hcju/isabelleps huggingface dataset:
sound_mode_use_def: shows "abs.sound_mode_use \<equiv> \<lambda>uu. case uu of (cms, mem) \<Rightarrow> list_all (\<lambda>cm. abs.locally_sound_mode_use (cm, mem)) cms \<and> abs.globally_sound_mode_use (cms, mem)"
bigo_plus_idemp: fixes f :: "'c \<Rightarrow> 'd" shows "O(f) + O(f) = O(f)"
monom_mult_poly: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and mc :: "'d monom \<times> 'c"   and p :: "('d monom \<times> 'c) list" shows "eval_poly \<alpha> (monom_mult_poly mc p) = eval_monomc \<alpha> mc * eval_poly \<alpha> p"
select_convs_1: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cLengthF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cLengthF"
oarrivemsg_simps_3: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and ii :: "nat set"   and ni :: "nat set"   and m :: "'e" shows "oarrivemsg P \<sigma> (ii\<not>ni:arrive(m)) = P \<sigma> m"
sup_top_left: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top :: "'a"   and x :: "'a" assumes "class.bounded_lattice_top inf less_eq less sup top" shows "sup top x = top"
span_clauses_1: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and S :: "'b set" assumes "module scale"   and "a \<in> S" shows "a \<in> module.span scale S"
monoid_comm_monoidI: assumes "\<And>x y. \<lbrakk>x \<in> carrier \<G>; y \<in> carrier \<G>\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" shows "Group.comm_monoid \<G>"
insert_remove: fixes A :: "'g set"   and x :: "'g" assumes "finite A" shows "card (insert x A) = Suc (card (A - {x}))"
NS_staticSecret_subset_parts_knows_Spy: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> ns_public" shows "(staticSecret A \<subseteq> parts (knows Spy evs)) = (A \<in> bad)"
simps_2: fixes x :: "'e"   and p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e"   and D :: "'f \<Rightarrow> 'h"   and E :: "'g \<Rightarrow> 'h" assumes "x \<notin> range p"   and "x \<in> range q" shows "con_comp_map D E p q x = Some (E (inv q x))"
cases: fixes s :: "Value \<Rightarrow> CValue \<Rightarrow> bool"   and a1 :: "Value"   and a2 :: "CValue" assumes "similar'_step s a1 a2" obtains "a1 = \<bottom>" and "a2 = \<bottom>"   | b :: "bool discr" where "a1 = B\<cdot>b" and "a2 = CB\<cdot>b"   | f :: "Value \<rightarrow> Value" and g :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" where "a1 = Fn\<cdot>f" and "a2 = CFn\<cdot>g" and "\<And>x y. s x (y\<cdot>C\<^sup>\<infinity>) \<Longrightarrow> s (f\<cdot>x) (g\<cdot>y\<cdot>C\<^sup>\<infinity>)"
dagger_unfoldl_eq: fixes x :: "('a \<times> 'a) set" shows "Id \<union> x O x\<^sup>* = x\<^sup>*"
restrict_current_def: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" shows "g \<upharpoonleft> \<Gamma> / f = (\<lambda>(x, y). g (x, y) * indicator (- RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)) x * indicator (- RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)) y)"
update_by_pair_ko_ntm_def: fixes ko :: "'a nat_term_order" shows "update_by_pair_ko_ntm ko \<equiv> comparator.update_by_pair (nat_term_compare_inv ko)"
corthogonal_inv_def: fixes A :: "'b mat" shows "corthogonal_inv A = mat_of_rows (dim_row A) (map vec_inv (cols A))"
IntD1: fixes c :: "'k"   and A :: "'k set"   and B :: "'k set" assumes "c \<in> A \<inter> B" shows "c \<in> A"
empt_invariant: fixes ks :: "int list"   and l :: "int" shows "dioph_invariant ks l (dioph_empt ks l)"
