hcju/isabelleps huggingface dataset:
induced_function_integral_restr_nonergodic_2: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "A \<in> sets M"   and "integrable M f" shows "integral\<^sup>L (restrict_space M A) (induced_function A f) = set_lebesgue_integral M (\<Union>n. (T ^^ n) --` A) f"
Act_eq_iff_perm_renaming: fixes f1 :: "'d"   and \<alpha>1 :: "'c"   and x1 :: "('a, 'b, 'c, 'd) FL_Formula.formula"   and f2 :: "'d"   and \<alpha>2 :: "'c"   and x2 :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "(FL_Formula.Act f1 \<alpha>1 x1 = FL_Formula.Act f2 \<alpha>2 x2) = (f1 = f2 \<and> (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2 \<and> supp p \<subseteq> bn \<alpha>1 \<union> p \<bullet> bn \<alpha>1))"
g_in_ik: shows "t \<in> ik\<^sub>s\<^sub>t S"
subdigraph_axioms: shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
awalk_decomp: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w :: "'a" assumes "awalk u p v"   and "w \<in> set (awalk_verts u p)" shows "\<exists>q r. p = q @ r \<and> awalk u q w \<and> awalk w r v"
ref_simps_3: fixes T :: "Arraytype"   and a :: "nat" shows "ref (arrLenLoc T a) = arrV T a"
l_get_dom_component_get_scdom_component\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_scdom_component\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_scdom_component is_strongly_scdom_component_safe is_weakly_scdom_component_safe get_owner_document get_disconnected_nodes get_disconnected_nodes_locs to_tree_order heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs 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"
frechet_derivative_times: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and g :: "'b \<Rightarrow> 'c" assumes "f differentiable at a"   and "g differentiable at a" shows "frechet_derivative (\<lambda>x. f x * g x) (at a) = (\<lambda>x. f a * frechet_derivative g (at a) x + frechet_derivative f (at a) x * g a)"
memb_4: fixes a :: "'a" assumes "a \<in> A" shows "q (a_to_b a) \<in> B"
dim_3: shows "dim_vec ((sylvester_mat p q)\<^sup>T *\<^sub>v v) = n + m"
castAbs_Abs_vsubst: fixes xs :: "'varSort"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and x :: "'var"   and x' :: "'var"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var" assumes "isInBar (xs, s)"   and "wls s X"   and "castAbs xs s (Abs xs x X) = (x', X')" shows "X #[y // x]_xs = X' #[y // x']_xs"
par_mono: fixes par :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and skip :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d\<^sub>1 :: "'b::refinement_lattice"   and c\<^sub>2 :: "'b::refinement_lattice"   and d\<^sub>2 :: "'b::refinement_lattice" assumes "par_distrib (par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (skip::'b::refinement_lattice)"   and "(c\<^sub>1::'b::refinement_lattice) \<sqsubseteq> (d\<^sub>1::'b::refinement_lattice)"   and "(c\<^sub>2::'b::refinement_lattice) \<sqsubseteq> (d\<^sub>2::'b::refinement_lattice)" shows "(par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c\<^sub>1::'b::refinement_lattice) (c\<^sub>2::'b::refinement_lattice) \<sqsubseteq> par (d\<^sub>1::'b::refinement_lattice) (d\<^sub>2::'b::refinement_lattice)"
insert_DiffM: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "add_mset x (M \<ominus> x) = M"
spmf_of_pmf_return_pmf: fixes x :: "'h" shows "spmf_of_pmf (return_pmf x) = return_spmf x"
