hcju/isabelleps huggingface dataset:
univ_predicate_impl_forall: fixes A :: "'a set"   and R :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool"   and X :: "'a set" assumes "equiv A R"   and "P respects R"   and "X \<in> A // R"   and "univ P X" shows "\<forall>x\<in>X. P x"
DirProds_is_group: fixes I :: "'c set"   and G :: "'c \<Rightarrow> ('d, 'e) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "Group.group (DirProds G I)"
complex_inner_1: fixes x :: "complex" shows "1 \<bullet> x = Re x"
line_in'_2: fixes y :: "real" assumes "y \<in> cbox 0 1" shows "x (h * y + t) \<in> X"
cases: fixes x :: "cell list" obtains "x = []"   | xs :: "cell list" where "x = Bk # xs__"   | xs :: "cell list" where "x = Oc # xs"
npow_0: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c" shows "x^\<^bsup>R 0\<^esup> = 1\<^sub>r\<^bsub>R\<^esub>"
min_key_val_in: fixes xs :: "('b, 'c) oalist_ntm"   and ko :: "'b nat_term_order" assumes "fst (list_of_oalist_ntm xs) \<noteq> []" shows "OAlist_min_key_val_ntm ko xs \<in> set (fst (list_of_oalist_ntm xs))"
degree_smult_not_zero_divisor_left: fixes c :: "'a"   and p :: "'a poly" assumes "\<not> zero_divisor c" shows "degree (smult c p) = degree p"
equal_nat_term_order_def: shows "equal_class.equal = (=)"
act: fixes p :: "net_tree" shows "opnet onp p \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U (net_tree_ips p) \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). castmsg (I \<sigma>) a \<and> (a \<noteq> \<tau> \<and> (\<forall>i d. a \<noteq> i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p. S (\<sigma> i) (\<sigma>' i))) \<and> (a = \<tau> \<or> (\<exists>i d. a = i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p. U (\<sigma> i) (\<sigma>' i)) \<and> (\<forall>i. i \<notin> net_tree_ips p \<longrightarrow> \<sigma>' i = \<sigma> i)))"
simps_1: fixes f :: "'fsym" shows "GE_wtFsym (Oldf f) = wtFsym f"
conv_callee_parallel_id_left: fixes callee :: "'h \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'h, 'e, 'f) gpv"   and s :: "'g"   and s' :: "'h" shows "CNV (1\<^sub>I \<ddagger>\<^sub>I callee) (s, s') = 1\<^sub>C |\<^sub>= CNV callee s'"
infsetsum_product: fixes A :: "'a set"   and B :: "'b set"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "countable A"   and "countable B"   and "f abs_summable_on A"   and "g abs_summable_on B" shows "(\<Sum>\<^sub>a(x, y)\<in>A \<times> B. f x * g y) = infsetsum f A * infsetsum g B"
distinct_mset_remove1_All: fixes C :: "'c multiset"   and L :: "'c" assumes "distinct_mset C" shows "remove1_mset L C = removeAll_mset L C"
dgrad_set_leE: fixes d :: "'c \<Rightarrow> nat"   and S :: "'c set"   and T :: "'c set"   and s :: "'c" assumes "dgrad_set_le d S T"   and "s \<in> S" obtains t :: "'c" where "t \<in> T" and "d s \<le> d t"
simps: fixes a :: "trm" shows "(a \<in> num) = (a = zer \<or> (\<exists>t. a = suc t \<and> t \<in> num))"
units_of_mult: fixes G :: "('d, 'e) monoid_scheme" shows "(\<otimes>\<^bsub>units_of G\<^esub>) = (\<otimes>\<^bsub>G\<^esub>)"
simps: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and xs :: "'c list" assumes "ring R" shows "ring.finite_extension R K xs = foldr (\<lambda>x K'. ring.simple_extension R K' x) xs K"
currentD_outside: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a" assumes "current \<Gamma> f"   and "\<not> edge \<Gamma> x y" shows "f (x, y) = 0"
shadow_root_host_dual: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and host :: "(_) element_ptr" assumes "h \<turnstile> get_host shadow_root_ptr \<rightarrow>\<^sub>r host" shows "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr"
