hcju/isabelleps huggingface dataset:
troeger_1: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "(x \<oplus> y)\<^sup>\<odot> = x\<^sup>\<odot> \<odot> matrix_idempotent_semiring.Rf y ((x \<oplus> y)\<^sup>\<odot>)"
strongBisimWeakBisim: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim> Q" shows "P \<approx> Q"
summable_imp_convergent: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f" shows "convergent f"
when_True: fixes X :: "'c \<Rightarrow> (unit \<times> 'c) set \<times> bool" shows "when True X = X"
continuous_imp_closed_map: fixes S :: "'b set"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "closedin (top_of_set S) U"   and "continuous_on S f"   and "f ` S = T"   and "compact S" shows "closedin (top_of_set T) (f ` U)"
map_of_map_restrict: fixes f :: "'e \<Rightarrow> 'f"   and ks :: "'e list" shows "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
emeasure_buffon_set_conv_buffon_set': shows "emeasure lborel (buffon_set l d) = 4 * emeasure lborel (buffon_set' l d)"
countable_Union: fixes X :: "'a set set"   and M :: "'a measure" assumes "countable X"   and "X \<subseteq> sets M" shows "\<Union> X \<in> sets M"
trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "a \<le>s b"   and "b \<le>s c" shows "a \<le>s c"
subst_cls_single: fixes L :: "'a literal"   and \<sigma> :: "'s" shows "{#L#} \<cdot> \<sigma> = {#L \<cdot>l \<sigma>#}"
MP_oalist_empty: fixes ko :: "'b nat_term_order" shows "MP_oalist (OAlist_empty_ntm ko) = 0"
l_new_shadow_root_get_disconnected_nodes_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_shadow_root_get_disconnected_nodes get_disconnected_nodes get_disconnected_nodes_locs \<equiv> (\<forall>new_shadow_root_ptr ptr' h h' r. cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr \<noteq> ptr' \<longrightarrow> h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr \<longrightarrow> h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<longrightarrow> r h h') \<and> (\<forall>h new_shadow_root_ptr h'. h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr \<longrightarrow> h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_disconnected_nodes (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr) \<rightarrow>\<^sub>r [])"
assms_1: shows "\<forall>i\<ge>low. P i \<longrightarrow> P (Suc i) \<and> Q i"
matchs_neq_length_none: fixes xs :: "term list"   and ys :: "'a list" assumes "length xs \<noteq> length ys" shows "matchs xs ys = None"
isnpolyh_Suc_const: fixes p :: "poly"   and n :: "nat" assumes "isnpolyh p (Suc n)" shows "isconstant p"
\<gamma>1'''s: shows "set (labels \<gamma>1''') \<subseteq> r \<down>s {fst \<alpha>_step, fst \<beta>_step}"
assms_4: shows "ideal_points (poincare_line u v) = {i1, i2}"
independent_explicit_module: fixes s :: "('a, 'b) vec set" shows "vec.independent s = (\<forall>t u v. finite t \<longrightarrow> t \<subseteq> s \<longrightarrow> (\<Sum>v\<in>t. u v *s v) = 0 \<longrightarrow> v \<in> t \<longrightarrow> u v = (0::'a))"
crel_vsI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "['d]" assumes "(state_dp_consistency.crel_vs R OO local.rel_state (=)) a b" shows "crel_vs R a b"
unfoldr_rbt_keys_generator_aux: fixes kts :: "('c \<times> ('c, 'd) rbt) list"   and t :: "('c, 'd) rbt" shows "list.unfoldr rbt_keys_generator (kts, t) = RBT_Impl.keys t @ concat (map (\<lambda>(k, t). k # RBT_Impl.keys t) kts)"
