub>1 \<noteq> opp p\<^sub>1" shows "p\<^sub>2 = opp p\<^sub>2"
aND_sort: fixes p :: "('b set set, 'c) Combinators list"   and l :: "'b set set set list" assumes "singleCombinators p"   and "allNetsDistinct p" shows "allNetsDistinct (FWNormalisationCore.sort p l)"
am5_lem: fixes y :: "'a"   and x :: "'a" shows "ar (ar y \<cdot> x) \<cdot> ar (y \<cdot> x) = ar x"
l23: shows "(- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<in> radical_sqrt"
delete_ran_inj_on: fixes j :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "j \<in> {0..<Suc n}"   and "f ` {0..<n} = {0..<Suc n} - {j}" shows "inj_on (delete_ran f j) {0..<n}"
simps_3: fixes a :: "'b"   and h :: "'b tree" shows "cost (Insert a) [h] = T_insert a h"
gt_imp_wt_ge: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "local.kbo_std.gt t s" shows "local.kbo_std.wt s \<le> local.kbo_std.wt t"
MakeFtorMor: fixes f :: "'a"   and F :: "('b, 'd, 'a, 'e, 'c, 'f, 'g) Functor_scheme" assumes "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "MakeFtor F ## f = F ## f"
lincomb_index: fixes i :: "nat"   and X :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "i < n"   and "X \<subseteq> carrier_vec n" shows "n.lincomb a X $ i = (\<Sum>x\<in>X. a x * x $ i)"
child_parent_dual: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "child \<in> set children"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r Some ptr"
TCd_consistency: assumes "infinite UNIV" shows "consistency {set G |G. \<not> (\<stileturn> G)}"
poly_cond_degree_0_imp_no_root: fixes p :: "int poly"   and x :: "'b" assumes "poly_cond p"   and "degree p = 0" shows "ipoly p x \<noteq> (0::'b)"
ucast_down_drop: fixes n :: "nat"   and w :: "'b word" assumes "source_size ucast = target_size ucast + n" shows "to_bl (ucast w) = drop n (to_bl w)"
simps_2: fixes a :: "('a, 'b) Node" shows "FingerTreeStruc.toList (Single a) = FingerTreeStruc.nodeToList a"
Rep: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and x :: "'a" assumes "type_definition Rep Abs A" shows "Rep x \<in> A"
ltln_of_dnf_def: fixes \<A> :: "'b ltln fset set" shows "ltln_of_dnf \<A> = Or\<^sub>n (And\<^sub>n ` fset ` \<A>)"
c: shows "consistent (Node w t\<^sub>1 t\<^sub>2)"
case_3: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f5 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f6 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f7 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x31 :: "poscheckvariant"   and x32 :: "('d, 'e) Term.term"   and x33 :: "('d, 'e) Term.term" shows "(case \<langle>x31: x32 \<doteq> x33\<rangle> of send\<langle>x\<rangle> \<Rightarrow> f1 x | receive\<langle>x\<rangle> \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle> \<Rightarrow> f3 x xa xb | insert\<langle>x,xa\<rangle> \<Rightarrow> f4 x xa | delete\<langle>x,xa\<rangle> \<Rightarrow> f5 x xa | \<langle>x: xa \<in> xb\<rangle> \<Rightarrow> f6 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa \<or>\<notin>: xb\<rangle> \<Rightarrow> f7 x xa xb) = f3 x31 x32 x33"
cases: fixes a :: "'a"   and A :: "('b, 'c, 'a, 'd) FSM_scheme" assumes "a \<in> nodes A" obtains p :: "'a" where "a = p__" and "p__ = initial A"   | p :: "'a" and aa :: "('b \<times> 'c) \<times> 'a" where "a = snd aa" and "p \<in> nodes A" and "snd aa \<in> succ A (fst aa) p"
mcdj: shows "m cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
