hcju/isabelleps huggingface dataset:
rComm1: fixes \<Psi>\<^sub>Q :: "'b"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* C" shows "Prop (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')"
construct_lhs_vector_def: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list" shows "w_vec p qs signs \<equiv> vec_of_list (map (\<lambda>w. rat_of_int (int (length (filter (\<lambda>v. v = w) (map (consistent_sign_vec_copr qs) (characterize_root_list_p p)))))) signs)"
awalk_verts_reachable_to: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and w :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v"   and "w \<in> set (pre_digraph.awalk_verts G u p)" shows "w \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
ABnd: shows "A \<in> \<int>\<^sub>m \<inter> Bounded_mat Bnd"
direct_decompD_2: fixes A :: "'a set"   and ss :: "'a set list" assumes "direct_decomp A ss" shows "inj_on sum_list (listset ss)"
simps_7: fixes \<phi> :: "atom fm"   and \<Gamma> :: "real list" shows "eval (ExQ \<phi>) \<Gamma> = (\<exists>x. eval \<phi> (x # \<Gamma>))"
intros: fixes s :: "'s2"   and tl :: "'tl2"   and s' :: "'s2" assumes "s -2-tl\<rightarrow> s'"   and "\<tau>move2 s tl s'" shows "s -\<tau>2\<rightarrow> s'"
simps_3: fixes M :: "'b trm"   and N :: "'b trm"   and s :: "('b \<times> 'b trm) list" shows "M \<cdot> N \<lhd> s = (M \<lhd> s) \<cdot> (N \<lhd> s)"
filtersD4: fixes F :: "'b set set"   and T :: "'b set set"   and A :: "'b set"   and B :: "'b set" assumes "F \<in> Filters\<^bsub>T\<^esub>"   and "A \<subseteq> B"   and "B \<subseteq> carrier\<^bsub>T\<^esub>"   and "A \<in> F" shows "B \<in> F"
integrable_const_bound: fixes f :: "'a \<Rightarrow> 'b"   and B :: "real"   and p :: "'a spmf" assumes "AE x in measure_spmf p. norm (f x) \<le> B"   and "f \<in> borel_measurable (measure_spmf p)" shows "integrable (measure_spmf p) f"
prems_2: shows "older_seniors x n = older_seniors x (Suc n)"
is_extensional: fixes f :: "'a" assumes "\<not> R'.C.arr f" shows "H\<^sub>R a' f = R'.C.null"
\<Delta>'''_sel_1: fixes x :: "'v"   and y :: "'v" shows "edge \<Delta>'''.\<Delta>''' x y = (edge \<Delta>''' x y \<and> 0 < capacity \<Delta>''' (x, y) \<and> y \<noteq> source \<Delta>''' \<and> x \<noteq> sink \<Delta>''')"
t1_space_nhds: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.t1_space open"   and "x \<noteq> y" shows "\<forall>\<^sub>F x in topological_space.nhds open x. x \<noteq> y"
assms: shows "matrix_p_algebra.dense (f \<odot> mtop)"
eq3: shows "int (card X) - int (poly_deg f) - int j - 1 gchoose (card X - Suc j) = (- 1) ^ (card X - Suc j) * (int (poly_deg f) - 1 gchoose (card X - Suc j))"
set_rel_simp: fixes R :: "('a \<times> 'b) set" shows "({}, {}) \<in> \<langle>R\<rangle>set_rel"
in_set_drop_conv_nth: fixes x :: "'b"   and n :: "nat"   and l :: "'b list" shows "(x \<in> set (drop n l)) = (\<exists>i\<ge>n. i < length l \<and> x = l ! i)"
sorted_wrt_nth_less: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and i :: "nat"   and j :: "nat" assumes "sorted_wrt P xs"   and "i < j"   and "j < length xs" shows "P (xs ! i) (xs ! j)"
comp_null_1: fixes f :: "'a" shows "hom'.comp hom'.null f = hom'.null"
cone_hull_empty_iff: fixes S :: "'c set" shows "(S = {}) = (cone hull S = {})"
CASE_3: shows "((ss, ce), e, ss', add_mset s' cc) \<in> gtrs"