solution_upd1: fixes c :: "'b"   and A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and p :: "nat"   and n :: "nat"   and x :: "nat \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "solution (A(p := \<lambda>j. A p j / c)) n x = solution A n x"
w_simps_4: shows "trg \<pi>.chine = src t\<^sub>0"
Conj\<^sub>\<alpha>: fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> set['a]" assumes "finite (supp tset\<^sub>\<alpha>)"   and "\<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (Conj\<^sub>\<alpha> tset\<^sub>\<alpha>)"
cSup_upper: fixes x :: "'k multiset"   and X :: "'k multiset set" assumes "x \<in> X"   and "subset_mset.bdd_above X" shows "x \<subseteq># Sup X"
assms: shows "z \<succeq> y \<and> y \<succeq> z"
VAs: fixes j :: "'sz" shows "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT (x2a j). case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)"
may_acquire_all_def: fixes ls :: "('a, 'b) locks"   and t :: "'b"   and ln :: "'a \<Rightarrow>f nat" shows "may_acquire_all ls t ln \<equiv> \<forall>l. 0 < ln $ l \<longrightarrow> may_lock (ls $ l) t"
eval_transcendental: fixes K :: "'a set"   and x :: "'a"   and p :: "'a list" assumes "(transcendental over K) x"   and "p \<in> carrier (K [X])"   and "local.eval p x = \<zero>" shows "p = []"
top_fun_def: shows "\<top> = (\<lambda>x. \<top>)"
coinduct: fixes X :: "'b \<Rightarrow> bool"   and x :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>xa y. x = xa \<and> r xa y \<and> (X y \<or> Inf.inf r y)" shows "Inf.inf r x"
chanEqClosed: fixes \<Psi> :: "'b"   and M :: "'a"   and N :: "'a"   and p :: "(name \<times> name) list" assumes "\<Psi> \<turnstile> M \<leftrightarrow> N" shows "(p \<bullet> \<Psi>) \<turnstile> (p \<bullet> M) \<leftrightarrow> (p \<bullet> N)"
assms: shows "abc_step_l (s, lm) (abc_fetch s B) = (sa, lma)"
reads_consistent_dropWhile: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list" shows "reads_consistent pending_write \<O> m (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) = reads_consistent True \<O> m (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
simps_4: fixes uu :: "('e set \<times> 'e set) list"   and vb :: "'h"   and v :: "('f, 'g, 'e) prot_fun"   and va :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" shows "equal_mod_timpls uu (Var vb) (Fun v va) = False"
that: fixes B :: "real" assumes "0 < B"   and "\<forall>\<^sub>F x in F. norm (f x) \<le> B" shows "thesis"
cod_simps_2: shows "gf.cod.leg0 = gf.leg0"
map_eq_Cons_D: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and y :: "'b"   and ys :: "'b list" assumes "map f xs = y # ys" shows "\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys"
PPs\<alpha>_of_step: fixes c :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr c m \<alpha> p m'" shows "set (PPV \<alpha>) \<subseteq> set (PPc c)"
indep_subset: fixes X :: "'a set"   and Y :: "'a set" assumes "indep_in \<E> X"   and "Y \<subseteq> X" shows "indep_in \<E> Y"
simps_13: fixes f :: "'c \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c llist" shows "lmap f (LCons x21 x22) = LCons (f x21) (lmap f x22)"
a_inv: fixes h :: "'c"   and R :: "('c, 'd) ring_scheme"   and H :: "'c set" assumes "h \<in> generate_field R H" shows "\<ominus>\<^bsub>R\<^esub> h \<in> generate_field R H"
fin_maintained_def: fixes R :: "('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph"   and G :: "('e, 'g) labeled_graph" shows "fin_maintained R G \<equiv> \<forall>F f. finite_graph F \<longrightarrow> subgraph F (fst R) \<longrightarrow> extensible (F, fst R) G f \<longrightarrow> graph_homomorphism F G f \<longrightarrow> extensible (F, snd R) G f"
