hcju/isabelleps huggingface dataset:
assms_3: shows "bijective (e * top) \<and> bijective (e[[top]])"
transpose_transpose: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.transpose (Finite_Cartesian_Product.transpose A) = A"
sinks_aux_union: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and V :: "'b set"   and xs :: "'c list" shows "sinks_aux I D (U \<union> V) xs = sinks_aux I D U xs \<union> sinks_aux I D V (ipurge_tr_aux I D U xs)"
nsumTail: fixes A :: "('c, 'd) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "aGroup A"   and "\<forall>j\<le>Suc n. f j \<in> carrier A" shows "\<Sigma>\<^sub>e A f Suc n = \<Sigma>\<^sub>e A (\<lambda>i. f (Suc i)) n \<plusminus>\<^bsub>A\<^esub> f 0"
simps_17: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x1 :: "'b" shows "(case Fo x1 of Fo x \<Rightarrow> f1 x | Z x \<Rightarrow> f2 x | Less x xa \<Rightarrow> f3 x xa | In x xa \<Rightarrow> f4 x xa) = f1 x1"
drop_bit_nat_eq: fixes n :: "nat"   and k :: "int" shows "drop_bit n (nat k) = nat (drop_bit n k)"
gupd_def: fixes r :: "oref"   and obj :: "obj" shows "gupd(r\<mapsto>obj) = case_st (\<lambda>g. st (g(r \<mapsto> obj)))"
qPsubst_preserves_qGood: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "qGoodEnv rho"   and "qGood X" shows "qGood X #[[rho]]"
div_write_new_val_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and i :: "sparc_operation \<times> inst_operand list"   and result :: "word32"   and temp_V :: "word1"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (div_write_new_val i result temp_V s1)) \<and> t2 = snd (fst (div_write_new_val i result temp_V s2))" shows "low_equal t1 t2"
view_ext_def: fixes ext :: "real_int"   and lan :: "nat_int"   and own :: "cars"   and more :: "'a" shows "\<lparr>ext = ext, lan = lan, own = own, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons view_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple ext (Record.iso_tuple_cons Record.tuple_iso_tuple lan own)) more"
p1span: fixes x :: "'a" shows "p1 x \<in> span T"
least_power_le: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "(f ^^ n) x = x"   and "0 < n" shows "least_power f x \<le> n"
idinfo_mpair_fst: fixes n :: "nat"   and X :: "msg"   and Y :: "msg"   and s :: "(agent \<times> msg) set" assumes "(Spy, \<lbrace>\<langle>n, X\<rangle>, Y\<rbrace>) \<in> s"   and "s\<^sub>0 \<Turnstile> s" shows "\<langle>n, X\<rangle> \<in> spied s"
subgroup_incl: fixes G :: "('c, 'd) monoid_scheme"   and I :: "'c set"   and J :: "'c set" assumes "Group.group G"   and "subgroup I G"   and "subgroup J G"   and "I \<subseteq> J" shows "subgroup I (G\<lparr>carrier := J\<rparr>)"
sel_3: fixes x1 :: "'b" shows "args (Var x1) = []"
f_def: fixes x :: "real" shows "f x = poly p x / x ^ degree p"
sum_complement_in_complement_product: fixes x :: "i"   and y :: "i" assumes "x \<oplus> y \<noteq> u" shows "P (\<midarrow> (x \<oplus> y)) (\<midarrow> x \<otimes> \<midarrow> y)"
subst_typ_nil: fixes T :: "typ" shows "subst_typ [] T = T"
prems_3: shows "\<I> \<turnstile>g gpv \<surd>"
irreducibleD: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and b :: "'a" assumes "Divisibility.irreducible G a"   and "Divisibility.properfactor G b a"   and "b \<in> carrier G" shows "b \<in> Units G"
