hcju/isabelleps huggingface dataset:
strong_delete_point_groups_index_zero: fixes G :: "'a set"   and x :: "'a"   and xa :: "'a"   and y :: "'a" assumes "G \<in> {b - {x} |b. b \<in># \<B> \<and> x \<in> b}"   and "xa \<in> G"   and "y \<in> G"   and "xa \<noteq> y" shows "str_del_point_blocks x index {xa, y} = 0"
Seq: fixes \<pi> :: "char list \<Rightarrow> com option"   and c\<^sub>1 :: "com"   and s\<^sub>1 :: "char list \<Rightarrow> int \<Rightarrow> int"   and s\<^sub>2 :: "char list \<Rightarrow> int \<Rightarrow> int"   and c\<^sub>2 :: "com"   and s\<^sub>3 :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (c\<^sub>1, s\<^sub>1) \<Rightarrow> s\<^sub>2"   and "\<pi>: (c\<^sub>2, s\<^sub>2) \<Rightarrow> s\<^sub>3" shows "\<pi>: (c\<^sub>1;; c\<^sub>2, s\<^sub>1) \<Rightarrow> s\<^sub>3"
parallel_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs \<parallel> ys) = (\<not> prefix xs ys \<and> \<not> prefix ys xs)"
refl_on_def': fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "refl_on A r = ((\<forall>(x, y)\<in>r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
nat_minus_add_max: fixes n :: "nat"   and m :: "nat" shows "n - m + m = ord_class.max n m"
simps_2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and a :: "'c"   and x :: "'d"   and xs :: "'d list" shows "foldl f a (x # xs) = foldl f (f a x) xs"
scast_nop1: fixes x :: "int" shows "SCAST('b \<rightarrow> 'b signed) (word_of_int x) = word_of_int x"
timpls_transformable_to_pred_inv_3: fixes A :: "'e set set"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term"   and i :: "nat" assumes "timpls_transformable_to_pred A (Fun f T) t"   and "i < length T" shows "timpls_transformable_to_pred A (T ! i) (args t ! i)"
inject_2: fixes x21 :: "POPLmark.type"   and x22 :: "trm"   and y21 :: "POPLmark.type"   and y22 :: "trm" shows "((\<lambda>:x21. x22) = (\<lambda>:y21. y22)) = (x21 = y21 \<and> x22 = y22)"
equivalent_matrices_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and B :: "(('a, 'd) vec, 'e) vec" shows "equivalent_matrices A B = (\<exists>P Q. invertible P \<and> invertible Q \<and> B = matrix_inv P ** A ** Q)"
z_def_2: fixes y :: "('a, 'b) insert_message" assumes "depends_on H y z" shows "y \<notin> H"
preSetpreList: fixes xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> 'b) list"   and C :: "acom"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "preList xs C l s" shows "preSet (set xs) C l s"
simE_3: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi" assumes "P \<leadsto><Rel> Q"   and "Q \<longmapsto> \<alpha> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
limit_step_pointwise: fixes x :: "'a"   and f :: "'a set \<Rightarrow> 'a set"   and X :: "'a set"   and y :: "'a" assumes "x \<in> limit f X"   and "pointwise f"   and "y \<in> f {x}" shows "y \<in> limit f X"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 (trg f) (trg f)"
keys_minus: fixes a :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys (- a) = keys a"
Red_I_without_red_F: fixes N :: "'f set" shows "Red_Red_I (N - Red_F N) = Red_Red_I N"
gen_lossless_gpv_unfold: shows "local.gen_lossless_gpv = F local.gen_lossless_gpv"
strong_induct: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "'a \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a" assumes " x1 \<longrightarrow>\<^sub>\<beta> x2"   and "\<And>s1 s2 t y. \<lbrakk> s1 \<longrightarrow>\<^sub>\<beta> s2; \<And>z. P z s1 s2\<rbrakk> \<Longrightarrow> P y (App s1 t) (App s2 t)"   and "\<And>s1 s2 t y. \<lbrakk> s1 \<longrightarrow>\<^sub>\<beta> s2; \<And>z. P z s1 s2\<rbrakk> \<Longrightarrow> P y (App t s1) (App t s2)"   and "\<And>s1 s2 a y. \<lbrakk>a \<sharp> y;  s1 \<longrightarrow>\<^sub>\<beta> s2; \<And>z. P z s1 s2\<rbrakk> \<Longrightarrow> P y (Lam [a].s1) (Lam [a].s2)"   and "\<And>a s2 s1 y. \<lbrakk>a \<sharp> y; a \<sharp> s2\<rbrakk> \<Longrightarrow> P y (App (Lam [a].s1) s2) (s1[a::=s2])" shows "P z x1 x2"