odd_nodes_set_def: fixes g :: "('a, 'b) graph" shows "odd_nodes_set g \<equiv> {v \<in> nodes g. odd (degree v g)}"
AE_support_countable: assumes "X.events = UNIV" shows "(AE x in distr M (count_space UNIV) (X 0). X.prob {x} \<noteq> 0) = (\<exists>S. countable S \<and> (AE x in distr M (count_space UNIV) (X 0). x \<in> S))"
le_funD: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "f \<le> g" shows "f x \<le> g x"
llist_of_stream_siterates: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "llist_of_stream (siterate f x) = iterates f x"
document_ptr_casts_commute2: fixes document_ptr :: "(_) document_ptr" shows "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^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 (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^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 document_ptr) = Some document_ptr"
intros_4: fixes s :: "'a"   and b :: "'a set"   and c2 :: "'a com"   and s' :: "'a"   and c1 :: "'a com" assumes "s \<notin> b"   and "Sem c2 s s'" shows "Sem (Cond b c1 c2) s s'"
finite_Psegments_less_eq2: fixes P :: "real \<Rightarrow> bool"   and a :: "real"   and c :: "real"   and b :: "real" assumes "finite_Psegments P a c"   and "a \<le> b" shows "finite_Psegments P b c"
cinner_Pair_0_2: fixes x :: "'c \<times> 'd"   and a :: "'c" shows "\<langle>x, (a, 0::'d)\<rangle> = \<langle>fst x, a\<rangle>"
add_sign_intros_4: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (plus a b)"
exhaust: fixes y :: "'d decision" obtains x1 :: "'d" where "y = allow x1"   | x2 :: "'d" where "y = deny x2"
cod_cod: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.VVV.cod (B.VVV.cod f) = B.VVV.cod f"
distinct_11: fixes x7 :: "'b ltln" shows "true\<^sub>n \<noteq> X\<^sub>n x7"
override_on_apply_notin: fixes a :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "a \<notin> A" shows "override_on f g A a = f a"
sub_still_allowed: fixes p :: "('c, 'b) fm"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'd"   and i :: "'d" assumes "\<forall>i. p = Nom i \<or> p = (\<^bold>\<diamond> Nom i) \<longrightarrow> i \<in> A" shows "sub f p = Nom i \<or> sub f p = (\<^bold>\<diamond> Nom i) \<longrightarrow> i \<in> f ` A"
assms_2: fixes x :: "'a" assumes "RETURN x \<le> m'" shows "f x = f' x"
assms_1: shows "pd_kleene_allegory_class.acyclic f"
sel_1: fixes x1 :: "'a"   and x2 :: "'b" shows "fst (x1, x2) = x1"
ChiRel_CompFun: fixes s :: "'a"   and a :: "('a, 'b, 'c) hierauto" assumes "s \<in> HAStates a" shows "ChiRel a `` {s} = \<Union> (States ` the (CompFun a s))"
exec_Normal_elim_cases_2: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Stuck\<rangle> \<Rightarrow> t" obtains "t = Stuck"
pm_pth_def: fixes m :: "('b, 'c) pointermap_scheme"   and p :: "nat" shows "pm_pth m p \<equiv> entries m ! p"
diff_numeral_special_4: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (uminus (numeral.numeral one plus m)) one = uminus (numeral.numeral one plus (m + num.One))"
results_gpv_inline_aux_1: fixes x :: "'b"   and s' :: "'s"   and y :: "('b, 'call, 'ret) gpv \<times> 's + ('b, 'call, 'ret) rpv \<times> ('ret \<times> 's, 'call', 'ret') gpv"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s" assumes "(x, s') \<in> results_gpv \<I>' (inline_aux callee y)"   and "y = Inl (gpv, s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "x \<in> results_gpv \<I> gpv \<and> I s'"
pm_powerprod_axioms: shows "pm_powerprod (\<preceq>) (\<prec>)"