valid: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration" assumes "computation channel trans send recv init final" shows "\<exists>t. distributed_system.trace channel trans send recv init t final"
hyps_3: fixes h\<^sub>0 :: "heap"   and l\<^sub>0 :: "sfields"   and sh\<^sub>0 :: "sheap" assumes "s\<^sub>1 = (h\<^sub>0, l\<^sub>0(V := None), sh\<^sub>0)" shows "P \<turnstile> \<langle>{V:T; e\<^sub>1},(h\<^sub>0, l\<^sub>0, sh\<^sub>0),b\<^sub>1\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2(V := l\<^sub>0 V), sh\<^sub>2),b\<^sub>2\<rangle>"
sets_rinss': fixes B :: "'b set"   and ys :: "'b list"   and t :: "('b \<times> real) tree" assumes "countable B"   and "set ys \<subseteq> B"   and "t \<in> trees (B \<times> UNIV)" shows "sets (rinss ys t B) = sets (tree_sigma (count_space B \<Otimes>\<^sub>M lborel))"
update_keys: fixes k :: "'c"   and v :: "'d"   and al :: "('c \<times> 'd) list" shows "map fst (update k v al) = (if k \<in> set (map fst al) then map fst al else map fst al @ [k])"
rem: shows "remdups_adj (x # xs) = (if x = y then remdups_adj (y # xs') else x # remdups_adj (y # xs'))"
accept_def: fixes m :: "'b"   and w :: "nat \<Rightarrow> 'L" shows "Automata.gba.accept (to_gba_ext m) w \<equiv> \<exists>r. gb_graph.is_acc_run (to_gba_ext m) r \<and> (\<forall>i. gba_L (to_gba_ext m) (r i) (w i))"
rel_gpv_conv_rel_gpv'': fixes A :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'b \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_gpv A C = rel_gpv'' A C (=)"
kildall_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and \<tau>s :: "'b list" shows "kildall r f step \<tau>s = fst (iter f step \<tau>s (unstables r step \<tau>s))"
conj_disj_distrib: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x AND (y OR z) = x AND y OR x AND z"
tendsto_Im: fixes g :: "'a \<Rightarrow> complex"   and a :: "complex"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Im (g x)) \<longlongrightarrow> Im a) F"
graph_union_edges: fixes G\<^sub>1 :: "('a, 'b) labeled_graph"   and G\<^sub>2 :: "('a, 'b) labeled_graph" shows "edges (graph_union G\<^sub>1 G\<^sub>2) = edges G\<^sub>1 \<union> edges G\<^sub>2"
INF_identity_eq: fixes A :: "'b set" shows "(\<Sqinter>x\<in>A. x) = \<Sqinter>A"
wf_rexp_ENC: fixes V :: "nat set"   and n :: "nat" assumes "finite V"   and "n = 0 \<or> (\<forall>v\<in>V. v < n)" shows "local.wf n (ENC n V)"
B_gets_good_key: fixes B :: "agent"   and A :: "agent"   and K :: "nat"   and NB :: "nat"   and evs :: "event list"   and NA :: "nat" assumes "Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>, Crypt K (Nonce NB)\<rbrace> \<in> set evs"   and "Says B Server \<lbrace>Agent B, Nonce NB, Crypt (shrK B) \<lbrace>Agent A, Nonce NA\<rbrace>\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "Key K \<notin> analz (knows Spy evs)"
remove_shadow_root_def: shows "remove_shadow_root = (\<lambda>element_ptr. Heap_Error_Monad.bind (get_shadow_root element_ptr) (case_option (error HierarchyRequestError) (\<lambda>shadow_root_ptr. Heap_Error_Monad.bind (get_child_nodes (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr)) (\<lambda>children. if children = [] then Heap_Error_Monad.bind (set_shadow_root element_ptr None) (\<lambda>_. delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr) else error HierarchyRequestError))))"
assms: fixes w :: "'a" assumes "w \<in> set (monom_vars_list m)" shows "f w = g w"