m_inv_monoid_consistent: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and H :: "'e set" assumes "Group.monoid G"   and "x \<in> Units (G\<lparr>carrier := H\<rparr>)"   and "submonoid H G" shows "inv\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> x = inv\<^bsub>G\<^esub> x"
sdom_neq: fixes G :: "nat graph_rec"   and n2 :: "nat"   and n1 :: "nat" assumes "cfg_doms G"   and "cfg_doms.reachable G n2"   and "cfg_doms.strict_dominate G n1 n2" shows "n1 \<noteq> n2"
IH: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set" assumes "as \<noteq> []"   and "slist \<noteq> []"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "slist \<in> state_set (state_list s as)" shows "\<exists>as'. prefix as' as \<and> exec_plan s as' = last slist \<and> length slist = Suc (length as')"
fps_numeral_nth: fixes n :: "num"   and i :: "nat" shows "fps_nth (numeral n) i = (if i = 0 then numeral n else (0::'a))"
div_mult_cancel: fixes a :: "'a"   and b :: "'a" shows "a div b * b = a - a mod b"
tail_sim_second: fixes s :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'a" assumes "tail s \<approx> tail t" shows "second s = second t"
PDG_path_Append: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b"   and control_dependence :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n'' :: "'b"   and n' :: "'b"   and n :: "'b" assumes "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit control_dependence"   and "PDG.PDG_path sourcenode targetnode valid_edge Def Use control_dependence n'' n'"   and "PDG.PDG_path sourcenode targetnode valid_edge Def Use control_dependence n n''" shows "PDG.PDG_path sourcenode targetnode valid_edge Def Use control_dependence n n'"
dom_comp: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "CopxC.seq g f" shows "CopxC.dom (g \<odot> f) = CopxC.dom f"
mat_det_hermitean_real: fixes A :: "complex \<times> complex \<times> complex \<times> complex" assumes "hermitean A" shows "is_real (mat_det A)"
Hermite_of_preserves_zero_rows: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "is_zero_row i (echelon_form_of A bezout)"   and "ass_function ass"   and "res_function res"   and "is_bezout_ext bezout" shows "is_zero_row i (Hermite_of A ass res bezout)"
I_intP_Cext: fixes p :: "'psym"   and al :: "univ list" assumes "polC p = Cext" shows "I_intP p al = intP p (map2 proj (parOf p) al)"
substitutes_onD: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set"   and a :: "'a"   and b :: "'a" assumes "substitutes_on A f"   and "B \<subseteq> A"   and "a \<in> A"   and "b \<in> A"   and "b \<notin> f (insert b B)" shows "b \<notin> f (insert a (insert b B))"
simps_4: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and v :: "'c match_expr"   and rs :: "'c rule list" shows "no_matching_Goto \<gamma> p (Rule v Drop # rs) = no_matching_Goto \<gamma> p rs"
padic_simps_4: fixes p :: "int"   and f :: "nat \<Rightarrow> int"   and g :: "nat \<Rightarrow> int"   and n :: "nat" shows "padic_add p f g n = f n \<oplus>\<^bsub>residue_ring (p ^ n)\<^esub> g n"
1: shows "deg R (T\<^bsub>a\<^esub> p) \<le> n"
br_Some_O_default_rel_eq: fixes d :: "'b"   and A :: "('a \<times> 'b) set" shows "br Some top O \<langle>A\<rangle>default_rel d = A"
zmset_frontier_empty: shows "zmset_of (mset_set (set_antichain (frontier {#}\<^sub>z))) = {#}\<^sub>z"
inject_3: fixes x5 :: "nat \<times> char list list"   and y5 :: "nat \<times> char list list" shows "(Ref x5 = Ref y5) = (x5 = y5)"
assms_2: shows "g maps\<^bsub>C\<^esub> Y to Z"
schroeder_5: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<le> z) = (y * - z\<^sup>T \<le> - x\<^sup>T)"