sel_6: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x4 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x5 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x6 :: "'b \<Rightarrow> 'b"   and x7 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x8 :: "'b \<Rightarrow> 'b"   and x9 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x10 :: "'b \<Rightarrow> 'b"   and x11 :: "'b \<Rightarrow> 'b"   and x12 :: "int \<Rightarrow> 'b"   and x13 :: "'b \<Rightarrow> int"   and x14 :: "'b \<Rightarrow> bool" shows "arith_ops_record.uminus (Arith_Ops_Record x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14) = x6"
strand_sem_append_stateful: fixes M :: "('fun, 'var) Term.term set"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set"   and A :: "('fun, 'var) stateful_strand_step list"   and B :: "('fun, 'var) stateful_strand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "\<lbrakk>M; D; A @ B\<rbrakk>\<^sub>s \<I> = (\<lbrakk>M; D; A\<rbrakk>\<^sub>s \<I> \<and> \<lbrakk>M \<union> (ik\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>); dbupd\<^sub>s\<^sub>s\<^sub>t A \<I> D; B\<rbrakk>\<^sub>s \<I>)"
validG: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/G\<^esub> P' c R,X"
gn: fixes x :: "'a"   and y :: "'a" shows "(x \<le> -' y) = (y \<le> - x)"
seq_invariant_ctermI: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "('f \<times> ('f, 'g, 'e, 'h) seqp, 'g seq_action) automaton"   and P :: "'f \<times> 'h \<Rightarrow> bool"   and I :: "'g seq_action \<Rightarrow> bool" assumes "wellformed \<Gamma>"   and "control_within \<Gamma> (init A)"   and "simple_labels \<Gamma>"   and "automaton.trans A = seqp_sos \<Gamma>"   and "\<And>\<xi> p l. \<lbrakk>(\<xi>, p) \<in> init A; l \<in> labels \<Gamma> p\<rbrakk> \<Longrightarrow> P (\<xi>, l)"   and "\<And>p l \<xi> a q l' \<xi>' pp. \<lbrakk>p \<in> cterms \<Gamma>; l \<in> labels \<Gamma> p; P (\<xi>, l); ((\<xi>, p), a, \<xi>', q) \<in> seqp_sos \<Gamma>; ((\<xi>, p), a, \<xi>', q) \<in> automaton.trans A; l' \<in> labels \<Gamma> q; (\<xi>, pp) \<in> reachable A I; p \<in> sterms \<Gamma> pp; (\<xi>', q) \<in> reachable A I; I a\<rbrakk> \<Longrightarrow> P (\<xi>', l')" shows "A \<TTurnstile> (I \<rightarrow>) onl \<Gamma> P"
le_infI1: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<le> x" shows "a \<sqinter> b \<le> x"
append_carrier_vec: fixes v :: "'b vec"   and n1 :: "nat"   and w :: "'b vec"   and n2 :: "nat" assumes "v \<in> carrier_vec n1"   and "w \<in> carrier_vec n2" shows "v @\<^sub>v w \<in> carrier_vec (n1 + n2)"
assms: shows "valid_unMultigraph.is_trail (del_unEdge x1 x2 x3 G) v ps v'"
q_def: fixes x :: "nat"   and y :: "nat" shows "q x y = pA (p x) y"
T_eq_prod: shows "T = (\<lambda>(x1, x2). K1.T x1 \<bind> (\<lambda>\<omega>1. K2.T x2 \<bind> (\<lambda>\<omega>2. return (stream_space (count_space UNIV)) (szip\<^sub>E x1 x2 (\<omega>1, \<omega>2)))))"
is_vals_map_Vals: fixes vs :: "'g val list" shows "is_vals (map Val vs) = True"
member_of_declC: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m member_of C" shows "G\<turnstile> mbr m declared_in declclass m"
CFG_edge_Uses_pred_equal: fixes a :: "'a"   and s :: "(('c \<Rightarrow> 'd option) \<times> 'e) list"   and s' :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" assumes "valid_edge a"   and "local.CFG.pred (kind a) s"   and "snd (hd s) = snd (hd s')"   and "\<forall>V\<in>Use (sourcenode a). local.CFG.state_val s V = local.CFG.state_val s' V"   and "length s = length s'" shows "local.CFG.pred (kind a) s'"
Int_greatest: fixes C :: "'c set"   and A :: "'c set"   and B :: "'c set" assumes "C \<subseteq> A"   and "C \<subseteq> B" shows "C \<subseteq> A \<inter> B"