shuffle_to_top_Field: fixes Y :: "'a set"   and d :: "'b" assumes "inj_on Xd Y" shows "Field (Pdds'.shuffle_to_top Y d) = Field (Pd_shuffle_to_top (insert d__ ds') Y d) \<union> Pdds'.dX Y d"
assms_3: fixes x :: "'a" assumes "x \<notin> A" shows "0 \<le> f x"
neg_inf_imp_neg_cyc: fixes f :: "'b \<Rightarrow> real"   and u :: "'a"   and v :: "'a" assumes "\<mu> f u v = - \<infinity>" shows "\<exists>p. awalk u p v \<and> (\<exists>w c. awalk w c w \<and> w \<in> set (awalk_verts u p) \<and> awalk_cost f c < 0)"
l_1: shows "(w \<circ> a) \<longlonglongrightarrow> liminf w"
assms_1: shows "a * x + r \<in> elts (a * y)"
delete_inorder: fixes k :: "nat"   and t :: "'a btree"   and x :: "'a" assumes "0 < k"   and "bal t"   and "root_order k t"   and "sorted_less (inorder t)" shows "inorder (delete k x t) = del_list x (inorder t)"
IH: fixes s :: "'s" shows "viterbi_ix_rec (IArray os) s t 0 = viterbi s t os"
homeomorphic_rel_frontiers_convex_bounded_sets: fixes S :: "'a set"   and T :: "'b set" assumes "convex S"   and "bounded S"   and "convex T"   and "bounded T"   and "aff_dim S = aff_dim T" shows "rel_frontier S homeomorphic rel_frontier T"
nonPrincipalID: fixes r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and A :: "('a, 'b) form"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" assumes "r \<in> Ax" shows "\<not> rightPrincipal r A R \<and> \<not> leftPrincipal r A R"
simps: fixes v :: "'a" shows "neighbors v = {u. {u, v} \<in> E}"
cos_pi_minus: fixes x :: "real" shows "cos (pi - x) = - cos x"
induced_arrowI_2: fixes \<chi>' :: "'j \<Rightarrow> 'c"   and a' :: "'c" assumes "\<chi>' \<in> D.cones a'" shows "D.cones_map (induced_arrow a' \<chi>') \<chi> = \<chi>'"
hyps: assumes "\<not> c \<le> d"   and "shift2_inv (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})))" shows "shift2_inv c (shift2 c (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)}))))"
eventually_at_left: fixes y :: "'b"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "y < x" shows "eventually P (at_left x) = (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
compl_le_swap1: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and y :: "'e"   and x :: "'e" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq y (uminus x)" shows "less_eq x (uminus y)"
lfp_transfer_bounded: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c"   and \<alpha> :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'd" assumes "P bot"   and "\<And>x. P x \<Longrightarrow> P (f x)"   and "\<And>M. (\<And>i. P (M i)) \<Longrightarrow> P (Sup (range M))"   and "\<And>M. \<lbrakk>incseq M; \<And>i. P (M i)\<rbrakk> \<Longrightarrow> \<alpha> (Sup (range M)) = (SUP i. \<alpha> (M i))"   and "sup_continuous f"   and "sup_continuous g"   and "\<And>x. \<lbrakk>P x; x \<le> lfp f\<rbrakk> \<Longrightarrow> \<alpha> (f x) = g (\<alpha> x)"   and "\<And>x. \<alpha> bot \<le> g x" shows "\<alpha> (lfp f) = lfp g"
fmlookup_cases: fixes y :: "'c \<Rightarrow> 'd option" assumes "y \<in> {m. finite (dom m)}" obtains x :: "('c, 'd) fmap" where "y = ($$) x"
gcd_add_dvd_right2: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a dvd c" shows "gcd a (b + c) = gcd a b"
Some_eq_map_of_iff: fixes xys :: "('a \<times> 'b) list"   and y :: "'b"   and x :: "'a" assumes "distinct (map fst xys)" shows "(Some y = Map.map_of xys x) = ((x, y) \<in> set xys)"