callee_invariant: fixes y :: "'b"   and s' :: "'s1"   and s :: "'s1"   and x :: "'a" assumes "(y, s') \<in> set_spmf (left s x)"   and "Il s"   and "x \<in> outs_\<I> \<I>l" shows "Il s'"
sprod_tau_nu_var: fixes x :: "'a"   and y :: "'a" shows "\<nu> (\<nu> x \<cdot> y) = \<nu> (x \<cdot> y)"
Gauss_Jordan_in_ij_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b" shows "Gauss_Jordan_in_ij A i j = (let n = LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n; interchange_A = interchange_rows A i n; A' = mult_row interchange_A i ((1::'a) / interchange_A $ i $ j) in \<chi>s. if s = i then A' $ s else row_add A' s i (- interchange_A $ s $ j) $ s)"
adjacent_preimage: fixes C :: "'a set"   and D :: "'a set" assumes "C \<in> folding.\<C> - f \<turnstile> folding.\<C>"   and "D \<in> folding.\<C> - f \<turnstile> folding.\<C>"   and "f ` C \<sim> f ` D" shows "C \<sim> D"
IH_prem1: fixes f :: "'a match_expr \<Rightarrow> 'a match_expr option"   and m :: "'a match_expr"   and m' :: "'a match_expr" assumes "f \<in> set fs"   and "normalized_nnf_match m"   and "f m = Some m'" shows "matches \<gamma> m' a p = matches \<gamma> m a p"
rep_eq: fixes x :: "uint32" shows "int_of_uint32 x = uint (Rep_uint32 x)"
simps_2: fixes p :: "('a, 'b) form"   and c :: "'a"   and cs :: "'a list" shows "vars_for_consts p (c # cs) = subc c (Var (length cs)) (vars_for_consts p cs)"
one_closed: shows "\<one> \<in> carrier (Ring_Divisibility.mult_of R)"
simps_2: fixes A :: "(('d, 'e) vec, 'f) vec"   and n :: "nat"   and i :: "'f"   and j :: "'e"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" shows "bezout_iterate A (Suc n) i j bezout = (if Suc n \<le> mod_type_class.to_nat i then A else bezout_iterate (bezout_matrix A i (mod_type_class.from_nat (Suc n)) j bezout ** A) n i j bezout)"
update_def: fixes v :: "'V"   and \<sigma> :: "'V set \<times> ('V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option)" shows "update v \<sigma> \<equiv> ASSERT (update_pre v \<sigma>) \<bind> (\<lambda>_. SPEC (update_spec v \<sigma>))"
prefix_Par_Int_skip1: fixes a :: "'a"   and A :: "'a set"   and P :: "'a process" assumes "a \<notin> A" shows "(a \<rightarrow> P \<lbrakk>A\<rbrakk> SKIP) = (a \<rightarrow> (P \<lbrakk>A\<rbrakk> SKIP))"
injflip: fixes A :: "('a \<times> 'b) set" shows "inj_on flip A"
Lp_D_3: fixes p :: "real"   and f :: "'a \<Rightarrow> real"   and M :: "'a measure" assumes "0 < p"   and "f \<in> lspace M (ennreal p)" shows "Norm (\<LL> (ennreal p) M) f = (LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p)"
C2: fixes \<phi> :: "'c"   and Q :: "('a, 'b, 'c) psi" assumes "(\<phi>, Q) mem CsQ" shows "\<exists>P. (\<phi>, P) mem CsP \<and> guarded P \<and> P \<doteq>\<^sub>c Q"
sum_nonneg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq zero (f x)" shows "less_eq zero (comm_monoid_add.sum plus zero f A)"
subset_butlast: fixes xs :: "'a list"   and ys :: "'a list"   and y :: "'a" assumes "set xs \<subseteq> set (ys @ [y])"   and "last xs = y"   and "distinct xs" shows "set (butlast xs) \<subseteq> set ys"
nat_pow_mult: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "x [^]\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> n \<otimes>\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> x [^]\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> m = x [^]\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> (n + m)"
span_scale: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and S :: "'c set"   and c :: "'b" assumes "module scale"   and "x \<in> module.span scale S" shows "scale c x \<in> module.span scale S"
