hcju/isabelleps huggingface dataset:
Jacobi_neg_right: fixes a :: "int"   and n :: "int" shows "Jacobi a (- n) = Jacobi a n"
det_I: shows "det (mat (1::'c)) = (1::'c)"
0: shows "lookup (unit_vec n i) [j] = (if i = j then 1::'b else (0::'b))"
emeasure_eq_measure2: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<in> fmeasurable M" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
set_shadow_root_writes: fixes element_ptr :: "(_) element_ptr"   and disc_nodes :: "(_) shadow_root_ptr option"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (set_shadow_root_locs element_ptr) (set_shadow_root element_ptr disc_nodes) h h'"
dg_prod_components_2: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V" shows "dg_prod I \<AA>\<lparr>Arr\<rparr> = (\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Arr\<rparr>)"
brnL_UN: fixes cl :: "('test, 'atom, 'choice) cmd list" assumes "properL cl" shows "{..<brnL cl (length cl)} = (\<Union>n<length cl. {brnL cl n..<brnL cl n + brn (cl ! n)})"
sorted_wrt_map_distr: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "sorted_wrt P (map f xs)" shows "sorted_wrt (\<lambda>x y. P (f x) (f y)) xs"
rlfm_l: fixes p :: "fm" assumes "qfree p" shows "isrlfm (rlfm p)"
unique_factorization_m_zero: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset" assumes "pl.unique_factorization_m f (c, fs)" shows "pl.M c \<noteq> 0"
hoare_conseq_left: fixes P' :: "'a set"   and R :: "('a \<times> 'a) set"   and Q :: "'a set"   and P :: "'a set" assumes "{P'} R {> Q}"   and "P \<subseteq> P'" shows "{P} R {> Q}"
fc_equiv_def: fixes a :: "'b"   and b :: "'b" shows "fc_equiv a b \<equiv> a \<in> face_cycle_set b"
carrierE: fixes x :: "'c" assumes "x \<in> manifold_eucl.dest.carrier" obtains c :: "('c, 'c) chart" where "c \<in> charts_eucl" and "x \<in> domain c"
S2_def: fixes h :: "'a"   and c :: "nat" shows "S2 h c = sample_uniform (order \<G>) \<bind> (\<lambda>z1. sample_uniform (order \<G>) \<bind> (\<lambda>z2. let a = \<^bold>g [^] z1 \<otimes> g' [^] z2 \<otimes> inv h [^] c in return_spmf (a, z1, z2)))"
objE: fixes a :: "'b" assumes "D.obj a" obtains "D.ide a" and "src\<^sub>D a = a" and "trg\<^sub>D a = a"
Rf_eq_Rf_diff_Rf: fixes N :: "'b literal multiset set" shows "standard_redundancy_criterion.Rf N = standard_redundancy_criterion.Rf (N - standard_redundancy_criterion.Rf N)"
False: shows "\<not> (M = - \<infinity> \<or> M = \<infinity> \<or> N = - \<infinity> \<or> N = \<infinity>)"
Crypt_used_imp_L_used: fixes F :: "msg"   and A :: "msg"   and B :: "msg"   and L :: "msg"   and K :: "msg"   and evs :: "event list" assumes "Crypt (priEK TTP) \<lbrace>F, A, B, L, K\<rbrace> \<in> used evs"   and "evs \<in> zg" shows "L \<in> used evs"
invertibleD_2: fixes A :: "(('d, 'e) vec, 'f) vec" assumes "invertible A" shows "A ** A\<^sup>-\<^sup>1 = mat (1::'d)"
Follows_union: fixes F :: "'b program"   and f' :: "'b \<Rightarrow> 'c multiset"   and f :: "'b \<Rightarrow> 'c multiset"   and g' :: "'b \<Rightarrow> 'c multiset"   and g :: "'b \<Rightarrow> 'c multiset" assumes "F \<in> f' Fols f"   and "F \<in> g' Fols g" shows "F \<in> (\<lambda>s. f' s + g' s) Fols (\<lambda>s. f s + g s)"
dom_comp: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "local.dom (g \<cdot> f) = local.dom f"
refl: fixes x :: "'b" assumes "x \<in> alts" shows "R i x x"
red_indI2: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "red F (tail p) q" shows "red F p (q + monomial (lc p) (lt p))"
cc_le: fixes j :: "nat" assumes "0 < j"   and "j < card X - 2" shows "aa j + bb j \<le> 2 + (aa (j + 1) + bb (j + 1) choose 2) + (\<Sum>i = j + 3..card X - 1. aa i + bb i choose Suc (i - j))"