one: fixes R :: "('a, 'b) ring_scheme"   and H :: "'a set" shows "\<one>\<^bsub>R\<^esub> \<in> generate_ring R H"
Bfun_def: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "Bfun f F = (\<exists>K>0. \<forall>\<^sub>F x in F. norm (f x) \<le> K)"
qTermLess_ingoreFirst_wf: shows "wf (inv_image qTermLess aux_qInit_ignoreFirst)"
establish_invarI_CB: fixes I :: "'v tarjan_state \<Rightarrow> bool" assumes "\<And>u v s. on_back_edge tarjan_params u v s = on_cross_edge tarjan_params u v s"   and "on_init tarjan_params \<le>\<^sub>n SPEC (\<lambda>x. I (empty_state x))"   and "\<And>s s' v0. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s = []; v0 \<in> V0; v0 \<notin> dom (discovered s); s' = new_root v0 s\<rbrakk> \<Longrightarrow> on_new_root tarjan_params v0 s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s\<rbrakk> \<Longrightarrow> on_finish tarjan_params u s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<in> dom (discovered s); discovered s' = discovered s; finished s' = finished s; stack s' = stack s; tree_edges s' = tree_edges s; counter s' = counter s; pending s' = pending s - {(u, v)}; cross_edges s' \<union> back_edges s' = cross_edges s \<union> back_edges s \<union> {(u, v)}; state.more s' = state.more s\<rbrakk> \<Longrightarrow> on_cross_edge tarjan_params u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<notin> dom (discovered s); s' = discover u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<rbrakk> \<Longrightarrow> on_discover tarjan_params u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))" shows "is_invar I"
fin: shows "finite (insert 0 (Suc ` {n. f n \<noteq> (0::'a)}))"
tabulate2_def: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a" shows "tabulate2 m n f = IArray.of_fun (\<lambda>i. IArray.of_fun (f i) n) m"
in_measure: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<Rightarrow> nat" shows "((x, y) \<in> measure f) = (f x < f y)"
fmset_perm_cong: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and bs :: "'c list" assumes "Group.monoid G"   and "as <~~> bs" shows "fmset G as = fmset G bs"
prime_card: assumes "class.prime_card TYPE('a)" shows "prime CARD('a)"
simps: fixes w1 :: "32 Word.word"   and w2 :: "32 Word.word" shows "wmin w1 w2 = (if w1 <s w2 then w1 else w2)"
simps: fixes ps :: "nat list" shows "prime_factorization_nat_main n jj is ps = (case is of [] \<Rightarrow> case next_candidates jj of (j, is) \<Rightarrow> prime_factorization_nat_main n j is ps | i # is \<Rightarrow> case divmod_nat n i of (n', m) \<Rightarrow> if m = 0 then case remove_prime_factor i n' (i # ps) of (n', ps') \<Rightarrow> if n' = 1 then ps' else prime_factorization_nat_main n' jj is ps' else if i * i \<le> n then prime_factorization_nat_main n jj is ps else n # ps)"
reds_steps: fixes c :: "cmd"   and s :: "char list \<Rightarrow> val option"   and c' :: "cmd"   and s' :: "char list \<Rightarrow> val option"   and prog :: "cmd"   and l :: "nat" assumes "\<langle>c,s\<rangle> \<rightarrow>* \<langle>c',s'\<rangle>"   and "labels prog l c" shows "\<exists>l'. prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto>* \<langle>c',s',l'\<rangle> \<and> labels prog l' c'"
Struct_axioms: shows "M.Struct wtFsym wtPsym arOf resOf (\<lambda>\<sigma>. D) intF intP"