map\<^sub>0_in_hhom: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<tau>\<^sub>0 a : src\<^sub>D (F a) \<rightarrow>\<^sub>D src\<^sub>D (G a)\<guillemotright>"
dom_restrict_def: fixes S :: "'b set"   and p :: "'b \<Rightarrow> 'c decision option" shows "S \<triangleleft> p \<equiv> \<lambda>x. if x \<in> S then p x else \<bottom>"
Rel: shows "MP_Rel f3 (of_int_poly (Mp f3))"
cs: shows "(path \<sigma> \<exclamdown> cs\<^bsup>path \<sigma>'\<^esup> j') = j"
identity\<^sub>1_def: shows "(\<^bold>=) \<equiv> \<lambda>F G. \<^bold>\<box>(\<^bold>\<forall>x. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>x\<^sup>P,G\<rbrace>)"
Uint_signed_def: fixes i :: "integer" shows "Uint_signed i = (if i < wivs_least_integer \<or> wivs_overflow_integer \<le> i then undefined Uint i else Uint i)"
worst_chain_bad: fixes x :: "'a" assumes "local.bad x" shows "local.inf_chain local.worst_chain"
openin_subset: fixes U :: "'b topology"   and S :: "'b set" assumes "openin U S" shows "S \<subseteq> topspace U"
dom_def_aux_var: fixes x :: "'a" shows "d x = 1' \<cdot> x ; x\<^sup>\<smile>"
isDiagonal_mat_def: fixes A :: "'a mat" shows "isDiagonal_mat A = (\<forall>i j. i \<noteq> j \<and> i < dim_row A \<and> j < dim_col A \<longrightarrow> A $$ (i, j) = (0::'a))"
remove_shadow_root_locs_def: shows "remove_shadow_root_locs \<equiv> l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_shadow_root_locs set_shadow_root_locs"
hoare2_valid_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "(\<Turnstile>\<^sub>2 {P} c {Q}) = (\<forall>s. P s < \<infinity> \<longrightarrow> (\<exists>t p. (c, s) \<Rightarrow> p \<Down> t \<and> enat p + Q t \<le> P s))"
simps_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and at :: "('d, 'f) atm" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "ProblemIkTpartG.gL (Neg at) = Neg (ProblemIkTpartG.gA at)"
per_def: fixes eff :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset \<Rightarrow> bool"   and rules :: "'a stream"   and S :: "'b set"   and r :: "'a" assumes "RuleSystem eff rules S" shows "RuleSystem.per eff rules S r \<equiv> \<forall>s r1 sl' s'. s \<in> S \<and> RuleSystem_Defs.enabled eff r s \<and> r1 \<in> i.R rules - {r} \<and> eff r1 s sl' \<and> s' |\<in>| sl' \<longrightarrow> RuleSystem_Defs.enabled eff r s'"
rev_upt_rec: fixes i :: "nat"   and j :: "nat" shows "[i>..j] = (if j < i then [i>..Suc j] @ [j] else [])"
SPRAY_nonevent: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and x :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "x \<notin> \<E>" shows "MinkowskiPrimitive.SPRAY \<P> x = {}"
characteristic_formula_card_aux: fixes P :: "'state" shows "|{distinguishing_formula P Q |Q. \<not> P =\<cdot> Q}| <o natLeq +c |UNIV|"
gauss_int_norm_mod_less_aux: fixes b :: "gauss_int"   and a :: "gauss_int" assumes "b \<noteq> 0" shows "2 * gauss_int_norm (a mod b) \<le> gauss_int_norm b"
insert_dominated: fixes gs :: "'a list"   and ffp :: "nat"   and xs :: "nat"   and x1 :: "'b \<Rightarrow> nat"   and x2 :: "'a \<Rightarrow> 'b" assumes "length gs \<le> ffp" shows "length gs + (max xs (Max (insert ffp (x1 ` x2 ` set gs))) - length gs) = max xs (Max (insert ffp (x1 ` x2 ` set gs)))"
rhs_less: shows "rhs < 2 ^ n\<^sup>2"
inv_comp_right_2: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "C.VV.iso (C.VV.comp g f)"   and "C.VV.iso f" shows "C.VV.iso g"