equivalent_on_lit: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and l :: "'b literal"   and I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" assumes "equivalent_on \<sigma> \<eta> (vars_of_lit l) I"   and "fo_interpretation I" shows "validate_ground_lit I (subst_lit l \<sigma>) = validate_ground_lit I (subst_lit l \<eta>)"
is_MST_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and t :: "'a ugraph" shows "is_MST w g t \<equiv> is_spanning_tree g t \<and> (\<forall>t'. is_spanning_tree g t' \<longrightarrow> weight w t \<le> weight w t')"
is_bisimulation_def: fixes R :: "('state, 'effect) L_state \<Rightarrow> ('state, 'effect) L_state \<Rightarrow> bool" shows "L_transform.is_bisimulation R \<equiv> symp R \<and> (\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<phi>. P \<turnstile>\<^sub>L \<phi> \<longrightarrow> Q \<turnstile>\<^sub>L \<phi>)) \<and> (\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<alpha> P'. bn \<alpha> \<sharp>* Q \<longrightarrow> P \<rightarrow>\<^sub>L \<langle>\<alpha>,P'\<rangle> \<longrightarrow> (\<exists>Q'. Q \<rightarrow>\<^sub>L \<langle>\<alpha>,Q'\<rangle> \<and> R P' Q')))"
Basis: fixes B :: "'b set"   and F :: "'b program" shows "B \<in> wens_set F B"
tell_writer_def: fixes return :: "'e \<times> 'd list \<Rightarrow> 'f"   and bind :: "'f \<Rightarrow> ('e \<times> 'd list \<Rightarrow> 'f) \<Rightarrow> 'f"   and w :: "'d"   and m :: "('d, 'e, 'f) writerT" shows "tell_writer return bind w m = WriterT (bind (run_writer m) (\<lambda>(a, ws). return (a, w # ws)))"
inf_vector_comp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y * top) * z = y * top \<sqinter> x * z"
LVar: fixes ln :: "lname" assumes "v = LVar ln" shows "thesis"
up_dist_sup: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.lattice inf less_eq less sup" shows "{ya. less_eq (sup x y) ya} = {y. less_eq x y} \<inter> {ya. less_eq y ya}"
S_fv_bvars_disj: shows "fv\<^sub>s\<^sub>t S \<inter> bvars\<^sub>s\<^sub>t S = {}"
dim_1: shows "dim_row (four_block_mat A B (0\<^sub>m m n) D) = n + m"
prime_divides: fixes a :: "'a"   and b :: "'a"   and p :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "prime G p"   and "p divides a \<otimes> b" shows "p divides a \<or> p divides b"
in_set_member: fixes x :: "'e"   and xs :: "'e list" shows "(x \<in> set xs) = List.member xs x"
valid_ref_valid_null_ref_simps_1: fixes r :: "'g"   and s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and p :: "'h process_name"   and ws :: "('e, 'f, 'g) mem_store_action list"   and w :: "('e, 'f, 'g) mem_store_action" shows "obj_at (\<lambda>s. True) r (s(sys := do_store_action w (s sys)\<lparr>mem_store_buffers := (mem_store_buffers (s sys))(p := ws)\<rparr>)) = obj_at (\<lambda>s. True) r s"
rel_fun_eq_onp_rel: fixes R :: "'a \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_fun (eq_onp R) S = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
iso_compose_rev: fixes f :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and g :: "'d \<Rightarrow> 'g"   and K :: "('g, 'h) monoid_scheme" assumes "f \<in> hom G H"   and "g \<in> hom H K"   and "g \<circ> f \<in> Group.iso G K" shows "f \<in> mon G H \<and> g \<in> epi H K"
terminal_path_empty: fixes v :: "'vertex" assumes "terminal_vertex v" shows "terminal_path v v []"
optimal_setI: fixes x :: "'b vec"   and S :: "'b vec set"   and c :: "'b vec" assumes "x \<in> S"   and "\<And>y. y \<in> S \<Longrightarrow> x \<bullet> c \<le> y \<bullet> c" shows "x \<in> optimal_set (\<lambda>x y. x \<bullet> c \<le> y \<bullet> c) S"
xor_funct_def: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "xor_funct A B = (let (a1, b1) = A; (a2, b2) = B in return_spmf (a1 \<oplus> a2, b1 \<oplus> b2))"