preserves_hom: fixes f :: "'j"   and a :: "'j"   and b :: "'j" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" shows "\<guillemotleft>A'.F.map f : A'.F.map a \<rightarrow> A'.F.map b\<guillemotright>"
less_eq_Assertion_def: fixes p :: "'c Assertion"   and q :: "'c Assertion" shows "(p \<le> q) = ({\<cdot> p } \<le> {\<cdot> q })"
le: fixes x :: "'n"   and k :: "real" assumes "x \<in> s" shows "norm (f k x) \<le> h x"
DiffE: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "c \<in> A - B" obtains "c \<in> A" and "c \<notin> B"
shortest_paths_locale_step1_def: fixes G :: "('c, 'd) pre_digraph"   and s :: "'c"   and num :: "'c \<Rightarrow> nat"   and parent_edge :: "'c \<Rightarrow> 'd option"   and dist :: "'c \<Rightarrow> ereal" shows "shortest_paths_locale_step1 G s num parent_edge dist \<equiv> (fin_digraph G \<and> s \<in> verts G \<and> dist s \<noteq> \<infinity>) \<and> (parent_edge s = None \<and> num s = 0) \<and> (\<forall>v. v \<in> verts G \<longrightarrow> v \<noteq> s \<longrightarrow> dist v \<noteq> \<infinity> \<longrightarrow> (\<exists>e\<in>arcs G. parent_edge v = Some e \<and> head G e = v \<and> dist (tail G e) \<noteq> \<infinity> \<and> num v = num (tail G e) + 1)) \<and> (\<forall>e. e \<in> arcs G \<longrightarrow> dist (tail G e) \<noteq> \<infinity> \<longrightarrow> dist (head G e) \<noteq> \<infinity>)"
roots_of_rat_poly_def: fixes p :: "rat poly" shows "roots_of_rat_poly p \<equiv> let (c, pis) = yun_factorization gcd_rat_poly p in concat (map (roots_of_rat_poly_main \<circ> fst) pis)"
length_signs_smash: fixes signs1 :: "'a list list"   and signs2 :: "'a list list" shows "length (signs_smash signs1 signs2) = length signs1 * length signs2"
dependent_live_vars_same_paths: fixes as :: "'edge list"   and n' :: "'node"   and V :: "'var" assumes "as \<noteq> [] \<longrightarrow> n' = last (targetnodes as)" shows "((V, as, as) \<in> dependent_live_vars n') = (V \<in> Use n' \<and> (\<forall>as' a as''. as = as' @ a # as'' \<longrightarrow> \<not> sourcenode a -{V}a # as''\<rightarrow>\<^sub>d\<^sub>d n'))"
aezfun: fixes x :: "('d, 'c) aezfun" shows "aezfun x \<in> aezfun_set"
distinct_3: fixes x1 :: "'d" shows "Le x1 \<noteq> \<infinity>"
sim_rel1_def: shows "sim_rel1 \<equiv> \<lambda>(c, n, ts) (ci, ni, tsi). ci = c mod N \<and> ni = n mod (k * N) \<and> tsi = map_ticket (\<lambda>t. t mod N) \<circ> ts"
simps_1: fixes x :: "'a"   and y :: "'a" shows "extT.Val x + extT.Val y = extT.Val (x + y)"
dom_char: fixes f :: "'a \<times> 'a" shows "VoV.dom f = (if VoV.arr f then VxV.dom f else VxV.null)"
simps_7: fixes a :: "Cooper.num" shows "bound0 (Eq a) = numbound0 a"
assms_2: shows "(R4 ===> R5 ===> R6) C C"
invfun_mem1: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and b :: "'d" assumes "f \<in> A \<rightarrow> B"   and "bij_to f A B"   and "b \<in> B" shows "(f\<inverse>\<^bsub>B,A\<^esub>) b \<in> A"
insert_sort_by_length: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a list" shows "length (insert_sort_by cmp l) = length l"
mismatchZero: fixes a :: "name"   and P :: "pi" shows "[a\<noteq>a]P \<approx> \<zero>"
degree_zero_imp_not_is_root: fixes p :: "'a list"   and x :: "'a" assumes "p \<in> carrier (poly_ring R)"   and "degree p = 0" shows "\<not> is_root p x"
equivalence_comp_dist_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "(top \<le> x \<and> x \<sqinter> x \<le> x) \<and> local.comp_inf.symmetric x" shows "x \<sqinter> y \<sqinter> (x \<sqinter> z) = x \<sqinter> (y \<sqinter> (x \<sqinter> z))"
obj_self_composable_2: fixes a :: "'a" assumes "obj a" shows "a \<star> a \<cong> a"
ordermap_insert: fixes \<alpha> :: "V"   and y :: "V"   and U :: "V set" assumes "Ord \<alpha>"   and "Ord y"   and "y \<le> \<alpha>"   and "U \<subseteq> elts \<alpha>" shows "ordermap (insert \<alpha> U) VWF y = ordermap U VWF y"