finprod_cong: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "A = B"   and "(f \<in> B \<rightarrow> carrier G) = True"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "finprod G f A = finprod G g B"
lunit_coherence2: fixes a :: "'c" assumes "C.ide a" shows "F \<l>\<^sub>C[a] \<cdot>\<^sub>D \<phi> (\<I>\<^sub>C, a) = \<l>\<^sub>1[F a]"
adds_fun_iff': fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "(f adds g) = (\<forall>x. \<exists>y. g x = f x + y)"
HFail_blocksOf: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p" shows "blocksOf s' q \<subseteq> blocksOf s q \<union> {dblock s' q}"
cINF_const: fixes A :: "'a set"   and c :: "'b multiset" assumes "A \<noteq> {}" shows "(INF x\<in>A. c) = c"
cases: fixes x :: "heap \<times> nat set" obtains h :: "heap" and as :: "nat set" where "x = (h, as)"
pr_loop_correct: fixes y :: "nat"   and x :: "nat"   and xs :: "nat list"   and n :: "nat"   and g :: "recf"   and gap :: "abc_inst list"   and gar :: "nat"   and gft :: "nat"   and f :: "recf"   and fft :: "nat"   and anything :: "nat list" assumes "y \<le> x"   and "length xs = n"   and "rec_ci g = (gap, gar, gft)"   and "\<forall>y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"   and "\<forall>y<x. \<forall>anything. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything}" shows "{\<lambda>nl. nl = xs @ (x - y) # rec_exec (Pr n f g) (xs @ [x - y]) # 0 \<up> (max (n + 3) (max fft gft) - Suc (Suc n)) @ y # anything} [Dec (max (n + 3) (max fft gft)) (length gap + 7)] [+] (gap [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)]) @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)] {\<lambda>nl. nl = xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ anything}"
simps_3: fixes x21 :: "'b"   and x22 :: "'b llist" shows "LCons x21 x22 \<noteq> LNil"
ca_lr_syn_def: fixes d :: "ValD"   and P :: "db" shows "d \<triangleleft> P \<equiv> (d, P) \<in> {(x, unProg Y) |x Y. (x, Y) \<in> unsynlr ca.delta}"
OneState_Conf: fixes S :: "'d"   and ST :: "('d, 'e, 'f) status"   and T :: "'d"   and SA :: "('d, 'e, 'f) seqauto" assumes "S \<in> Conf ST"   and "T \<in> Conf ST"   and "S \<in> States SA"   and "T \<in> States SA"   and "SA \<in> SAs (HA ST)" shows "T = S"
a_closure': fixes x :: "'a" shows "ans_d (ad x) = ad x"
less_le_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x < y"   and "y \<le> z" shows "x < z"
Special_ax_imp_special_axioms: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "x \<in> Special_ax" shows "\<exists>A. x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<and> A \<in> special_axioms"
rep_eq: fixes x :: "nat"   and xa :: "float"   and xb :: "nat" shows "real_of_float (power_up_fl x xa xb) = power_up x (real_of_float xa) xb"
exists_least_iff: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool" assumes "class.wellorder less_eq less" shows "(\<exists>n. P n) = (\<exists>n. P n \<and> (\<forall>m. less m n \<longrightarrow> \<not> P m))"
simps_5: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and a :: "'d" shows "(case Def a of \<bottom> \<Rightarrow> f1 | Def x \<Rightarrow> f2 x) = f2 a"
lin_a: shows "class.linorder (\<sqsubseteq>\<^sub>a) (\<sqsubset>\<^sub>a)"