Assign2: fixes a :: "aexp"   and x :: "char list" assumes "sec a \<le> sec x" shows "\<turnstile> x ::= a : sec x"
subset_insertI2: fixes A :: "'b set"   and B :: "'b set"   and b :: "'b" assumes "A \<subseteq> B" shows "A \<subseteq> b \<triangleright> B"
below: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>xa. \<lbrakk>xa = x; xa \<notin> S\<rbrakk> \<Longrightarrow> f xa \<sqsubseteq> g xa"   and "\<And>xa. \<lbrakk>xa \<in> S; xa \<noteq> x\<rbrakk> \<Longrightarrow> f xa \<sqsubseteq> g xa"   and "\<And>xa. \<lbrakk>xa \<notin> S; xa \<noteq> x\<rbrakk> \<Longrightarrow> f xa \<sqsubseteq> g xa" shows "f \<sqsubseteq> g"
sbh_step_preserves_read_tmps_bound: fixes "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<S> :: "(nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "'a memref list"   and m' :: "nat \<Rightarrow> nat"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<S>' :: "(nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat" assumes "(is, \<theta>, sb, m, \<D>, \<O>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<theta>', sb', m', \<D>', \<O>', \<S>')"   and "\<forall>i\<in>load_tmps is. i < n"   and "\<forall>i\<in>read_tmps sb. i < n" shows "\<forall>i\<in>read_tmps sb'. i < n"
changes_Cons: fixes x :: "'b"   and xs :: "'b list" shows "changes (x # xs) = (if filter (\<lambda>x. x \<noteq> (0::'b)) xs = [] then 0 else if x * hd (filter (\<lambda>x. x \<noteq> (0::'b)) xs) < (0::'b) then 1 + changes xs else changes xs)"
prems_1: assumes "(q, q') \<in> (R - UNIV \<times> S)\<^sup>*" shows "P"
vars\<^sub>l_def: fixes l :: "fterm literal" shows "vars\<^sub>l l = vars\<^sub>t\<^sub>s (get_terms l)"
qp2: shows "is_quantum_predicate (adjoint (tensor_P mat_Ph (1\<^sub>m K)) * (adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1)) * tensor_P mat_Ph (1\<^sub>m K))"
in_restricted_global_in_unrestricted_global_1: fixes r' :: "'a"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a" assumes "r' \<in> RID\<^sub>G (s(r := None))" shows "r' \<in> RID\<^sub>G s"
cancel_sym: fixes a :: "bool \<times> 'd"   and b :: "bool \<times> 'd" assumes "canceling a b" shows "canceling b a"
asubg_test: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set" assumes "aGroup A"   and "H \<subseteq> carrier A"   and "H \<noteq> {}"   and "\<forall>a\<in>H. \<forall>b\<in>H. a \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> b \<in> H" shows "A +> H"
hyps: fixes p :: "dir list"   and q :: "dir list"   and t :: "('a, 'b) tm" assumes "length p + length q < length pa__ + length qa__"   and "length p \<le> length q \<Longrightarrow> take (length p) q \<noteq> p"   and "length q \<le> length p \<Longrightarrow> take (length q) p \<noteq> q" shows "emb_step_at q d2 (emb_step_at p d1 t) = emb_step_at p d1 (emb_step_at q d2 t)"
of_nat_0_eq_iff: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(zero = semiring_1.of_nat one plus zero n) = (0 = n)"
assms_3: shows "(r, res') \<in> set_spmf (run_resource res q)"
rbt_def: fixes t :: "('c \<times> color) tree" shows "rbt t = (invc t \<and> invh t \<and> color t = Black)"
facetrel_diff_vertex: fixes v :: "'b"   and x :: "'b set" assumes "v \<in> x" shows "x - {v} \<lhd> x"
lhs_rhs: shows "mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('a) n (\<lambda>k. B\<^sup>T $$ (i, k) \<cdot>\<^sub>v Matrix.row A\<^sup>T k) {0..<m}) = mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('a) n (\<lambda>k. B $$ (k, i) \<cdot>\<^sub>v col A k) {0..<m})"
distinct_41: fixes x4 :: "nat"   and x7 :: "'e" shows "\<not>unicast x4 \<noteq> receive x7"