asIMOD_igAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and xs :: "'varSort"   and x :: "'var"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b" shows "igAbs (asIMOD SEM) xs x X = (\<lambda>val. if sWlsVal SEM val then sAbs xs (\<lambda>sX. if sWls SEM (asSort xs) sX then X (val (x := sX)_xs) else sDummy SEM (SOME s. sWls SEM s (X val))) else undefined)"
last_conv_nth_default: fixes xs :: "'f list"   and dflt :: "'f" assumes "xs \<noteq> []" shows "last xs = nth_default dflt xs (length xs - 1)"
r_parallel''_2: fixes i :: "nat"   and x :: "nat"   and j :: "nat" assumes "eval r_phi [i, x] \<down> \<and> eval r_phi [j, x] \<up>" shows "eval r_parallel [i, j, x] \<down>= prod_encode (0, the (eval r_phi [i, x]))"
trivial_empty_or_singleton: fixes X :: "'c set" shows "trivial X = (X = {} \<or> X = {the_elem X})"
a: shows "\<forall>x'\<in>dual ` A. P x'"
RS1n_assoc: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" assumes "x \<noteq> DenyAll" shows "removeShadowRules1_alternative xs @ [x] = removeShadowRules1_alternative (xs @ [x])"
hyps_2: fixes T\<^sub>1 :: "POPLmark.type"   and T\<^sub>2 :: "POPLmark.type" assumes "S = T\<^sub>1 \<rightarrow> T\<^sub>2"   and "t \<in> value" shows "\<exists>ta S. t = (\<lambda>:S. ta)"
hyps: fixes y :: "'a" assumes "(y, a) \<in> r\<inverse>" shows "P y"
baEA: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and \<phi> :: "'cmp \<Rightarrow> bool" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "eval c t t' n [\<phi>]\<^sub>b" shows "\<phi> (\<sigma>\<^bsub>c\<^esub>t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>)"
ComplI: fixes c :: "'c"   and A :: "'c set" assumes "s2p A c \<Longrightarrow> False" shows "s2p (- A) c"
distinct_6: fixes name1 :: "name"   and name2 :: "name" shows "\<tau> \<noteq> name1[name2]"
good_imp_wf_ruleset: fixes rs :: "'c rule list"   and \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d" assumes "good_ruleset rs" shows "wf_ruleset \<gamma> p rs"
intro: fixes L :: "('c, 'd) gorder_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier L; y \<in> carrier L\<rbrakk> \<Longrightarrow> \<exists>s. is_glb L s {x, y}" shows "lower_semilattice_axioms L"
types_preserved_binop_relop: fixes v1 :: "v"   and v2 :: "v"   and e :: "b_e"   and v' :: "v"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and t :: "t"   and iop :: "binop_i"   and fop :: "binop_f"   and irop :: "relop_i"   and frop :: "relop_f" assumes "\<lparr>[$C v1, $C v2, $e]\<rparr> \<leadsto> \<lparr>[$C v']\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [$C v1, $C v2, $e] : ts _> ts'"   and "e = Binop_i t iop \<or> e = Binop_f t fop \<or> e = Relop_i t irop \<or> e = Relop_f t frop" shows "\<S>\<bullet>\<C> \<turnstile> [$C v'] : ts _> ts'"
coeff_reflect_poly: fixes p :: "'b poly"   and n :: "nat" shows "coeff (reflect_poly p) n = (if degree p < n then 0::'b else coeff p (degree p - n))"
fls_pow_subdegree_ge: fixes f :: "'a fls"   and n :: "nat" assumes "f ^ n \<noteq> 0" shows "int n * fls_subdegree f \<le> fls_subdegree (f ^ n)"
Max_insert: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Max less_eq (insert x A) = ord.max less_eq x (linorder.Max less_eq A)"
assms_4: shows "real c < W B\<^sub>1 + w u"
Compl_eq_Diff_UNIV: fixes A :: "'a set" shows "- A = UNIV - A"
distinct_6: fixes x41 :: "'d"   and x42 :: "'c"   and x43 :: "('a, 'b, 'c, 'd) Tree"   and x1 :: "('a, 'b, 'c, 'd) Tree set['a]" shows "tAct x41 x42 x43 \<noteq> tConj x1"