proots_pcompose: fixes p :: "'a poly"   and q :: "'a poly"   and s :: "'a set" assumes "p \<noteq> 0"   and "degree q = 1" shows "proots_count (p \<circ>\<^sub>p q) s = proots_count p (poly q ` s)"
suminf_ennreal: fixes f :: "nat \<Rightarrow> real" assumes "\<And>i. 0 \<le> f i"   and "(\<Sum>i. ennreal (f i)) \<noteq> \<top>" shows "(\<Sum>i. ennreal (f i)) = ennreal (\<Sum>i. f i)"
eval_eqI: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.VPar t u"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>" shows "\<lbrace>t\<rbrace> = \<lbrace>u\<rbrace>"
inconsistent_updates: fixes t1 :: "transition"   and t2 :: "transition"   and r :: "nat \<Rightarrow>f value option" assumes "\<exists>p2 p1. (\<exists>i. posterior_separate (Arity t1) (Guards t1) (Updates t2) i r = Some p2 \<and> posterior_separate (Arity t1) (Guards t1) (Updates t1) i r = Some p1) \<and> (\<exists>r' P. P (p2 $ r') \<and> (\<exists>y. p1 $ r' = Some y) \<and> \<not> P (p1 $ r'))" shows "\<not> subsumes t2 r t1"
rel_Grp: fixes A :: "'i set"   and f :: "'i \<Rightarrow> 'j" shows "rel_fun (=) (BNF_Def.Grp A f) = BNF_Def.Grp {x. range x \<subseteq> A} ((\<circ>) f)"
inv_part_0_Suc: fixes m_j :: "nat"   and p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat" assumes "m_j < n" shows "local.inv_part p A 0 m_j = local.inv_part p A n (Suc m_j)"
scaleR_vec_def: shows "(*\<^sub>R) \<equiv> \<lambda>r x. \<chi>i. r *\<^sub>R x $ i"
2: shows "prv (imp (cnj (eqv \<phi> \<chi>) (eqv \<chi> \<psi>)) (imp \<psi> \<phi>))"
s: fixes z :: "complex" assumes "z \<in> cball \<xi> s - {\<xi>}" shows "f z \<noteq> 0"
ipresIGFreshAll_def: fixes h :: "'w \<Rightarrow> 'x"   and hA :: "'y \<Rightarrow> 'z"   and MOD :: "('aa, 'ab, 'ac, 'ad, 'ae, 'af, 'w, 'y, 'ag) model_scheme"   and MOD' :: "('ah, 'ai, 'ac, 'aj, 'ak, 'af, 'x, 'z, 'al) model_scheme" shows "ipresIGFreshAll h hA MOD MOD' \<equiv> ipresIGFresh h MOD MOD' \<and> ipresIGFreshAbs hA MOD MOD'"
strategyproofness_2: shows "pmf (sds R29) d + pmf (sds R29) c \<le> pmf (sds R36) d + pmf (sds R36) c"
E'_def: fixes x :: "'a"   and y :: "'a" shows "E' x y \<equiv> x \<rightarrow> y \<and> V x \<and> V y"
A: shows "D x \<in> sinks I D u xs \<longrightarrow> (u, D x) \<in> I \<or> (\<exists>v\<in>sinks I D u xs. (v, D x) \<in> I)"
k_2: shows "x - real_of_int (k * 2) * pi \<le> pi"
simps_2: fixes A :: "real mat"   and m :: "real mat convnet" shows "tensors_from_net (Conv A m) = mat_tensorlist_mult A (tensors_from_net m) (input_sizes m)"
qTermLess_def: shows "qTermLess \<equiv> {uu_. \<exists>X delta inp binp i. uu_ = (termIn X, termIn (qOp delta inp binp)) \<and> inp i = Some X} \<union> {uu_. \<exists>A delta inp binp i. uu_ = (absIn A, termIn (qOp delta inp binp)) \<and> binp i = Some A} \<union> {(termIn X, absIn (qAbs xs x X)) |X xs x. True}"
wb_lens_axioms_def: fixes x :: "'b \<Longrightarrow> 'c" shows "wb_lens_axioms x \<equiv> \<forall>\<sigma>. put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
Bernstein_changes_eq_rescale: fixes c :: "real"   and d :: "real"   and P :: "real poly"   and p :: "nat" assumes "c \<noteq> d"   and "degree P \<le> p" shows "Bernstein_changes p c d P = Bernstein_changes_01 p (P \<circ>\<^sub>p [:c, 1:] \<circ>\<^sub>p [:0, d - c:])"
is_min: shows "is_minimal_basis (set (a # xs @ ys))"