indicator_times: fixes A :: "'b set"   and B :: "'c set"   and x :: "'b \<times> 'c" shows "indicator (A \<times> B) x = indicator A (fst x) * indicator B (snd x)"
WHILET_def: shows "cWHILET \<equiv> cWHILEIT (\<lambda>_. True)"
Lambert_W'_exp_times_self: fixes x :: "real" assumes "x \<le> - 1" shows "Lambert_W' (x * exp x) = x"
inf_le1: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) x"
rank_ins2: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.rank_invar bq" shows "SkewBinomialHeapStruc.rank t \<le> SkewBinomialHeapStruc.rank (hd (SkewBinomialHeapStruc.ins t bq)) \<or> SkewBinomialHeapStruc.rank (hd (SkewBinomialHeapStruc.ins t bq)) = SkewBinomialHeapStruc.rank (hd bq) \<and> bq \<noteq> []"
gen_lucas_lehmer_sequence_Suc': fixes a :: "int"   and n :: "nat" shows "gen_lucas_lehmer_sequence a (Suc n) = gen_lucas_lehmer_sequence (a\<^sup>2 - 2) n"
relation_refines: fixes a :: "'a"   and b :: "'a" assumes "a \<rightarrow> b" shows "ord b a"
prems_2: shows "has_sort (osig (sig \<Theta>)) (Tv idn S') S"
comp_arr_dom: fixes f :: "'b"   and a :: "'b" assumes "B.arr f"   and "B.dom f = a" shows "B f a = f"
legs_form_cospan_2: shows "C.cospan g.cod.leg0 f.cod.leg1"
monotone_ltake2: fixes n :: "enat" shows "monotone lprefix lprefix (ltake n)"
lossless_bind_spmf: fixes p :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "lossless_spmf (p \<bind> f) = (lossless_spmf p \<and> (\<forall>x\<in>set_spmf p. lossless_spmf (f x)))"
inverts_components: fixes a :: "'a" assumes "\<tau>.A.ide a" shows "\<tau>.B.inverse_arrows (\<tau> a) (\<tau>'.map a)"
permutes_implies_inv_image_on_eq: fixes p :: "'c \<Rightarrow> 'c"   and B :: "'c set"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "p permutes B" shows "(\<lambda>b. {x \<in> A. p (f x) = b}) ` B = (\<lambda>b. {x \<in> A. f x = b}) ` B"
in_new_spairsD_1: fixes a :: "'t \<Rightarrow>\<^sub>0 'b"   and b :: "'t \<Rightarrow>\<^sub>0 'b"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "Inl (a, b) \<in> set (new_spairs bs p)" shows "a = p"
mem_box_2: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "(x \<in> cbox a b) = (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
app: fixes t :: "'a"   and u :: "'a" shows "P (app t u) = (P t \<or> P u)"
scaleR: fixes b :: "'a \<Rightarrow> real"   and r :: "real" assumes "\<forall>x. x \<in> diff_fun_space \<longrightarrow> X x \<in> UNIV"   and "b \<in> diff_fun_space" shows "X (r *\<^sub>R b) = r *\<^sub>R X b"
poly_eval_zero: fixes a :: "'d \<Rightarrow> 'c" shows "poly_eval a 0 = (0::'c)"
C6: shows "rel_weakly_respects_barbs Rel (STCalWB SWB TWB)"
IH_2: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "(X, Y) \<in> qSwapped" shows "(\<forall>z\<in>V. \<forall>zs\<in>fZs z. qFresh zs z Y) \<longrightarrow> (\<exists>X'. Y #= X' \<and> (\<forall>z\<in>V. \<forall>zs\<in>fZs z. qAFresh zs z X'))"
u_simps_3: shows "trg r\<^sub>0 = src t"
tauLaw2CongSimLeft: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "\<And>\<Psi> P. (\<Psi>, P, P) \<in> Rel"   and "\<And>\<Psi> P Q R S. \<lbrakk>\<Psi> \<rhd> P \<sim> Q; (\<Psi>, Q, R) \<in> Rel; \<Psi> \<rhd> R \<sim> S\<rbrakk> \<Longrightarrow> (\<Psi>, P, S) \<in> Rel" shows "\<Psi> \<rhd> P \<oplus>\<^sub>\<top> \<tau>.P \<leadsto>\<guillemotleft>Rel\<guillemotright> \<tau>.P"
update_def: fixes F :: "'a Abs_State.st"   and x :: "char list"   and y :: "'a" shows "update F x y = FunDom ((fun F)(x := y)) (if x \<in> set (Abs_State.dom F) then Abs_State.dom F else x # Abs_State.dom F)"