intros: fixes \<tau>move :: "('e, 'f, 'c, 'd, 'g, 'h) \<tau>moves"   and x :: "'c"   and m :: "'d"   and ta :: "('e, 'f, 'c, 'd, 'g, 'h) thread_action"   and x' :: "'c"   and m' :: "'d" assumes "\<tau>move (x, m) ta (x', m')" shows "\<tau>multithreaded.init_fin_\<tau>move \<tau>move ((Running, x), m) (convert_TA_initial (convert_obs_initial ta)) ((Running, x'), m')"
gt: shows "s0 \<bullet> (1::'a) < s \<bullet> (1::'a)"
idem: fixes a :: "'k multiset" shows "a \<inter># a = a"
assms: shows "GEN_OP le (\<le>) (A \<rightarrow> A \<rightarrow> bool_rel)"
Says_K5: fixes servK :: "nat"   and A :: "agent"   and T3 :: "nat"   and evs :: "event list"   and authK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Crypt servK \<lbrace>Agent A, Number T3\<rbrace> \<in> parts (knows Spy evs)"   and "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, servTicket\<rbrace> \<in> set evs"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbV" shows "\<exists>ST. Says A B \<lbrace>ST, Crypt servK \<lbrace>Agent A, Number T3\<rbrace>\<rbrace> \<in> set evs"
oth_class_taut_9_b: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>& \<psi> \<^bold>\<rightarrow> \<psi> in v]"
avg_proj_onto_right: fixes W :: "'v set"   and U :: "'v set" assumes "VectorSpace (\<sharp>\<cdot>) W"   and "RModule FG (\<cdot>) U \<and> U \<subseteq> V"   and "add_independentS [W, U]"   and "V = W \<oplus> U" shows "((\<sharp>\<cdot>) ((1::'f) / ordG) \<circ> (\<Sum>g\<in>G. (*\<cdot>) (- g) \<circ> (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>[W, U]. G) then W\<oplus>U\<leftarrow> a ! 1 else (0::'v)) \<circ> (*\<cdot>) g)) ` V = U"
order_trans: fixes x :: "'d"   and y :: "'d"   and z :: "'d" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
code: fixes gpv :: "('b, 'd, 'e) gpv" shows "gpv_stop gpv = GPV (map_spmf (map_generat Some id (\<lambda>rpv input. case input of None \<Rightarrow> Generative_Probabilistic_Value.Done None | Some input' \<Rightarrow> gpv_stop (rpv input'))) (the_gpv gpv))"
prefix_msg_in_history: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and es :: "'c event list"   and i :: "nat"   and m :: "'c" assumes "network history msg_id"   and "node_histories.prefix_of_node_history history es i"   and "m \<in> set (trivial_network.node_deliver_messages es)" shows "Deliver m \<in> set (history i)"
Un_tablesD: fixes x :: "'a"   and ts :: "('b \<Rightarrow> 'a set) set"   and k :: "'b" assumes "x \<in> Un_tables ts k" shows "\<exists>t. t \<in> ts \<and> x \<in> t k"
norm_snd_le: fixes y :: "'a"   and x :: "'b" shows "norm y \<le> norm (x, y)"
Suc: fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<Sum>ms | set ms \<subseteq> S \<and> length ms = n \<and> (\<forall>i<n. P i (ms ! i)). \<Prod>x<n. f (ms ! x)) = (\<Prod>i<n. sum f {m \<in> S. P i m})"
snoc': fixes xs :: "'a list"   and x :: "'a" assumes "xs \<noteq> []"   and "P xs" shows "P (xs @ [x])"
simps_2: fixes a :: "'a" shows "poly_uminus_inner [a] = [:a:]"
stack_ss_tj_stack: shows "set (stack s) \<subseteq> set (tj_stack s)"
inv_is_inverse: fixes f :: "'d" assumes "D.iso f" shows "D.inverse_arrows f (D.inv f)"
ord_resolve_rename_lifting: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and M :: "'a literal multiset set"   and CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset" assumes "\<And>\<rho> C. is_renaming \<rho> \<Longrightarrow> S (C \<cdot> \<rho>) = S C \<cdot> \<rho>"   and "ord_resolve (S_M S M) CAs DA AAs As \<sigma> E"   and "selection S"   and "{DA} \<union> set CAs \<subseteq> grounding_of_clss M" obtains \<eta>s :: "'s list" and \<eta> :: "'s" and \<eta>2 :: "'s" and CAs0 :: "'a literal multiset list" and DA0 :: "'a literal multiset" and AAs0 :: "'a multiset list" and As0 :: "'a list" and E0 :: "'a literal multiset" and \<tau> :: "'s" where "is_ground_subst \<eta>" and "is_ground_subst_list \<eta>s" and "is_ground_subst \<eta>2" and "ord_resolve_rename S CAs0 DA0 AAs0 As0 \<tau> E0" and "CAs0 \<cdot>\<cdot>cl \<eta>s = CAs" and "DA0 \<cdot> \<eta> = DA" and "E0 \<cdot> \<eta>2 = E" and "{DA0} \<union> set CAs0 \<subseteq> M" and "length CAs0 = length CAs" and "length \<eta>s = length CAs"