nth_append: fixes xs :: "'b list"   and ys :: "'b list"   and n :: "nat" shows "(xs @ ys) ! n = (if n < length xs then xs ! n else ys ! (n - length xs))"
tdghm_NTMap_vsv: shows "vsv (\<NN>\<lparr>NTMap\<rparr>)"
measurable_on_limit: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and S :: "'a set"   and N :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>n. f n measurable_on S"   and "negligible N"   and "\<And>x. x \<in> S - N \<Longrightarrow> (\<lambda>n. f n x) \<longlonglongrightarrow> g x" shows "g measurable_on S"
Append_def: shows "Append \<equiv> \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>Safe\<rbrace> \<acute>ind := 0;; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>Safe \<and> \<acute>ind = 0\<rbrace>  WHILE \<acute>ind < length \<acute>M INV \<lbrace>\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind \<le> length \<acute>M\<rbrace>  DO \<lbrace>\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind < length \<acute>M\<rbrace>  IF \<acute>M ! \<acute>ind = Black THEN \<lbrace>\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind < length \<acute>M \<and> \<acute>M ! \<acute>ind = Black\<rbrace> \<acute>M := \<acute>M[\<acute>ind := White] ELSE \<lbrace>\<acute>Proper \<and> \<acute>AppendInv \<acute>ind \<and> \<acute>ind < length \<acute>M \<and> \<acute>ind \<notin> Reach \<acute>E\<rbrace> \<acute>E := Append_to_free (\<acute>ind, \<acute>E) FI;; \<lbrace>\<acute>Proper \<and> \<acute>AppendInv (\<acute>ind + 1) \<and> \<acute>ind < length \<acute>M\<rbrace> \<acute>ind := \<acute>ind + 1 OD"
infinite_solution_set_homogeneous_eq: fixes A :: "(('b, 'c) vec, 'd) vec" shows "infinite (solution_set A 0) = (0 < vec.dim (solution_set A 0))"
simps_3: fixes b :: "nat"   and \<phi> :: "'c MFOTL.formula" shows "MFOTL.fvi b (formula.Neg \<phi>) = MFOTL.fvi b \<phi>"
MOST_rev_mp: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<forall>\<^sub>\<infinity>x. P x"   and "\<forall>\<^sub>\<infinity>x. P x \<longrightarrow> Q x" shows "\<forall>\<^sub>\<infinity>x. Q x"
approximation_preproc_push_neg_1: fixes a :: "real"   and b :: "real" shows "(\<not> a < b) = (b \<le> a)"
iso_test_def: fixes g\<^sub>1 :: "'a list list"   and g\<^sub>2 :: "'b list list" shows "iso_test g\<^sub>1 g\<^sub>2 = (pr_iso_test g\<^sub>1 g\<^sub>2 \<or> pr_iso_test g\<^sub>1 (map rev g\<^sub>2))"
prim_no_rotate: fixes w :: "'b list"   and n :: "nat" assumes "primitive w"   and "0 < n"   and "n < \<^bold>|w\<^bold>|" shows "rotate n w \<noteq> w"
upt_append: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "j < k" shows "[i..<j] @ [j..<k] = [i..<k]"
st_vec: fixes x :: "'a st_vec" shows "st_vec x \<in> {v. non_neg_vec v \<and> stoch_vec v}"
wf_strictly_generalizes_atm: shows "wfP strictly_generalizes_atm"
unfold_optimisation_correct_M: fixes \<pi>\<^sub>\<A> :: "'a ltl \<Rightarrow> nat option"   and \<phi> :: "'a ltl"   and \<pi>\<^sub>\<UU> :: "'a ltl \<Rightarrow> nat option" assumes "dom \<pi>\<^sub>\<A> \<subseteq> \<^bold>G \<phi>"   and "dom \<pi>\<^sub>\<UU> = dom \<pi>\<^sub>\<A>"   and "\<And>\<chi>. \<chi> \<in> dom \<pi>\<^sub>\<A> \<Longrightarrow> \<pi>\<^sub>\<A> \<chi> = mojmir_def.smallest_accepting_rank \<Sigma> \<up>af\<^sub>G (Abs (theG \<chi>)) w {q. dom \<pi>\<^sub>\<A> \<up>\<Turnstile>\<^sub>P q}"   and "\<And>\<chi>. \<chi> \<in> dom \<pi>\<^sub>\<UU> \<Longrightarrow> \<pi>\<^sub>\<UU> \<chi> = mojmir_def.smallest_accepting_rank \<Sigma> \<up>af\<^sub>G\<^sub>\<UU> (Abs (Unf\<^sub>G (theG \<chi>))) w {q. dom \<pi>\<^sub>\<UU> \<up>\<Turnstile>\<^sub>P q}" shows "accepting_pair\<^sub>R (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) (M_fin \<pi>\<^sub>\<A>, UNIV) w = accepting_pair\<^sub>R (\<delta>\<^sub>\<UU> \<Sigma>) (\<iota>\<^sub>\<UU> \<phi>) (M\<^sub>\<UU>_fin \<pi>\<^sub>\<UU>, UNIV) w"