ctr: fixes g1 :: "'d \<Rightarrow> 'c"   and g2 :: "'d \<Rightarrow> 'd"   and a :: "'d" shows "unfold_stream g1 g2 a = g1 a ## unfold_stream g1 g2 (g2 a)"
dflt_cmp_2inv: fixes cmp :: "'a \<Rightarrow> 'b \<Rightarrow> comp_res" shows "dflt_cmp (comp2le cmp) (comp2lt cmp) = cmp"
integral_eq_zero_AE: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "AE x in M. f x = (0::'c)" shows "integral\<^sup>L M f = (0::'c)"
b_2: shows "q = 0 \<or> b' \<in> sqrt_irrat"
mersenne_mod_less': fixes k :: "int"   and n :: "nat" assumes "k \<le> 5 * 2 ^ n" shows "mersenne_mod k n < 2 ^ n + 5"
SMP_D: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "t \<in> SMP (set M)"   and "is_Fun t"   and "finite_SMP_representation arity Ana \<Gamma> M" shows "(\<exists>f. t = Fun f []) \<and> (\<exists>m\<in>set M. \<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t = m \<cdot> \<delta>) \<or> (\<exists>m\<in>set M. \<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t = m \<cdot> \<delta> \<and> is_Fun m)"
sup_state_opt_any_None: fixes P :: "'b prog"   and s :: "(ty list \<times> ty err list) option" shows "P \<turnstile> s \<le>' None = (s = None)"
return_wp_rule: fixes Q :: "'b \<Rightarrow> assn"   and x :: "'b" shows "<Q x> return x <Q>"
tendsto_mult: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and b :: "'c" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x * g x) \<longlongrightarrow> a * b) F"
create_opt_solutions_correct: fixes X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "x \<in> max_lp A b c"
rb_spp_aux_2: fixes fs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and rword_strict :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and dgrad :: "'a \<Rightarrow> nat"   and z :: "nat" assumes "is_strict_rewrite_ord fs rword_strict"   and "dickson_grading dgrad"   and "distinct fs"   and "0 \<notin> set fs" shows "ideal (set (map snd (fst (fst (rb_spp_aux fs rword_strict (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z)))))) = ideal (set fs)"
composability: fixes middle :: "nat \<Rightarrow> ('h + 'i, 'j + 'k) resource"   and ideal :: "nat \<Rightarrow> ('l + 'i, 'm + 'k) resource"   and sim_inner :: "nat \<Rightarrow> ('h, 'j, 'l, 'm) converter"   and \<I>_middle :: "nat \<Rightarrow> ('h, 'j) \<I>"   and \<I>_inner :: "nat \<Rightarrow> ('l, 'm) \<I>"   and \<I>_common :: "nat \<Rightarrow> ('i, 'k) \<I>"   and bound_inner :: "nat \<Rightarrow> enat"   and lossless_inner :: "bool"   and w1 :: "nat \<Rightarrow> ('l \<Rightarrow> 'h) \<times> ('j \<Rightarrow> 'm)"   and real :: "nat \<Rightarrow> ('n + 'i, 'o + 'k) resource"   and sim_outer :: "nat \<Rightarrow> ('n, 'o, 'h, 'j) converter"   and \<I>_real :: "nat \<Rightarrow> ('n, 'o) \<I>"   and bound_outer :: "nat \<Rightarrow> enat"   and lossless_outer :: "bool"   and w2 :: "nat \<Rightarrow> ('h \<Rightarrow> 'n) \<times> ('o \<Rightarrow> 'j)"   and bound_sim :: "nat \<Rightarrow> enat" assumes "constructive_security middle ideal sim_inner \<I>_middle \<I>_inner \<I>_common bound_inner lossless_inner w1"   and "constructive_security real middle sim_outer \<I>_real \<I>_middle \<I>_common bound_outer lossless_outer w2"   and "\<And>\<eta>. interaction_any_bounded_converter (sim_outer \<eta>) (bound_sim \<eta>)"   and "\<And>\<eta>. bound_outer \<eta> * max (bound_sim \<eta>) 1 \<le> bound_inner \<eta>"   and "\<And>\<eta>. lossless_inner \<Longrightarrow> plossless_converter (\<I>_real \<eta>) (\<I>_middle \<eta>) (sim_outer \<eta>)" shows "constructive_security real ideal (\<lambda>\<eta>. sim_outer \<eta> \<odot> sim_inner \<eta>) \<I>_real \<I>_inner \<I>_common bound_outer (lossless_outer \<or> lossless_inner) (\<lambda>\<eta>. w1 \<eta> \<circ>\<^sub>w w2 \<eta>)"