topology_inverse: fixes y :: "'c set \<Rightarrow> bool" assumes "y \<in> {L. istopology L}" shows "openin (topology y) = y"
intro: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "category A"   and "category B"   and "functor A B F"   and "functor A B G"   and "functor A B H"   and "natural_transformation A B F G \<sigma>"   and "natural_transformation A B G H \<tau>" shows "vertical_composite A B F G H \<sigma> \<tau>"
paint_invc2: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and c :: "tcolor" assumes "invc2 t" shows "invc2 (paint c t)"
union_insert_0: fixes A :: "nat \<Rightarrow> 'b set" shows "\<Union> (range A) = A 0 \<union> \<Union> (A ` {1..})"
realrel_refl: fixes X :: "nat \<Rightarrow> rat" assumes "Real.cauchy X" shows "realrel X X"
theFTOne_def: fixes cl :: "('a, 'b, 'c) cmd list"   and dl :: "('d, 'e, 'f) cmd list" shows "theFTOne cl dl \<equiv> theFT cl \<union> theFT dl"
sorted_map_of_Some_eq: fixes xs :: "('c \<times> 'd) list"   and k :: "'c"   and v :: "'d" assumes "sorted1 xs" shows "(map_of xs k = Some v) = ((k, v) \<in> set xs)"
perp_foot_def: fixes p :: "proj2"   and l :: "proj2_line" shows "perp_foot p l \<equiv> proj2_intersection l (drop_perp p l)"
dfs_eq_rtrancl: fixes y :: "nat"   and x :: "nat" assumes "dfa_is_node A y"   and "dfa_is_node A x" shows "rquot_memb y (dfs (rquot_empt A) [x]) = ((x, y) \<in> (succsr (rquot_succs A n))\<^sup>*)"
subset_insert: fixes x :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "x \<notin> A" shows "(A \<subseteq> insert x B) = (A \<subseteq> B)"
push_relabel_term_imp_maxflow: assumes "\<forall>(u, v)\<in>cf.E. \<not> push_precond f l (u, v)"   and "\<forall>u. \<not> relabel_precond f l u" shows "isMaxFlow f"
exhaust: fixes y :: "'d rexp" obtains x1 :: "'d" where "y = Atom x1"   | x21 :: "'d rexp" and x22 :: "'d rexp" where "y = Alt x21 x22"   | x31 :: "'d rexp" and x32 :: "'d rexp" where "y = Conc x31 x32"   | x4 :: "'d rexp" where "y = Star x4"
prefix_length_less: fixes xs :: "'a list"   and ys :: "'a list" assumes "strict_prefix xs ys" shows "length xs < length ys"
rec_2: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "rec_latin_type f1 f2 f3 Col = f2"
CCollect_ext_def: fixes f :: "'d \<Rightarrow> 'c option"   and p :: "'d \<Rightarrow> bool option" shows "CCollect_ext f p = CCollect p \<bind> (\<lambda>xs. cset_mapM (f `\<^sub>c xs))"
ln_lower_1_def: shows "ln_lower_1 \<equiv> \<lambda>x. 1 - inverse x"
fair_run_is_run: fixes s :: "nat \<Rightarrow> 'a" assumes "is_fair_run s" shows "is_run s"
inner_Basis_eq_zeroI: fixes a :: "'a" assumes "local.inner_Basis a a = 0"   and "finite B"   and "a \<in> S" shows "a = (0::'a)"
assms_2: shows "le_vec (A *v y) (mu *s y)"
iso_Sup_supdistl: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "prod.mono f" shows "Sup \<circ> (`) f \<le> f \<circ> Sup_prod"
assms: shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"
l2_eq: fixes a :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list" assumes "length a = length b" shows "l2 a b = (\<Prod>d<length a. l2_\<phi> (a ! d) (b ! d))"