mod_in_totatives: fixes n :: "nat"   and a :: "nat" assumes "1 < n" shows "(a mod n \<in> totatives n) = coprime a n"
simps_3: fixes x21 :: "'c"   and x22 :: "'c llist" shows "LCons x21 x22 \<noteq> LNil"
in_elts_E_less: fixes x' :: "V"   and k' :: "nat"   and x :: "V"   and k :: "nat" assumes "x' \<in> elts (E k')"   and "x \<in> elts (E k)"   and "k' < k"   and "k < length \<alpha>s" shows "x' < x"
elims: fixes x :: "\<langle>'d\<rangle>\<^sub>\<bottom>"   and y :: "'d" assumes "\<lceil>x\<rceil> = y" obtains v :: "'d" where "x = \<lfloor>v\<rfloor>" and "y = v"   | "x = None" and "y = undefined"
target: fixes T :: "'procT" shows "TargetTerm T \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R TargetTerm T"
Image_within_domain': fixes x :: "'c"   and R :: "('c \<times> 'd) set" shows "(x \<in> Domain R) = (R `` {x} \<noteq> {})"
assms_1: shows "A \<subseteq> {i. i < 2 ^ n}"
\<pi>'_def: fixes x :: "'a" shows "\<pi>' x = (if x \<in> agents then (unidx \<circ> inv f \<circ> idx) x else x)"
CSP1_R1_commute: shows "CSP1 \<circ> R1 = R1 \<circ> CSP1"
intFF_def: shows "intFF \<equiv> InfModel.intFF GE_arOf GE_resOf intTI intFI"
iIN_mult: fixes n :: "nat"   and d :: "nat"   and k :: "nat" shows "[n\<dots>,d] \<otimes> k = [ n * k, mod k, d ]"
distinct_5: fixes x31 :: "'a"   and x32 :: "'a ifex"   and x33 :: "'a ifex" shows "Falseif \<noteq> IF x31 x32 x33"
ok2: shows "typ_ok \<Theta> (\<tau> \<rightarrow> \<tau>')"
natfun_bigoE: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and n0 :: "nat" assumes "f \<in> O(g)"   and "\<And>n. n0 \<le> n \<Longrightarrow> g n \<noteq> (0::'a)" obtains c :: "real" where "0 < c" and "\<And>n. n0 \<le> n \<Longrightarrow> norm (f n) \<le> c * norm (g n)"
LIM_isCont_iff: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a" shows "f \<midarrow>a\<rightarrow> f a = (\<lambda>h. f (a + h)) \<midarrow>0::'a\<rightarrow> f a"
stability: fixes x :: "('b com option \<times> 'b) list"   and p :: "'b set"   and rely :: "('b \<times> 'b) set"   and j :: "nat"   and k :: "nat" assumes "x \<in> cptn"   and "stable p rely"   and "j \<le> k"   and "k < length x"   and "snd (x ! j) \<in> p"   and "\<And>i. \<lbrakk>Suc i < length x; x ! i -e\<rightarrow> x ! Suc i\<rbrakk> \<Longrightarrow> (snd (x ! i), snd (x ! Suc i)) \<in> rely"   and "\<And>i. j \<le> i \<and> i < k \<Longrightarrow> x ! i -e\<rightarrow> x ! Suc i" shows "snd (x ! k) \<in> p \<and> fst (x ! j) = fst (x ! k)"
axioms_2: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "Vector_Spaces.linear s1 s2 f" shows "vector_space s2"
Sup_upto_llist_def: fixes Xs :: "'a set llist"   and j :: "enat" shows "Sup_upto_llist Xs j = \<Union> (lnth Xs ` {i. enat i < llength Xs \<and> enat i \<le> j})"
eint_0_iff_1: fixes x :: "int" shows "(eint x = 0) = (x = 0)"
gi_is_fi_minus_sum_mu_gso: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "fs.gs.gso i = map of_int_hom.vec_hom fs ! i - gs.M.sumlist (map (\<lambda>j. fs.gs.\<mu> i j \<cdot>\<^sub>v fs.gs.gso j) [0..<i])"
ParComp: fixes P :: "pi"   and Q :: "pi"   and R :: "pi"   and T :: "pi" assumes "(P, Q) \<in> Rel"   and "(R, T) \<in> Rel'" shows "(P \<parallel> R, Q \<parallel> T) \<in> Rel'"
elims_2: fixes D :: "ast_domain"   and ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and x :: "'a atom formula" assumes "ast_domain.wf_fmla_atom D ty_ent x" obtains a :: "predicate" and vs :: "'a list" where "x = Atom (predAtm a vs)" and "ast_domain.wf_pred_atom D ty_ent (a, vs)"
succ_of_rel_def: fixes s :: "('d \<times> 'e) set" shows "succ_of_rel s \<equiv> \<lambda>a. {b. (a, b) \<in> s}"