document_ptr_kinds_commutes: fixes document_ptr :: "(_) document_ptr"   and h :: "(_) heap" shows "(cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^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 document_ptr |\<in>| object_ptr_kinds h) = (document_ptr |\<in>| document_ptr_kinds h)"
cutoff_internal': fixes T :: "tree"   and red :: "bool list \<Rightarrow> bool"   and ds :: "bool list" assumes "anybranch T (\<lambda>p. red (ds @ p))" shows "anyinternal (cutoff red ds T) (\<lambda>p. \<not> red (ds @ p))"
evalOP_sound: fixes P :: "db"   and V :: "db"   and \<rho> :: "nat \<rightarrow> ValD" assumes "P \<Down> V" shows "evalDdb P\<cdot>\<rho> = evalDdb V\<cdot>\<rho>"
simps_2: fixes r :: "'b llist \<rightarrow> 'b llist \<rightarrow> 'b llist"   and x :: "'b"   and xs :: "'b llist" shows "lrev_body3\<cdot>r\<cdot>(x :@ xs) = r\<cdot>xs oo list2H\<cdot>(x :@ lnil)"
left_idem: fixes a :: "'a"   and b :: "'a" shows "lcm a (lcm a b) = lcm a b"
LT6: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<diamond>F \<longrightarrow> (F \<leadsto> G) \<longrightarrow> \<diamond>G"
rational_algebraic_int_is_int: fixes x :: "'a" assumes "algebraic_int x"   and "x \<in> \<rat>" shows "x \<in> \<int>"
nat_mult_eq_cancel_disj: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(k * m = k * n) = (k = 0 \<or> m = n)"
Bij_imp_funcset: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "f \<in> Bij S" shows "f \<in> S \<rightarrow> S"
proots_ball_card_def: fixes p :: "complex poly"   and z0 :: "complex"   and r :: "real" shows "proots_ball_card p z0 r = card (proots_within p (ball z0 r))"
power_up_mono: fixes a :: "real"   and b :: "real"   and n :: "nat"   and prec :: "nat" assumes "0 \<le> a \<and> a \<le> b \<or> odd n \<and> a \<le> b \<or> even n \<and> a \<le> 0 \<and> b \<le> a" shows "power_up prec a n \<le> power_up prec b n"
evContainsExSuc_containsEx: fixes fs :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and body :: "formula" assumes "branch subs (pseq fs) f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "EV (contains f (Suc i, FAll Neg body))" shows "EV (contains f (i, FAll Neg body))"
in_set_concat_idx: fixes x :: "'d"   and xss :: "'d list list" assumes "x \<in> set (concat xss)" shows "\<exists>i<length xss. x \<in> set (xss ! i)"
simps_1: fixes cfg :: "('p, 'v, 's) configuration"   and fStepCfg :: "('p, 'v, 's) configuration list \<Rightarrow> ('p, 'v) message list \<Rightarrow> ('p, 'v, 's) configuration list"   and fStepMsg :: "('p, 'v, 's) configuration list \<Rightarrow> ('p, 'v) message list \<Rightarrow> ('p, 'v) message list" shows "infiniteExecutionCfg cfg fStepCfg fStepMsg 0 = [cfg]"
push_bit_int_def: fixes n :: "nat"   and k :: "int" shows "push_bit n k = k * 2 ^ n"
ed_ou_not_discrete_card: fixes X :: "'b set" assumes "ed_ou X"   and "\<not> discrete X" shows "card {f. CK_nf_pos f} = 4"
pr_sesk_passwd: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat"   and run :: "nat"   and X :: "agent"   and K :: "rat"   and m :: "nat" assumes "(evs, S, A, U) \<in> protocol"   and "Says n run 5 X (Card n) (Crypt (sesK K) (Passwd m)) \<in> set evs" shows "Key (sesK K) \<in> U"
assms_3: fixes a :: "'b" assumes "QI a" shows "PI a"
inverse_positive_iff_positive: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (inverse a) = less zero a"
small_circle_def: shows "small_circle = part_circlepath 0 r pi 0"