comm_monoid_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" shows "comm_monoid f z \<equiv> abel_semigroup f \<and> comm_monoid_axioms f z"
ptoq: shows "(red (G - {p}))\<^sup>*\<^sup>* p q"
IH_2: fixes w :: "nat \<Rightarrow> real" shows "remove_weights (insert_weights s m2 w) = m2"
af_G: fixes \<phi> :: "'a ltl"   and w :: "'a set list" shows "af (G \<phi>) w = And (G \<phi> # map (af \<phi>) (suffixes w))"
cases: fixes a1 :: "(name, value) fmap"   and a2 :: "sterm"   and a3 :: "value" assumes "a1 \<turnstile>\<^sub>v a2 \<down> a3" obtains name :: "name" and \<Gamma> :: "(name, value) fmap" and val :: "value" where "a1 = \<Gamma>''__" and "a2 = Sconst name__" and "a3 = val__" and "name__ |\<notin>| C" and "fmlookup \<Gamma>''__ name__ = Some val__"   | \<Gamma> :: "(name, value) fmap" and name :: "name" and val :: "value" where "a1 = \<Gamma>'''__" and "a2 = Svar namea__" and "a3 = vala__" and "fmlookup \<Gamma>'''__ namea__ = Some vala__"   | \<Gamma> :: "(name, value) fmap" and cs :: "(Term_Class.term \<times> sterm) list" where "a1 = \<Gamma>''''__" and "a2 = Sabs csa__" and "a3 = Vabs csa__ \<Gamma>''''__"   | \<Gamma> :: "(name, value) fmap" and t :: "sterm" and cs :: "(Term_Class.term \<times> sterm) list" and \<Gamma>' :: "(name, value) fmap" and u :: "sterm" and u' :: "value" and env :: "(name, value) fmap" and uu_ :: "Term_Class.term" and rhs :: "sterm" and val :: "value" where "a1 = \<Gamma>'''''__" and "a2 = t__ $\<^sub>s u__" and "a3 = valb__" and "\<Gamma>'''''__ \<turnstile>\<^sub>v t__ \<down> Vabs csb__ \<Gamma>''''''__" and "\<Gamma>'''''__ \<turnstile>\<^sub>v u__ \<down> u'a__" and "vfind_match csb__ u'a__ = Some (enva__, uu_, rhsa__)" and "\<Gamma>''''''__ ++\<^sub>f enva__ \<turnstile>\<^sub>v rhsa__ \<down> valb__"   | \<Gamma> :: "(name, value) fmap" and t :: "sterm" and css :: "(name, (Term_Class.term \<times> sterm) list) fmap" and name :: "name" and \<Gamma>' :: "(name, value) fmap" and cs :: "(Term_Class.term \<times> sterm) list" and u :: "sterm" and u' :: "value" and env :: "(name, value) fmap" and uv_ :: "Term_Class.term" and rhs :: "sterm" and val :: "value" where "a1 = \<Gamma>'''''''__" and "a2 = t $\<^sub>s u" and "a3 = val" and "\<Gamma>'''''''__ \<turnstile>\<^sub>v t \<down> Vrecabs css nameb__ \<Gamma>'" and "fmlookup css nameb__ = Some cs" and "\<Gamma>'''''''__ \<turnstile>\<^sub>v u \<down> u'" and "vfind_match cs u' = Some (env, uv_, rhs)" and "\<Gamma>' ++\<^sub>f mk_rec_env css \<Gamma>' ++\<^sub>f env \<turnstile>\<^sub>v rhs \<down> val"   | name :: "name" and \<Gamma> :: "(name, value) fmap" and ts :: "sterm list" and us :: "value list" where "a1 = \<Gamma>" and "a2 = name $$ ts" and "a3 = Vconstr name us" and "name |\<in>| C" and "list_all2 (veval' \<Gamma>) ts us"
dom_char: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2" shows "partial_magma.dom (product_category.comp C1 C2) f = (if partial_magma.arr C1 (fst f) \<and> partial_magma.arr C2 (snd f) then (partial_magma.dom C1 (fst f), partial_magma.dom C2 (snd f)) else (partial_magma.null C1, partial_magma.null C2))"
Affine_aform_of_ivl: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "Affine (aform_of_ivl a b) = {a..b}"