interrupt_actions_ok_append: fixes "is" :: "'b set"   and ias :: "'b interrupt_action list"   and ias' :: "'b interrupt_action list" shows "interrupt_actions_ok is (ias @ ias') = (interrupt_actions_ok is ias \<and> interrupt_actions_ok (redT_updIs is ias) ias')"
subspace_sum: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "module scale"   and "module.subspace scale A"   and "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A" shows "sum f B \<in> A"
case_4: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case Public of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f4"
9414: fixes x :: "'a"   and y :: "'a" shows "- - (- x \<sqinter> y) \<sqinter> y = - x \<sqinter> y"
simple_rotations_inj: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations" shows "inj r"
set_intros_2: fixes yb :: "('e, 'f) T\<^sub>m"   and x :: "('e, 'f, ('e, 'f) T\<^sub>m, 'f T\<^sub>h) F\<^sub>m"   and yc :: "'e" assumes "yb \<in> set3_F\<^sub>m x"   and "yc \<in> set1_T\<^sub>m yb" shows "yc \<in> set1_T\<^sub>m (T\<^sub>m x)"
simps: fixes t :: "'a" shows "strip_comb t = (case unapp t of None \<Rightarrow> (t, []) | Some (t, u) \<Rightarrow> let (f, args) = strip_comb t in (f, args @ [u]))"
FL_valid_Tree\<^sub>\<alpha>_Pred\<^sub>\<alpha>: fixes P :: "'state"   and f :: "'effect"   and \<phi> :: "'pred" shows "FL_valid_Tree\<^sub>\<alpha> P (Pred\<^sub>\<alpha> f \<phi>) = \<langle>f\<rangle>P \<turnstile> \<phi>"
bv_Nils: shows "[] \<preceq>\<^sub>b [] = True"
get_dom_component_no_overlap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and c' :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c'" shows "set c \<inter> set c' = {} \<or> c = c'"
simps_3: fixes b :: "nat"   and \<phi> :: "'b formula" shows "fvi b (Neg \<phi>) = fvi b \<phi>"
contr_graph_slim_eq: shows "gen_contr_graph (with_proj slim) (verts3 (with_proj G)) = gen_contr_graph (with_proj G) (verts3 (with_proj G))"
degree_pdevs_of_ivl_le: fixes l :: "'a"   and u :: "'a" shows "degree (pdevs_of_ivl l u) \<le> DIM('a)"
ereal_SUP_uminus: fixes f :: "'b \<Rightarrow> ereal"   and R :: "'b set" shows "(SUP i\<in>R. - f i) = - Inf (f ` R)"
nth_list_update: fixes i :: "nat"   and xs :: "'k list"   and x :: "'k"   and j :: "nat" assumes "i < length xs" shows "xs[i := x] ! j = (if i = j then x else xs ! j)"
lift_SD_iff_agent: fixes p :: "'b pmf"   and q :: "'b pmf"   and i :: "'a" assumes "p \<in> lotteries_on alts"   and "q \<in> lotteries_on alts"   and "i \<in> agents" shows "SD (R' i) p q = SD (R i) p q"
map_of_map_Pair_const: fixes v :: "'c"   and xs :: "'b list" shows "map_of (map (\<lambda>x. (x, v)) xs) = (\<lambda>x. if x \<in> set xs then Some v else None)"
left_inverse: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "group_ow U f z inverse"   and "a \<in> U" shows "f (inverse a) a = z"
assms: fixes x :: "'a" shows "rel_option R (fmlookup m x) (fmlookup n x)"
simps: fixes a :: "'a list"   and A :: "'a set" shows "(a \<in> lists A) = (a = [] \<or> (\<exists>aa l. a = aa # l \<and> aa \<in> A \<and> l \<in> lists A))"
set_child_nodes_writes: fixes ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (Shadow_DOM.set_child_nodes_locs ptr) (Shadow_DOM.set_child_nodes ptr children) h h'"
f_eq: fixes a :: "int"   and b :: "int" shows "f (2 * a) + 2 * f b = f (f (a + b))"
rcsTr5: fixes H :: "'a set"   and a :: "'a"   and b :: "'a"   and x :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "b \<cdot> \<rho> a \<in> H"   and "x \<in> H \<bullet> a" shows "x \<cdot> \<rho> b \<in> H"
test_not: fixes test :: "'c \<Rightarrow> 'd"   and not :: "'d \<Rightarrow> 'd"   and p :: "'c" assumes "dioid_tests test not" shows "test (- p) = not (test p)"