keys_lowerPowers: fixes b :: "nat"   and i :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 'a" shows "keys (lowerPowers b i m) = inc_above b i ` keys m"
forget_subst_tm: fixes a :: "name"   and tm :: "tm"   and x :: "tm" assumes "atom a \<sharp> tm" shows "subst a x tm = tm"
alpha_Tree_induct: fixes t1 :: "('b, 'c, 'd) Tree"   and t2 :: "('b, 'c, 'd) Tree"   and P :: "('b, 'c, 'd) Tree \<Rightarrow> ('b, 'c, 'd) Tree \<Rightarrow> bool" assumes "t1 =\<^sub>\<alpha> t2"   and "\<And>tset1 tset2. \<lbrakk>\<And>a b. \<lbrakk>a \<in> set_bset tset1; b \<in> set_bset tset2; a =\<^sub>\<alpha> b\<rbrakk> \<Longrightarrow> P a b; rel_bset (=\<^sub>\<alpha>) tset1 tset2\<rbrakk> \<Longrightarrow> P (tConj tset1) (tConj tset2)"   and "\<And>t1 t2. \<lbrakk>t1 =\<^sub>\<alpha> t2; P t1 t2\<rbrakk> \<Longrightarrow> P (tNot t1) (tNot t2)"   and "\<And>\<phi>. P (tPred \<phi>) (tPred \<phi>)"   and "\<And>\<alpha>1 t1 \<alpha>2 t2. \<lbrakk>\<And>p. p \<bullet> t1 =\<^sub>\<alpha> t2 \<Longrightarrow> P (p \<bullet> t1) t2; \<And>a b. (a \<rightleftharpoons> b) \<bullet> t1 =\<^sub>\<alpha> t1 \<Longrightarrow> P ((a \<rightleftharpoons> b) \<bullet> t1) t1; \<And>a b. (a \<rightleftharpoons> b) \<bullet> t2 =\<^sub>\<alpha> t2 \<Longrightarrow> P ((a \<rightleftharpoons> b) \<bullet> t2) t2; \<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)\<rbrakk> \<Longrightarrow> P (tAct \<alpha>1 t1) (tAct \<alpha>2 t2)" shows "P t1 t2"
top_def: shows "DRA_Models.top \<equiv> \<lambda>x. UNIV"
Units_def: shows "Units = {x. \<exists>k. (1::'a) = x * k}"
dual_MonoTran_def: shows "dual \<equiv> map_fun Rep_MonoTran Abs_MonoTran dual_fun"
le_infI: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and a :: "'b::type"   and b :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (a::'b::type)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (b::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (b::'b::type))"
CR_join_left_I: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "CR r"   and "(x, y) \<in> r\<^sup>*"   and "(x, z) \<in> r\<^sup>\<down>" shows "(y, z) \<in> r\<^sup>\<down>"
simps_1: fixes step :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and A :: "('a \<Rightarrow> 'b pmf) \<times> ('a \<times> 'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) pmf)"   and s :: "('a \<times> 'b) pmf" shows "On_Off.config'_rand step A s [] = s"
ts_c_def_1: shows "tn _> tm = tfn _> tfm"
ipresSwapAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'g, 'h, 'var, 'i, 'd, 'j) model_scheme" shows "ipresSwapAbs hA MOD \<equiv> \<forall>zs z1 z2 us s A. wlsAbs (us, s) A \<longrightarrow> hA (A $[z1 \<and> z2]_zs) = igSwapAbs MOD zs z1 z2 (hA A)"
HRB_slice_cases: fixes x :: "'node SDG_node"   and S :: "'node SDG_node set"   and P :: "'node SDG_node \<Rightarrow> bool" assumes "x \<in> HRB_slice S"   and "\<And>n nx. \<lbrakk>n \<in> sum_SDG_slice1 nx; nx \<in> S\<rbrakk> \<Longrightarrow> P n"   and "\<And>nx n' n'' p n. \<lbrakk>n' \<in> sum_SDG_slice1 nx; n'' s-p\<rightarrow>\<^bsub>ret\<^esub> CFG_node (parent_node n'); n \<in> sum_SDG_slice2 n'; nx \<in> S\<rbrakk> \<Longrightarrow> P n" shows "P x"