emeasure_eq_1_AE: fixes S :: "'s set" assumes "S \<in> stat.events"   and "AE x in stat C. x \<in> S" shows "emeasure (stat C) S = 1"
PK_H: shows "(P, K) = local.reduce_column div_op H"
iso_iff: shows "(f \<in> Group.iso A B) = (carrier B \<subseteq> f ` carrier A \<and> (\<forall>x\<in>carrier A. f x = \<one>\<^bsub>B\<^esub> \<longrightarrow> x = \<one>\<^bsub>A\<^esub>))"
termi_pr: fixes x :: "nat"   and g :: "recf"   and xs :: "nat list"   and n :: "nat"   and f :: "recf" assumes "\<forall>y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"   and "terminate f xs"   and "length xs = n" shows "terminate (Pr n f g) (xs @ [x])"
elem_index_def: shows "elem_index = (SOME f. inj f)"
isomorphic_to_left_adjoint_implies_isomorphic_right_adjoint: fixes f :: "'a"   and h :: "'a" assumes "is_left_adjoint f"   and "f \<cong> h" shows "f\<^sup>* \<cong> h\<^sup>*"
flincomb': shows "(\<bullet>\<sharp>\<star>) \<equiv> scalar_mult.lincomb (aezfun_scalar_mult.fsmult (\<star>))"
lens_indep_comm: fixes x :: "'c \<Longrightarrow> 'd"   and y :: "'e \<Longrightarrow> 'd"   and \<sigma> :: "'d"   and v :: "'e"   and u :: "'c" assumes "x \<bowtie> y" shows "put\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) u = put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) v"
subcls_asym2: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> C \<preceq>\<^sup>* D"   and "wf_prog wf_md P"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "C = D"
starfun_abs_exp_cancel: fixes x :: "real star" shows "\<bar>(*f* exp) x\<bar> = (*f* exp) x"
div_mult_mono: fixes a :: "nat"   and d :: "nat"   and b :: "nat" assumes "0 < a"   and "a \<le> d" shows "a * b div d \<le> b"
assms_2: shows "F \<subseteq> P[insert x X]"
assms_2: shows "\<forall>\<^sub>F x in F. f x = f' x"
pac_ideal_mult: fixes p :: "int mpoly"   and A :: "int mpoly multiset"   and q :: "int mpoly" assumes "p \<in># A" shows "p * q \<in> pac_ideal (set_mset A)"
mgp_TameEnum: fixes g :: "graph"   and p :: "nat" assumes "g \<in> TameEnum\<^bsub>p\<^esub>" shows "minGraphProps g"
power_up_le: fixes x :: "real"   and xa :: "real"   and n :: "nat"   and p :: "nat" assumes "x \<le> xa ^ n"   and "0 \<le> xa" shows "x \<le> power_up p xa n"
UN_extend_simps_8: fixes B :: "'d \<Rightarrow> 'c set"   and A :: "'d set set" shows "(\<Union>y\<in>A. \<Union> (B ` y)) = \<Union> (B ` \<Union> A)"
tosfw_2: shows "\<forall>r\<in>set (to_simple_firewall_without_interfaces ipassmt rtblo rs). iiface (match_sel r) = ifaceAny \<and> oiface (match_sel r) = ifaceAny"
key_map_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and \<pp> :: "'c set" shows "key_map R addition multiplication zero unit \<pp> \<equiv> comm_ring R addition multiplication zero unit \<and> key_map_axioms R addition multiplication zero unit \<pp>"
lc_subst1: fixes s :: "(nat \<times> form) list"   and f :: "form" shows "lc (subst1 s f) \<subseteq> lc f \<union> \<Union> (lc ` snd ` set s)"
empty: fixes dxs :: "'a odlist" assumes "dxs = ODList.empty" shows "P dxs"
paths_Cons_nxt: fixes t :: "'b ttree"   and x :: "'b"   and xs :: "'b list" assumes "possible t x"   and "xs \<in> paths (nxt t x)" shows "x # xs \<in> paths t"
simps_2: fixes y :: "'b"   and P :: "'b Predicate.pred"   and x :: "'b" shows "Predicate.member (seq.Insert y P) x = (x = y \<or> pred.eval P x)"
\<rho>'_ide_simp: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.\<rho>' a = \<F>C.runit' a"