lin_dep: shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs_init)"
rem_implicit_pres_ops_lookup_op_Some_2: fixes vars :: "(char list \<times> nat option \<times> char list list) list"   and init :: "nat list"   and goal :: "(nat \<times> nat) list"   and ops :: "(char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "ast_problem.lookup_operator (rem_implicit_pres_ops (vars, init, goal, ops)) name = Some op" shows "\<exists>op'. ast_problem.lookup_operator (vars, init, goal, ops) name = Some op' \<and> op = rem_implicit_pres op'"
not_odd_v': shows "v' \<notin> odd_nodes_set (del_unEdge v w v' G)"
abs_eq: fixes xa :: "'a ltl"   and x :: "'a set" shows "\<up>step (Abs xa) x = Abs (step xa x)"
case_transfer: fixes R2 :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and R1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun (rel_fun R2 S) (rel_fun (rel_fun R1 (rel_fun (tllist_all2 R1 R2) S)) (rel_fun (tllist_all2 R1 R2) S)) case_tllist case_tllist"
sound_nneg: fixes P :: "'c \<Rightarrow> real" assumes "sound P" shows "nneg P"
pseudo_mod_zero_2: fixes f :: "'b poly" shows "pseudo_mod f 0 = f"
mult_fract_cancel: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c \<noteq> (0::'b)" shows "Fraction_Field.Fract (c * a) (c * b) = Fraction_Field.Fract a b"
pr_in_hom_1: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<guillemotleft>\<pp>\<^sub>0[a, b] : a \<otimes> b \<rightarrow> b\<guillemotright>"
norm_vec_def: fixes x :: "('b, 'c) vec" shows "\<parallel>x\<parallel> = L2_set (\<lambda>i. \<parallel>x $ i\<parallel>) UNIV"
path_from_to_ends: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a"   and ys :: "'a list" assumes "v \<leadsto>(xs @ w # ys)\<leadsto> w" shows "ys = []"
iT_Plus_neg_def: fixes I :: "nat set"   and k :: "nat" shows "I \<oplus>- k \<equiv> {x. x + k \<in> I}"
eval_rest: shows "P,E \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
h4: shows "st_in t = init_state \<and> req t = [init] \<longrightarrow> ack t = [call] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = call"
robbins2: fixes y :: "'a"   and x :: "'a" shows "y = - (- (- x \<squnion> y) \<squnion> - (x \<squnion> y))"
sign_extend_bitwise_disj: fixes i :: "nat"   and w :: "'b word"   and e :: "nat" assumes "i < size w" shows "sign_extend e w !! i = (i \<le> e \<and> w !! i \<or> e \<le> i \<and> w !! e)"
fresh_context: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list" assumes "x \<sharp> \<Gamma>" shows "\<nexists>\<tau>. (x, \<tau>) \<in> set \<Gamma>"
ereal_minus_real_tendsto_MInf: shows "(\<lambda>x. ereal (- real x)) \<longlonglongrightarrow> - \<infinity>"
delayed_other_s_stop_eq: shows "delayed_safe_distance.other.s_stop = other.s_stop"
lim_2: shows "(h \<longlongrightarrow> c) net"
countable_UN': fixes X :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and M :: "'d measure" assumes "countable X"   and "A ` X \<subseteq> sets M" shows "\<Union> (A ` X) \<in> sets M"
upper_asymptotic_density_in_01_1: fixes A :: "nat set" shows "ereal (upper_asymptotic_density A) = limsup (\<lambda>x. ereal (real (card (A \<inter> {..<x})) / real x))"
simps_9: fixes f :: "'c \<Rightarrow> 'a"   and p :: "('c, 'b) form" shows "psubst f (Exists p) = Exists (psubst f p)"
GSubspace_is_FinGroupRep: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and U :: "'c set" assumes "FinGroupRepresentation G smult V"   and "RModule (aezfun_setspan G) smult U \<and> U \<subseteq> V" shows "FinGroupRepresentation G smult U"
unsigned_def: shows "unsigned \<equiv> map_fun Word.rep id (of_nat \<circ> nat \<circ> take_bit LENGTH('c))"