fixp_preserves_mono1: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lub :: "'c set \<Rightarrow> 'c"   and F :: "('d \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "partial_function_definitions leq lub"   and "\<And>x. monotone (fun_ord leq) leq (\<lambda>f. F f x)"   and "f \<equiv> ccpo.fixp (fun_lub lub) (fun_ord leq) F"   and "\<And>f. monotone ord leq f \<Longrightarrow> monotone ord leq (F f)" shows "monotone ord leq f"
Tree\<^sub>\<alpha>_free_1: fixes tset\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha> set['e]"   and t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "Conj\<^sub>\<alpha> tset\<^sub>\<alpha> \<noteq> Not\<^sub>\<alpha> t\<^sub>\<alpha>"
P2_real_model_def: fixes M :: "'grp \<times> 'grp"   and \<sigma> :: "bool"   and z :: "'e"   and \<A> :: "(bool \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp \<times> 'grp) \<times> nat) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> 'd spmf)" shows "P2_real_model M \<sigma> z \<A> = (let (x0, x1) = M; (\<A>1, \<A>2, \<A>3) = \<A> in \<A>1 \<sigma> z \<bind> (\<lambda>((h0, h1, a, b0, b1), s). assert_spmf (h0 \<in> carrier \<G> \<and> h1 \<in> carrier \<G> \<and> a \<in> carrier \<G> \<and> b0 \<in> carrier \<G> \<and> b1 \<in> carrier \<G>) \<bind> (\<lambda>_. \<A>2 (h0, h1, a, b0, b1) s \<bind> (\<lambda>(((in1, in2, in3), r), s'). let (h, a, b) = (h0 \<otimes> inv h1, a, b0 \<otimes> inv b1) in funct_DH_ZK (h, a, b) ((in1, in2, in3), r) \<bind> (\<lambda>(out_zk_funct, uu_). assert_spmf out_zk_funct \<bind> (\<lambda>_. sample_uniform (order \<G>) \<bind> (\<lambda>u0. sample_uniform (order \<G>) \<bind> (\<lambda>u1. sample_uniform (order \<G>) \<bind> (\<lambda>v0. sample_uniform (order \<G>) \<bind> (\<lambda>v1. let z0 = b0 [^] u0 \<otimes> h0 [^] v0 \<otimes> x0; w0 = a [^] u0 \<otimes> \<^bold>g [^] v0; e0 = (w0, z0); z1 = (b1 \<otimes> inv \<^bold>g) [^] u1 \<otimes> h1 [^] v1 \<otimes> x1; w1 = a [^] u1 \<otimes> \<^bold>g [^] v1; e1 = (w1, z1) in \<A>3 e0 e1 s' \<bind> (\<lambda>out. return_spmf ((), out))))))))))))"
abs_summable_on_insert_iff: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and A :: "'c set" shows "(f abs_summable_on insert x A) = (f abs_summable_on A)"
fmapU_maybe_simps_3: fixes f :: "udom \<rightarrow> udom"   and x :: "udom" shows "fmapU\<cdot>f\<cdot>(Just\<cdot>x) = Just\<cdot>(f\<cdot>x)"
suc1: fixes A :: "nat"   and k :: "nat"   and a :: "nat" assumes "A \<equiv> k + a" shows "Suc A \<equiv> k + Suc a"
starlike_imp_contractible_gen: fixes S :: "'b set"   and P :: "('b \<Rightarrow> 'b) \<Rightarrow> bool" assumes "starlike S"   and "\<And>a T. \<lbrakk>a \<in> S; 0 \<le> T; T \<le> 1\<rbrakk> \<Longrightarrow> P (\<lambda>x. (1 - T) *\<^sub>R x + T *\<^sub>R a)" obtains a :: "'b" where "homotopic_with_canon P S S (\<lambda>x. x) (\<lambda>x. a)"
match_fv_cong: fixes r :: "'a Regex.regex"   and test :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and test' :: "nat \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>i x. x \<in> regex.atms r \<Longrightarrow> test i x = test' i x" shows "Regex.match test r = Regex.match test' r"
subspace_hom_deg_set: fixes A :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and z :: "nat" assumes "phull.subspace A" shows "phull.subspace (hom_deg_set z A)"
permute_flip_cancel2: fixes a :: "'b"   and b :: "'b"   and x :: "'a" shows "(a \<leftrightarrow> b) \<bullet> (b \<leftrightarrow> a) \<bullet> x = x"