distinct_782: fixes x23 :: "v"   and x191 :: "t"   and x192 :: "(tp \<times> sx) option"   and x193 :: "nat"   and x194 :: "nat" shows "C x23 \<noteq> Load x191 x192 x193 x194"
rExt: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "(\<Psi>, P, Q) \<in> X" shows "(\<Psi> \<otimes> \<Psi>', P, Q) \<in> X \<union> weakenBisim"
finfun_def: shows "finfun = {f. \<exists>b. finite {a. f a \<noteq> b}}"
identify_block_cong: fixes k :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and i :: "nat" assumes "local.same_upto k A B"   and "k < n"   and "i < k" shows "identify_block A i = identify_block B i"
q\<^sub>0_def: shows "q\<^sub>0 \<equiv> \<^bold>\<exists>x. \<lparr>E!,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>\<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>"
ord_eq_less_trans: fixes a :: "'g"   and b :: "'g"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and c :: "'g" assumes "a = b"   and "less b c" shows "less a c"
map_add_dom_app_simps_3: fixes m :: "'a"   and l2 :: "'a \<Rightarrow> 'b option"   and l1 :: "'a \<Rightarrow> 'b option" assumes "m \<notin> dom l2" shows "(l1 ++ l2) m = l1 m"
arc_def: fixes g :: "real \<Rightarrow> 'c" shows "arc g = (path g \<and> inj_on g {0..1})"
uniform_count_measure_def: fixes A :: "'a set" shows "uniform_count_measure A = point_measure A (\<lambda>x. ennreal (1 / real (card A)))"
Max_by_eqI: fixes A :: "'c set"   and cmp_f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> cmp_f y \<le> cmp_f x"   and "x \<in> A"   and "inj_on cmp_f A" shows "Max_by cmp_f A = x"
types_preserved_drop: fixes v :: "v"   and e :: "b_e"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "\<lparr>[$C v, $e]\<rparr> \<leadsto> \<lparr>[]\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [$C v, $e] : ts _> ts'"   and "e = Drop" shows "\<S>\<bullet>\<C> \<turnstile> [] : ts _> ts'"
ex_fresh_variable: fixes A :: "type_scheme list"   and A' :: "type_scheme list"   and t :: "typ"   and t' :: "typ" shows "\<exists>n. new_tv n A \<and> new_tv n A' \<and> new_tv n t \<and> new_tv n t'"
diff: fixes A :: "(('a, 'c) vec, 'b) vec"   and x :: "('a, 'c) vec"   and y :: "('a, 'c) vec" shows "A *v (x - y) = A *v x - A *v y"
fds_nth_one_Suc_0: shows "fds_nth 1 (Suc 0) = (1::'a)"
g'_def: fixes n :: "nat"   and y :: "nat" shows "g' n y = (if y \<le> n then g y else if y = Suc n then \<Sum>k. g (k + (n + 1)) else 0)"
fold_graph_finite: fixes z :: "'a"   and B :: "'a set"   and y :: "'a" assumes "fold_graph f z B y" shows "finite B"
one_poly_eq_simps_1: shows "(1 = [:1::'a:]) = True"
Mismatch_2: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and a :: "name"   and b :: "name" assumes "P \<Longrightarrow>\<alpha> \<prec> P'"   and "a \<noteq> b" shows "[a\<noteq>b]P \<Longrightarrow>\<alpha> \<prec> P'"
discI_1: fixes literal :: "'a literal"   and x11 :: "char list"   and x12 :: "'a list" assumes "literal = Pos x11 x12" shows "sign literal"
pfinite_converterI: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter" assumes "\<And>a. a \<in> outs_\<I> \<I> \<Longrightarrow> pfinite_gpv \<I>' (run_converter conv a)"   and "\<And>a b conv'. \<lbrakk>a \<in> outs_\<I> \<I>; (b, conv') \<in> results_gpv \<I>' (run_converter conv a)\<rbrakk> \<Longrightarrow> pfinite_converter \<I> \<I>' conv'" shows "pfinite_converter \<I> \<I>' conv"
map_id: fixes t :: "'e set['f]" shows "map_bset id t = t"
Ex_list_of_length: fixes n :: "nat" shows "\<exists>xs. \<^bold>|xs\<^bold>| = n"