l_get_dom_component_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and create_document :: "((_) heap, exception, (_) document_ptr) prog" shows "l_get_dom_component_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr heap_is_wellformed parent_child_rel type_wf known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_element_by_id get_elements_by_tag_name create_document \<equiv> l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name \<and> l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_document"
union_open: fixes M :: "'a set set" assumes "\<forall>m\<in>M. m open" shows "\<Union> M open"
enc_weakly_respects_barb_set_iff_source_target_rel: fixes Barbs :: "'barbs set" shows "((\<forall>S x. x \<in> Barbs \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x)) \<longrightarrow> x \<in> Barbs \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x))) \<and> (\<forall>S x. x \<in> Barbs \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x)) \<longrightarrow> x \<in> Barbs \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x)))) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_weakly_respects_barb_set Rel (STCalWB SWB TWB) Barbs)"
clique_def: fixes V :: "'b set"   and E :: "'b set set" shows "clique V E = (\<forall>v\<in>V. \<forall>w\<in>V. v \<noteq> w \<longrightarrow> {v, w} \<in> E)"
parents_single: fixes d :: "nat"   and b :: "(nat \<times> int) list" shows "parents d b b = {b}"
one_ge_zero: shows "(0::'a) \<le> (1::'a)"
1: fixes s :: "'c"   and s' :: "'b set"   and c :: "'g \<Rightarrow> bool"   and R\<sigma> :: "('g \<times> 'h set) set"   and fa__ :: "'a \<Rightarrow> 'g \<Rightarrow> 'g"   and f :: "'b \<Rightarrow> 'h"   and \<sigma> :: "'g" assumes "(s, s') \<in> \<langle>Rk\<rangle>Rs1"   and "(c, \<lambda>_. True) \<in> R\<sigma> \<rightarrow> bool_rel"   and "\<And>x y. \<lbrakk>(x, y) \<in> Rk; y \<in> s'\<rbrakk> \<Longrightarrow> (fa__ x, insert (f y)) \<in> R\<sigma> \<rightarrow> R\<sigma>"   and "(\<sigma>, {}) \<in> R\<sigma>" shows "(foldli (it1 s) c fa__ \<sigma>, f ` s') \<in> R\<sigma>"
R_ge: fixes \<alpha> :: "complex" assumes "\<alpha> \<in> Roots'" shows "cmod \<alpha> \<le> R"
exec_gpv_oracle_bisim_bad_full: fixes X :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and s1 :: "'a"   and s2 :: "'b"   and bad1 :: "'a \<Rightarrow> bool"   and bad2 :: "'b \<Rightarrow> bool"   and oracle1 :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf"   and oracle2 :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and gpv :: "('e, 'c, 'd) gpv"   and R :: "'e \<times> 'a \<Rightarrow> 'e \<times> 'b \<Rightarrow> bool" assumes "X s1 s2"   and "bad1 s1 = bad2 s2"   and "\<And>s1 s2 x. X s1 s2 \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (\<not> bad2 s2' \<longrightarrow> a = b \<and> X s1' s2')) (oracle1 s1 x) (oracle2 s2 x)"   and "callee_invariant oracle1 bad1"   and "callee_invariant oracle2 bad2"   and "\<And>s1 x. bad1 s1 \<Longrightarrow> lossless_spmf (oracle1 s1 x)"   and "\<And>s2 x. bad2 s2 \<Longrightarrow> lossless_spmf (oracle2 s2 x)"   and "lossless_gpv \<I>_full gpv"   and "\<And>a s1 b s2. \<lbrakk>bad1 s1 = bad2 s2; \<not> bad2 s2 \<Longrightarrow> a = b \<and> X s1 s2\<rbrakk> \<Longrightarrow> R (a, s1) (b, s2)" shows "rel_spmf R (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