PfP_Single_I: fixes Aa :: "'a"   and A :: "tm" assumes "{} \<turnstile> PrfP (Eats Zero (HPair Zero \<guillemotleft>Aa\<guillemotright>)) Zero A" shows "{} \<turnstile> PfP A"
sq_mtx_minus_ith: fixes A :: "'b sq_mtx"   and B :: "'b sq_mtx"   and i :: "'b" shows "(A - B) $$ i = A $$ i - B $$ i"
norm_optim: fixes n :: "nat"   and xs :: "('dyn, 'b, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr list"   and \<Sigma>1 :: "Unboxed.type option list"   and ys :: "('dyn, 'b, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr list"   and \<Sigma>2 :: "Unboxed.type option list" assumes "optim n xs \<Sigma>1 = Ok (ys, \<Sigma>2)" shows "map norm_instr ys = map norm_instr xs"
intro: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" assumes "heap_is_wellformed = l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_heap_is_wellformed get_child_nodes get_disconnected_nodes"   and "parent_child_rel = l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_parent_child_rel get_child_nodes" shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_disconnected_nodes heap_is_wellformed parent_child_rel"
elims_3: fixes x :: "'b literal" assumes "\<not> negative_literal x" obtains xa :: "'b equation" where "x = Pos xa"
assms_4: fixes x :: "'a" assumes "x \<in> space M" shows "has_density (f x) N (g x)"
square_reflcl: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "square R S T R\<^sup>=\<^sup>="   and "S \<le> T" shows "square R\<^sup>=\<^sup>= S T R\<^sup>=\<^sup>="
i_less_m: shows "mod_type_class.from_nat i \<le> m"
cases: fixes p :: "'b \<times> 'c"   and a :: "'c" assumes "sndsp p a" obtains "a = snd p"
adjoint_equivalence_in_bicategory_axioms: shows "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
assms_2: shows "all_normalized_and_ex_tcsigs (set cs) (translate_ars arss)"
less_eq_V_def: fixes x :: "V"   and y :: "V" shows "x \<le> y \<equiv> elts x \<subseteq> elts y"
simps_2: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "upd_lock l t Unlock = unlock_lock l"
LtK_Enc: fixes X :: "ltkey"   and X' :: "msg"   and Y' :: "msg" shows "LtK X \<noteq> Enc X' Y'"
ipresVar_def: fixes h :: "('h, 'i, 'varSort, 'var, 'j) qTerm set \<Rightarrow> 'k"   and MOD :: "('l, 'm, 'varSort, 'n, 'o, 'var, 'k, 'p, 'q) model_scheme" shows "ipresVar h MOD \<equiv> \<forall>xs x. h (Var xs x) = igVar MOD xs x"
less_eq_preal_def: fixes x :: "preal"   and y :: "preal" shows "x \<le> y \<equiv> x + y = y"
domains_comp: fixes g :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" assumes "seq g f" shows "domains (g \<cdot> f) = domains f"
choose_two_int: fixes x :: "int" assumes "0 \<le> x" shows "int (nat x choose 2) = x * (x - 1) div 2"
le_diff_eq: fixes plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and zero :: "'r"   and minus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and uminus :: "'r \<Rightarrow> 'r"   and less_eq :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and less :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and a :: "'r"   and c :: "'r"   and b :: "'r" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a (minus c b) = less_eq (plus a b) c"
is_Read\<^sub>s\<^sub>b_simps_3: fixes p :: "'b"   and p' :: "'b"   and "is" :: "instr list" shows "is_Read\<^sub>s\<^sub>b (Prog\<^sub>s\<^sub>b p p' is) = False"
less_name_code_1: fixes xs :: "char list" shows "(Name xs < Name []) = False"