hom_one: fixes p :: "'b poly" shows "1 \<circ>\<^sub>p p = 1"
init_red_sync: fixes e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and e\<^sub>1 :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and e' :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow> \<langle>e\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle>"   and "init_exp_of e\<^sub>0 = \<lfloor>e\<rfloor>"   and "e\<^sub>1 \<noteq> e" shows "P \<turnstile> \<langle>init_switch e\<^sub>0 e',s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow> \<langle>init_switch e\<^sub>1 e',s\<^sub>1,b\<^sub>1\<rangle>"
fin_ex_inf_eq_limit: fixes A :: "'c set"   and w :: "nat \<Rightarrow> 'c" assumes "finite A" shows "(\<exists>\<^sub>\<infinity>i. w i \<in> A) = (limit w \<inter> A \<noteq> {})"
induct: fixes P :: "'b wordinterval \<Rightarrow> bool"   and a0 :: "'b wordinterval" assumes "\<And>rs. (\<And>x y x2. \<lbrakk>\<not> wordinterval_empty rs; (x, y) = wordinterval_CIDR_split1 rs; x = Some x2\<rbrakk> \<Longrightarrow> P y) \<Longrightarrow> P rs" shows "P a0"
eqvtI: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and perm :: "(name \<times> name) list" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "Rel \<subseteq> Rel'"   and "eqvt Rel'" shows "perm \<bullet> P \<leadsto>\<^sup>^<Rel'> perm \<bullet> Q"
rcos_module_imp: fixes x :: "'a"   and x' :: "'a" assumes "Group.group G"   and "x \<in> carrier G"   and "x' \<in> derived G (carrier G) #> x" shows "x' \<otimes> inv x \<in> derived G (carrier G)"
llength_ltakeWhile_eq_infinity: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "(llength (ltakeWhile P xs) = \<infinity>) = (\<not> lfinite xs \<and> ltakeWhile P xs = xs)"
valid_plan_def: fixes \<pi>s :: "char list list" shows "valid_plan \<pi>s \<equiv> \<exists>s'\<in>G. path_to I \<pi>s s'"
ref_rule: fixes x :: "'b" shows "<emp> ref x <\<lambda>r. r \<mapsto>\<^sub>r x>"
singletonI: fixes a :: "'k" shows "a \<in> {a}"
preserves_cod: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.cod (\<psi>.map f) = hom'.map (hom'.cod f)"
simps_1: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "stake 0 \<sigma> = []"
uadmit_dterm_adjoint: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<theta> :: "('sf, 'sz) trm"   and U :: "('sz + 'sz) set"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and I :: "('sf, 'sc, 'sz) interp" assumes "TUadmit \<sigma> \<theta> U"   and "Vagree \<nu> \<omega> (- U)"   and "\<And>f f'. SFunctions \<sigma> f = Some f' \<Longrightarrow> dfree f'"   and "\<And>f f'. SPredicates \<sigma> f = Some f' \<Longrightarrow> fsafe f'"   and "dsafe \<theta>"   and "is_interp I" shows "dterm_sem (local.adjoint I \<sigma> \<nu>) \<theta> = dterm_sem (local.adjoint I \<sigma> \<omega>) \<theta>"
inducts: fixes x1 :: "('procS, 'procT) Proc"   and x2 :: "('procS, 'procT) Proc"   and P :: "('procS, 'procT) Proc \<Rightarrow> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "indRelp x1 x2"   and "\<And>S. P (SourceTerm S) (TargetTerm (\<lbrakk>S\<rbrakk>))"   and "\<And>S. P (TargetTerm (\<lbrakk>S\<rbrakk>)) (SourceTerm S)" shows "P x1 x2"
assms_1: shows "\<forall>y\<in>set aa. y \<le> maxne0_impl aaa b"