ptr_kinds_M_ok: fixes h :: "'x"   and ptr_kinds :: "'x \<Rightarrow> 'y fset" shows "h \<turnstile> ok l_ptr_kinds_M.a_ptr_kinds_M ptr_kinds"
dfree_Times: fixes \<theta>\<^sub>1 :: "('a, 'b) trm"   and \<theta>\<^sub>2 :: "('a, 'b) trm" assumes "dfree \<theta>\<^sub>1"   and "dfree \<theta>\<^sub>2" shows "dfree (Times \<theta>\<^sub>1 \<theta>\<^sub>2)"
list_minus_butlast_eq_butlast_list: fixes t :: "'b list"   and s :: "'b list" assumes "length t = length s"   and "strict_prefix (butlast s) t" shows "t - butlast s = [last t]"
unat_shiftr_less_t2n: fixes x :: "'b word"   and n :: "nat"   and m :: "nat" assumes "unat x < 2 ^ (n + m)" shows "unat (x >> n) < 2 ^ m"
elims: fixes x :: "Core.term"   and y :: "Core.term" assumes "strip_all_single_body x = y" obtains alla :: "String.literal" and S :: "typ" and T :: "typ" and t :: "Core.term" where "x = Ct alla S $ Abs T t" and "y = (if alla = STR ''Pure.all'' \<and> S = (T \<rightarrow> propT) \<rightarrow> propT then t else Ct alla S $ Abs T t)"   | v :: "String.literal" and va :: "typ" where "x = Ct v__ va__" and "y = Ct v__ va__"   | v :: "variable" and va :: "typ" where "x = Fv vb__ vaa__" and "y = Fv vb__ vaa__"   | v :: "nat" where "x = Bv vc__" and "y = Bv vc__"   | v :: "typ" and va :: "Core.term" where "x = Abs vd__ vab__" and "y = Abs vd__ vab__"   | vb :: "variable" and vc :: "typ" and va :: "Core.term" where "x = Fv vba__ vca__ $ vac__" and "y = Fv vba__ vca__ $ vac__"   | vb :: "nat" and va :: "Core.term" where "x = Bv vbb__ $ vad__" and "y = Bv vbb__ $ vad__"   | vb :: "typ" and vc :: "Core.term" and va :: "Core.term" where "x = Abs vbc__ vcb__ $ vae__" and "y = Abs vbc__ vcb__ $ vae__"   | vb :: "Core.term" and vc :: "Core.term" and va :: "Core.term" where "x = vbd__ $ vcc__ $ va" and "y = vbd__ $ vcc__ $ va"   | v :: "Core.term" and vb :: "String.literal" and vc :: "typ" where "x = ve__ $ Ct vbe__ vcd__" and "y = ve__ $ Ct vbe__ vcd__"   | v :: "Core.term" and vb :: "variable" and vc :: "typ" where "x = vf__ $ Fv vbf__ vce__" and "y = vf__ $ Fv vbf__ vce__"   | v :: "Core.term" and vb :: "nat" where "x = vg__ $ Bv vbg__" and "y = vg__ $ Bv vbg__"   | v :: "Core.term" and vb :: "Core.term" and vc :: "Core.term" where "x = v $ (vb $ vc)" and "y = v $ (vb $ vc)"
Fun_ide: fixes a :: "'s" assumes "S.ide a" shows "S.Fun a = (\<lambda>x\<in>S.set a. x)"
invimage_of_vempty: fixes r :: "V" shows "r -`\<^sub>\<circ> 0 = 0"
assms_2: shows "vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2"
case_prodI2: fixes p :: "'a \<times> 'b"   and c :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
dgrad_p_set_varnum: fixes X :: "'x set" shows "i.punit.dgrad_p_set (varnum X) 0 = P[X]"
y_def: shows "y = (1 - u) *\<^sub>R a + u *\<^sub>R b"
f_def: shows "f \<equiv> \<lambda>i x. real_of_int \<lfloor>enn2real (min (of_nat i) (u x)) * 2 ^ i\<rfloor> / 2 ^ i"
term_ok_def: fixes \<Theta> :: "((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)) \<times> Core.term set"   and t :: "Core.term" shows "term_ok \<Theta> t \<equiv> wt_term (sig \<Theta>) t"
remove: fixes x :: "'a"   and A :: "'a multiset" assumes "x \<in># A" shows "\<Sum>\<^sub># A = x + \<Sum>\<^sub># (A - {#x#})"
assms: fixes a :: "'a" assumes "a \<in> ds \<pi> \<sigma>" shows "a \<notin> fvs M"
attractor_in_V: fixes W :: "'a set"   and p :: "Player" assumes "W \<subseteq> V\<^bsub>G'\<^esub>" shows "G'.attractor p W \<subseteq> V\<^bsub>G'\<^esub>"
counit_simps_3: shows "cod \<i>[trg \<mu>] = trg (trg \<mu>)"
IH: assumes "xs \<noteq> []" shows "list_max xs \<in> set xs"