get_proc_intra: fixes a :: "'a" assumes "valid_edge a"   and "intra_kind (kind a)" shows "get_proc (sourcenode a) = get_proc (targetnode a)"
the_elem_image_unique: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "A \<noteq> {}"   and "\<And>y. y \<in> A \<Longrightarrow> f y = f x" shows "the_elem (f ` A) = f x"
int_pow_diff: fixes x :: "'a"   and n :: "int"   and m :: "int" assumes "x \<in> carrier \<G>" shows "x [^] (n - m) = x [^] n \<otimes> inv (x [^] m)"
simps_3: fixes v :: "'a"   and va :: "'a list"   and q :: "'a"   and qs :: "'a list" shows "p_fusion (v # va) (q # qs) = (v # va) @ qs"
in_set_inl_nonzero: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) pdevs" assumes "x \<in> set (inl xs)" shows "x \<noteq> 0"
CASAddr: fixes a :: "expr"   and \<theta> :: "nat \<Rightarrow> nat option"   and c\<^sub>e :: "expr"   and s\<^sub>e :: "expr"   and A :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and L :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and R :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and W :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and t :: "nat" assumes "\<forall>sop. a \<noteq> Tmp sop" shows "\<theta>\<turnstile> (CAS a c\<^sub>e s\<^sub>e A L R W, t) \<rightarrow>\<^sub>p ((CAS (Tmp (eval_expr t a)) c\<^sub>e s\<^sub>e A L R W, t + used_tmps a), issue_expr t a)"
back_shift: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and I :: "nat set" assumes "PL_Indis cval indis" shows "PL_Indis.back cl n (PL_Indis.shift cl n I) = I"
eq_monom_sum_var_list: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('c \<times> nat) list"   and n :: "('c \<times> nat) list" assumes "class.linorder less_eq less"   and "linorder.monom_inv less_eq m"   and "linorder.monom_inv less_eq n" shows "(m = n) = (\<forall>x. linorder.sum_var_list m x = linorder.sum_var_list n x)"
one_eq_neg_one: assumes "\<one> = \<ominus> \<one>" shows "int p = 2"
preserves_reflects_arr: fixes f :: "'a" shows "C.arr (src\<^sub>C f) = C.arr f"
useful_tautologies_10: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<psi> \<^bold>\<rightarrow> (\<phi> \<^bold>\<rightarrow> \<psi> \<^bold>\<rightarrow> \<^bold>\<not>\<phi>) in v]"
a_add_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<oplus> (y \<oplus> z) = x \<oplus> y \<oplus> z"
E_def: shows "cf.E \<equiv> {(u, v). cf (u, v) \<noteq> (0::'a)}"
derived_is_wellformed: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int"   and block :: "'a set"   and b :: "'a set" assumes "bibd_block_transformations point_set block_collection u_block_size index block"   and "b \<in># bibd_block_transformations.derived_blocks block_collection block" shows "b \<subseteq> block"
iso_sym: fixes H :: "('e, 'f) monoid_scheme" assumes "G \<cong> H" shows "H \<cong> G"
greedy_path_deadend_v0: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and v :: "'a"   and P :: "'a llist" assumes "greedy_conforming_path p \<sigma> \<sigma>' v0 = LCons v P" shows "(P = LNil) = deadend v0"
proj_fm: fixes x :: "'a"   and m :: "'a \<Rightarrow> 'd" assumes "x \<in> J" shows "(fm m)\<^sub>F (f x) = m x"
simps_2: fixes name :: "name" shows "pterm_to_sterm (Pvar name) = Svar name"
LIMSEQ_abs_realpow_zero2: fixes c :: "real" assumes "\<bar>c\<bar> < 1" shows "(^) c \<longlonglongrightarrow> 0"
nat_to_ce_set_lm_1: fixes n :: "nat" shows "nat_to_ce_set n = {x. \<exists>y. nat_to_pr n (c_pair x y) = 0}"