elims_3: fixes x :: "ereal"   and xa :: "ereal" assumes "\<not> x < xa" obtains xb :: "real" and y :: "real" where "x = ereal xb" and "xa = ereal y" and "\<not> xb < y"   | a :: "ereal" where "x = \<infinity>" and "xa = aa__"   | a :: "ereal" where "x = a" and "xa = - \<infinity>"
degree_to_poly: fixes a :: "'a" assumes "a \<in> carrier R" shows "deg R (to_polynomial R a) = 0"
wens_single_finite_subset_wens_single: fixes act :: "('b \<times> 'b) set"   and B :: "'b set"   and k :: "nat" shows "wens_single_finite act B k \<subseteq> wens_single act B"
trms\<^sub>s\<^sub>s\<^sub>t_subst_cons: fixes a :: "('d, 'e) stateful_strand_step"   and A :: "('d, 'e) stateful_strand_step list"   and \<delta> :: "'e \<Rightarrow> ('d, 'e) Term.term" shows "trms\<^sub>s\<^sub>s\<^sub>t (a # A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>) = trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) \<union> trms\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)"
size_4: fixes x21 :: "'e"   and x22 :: "'e list" shows "length (x21 # x22) = length x22 + Suc 0"
diff_fun_onI: fixes A :: "'a set"   and W :: "'a set"   and f' :: "'a \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'c" assumes "A \<subseteq> W"   and "W \<subseteq> carrier"   and "open W"   and "diff_fun k (charts_submanifold W) f'"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' x" shows "diff_fun_on A f"
summable_mult: fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "summable f" shows "summable (\<lambda>n. c * f n)"
G1_nodes: shows "G1_nodes = {n. \<exists>ps. valid_graph.is_path (del_unEdge v w v' G) n ps v}"
dghm_HomDom: shows "\<GG>\<lparr>HomDom\<rparr> = op_dg \<BB>"
am_is_in_V_impl_def: fixes am :: "'a list Heap.array"   and u :: "nat" shows "am_is_in_V_impl am u \<equiv> Array.nth am u \<bind> (\<lambda>amu. return (\<not> is_Nil amu))"
nn_integral_has_integral_lebesgue: fixes f :: "'b \<Rightarrow> real"   and I :: "real"   and \<Omega> :: "'b set" assumes "\<And>x. 0 \<le> f x"   and "(f has_integral I) \<Omega>" shows "\<integral>\<^sup>+ x. ennreal (indicat_real \<Omega> x * f x) \<partial>lborel = ennreal I"
disjE: fixes cid :: "'bid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'BB"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'BB) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'BB) \<Rightarrow> nat \<Rightarrow> bool" assumes "pb.eval cid t t' n (\<gamma> \<or>\<^sup>b \<gamma>')" shows "pb.eval cid t t' n \<gamma> \<or> pb.eval cid t t' n \<gamma>'"
prefixI': fixes vs :: "nat list"   and n :: "nat"   and f :: "nat \<Rightarrow> nat option" assumes "length vs = Suc n"   and "\<And>x. x < Suc n \<Longrightarrow> f x \<down>= vs ! x" shows "prefix f n = vs"
a_in_oa: shows "a \<in> out_arcs G (tail G a)"
prep: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm" shows "Ifm bbs bs (prep p) = Ifm bbs bs p"
singular_relcycle_cmul: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and k :: "int" assumes "singular_relcycle p X S c" shows "singular_relcycle p X S (frag_cmul k c)"
simps_5: fixes uu :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and C :: "acom" shows "strip ({uu} WHILE b DO C) = WHILE b DO strip C"
left_diff_repeat_mset_distrib': fixes i :: "nat"   and j :: "nat"   and u :: "'b multiset" shows "repeat_mset (i - j) u = repeat_mset i u - repeat_mset j u"
isomorphic_expansion: fixes a :: "'c"   and a' :: "'c"   and g :: "'d" assumes "C.obj a"   and "C.obj a'"   and "\<guillemotleft>g : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 a'\<guillemotright>"   and "D.ide g" shows "D.isomorphic (\<tau>\<^sub>0 a' \<star>\<^sub>D (\<sigma>'.map\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<sigma>'.map\<^sub>0 a) g"