in_Inf_FL_imp_to_F_in_Inf_F: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Q :: "'c set"   and entails_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G_q :: "'c \<Rightarrow> 'b inference set"   and Red_I_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b inference set"   and Red_F_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set"   and \<G>_F_q :: "'c \<Rightarrow> 'a \<Rightarrow> 'b set"   and \<G>_I_q :: "'c \<Rightarrow> 'a inference \<Rightarrow> 'b inference set option"   and Inf_FL :: "('a \<times> 'd) inference set"   and \<iota> :: "('a \<times> 'd) inference" assumes "labeled_lifting_intersection Bot_F Inf_F Bot_G Q entails_q Inf_G_q Red_I_q Red_F_q \<G>_F_q \<G>_I_q Inf_FL"   and "\<iota> \<in> Inf_FL" shows "labeled_lifting_intersection.to_F \<iota> \<in> Inf_F"
induct: fixes P :: "MIR.num \<Rightarrow> bool"   and a0 :: "MIR.num" assumes "\<And>c. P (C c)"   and "\<And>n c b. P b \<Longrightarrow> P (CN n c b)"   and "\<And>c a b. P b \<Longrightarrow> P (CF c a b)"   and "\<And>v. P (Bound v)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Floor v)" shows "P a0"
Gp_rcs: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "H \<subseteq> K"   and "x \<in> K" shows "H \<bullet>\<^bsub>\<natural>K\<^esub> x = H \<bullet> x"
and_R1_def: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "and_R1 A B = (let (a1, a2) = A; (b1, b2) = B in coin_spmf \<bind> (\<lambda>\<sigma>. let s0 = \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> False); s1 = \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> True); s2 = \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> False); s3 = \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> True) in R1_14_OT (s0, s1, s2, s3) (b1, b2) \<bind> (\<lambda>V. protocol_14_OT (s0, s1, s2, s3) (b1, b2) \<bind> (\<lambda>(uu_, s). return_spmf (((a1, a2), \<sigma>, V), \<sigma>, s)))))"
SReal_complete: fixes P :: "real star set"   and Y :: "real star" assumes "isUb \<real> P Y"   and "P \<subseteq> \<real>"   and "P \<noteq> {}" shows "\<exists>t. isLub \<real> P t"
simps: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat" shows "gram_schmidt_fs.gso n fs i = fs ! i + abelian_monoid.sumlist (module_vec TYPE('a) n) (map (\<lambda>j. - gram_schmidt_fs.\<mu> n fs i j \<cdot>\<^sub>v gram_schmidt_fs.gso n fs j) [0..<i])"
merges_preserves_type: fixes xs :: "'a list"   and n :: "nat"   and ps :: "(nat \<times> 'a) list" assumes "xs \<in> list n A"   and "\<forall>(p, x)\<in>set ps. p < n \<and> x \<in> A" shows "merges f ps xs \<in> list n A"
intro: fixes insert_before :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> (_) node_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and insert_before_locs :: "(_) object_ptr \<Rightarrow> (_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "insert_before = l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before get_parent get_child_nodes set_child_nodes get_ancestors adopt_node set_disconnected_nodes get_disconnected_nodes get_owner_document"   and "insert_before_locs = l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before_locs set_child_nodes_locs adopt_node_locs set_disconnected_nodes_locs" shows "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_child_nodes set_child_nodes set_child_nodes_locs get_ancestors adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs get_disconnected_nodes get_owner_document insert_before insert_before_locs"
eq: shows "lookup (row_to_poly (pps_to_list S) (row A i)) (pps_to_list S ! f i) = A $$ (i, f i)"
4: shows "v ||| l' ## sscan (count cs) v l' = sdrop (length u) (w ||| k ## sscan (count cs) w k)"