\<E>: shows "\<E> (TER g) - A \<Gamma> = \<E> (TER f) - A \<Gamma>"
path_component_of_eq_overlap: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "(path_component_of X x = path_component_of X y) = (x \<notin> topspace X \<and> y \<notin> topspace X \<or> path_component_of_set X x \<inter> path_component_of_set X y \<noteq> {})"
assms: shows "GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj n A)) (trms\<^sub>l\<^sub>s\<^sub>t (proj m A)) Sec"
smallstepI_ps_emptyD_1: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> I\<rightarrow> \<lless>[], e'\<ggreater>" shows "\<lless>[], e\<ggreater> I\<rightarrow> \<lless>[], e'\<ggreater>"
comp_corr: fixes l :: "nat"   and s :: "item list"   and e :: "expr"   and cs :: "instr list" assumes "isFresh l s" shows "exec (cmp l e @ cs) s = conv cs s (eval e)"
group_card_non_zero_groups_of_size_min: fixes g :: "'a set"   and a :: "nat" assumes "g \<in> \<G>"   and "card g = a" shows "1 \<le> groups_of_size a"
SUPSEQ_union: fixes A :: "'b list set"   and B :: "'b list set" shows "SUPSEQ (A \<union> B) = SUPSEQ A \<union> SUPSEQ B"
r_one: fixes x :: "int" assumes "x \<in> carrier R" shows "x \<oplus> \<zero> = x"
eq_dual_iff: fixes x :: "'d"   and y :: "'d" shows "(dual x = dual y) = (x = y)"
tt': shows "t \<^bsub>\<^esub>\<bowtie>\<^bsub>a\<^esub> t' \<in> SPR.jkbpC"
le_N: fixes a :: "'a" assumes "a \<in> set as"   and "norm a = 1" shows "length (filter ((=) a) as) \<le> N"
b_accessible_is_accs: fixes q :: "'a"   and TA :: "('a, 'b, 'c) tree_automaton_rec_scheme" assumes "q \<in> b_accessible (ta_rules TA)" obtains t :: "'b tree" where "accs (ta_rules TA) t q"
iso_assoc: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "local.iso \<a>[f, g, h]"
homotopic_loops_refl: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c" shows "homotopic_loops s p p = (path p \<and> path_image p \<subseteq> s \<and> pathfinish p = pathstart p)"
nxt_many_calls: fixes x :: "'b"   and x' :: "'b" shows "nxt (many_calls x) x' = (if x' = x then many_calls x else TTree.empty)"
dim_col_transfer_rule: fixes A :: "'a mat"   and A' :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" assumes "HMA_M A A'" shows "dim_col A = CARD('b)"
dist: fixes L :: "(nat \<times> int \<times> nat) list" assumes "fromlist L" shows "distinct (map \<alpha> L)"
IH: assumes "j < Array.length h a" shows "Array.length h a = Array.length h' a \<and> Ref.get h' i \<le> j \<and> mset ((Array.get h a)[j := key]) = mset ((Array.get h' a)[Ref.get h' i := key]) \<and> ini h a j = both h' a j i \<and> sorted (both h' a j i) \<and> (\<forall>a\<in>set (right h' a j i). key < a)"
eqvt: fixes p :: "perm"   and x :: "'a"   and y :: "'b"   and z :: "'c" shows "p \<bullet> f x y z = f (p \<bullet> x) (p \<bullet> y) (p \<bullet> z)"
subst_lsst_unlabel: fixes A :: "('c strand_label \<times> ('a, 'b) stateful_strand_step) list"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "unlabel (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) = unlabel A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>"
inversions_imp_less_2: fixes z :: "nat \<times> nat"   and xs :: "'a list" assumes "z \<in> inversions xs" shows "snd z < length xs"
subset_T: fixes t1 :: "real" assumes "t1 \<in> T" shows "{t0--t1} \<subseteq> T"
rel_distinct_1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and y21 :: "'b"   and y22 :: "'b list" shows "\<not> list_all2 R [] (y21 # y22)"
Right_Cosets_cardinality: fixes x :: "'a"   and y :: "'a" assumes "x \<in> G"   and "y \<in> G" shows "card (H |\<cdot> x) = card (H |\<cdot> y)"