eq_refl: fixes x :: "'e"   and y :: "'e" assumes "x = y" shows "x \<le> y"
sum_Un_nat: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> nat" assumes "finite A"   and "finite B" shows "sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
ptr_kinds_ptr_kinds_M: fixes ptr :: "(_) character_data_ptr"   and h :: "(_) heap" shows "(ptr \<in> set |h \<turnstile> character_data_ptr_kinds_M|\<^sub>r) = (ptr |\<in>| character_data_ptr_kinds h)"
simps_1: fixes uu :: "'q \<Rightarrow> 'r fundef option"   and \<Sigma> :: "Unboxed.type option list" shows "sp uu [] \<Sigma> = Ok \<Sigma>"
bres_interchange: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<rightarrow> y) \<cdot> z \<le> x \<rightarrow> y \<cdot> z"
1535: fixes x :: "'a" shows "- x \<squnion> - - - - x = top"
assms_16: fixes \<phi> :: "'a formula"   and \<psi> :: "'a formula"   and I :: "\<I>" assumes "fv (Neg \<phi>) \<subseteq> fv \<psi>"   and "\<not> safe_formula (Neg \<phi>)"   and "P \<phi>"   and "P \<psi>" shows "P (Until (Neg \<phi>) I \<psi>)"
distinct_8: fixes x31 :: "char list"   and x32 :: "char list list"   and x21 :: "char list"   and x22 :: "char list list" shows "StepC2 x31 x32 \<noteq> StepC x21 x22"
rep_eq: fixes x :: "'k multiset set" shows "count (Inf x) = (\<lambda>i. if count ` x = {} then 0 else INF f\<in>count ` x. f i)"
eval_monom_uminus: fixes c :: "real"   and es :: "real list"   and basis :: "(real \<Rightarrow> real) list"   and x :: "real" shows "eval_monom (- c, es) basis x = - eval_monom (c, es) basis x"
intro: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "semiring_ow U plus times"   and "cancel_comm_monoid_add_ow U plus minus zero" shows "semiring_0_cancel_ow U plus minus zero times"
take_bit_of_nat: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n (semiring_1.of_nat one plus zero m) = semiring_1.of_nat one plus zero (take_bit n m)"
put_eq: fixes \<sigma> :: "'b"   and k :: "'a"   and u :: "'a"   and \<rho> :: "'b"   and v :: "'a" assumes "get \<sigma> = k"   and "put \<sigma> u = put \<rho> v" shows "put \<rho> k = \<sigma>"
scale_left_distrib: fixes a :: "'a"   and b :: "'a"   and x :: "'c" shows "s2 (a + b) x = s2 a x + s2 b x"
dom_ssubstoption_lem: fixes z :: "char list"   and u :: "sterm"   and f :: "'a \<Rightarrow> sterm option" shows "dom (\<lambda>l. ssubst_option z u (f l)) = dom f"
simps: fixes l :: "name"   and sl :: "name"   and s :: "tm"   and k :: "tm"   and j :: "tm"   and y :: "tm" assumes "atom l \<sharp> (sl, s, k, j)"   and "atom sl \<sharp> (s, j)" shows "SeqHaddP s j k y = LstSeqP s k y AND HPair Zero j IN s AND All2 l k (SyntaxN.Ex sl (HPair (Var l) (Var sl) IN s AND HPair (SUCC (Var l)) (SUCC (Var sl)) IN s))"
r12: shows "generalized_sfw (lr_of_tran_s1 rt) p = Some (r1, oif) \<and> generalized_sfw (map simple_rule_dtor fw) p = Some (r2, Accept) \<and> Some r = simple_match_and r1 r2"
C_restr_below: fixes r :: "C"   and x :: "C \<rightarrow> 'a" shows "x|\<^bsub>r\<^esub> \<sqsubseteq> x"
aperiodic_eventually_recurrent: fixes C :: "'a set" shows "K1.aperiodic C = (C \<in> UNIV // K1.communicating \<and> (\<forall>x\<in>C. \<forall>\<^sub>F m in sequentially. 0 < K1.p x x m))"
arctan_ubound: fixes y :: "real" shows "arctan y < pi / 2"
simps_1: fixes c :: "tcolor" shows "paint c \<langle>\<rangle> = \<langle>\<rangle>"