simps: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set" shows "orthogonal \<Delta> f S = ((\<forall>x xa. edge \<Delta> x xa \<longrightarrow> x \<in> S \<longrightarrow> xa \<notin> S \<longrightarrow> f (x, xa) = capacity \<Delta> (x, xa)) \<and> (\<forall>x xa. edge \<Delta> x xa \<longrightarrow> x \<notin> S \<longrightarrow> xa \<in> S \<longrightarrow> f (x, xa) = 0))"
add_le_cancel_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less_eq (plus c a) (plus c b) = less_eq a b"
packet_independent_\<alpha>_def: fixes \<alpha> :: "action \<Rightarrow> 'c \<Rightarrow> bool" shows "packet_independent_\<alpha> \<alpha> = (\<forall>a p1 p2. a = Accept \<or> a = Drop \<longrightarrow> \<alpha> a p1 = \<alpha> a p2)"
assms: shows "run_one_step d i (s, vs, ves, $Tee_local x16) = (s', vs', res)"
scaleR_minus1_left: fixes x :: "'b" shows "- 1 *\<^sub>R x = - x"
insert_before_child_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and node :: "(_) node_ptr"   and reference_child :: "(_) node_ptr option" assumes "h \<turnstile> ok insert_before ptr node reference_child" shows "node |\<in>| node_ptr_kinds h"
c_remove_def: shows "c_remove = (\<lambda>x u. if c_in x u = 0 then u else u - 2 ^ x)"
cobounded1: fixes a :: "nat"   and b :: "nat" shows "gcd a b dvd a"
tNode_2: fixes p :: "ref" assumes "x \<in> set_of rt"   and "x \<noteq> Null"   and "low x \<noteq> Null"   and "Dag p low high rt" shows "low x \<in> set_of rt"
INT_iff: fixes b :: "'d"   and B :: "'e \<Rightarrow> 'd set"   and A :: "'e set" shows "(b \<in> \<Inter> (B ` A)) = (\<forall>x\<in>A. b \<in> B x)"
rel_witness_converter2: fixes A :: "'a \<Rightarrow> 'i \<Rightarrow> bool"   and A' :: "'i \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'j \<Rightarrow> bool"   and R' :: "'j \<Rightarrow> 'h \<Rightarrow> bool"   and conv :: "('a, 'c, 'e, 'g) converter"   and conv' :: "('b, 'd, 'f, 'h) converter" assumes "rel_converter (A OO A') B C (R OO R') conv conv'"   and "left_unique A"   and "right_total A"   and "right_unique A'"   and "left_total A'"   and "left_unique R"   and "right_total R"   and "right_unique R'"   and "left_total R'" shows "rel_converter A' (\<lambda>(b, c') c. c = c' \<and> B b c') (\<lambda>(c, d') d. d = d' \<and> C c d') R' (rel_witness_converter A A' B C R R' (conv, conv')) conv'"
exhaust: fixes y :: "'g option" obtains "y = None"   | x2 :: "'g" where "y = Some x2"
lub_range_shift: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'b"   and j :: "nat" assumes "class.cpo below"   and "po.chain below Y" shows "po.lub below (range (\<lambda>i. Y (i + j))) = po.lub below (range Y)"
rec: fixes f :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> 'b"   and x1 :: "bool"   and x2 :: "bool"   and x3 :: "bool"   and x4 :: "bool"   and x5 :: "bool"   and x6 :: "bool"   and x7 :: "bool"   and x8 :: "bool" shows "rec_char f (Char x1 x2 x3 x4 x5 x6 x7 x8) = f x1 x2 x3 x4 x5 x6 x7 x8"
is_functor_if_value_is_ide: assumes "B.ide g" shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) local.map"
MREF: shows "(m, m') \<in> \<langle>Rk, Rv\<rangle>Rm"
card_permutations_of_multiset_remove_aux: fixes x :: "'b"   and A :: "'b multiset" assumes "x \<in># A" shows "card (permutations_of_multiset A) * count A x = size A * card (permutations_of_multiset (A - {#x#}))"
exhaust: fixes y :: "'a decision" obtains x1 :: "'a" where "y = allow x1"   | x2 :: "'a" where "y = deny x2"