step: shows "\<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s')"
inv_solve_left: fixes a :: "Vertex \<Rightarrow> Vertex"   and b :: "Vertex \<Rightarrow> Vertex"   and c :: "Vertex \<Rightarrow> Vertex" assumes "a \<in> carrier tetrahedral_group"   and "b \<in> carrier tetrahedral_group"   and "c \<in> carrier tetrahedral_group" shows "(a = inv\<^bsub>tetrahedral_group\<^esub> b \<otimes>\<^bsub>tetrahedral_group\<^esub> c) = (c = b \<otimes>\<^bsub>tetrahedral_group\<^esub> a)"
\<E>_sequential_completion: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and r :: "nat" assumes "E \<in> \<E>"   and "P \<turnstile> (E, ws) \<surd>"   and "\<And>a. \<lbrakk>a < r; a \<in> read_actions E\<rbrakk> \<Longrightarrow> P,E \<turnstile> a \<leadsto>mrw ws a" shows "\<exists>E'\<in>\<E>. \<exists>ws'. P \<turnstile> (E', ws') \<surd> \<and> ltake (enat r) E = ltake (enat r) E' \<and> sequentially_consistent P (E', ws') \<and> action_tid E r = action_tid E' r \<and> action_obs E r \<approx> action_obs E' r \<and> (r \<in> actions E \<longrightarrow> r \<in> actions E')"
proj_def: fixes \<sigma> :: "'tp"   and a :: "univ" shows "proj \<sigma> a \<equiv> if Ik_intT \<sigma> a then a else Ik.pickT \<sigma>"
to_cnf2_not_0: fixes x :: "ordinal" assumes "0 < x" shows "to_cnf2 x = (oLog \<omega> x, inv ordinal_of_nat (x div \<omega> ** oLog \<omega> x)) # to_cnf2 (x mod \<omega> ** oLog \<omega> x)"
labeled_tiebreaker_lifting_axioms: shows "labeled_tiebreaker_lifting Bot_F Inf_F Bot_G (\<Turnstile>G) Inf_G Red_I_G Red_F_G \<G>_F \<G>_I (\<sqsubset>) Inf_FL"
yz: shows "infdist w Gyz \<le> 4 * deltaG TYPE('a)"
arr_cod: fixes f :: "'a \<times> 'a" assumes "CC.arr f" shows "CC.arr (CC.cod f)"
strong_fusion: fixes F :: "i \<Rightarrow> bool" assumes "GEM1"   and "\<exists>x. F x" shows "\<exists>x. (\<forall>y. F y \<longrightarrow> P y x) \<and> (\<forall>y. P y x \<longrightarrow> (\<exists>z. F z \<and> O y z))"
section_def: fixes f :: "'a \<times> 'a" shows "VV.section f \<equiv> \<exists>g. VV.ide (VV.comp g f)"
Suc: shows "\<Union> (extend A S f ` {..m}) = extend A S f m"
map_nth_Joints: fixes xs :: "'b list"   and XS :: "('b \<times> 'b pdevs) list"   and "is" :: "nat list" assumes "xs \<in> Joints XS"   and "\<And>i. i \<in> set is \<Longrightarrow> i < length XS" shows "map ((!) xs) is @ xs \<in> Joints (map ((!) XS) is @ XS)"
choose_arc_bot_iff: fixes x :: "'a" shows "(choose_arc x = bot) = (x = bot)"
sfs: assumes "M = clinit" shows "sh C = \<lfloor>(sfs, Processing)\<rfloor>"
simps_2: fixes \<sigma> :: "'tp" shows "GE_wtFsym (Wit \<sigma>) = (\<not> isRes \<sigma> \<or> protCl \<sigma>)"
lsz: shows "\<forall>x\<in>set ls. x \<sqsubseteq>\<^bsub>r\<^esub> z"
comp_red_basis_adds: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "p \<in> set xs"   and "p \<noteq> 0" obtains q :: "'t \<Rightarrow>\<^sub>0 'c" where "q \<in> set (comp_red_basis xs)" and "lt q adds\<^sub>t lt p"
natural_transformation_axioms: shows "natural_transformation B D (K \<circ> F) (L \<circ> G) (\<sigma> \<circ> \<tau>)"
lincomb_list_eq_0: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and vs :: "'d list"   and c :: "nat \<Rightarrow> 'b" assumes "Module.module R M"   and "\<And>i. i < length vs \<Longrightarrow> c i \<odot>\<^bsub>M\<^esub> vs ! i = \<zero>\<^bsub>M\<^esub>" shows "module.lincomb_list M c vs = \<zero>\<^bsub>M\<^esub>"