birkhoff_sum_small_asymp_lemma: fixes f :: "'a \<Rightarrow> real"   and e :: "real" assumes "integrable M f"   and "AE x in M. real_cond_exp M Invariants f x = 0"   and "0 < e" shows "AE x in M. infinite {n. birkhoff_sum f n x \<in> {0..e}}"
writeCArray_def: fixes array :: "nat \<Rightarrow> 'b"   and index :: "nat"   and "value" :: "'b" shows "writeCArray array index value \<equiv> array(index := value)"
intros: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<epsilon> :: "ennreal" assumes "hindrance_by \<Gamma> f \<epsilon>"   and "current \<Gamma> f"   and "wave \<Gamma> f" shows "hindered_by \<Gamma> \<epsilon>"
assms_3: shows "INIT_class e\<^sub>0 = \<lfloor>C\<rfloor>"
xx_3: shows "xb \<in> set (if fst (dports r) = 0 \<and> snd (dports r) = - 1 then [None] else if fst (dports r) \<le> snd (dports r) then map (\<lambda>x. Some (pfxm_prefix x, ~~ pfxm_mask x)) (wordinterval_CIDR_split_prefixmatch (WordInterval (fst (dports r)) (snd (dports r)))) else [])"
prems_5: shows "fmrestrict_set vs s' = fmrestrict_set vs s"
mat_gt_arc_imp_mat_ge: fixes A :: "'a mat"   and B :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "mat_gt_arc A B"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A \<ge>\<^sub>m B"
induct: fixes P :: "Core.term \<times> Core.term list \<Rightarrow> bool"   and a0 :: "Core.term \<times> Core.term list" assumes "\<And>f t ts. P (f, t # ts) \<Longrightarrow> P (f $ t, ts)"   and "\<And>vb vc va. P (Ct vb vc, va)"   and "\<And>vb vc va. P (Fv vb vc, va)"   and "\<And>vb va. P (Bv vb, va)"   and "\<And>vb vc va. P (Abs vb vc, va)" shows "P a0"
elims_2: fixes x :: "('b \<times> nat) tree"   and xa :: "('b \<times> nat) tree" assumes "single x xa" obtains t1 :: "('b \<times> nat) tree" and t2 :: "('b \<times> nat) tree" where "x = t1" and "xa = t2" and "10 * (size_wbt t1 + 1) < 14 * (size_wbt t2 + 1)"
refine_trans: fixes a :: "'b::order"   and b :: "'b::order"   and c :: "'b::order" assumes "(a::'b::order) \<le> (b::'b::order)"   and "(b::'b::order) \<le> (c::'b::order)" shows "(a::'b::order) \<le> (c::'b::order)"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "??.Quickcheck_Narrowing.narrowing_type list list \<Rightarrow> 'a"   and narrowing_type :: "??.Quickcheck_Narrowing.narrowing_type" shows "P (case narrowing_type of Narrowing_sum_of_products x \<Rightarrow> f x) = (\<forall>x. narrowing_type = Narrowing_sum_of_products x \<longrightarrow> P (f x))"
release_shared_exchange_weak: fixes \<O> :: "nat set"   and sb :: "'b memref list"   and \<S>' :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and \<R> :: "nat \<Rightarrow> bool option" assumes "\<forall>a\<in>\<O> \<union> all_acquired sb. \<S>' a = \<S> a"   and "weak_sharing_consistent \<O> sb" shows "release sb (dom \<S>') \<R> = release sb (dom \<S>) \<R>"
simps: fixes a :: "'a \<times> 'b"   and A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "(a \<in> oreachable A S U) = ((\<exists>s. a = s \<and> s \<in> init A) \<or> (\<exists>s aa s'. a = s' \<and> s \<in> oreachable A S U \<and> (s, aa, s') \<in> automaton.trans A \<and> S (fst s) (fst s') aa) \<or> (\<exists>s \<sigma>'. a = (\<sigma>', snd s) \<and> s \<in> oreachable A S U \<and> U (fst s) \<sigma>'))"
ballE: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>x\<in>A. P x" obtains "P x"   | "x \<notin> A"
fps_conv_radius_def: fixes f :: "'b fps" shows "fps_conv_radius f = conv_radius (fps_nth f)"
is_Read\<^sub>s\<^sub>b_def: fixes r :: "'b memref" shows "is_Read\<^sub>s\<^sub>b r = (case r of Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> True | _ \<Rightarrow> False)"