hlog_inverse: fixes a :: "real star"   and x :: "real star" assumes "0 < a"   and "a \<noteq> 1"   and "0 < x" shows "hlog a (inverse x) = - hlog a x"
Acc: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and va :: "Term.var"   and T :: "ty" assumes "E,dt\<Turnstile>va\<Colon>=T" shows "E,dt\<Turnstile>Acc va\<Colon>-T"
intros: fixes \<delta> :: "('a, 'b) ta_rule set"   and rcm :: "('a, 'b) ta_rule \<Rightarrow> nat option"   and Q :: "'a \<Rightarrow> 'b tree option" assumes "\<And>r. r \<in> \<delta> \<Longrightarrow> rcm r = Some (card (set (rhsq r)))"   and "Q \<in> brw_iq \<delta>" shows "(Q, br_iq \<delta>, rcm) \<in> brw_initial \<delta>"
obtain_x: shows "x \<in> PLS s as \<and> x = Inf (PLS s as)"
Infinite_cong: fixes X :: "'a set"   and Y :: "'a set" assumes "X = Y" shows "infinite X = infinite Y"
r: fixes b :: "trm" assumes "SN b" shows "P a b"
cons: fixes extra :: "'b set" assumes "finite extra" shows "consistent A ({(witness_list ps (extra \<union> used), a)} \<union> ({(ps, a)} \<union> S))"
trg_\<gamma>_eq: shows "trg \<gamma> = trg w"
g: fixes x :: "'c" shows "lfp.mono_body (\<lambda>f. G f x)"
induct: fixes x1 :: "bool"   and x2 :: "v sem_env"   and x3 :: "'b state"   and x4 :: "top0 list"   and x5 :: "'b state \<times> (v sem_env, v) result"   and P :: "bool \<Rightarrow> v sem_env \<Rightarrow> 'b state \<Rightarrow> top0 list \<Rightarrow> 'b state \<times> (v sem_env, v) result \<Rightarrow> bool" assumes "evaluate_prog x1 x2 x3 x4 x5"   and "\<And>ck env s. P ck env s [] (s, Rval (make_sem_env nsEmpty nsEmpty))"   and "\<And>ck s1 s2 s3 env top0 tops new_env r. evaluate_top ck env s1 top0 (s2, Rval new_env) \<and> evaluate_prog ck (extend_dec_env new_env env) s2 tops (s3, r) \<and> P ck (extend_dec_env new_env env) s2 tops (s3, r) \<Longrightarrow> P ck env s1 (top0 # tops) (s3, combine_dec_result new_env r)"   and "\<And>ck s1 s2 env top0 tops err. evaluate_top ck env s1 top0 (s2, Rerr err) \<Longrightarrow> P ck env s1 (top0 # tops) (s2, Rerr err)" shows "P x1 x2 x3 x4 x5"
psubstEnv_updEnv: fixes rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" shows "rho [x \<leftarrow> X]_xs &[rho'] = rho &[rho'] [x \<leftarrow> X #[rho']]_xs"
finsum_closed: fixes f :: "'c \<Rightarrow> 'a vec"   and A :: "'c set" assumes "f \<in> A \<rightarrow> carrier_vec n" shows "finsum V f A \<in> carrier_vec n"
arr_dom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.arr C (partial_magma.dom C f)"
\<A>_models_\<Phi>\<^sub>G: shows "\<A> \<Turnstile> (\<Phi>\<^sub>G \<Pi>) 0"
pbij_vwb_is_bij_lens: fixes X :: "'c \<Longrightarrow> 'd" assumes "pbij_lens X"   and "vwb_lens X" shows "bij_lens X"
iso_ret_no_ret: fixes p :: "'b"   and u :: "'a"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and p' :: "'b"   and c' :: "'a list multiset" assumes "isolated_ret fg p"   and "proc_of fg u = p"   and "u \<noteq> return fg p"   and "(([u], c), w, [return fg p'], c') \<in> trcl (trss fg)" shows "False"
beta_star_preserves_term_ok': fixes r :: "Core.term"   and s :: "Core.term"   and \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" assumes "r \<rightarrow>\<^sub>\<beta>\<^sup>* s"   and "term_ok' \<Sigma> r" shows "term_ok' \<Sigma> s"
assms_1: shows "\<forall>S\<in>I. convex S"