get_offending_flows_def: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" shows "get_offending_flows M G = (\<Union>m\<in>set M. c_offending_flows m G)"
weak_reduction_bisimulation_and_closures_1: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_bisimulation Rel Cal" shows "weak_reduction_bisimulation (Rel\<^sup>=) Cal"
simps: fixes l :: "'b"   and sl :: "'a list" shows "ssort' l sl = (if is_empty l then sl else let (m, l') = remove_max l in ssort' l' (m # sl))"
map_poly_simps: fixes f :: "'c \<Rightarrow> 'b"   and c :: "'c"   and p :: "'c poly" shows "map_poly f (pCons c p) = (if c = (0::'c) \<and> p = 0 then 0 else pCons (f c) (map_poly f p))"
assms_5: fixes q :: "'a list"   and r :: "'a list" assumes "q \<in> carrier (K [X])"   and "r \<in> carrier (K [X])"   and "p = q \<otimes>\<^bsub>K [X]\<^esub> r" shows "q \<in> Units (K [X]) \<or> r \<in> Units (K [X])"
nth_rotate_conv_nth1_conv_nth: fixes m :: "nat"   and xs :: "'a list" assumes "m < length xs" shows "rotate1 xs ! m = xs ! (Suc m mod length xs)"
invariant_def: fixes M :: "'a set" shows "invariant M = (\<forall>x\<in>M. trapped x M)"
assms: shows "qGood (X #[[x \<and> y]]_zs)"
mset_heap_empty_iff: fixes h :: "'b heap" shows "(mset_heap h = {#}) = (h = Empty)"
ennreal_enn2real_if: fixes r :: "ennreal" shows "ennreal (enn2real r) = (if r = top then 0 else r)"
iT_Plus_finite_iff: fixes I :: "nat set"   and k :: "nat" shows "finite (I \<oplus> k) = finite I"
beta_coerce_cfun: fixes f :: "'c \<rightarrow> 'd"   and x :: "'b" shows "COERCE('c \<rightarrow> 'd, 'b \<rightarrow> 'a)\<cdot>f\<cdot>x = COERCE('d, 'a)\<cdot>(f\<cdot>(COERCE('b, 'c)\<cdot>x))"
n\<^sub>0'_def: shows "n\<^sub>0' = max n\<^sub>0 (nat \<lceil>(max 0 b + max 0 d + 2) / (1 - a - c) + 1\<rceil>)"
map_spmf_eq_map_spmfI: fixes f :: "'b \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and p :: "'b spmf"   and q :: "'d spmf" assumes "rel_spmf (\<lambda>x y. f x = g y) p q" shows "map_spmf f p = map_spmf g q"
set_bd_2: fixes x :: "('e, 'd, 'f) F2" shows "|F2set2 x| \<le>o bd_F2"
dual_star_MonoTran_def: fixes x :: "'b MonoTran" shows "x ^ \<otimes> = ((x ^ o) ^ *) ^ o"
disjSD_no_p_in_both: fixes x :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and y :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "disjSD_2 x y"   and "\<not> member DenyAll x"   and "\<not> member DenyAll y"   and "p \<in> dom (Cp x)"   and "p \<in> dom (Cp y)" shows "False"
ge: fixes x :: "real" assumes "0 < x"   and "x < 1" shows "0 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z))"
harm_mono: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "harm m \<le> harm n"
alw: shows "epath steps \<and> fst (shd steps) \<in> S"
distinct_29: fixes x51 :: "'b formula"   and x52 :: "'b formula"   and x61 :: "'b formula"   and x62 :: "'b formula" shows "(x51::'b formula) \<^bold>\<or> (x52::'b formula) \<noteq> (x61::'b formula) \<^bold>\<rightarrow> (x62::'b formula)"
lstrict_prefix_lappend_conv: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lstrict_prefix xs (lappend xs ys) = (lfinite xs \<and> \<not> lnull ys)"
all_not_in_conv: fixes A :: "'f set" shows "(\<forall>x. x \<notin> A) = (A = {})"