tail_of_expansion1: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and n :: "nat" assumes "Ring R"   and "x \<in> carrier R" shows "(1\<^sub>r\<^bsub>R\<^esub> \<plusminus>\<^bsub>R\<^esub> x)^\<^bsup>R Suc n\<^esup> = x \<cdot>\<^sub>r\<^bsub>R\<^esub> \<Sigma>\<^sub>e R (\<lambda>i.  \<^bsub>Suc n\<^esub>C\<^bsub>Suc i\<^esub> \<times>\<^bsub>R\<^esub> x^\<^bsup>R i\<^esup>) n \<plusminus>\<^bsub>R\<^esub> 1\<^sub>r\<^bsub>R\<^esub>"
tauLaw1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> \<tau>.P \<approx> P"
widen_subcls: fixes P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal" assumes "P \<turnstile> C \<preceq>\<^sup>* D" shows "P \<turnstile> Class C \<le> Class D"
strand_fv_subst_subset_if_subst_elim: fixes \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and v :: "'c"   and S :: "('d, 'c) strand_step list" assumes "subst_elim \<delta> v"   and "v \<in> fv\<^sub>s\<^sub>t S \<or> bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "v \<notin> fv\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>)"
in_Basis_index_Basis_list: fixes i :: "'e" assumes "i \<in> Basis" shows "i = Basis_list ! index Basis_list i"
card_Union_disjoint: fixes C :: "'c set set" assumes "disjoint C"   and "\<And>A. A \<in> C \<Longrightarrow> finite A" shows "card (\<Union> C) = sum card C"
assms: shows "list_all (\<lambda>f. \<tau> \<Turnstile> \<upsilon> f) l"
union_diff_inter_eq_sup: fixes A :: "'k multiset"   and B :: "'k multiset" shows "A + B - A \<inter># B = A \<union># B"
step_5: shows "cbelow_on (dgrad_p_set d m) (\<prec>\<^sub>p) (z + r) (\<lambda>a b. red F a b \<or> red F b a) (p + r) (q' + r)"
DBM_val_bounded_def: fixes v :: "'b \<Rightarrow> nat"   and u :: "'b \<Rightarrow> 'c"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat" shows "u \<turnstile>\<^bsub>v,n\<^esub> m \<equiv> Le (0::'c) \<preceq> m 0 0 \<and> (\<forall>c. v c \<le> n \<longrightarrow> dbm_entry_val u None (Some c) (m 0 (v c)) \<and> dbm_entry_val u (Some c) None (m (v c) 0)) \<and> (\<forall>c1 c2. v c1 \<le> n \<and> v c2 \<le> n \<longrightarrow> dbm_entry_val u (Some c1) (Some c2) (m (v c1) (v c2)))"
\<sigma>_wf_trm: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<sigma>)"
greatest: fixes d :: "'a" assumes "d dvd a"   and "d dvd b" shows "d dvd c"
x2ak: shows "\<forall>k<x1. x2a k \<in> carrier_mat d d"
distinct_35: fixes x31 :: "('a, 'b, 'c, 'd) seqp"   and x32 :: "('a, 'b, 'c, 'd) seqp"   and x41 :: "'d"   and x42 :: "'a \<Rightarrow> nat"   and x43 :: "'a \<Rightarrow> 'b"   and x44 :: "('a, 'b, 'c, 'd) seqp"   and x45 :: "('a, 'b, 'c, 'd) seqp" shows "x31 \<oplus> x32 \<noteq> {x41}unicast(x42, x43) . x44 \<triangleright> x45"
Ssegment_Ssegment: fixes D :: "'c Order"   and b :: "'c"   and a :: "'c" assumes "Order D"   and "b \<in> carrier D"   and "a \<in> Ssegment D b" shows "Ssegment (SIod D (Ssegment D b)) a = Ssegment D a"
qSwap_preserves_qGood1: fixes X :: "('e, 'f, 'varSort, 'var, 'g) qTerm"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "qGood X" shows "qGood (X #[[x \<and> y]]_zs)"
add_strict_increasing2: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "{#} \<subseteq># a"   and "b \<subset># c" shows "b \<subset># a + c"
lcs_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "lcs f g = (\<lambda>x. max (f x) (g x))"
prems_9: shows "y \<notin> snd ` set (tx # txs)"
assms_2: shows "repJ ** M ** Finite_Cartesian_Product.transpose repJ = k *\<^sub>R M"
R: shows "UNlift01 cl dl P F ` UNpart1 cl dl P = UNpart1 dl cl (\<lambda>n. F n ` P n)"