hcju/isabelleps huggingface dataset:
is_closedI: fixes A :: "'a set" assumes "\<And>x y. \<lbrakk>x .= y; x \<in> A; y \<in> carrier S\<rbrakk> \<Longrightarrow> y \<in> A"   and "A \<subseteq> carrier S" shows "is_closed A"
hyps_3: shows " P \<longrightarrow>\<^sub>1 P'"
RedCast: fixes h :: "heap"   and a :: "nat"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list"   and P :: "(char list list \<times> char list exp) prog"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool" assumes "h a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "P \<turnstile> \<langle>Cast C (addr a),(h, l, sh),b\<rangle> \<rightarrow> \<langle>addr a,(h, l, sh),b\<rangle>"
update_def: fixes x :: "'c"   and y :: "'d"   and t :: "(('c \<times> 'd) \<times> color) tree" shows "update x y t = paint Black (upd x y t)"
finite_measure_mono: fixes A :: "real set"   and B :: "real set"   and n :: "nat" assumes "A \<subseteq> B"   and "B \<in> Mn.events n" shows "Mn.prob n A \<le> Mn.prob n B"
heap_is_wellformed_children_disc_nodes: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "node_ptr |\<in>| node_ptr_kinds h"   and "\<not> (\<exists>parent\<in>fset (object_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_child_nodes parent|\<^sub>r)" shows "\<exists>document_ptr\<in>fset (document_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r"
wf_subcls1: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "wf_prog wf_md P" shows "wfP (subcls1 P)\<inverse>\<inverse>"
diff_union_cancelR: fixes M :: "'b multiset"   and N :: "'b multiset" shows "M + N - N = M"
map\<^sub>0_simps_1: fixes a :: "'b" assumes "B.obj a" shows "C.obj (F.map\<^sub>0 a)"
scaleR_diff_left: fixes a :: "real"   and b :: "real"   and x :: "'b" shows "(a - b) *\<^sub>R x = a *\<^sub>R x - b *\<^sub>R x"
ldistinct_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "ldistinct (lappend xs ys) = (ldistinct xs \<and> (lfinite xs \<longrightarrow> ldistinct ys \<and> lset xs \<inter> lset ys = {}))"
fct_gen_subalgebra_is_subalgebra: fixes X :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure" assumes "X \<in> M \<rightarrow>\<^sub>M N" shows "subalgebra M (fct_gen_subalgebra M N X)"
C_runs: fixes xs :: "'b list" shows "C_runs xs \<le> length xs - 1"
hyps_1: shows "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> Rs \<prec> C"
ord_def: fixes n :: "'b"   and a :: "'b" shows "ord n a = (if coprime n a then LEAST d. 0 < d \<and> [a ^ d = 1::'b] (mod n) else 0)"
Jacobi_def: fixes a :: "int"   and n :: "int" shows "Jacobi a n = (if n = 0 then 0 else \<Prod>\<^sub># (image_mset (Legendre a) (prime_factorization n)))"
3: fixes x :: "'a" assumes "x \<in> A" shows "e \<le> f x"
BseqI': fixes X :: "nat \<Rightarrow> 'a"   and K :: "real" assumes "\<And>n. norm (X n) \<le> K" shows "Bseq X"
distinct_85: fixes pi1 :: "pi"   and pi2 :: "pi"   and name' :: "name"   and pi' :: "pi" shows "pi1 \<parallel> pi2 \<noteq> <\<nu>name'>pi'"
r_2: shows "q = p \<otimes>\<^bsub>K [X]\<^esub> r"
in_set_merge_coeffs0D: fixes a :: "char list list"   and b :: "int"   and p :: "(char list list \<times> int) list" assumes "(a, b) \<in> set (merge_coeffs0 p)" shows "\<exists>b. (a, b) \<in> set p"
fps_mult_equiv: fixes f :: "'a fps"   and g :: "'a fps"   and p :: "'a"   and h :: "'a fps" assumes "(f, g) \<in> fpsmodrel p" shows "(f * h, g * h) \<in> fpsmodrel p"
list_cdh'_def: fixes \<G> :: "nat \<Rightarrow> 'b cyclic_group" shows "list_cdh' \<G> \<equiv> \<forall>\<eta>. cyclic_group (\<G> \<eta>)"
polynomial_decomp_alien_var_1: fixes q :: "int mpoly"   and A :: "int mpoly"   and x' :: "nat"   and b :: "int mpoly" assumes "q = A * MPoly_Type.monom (monomial (Suc 0) x') 1 + b"   and "x' \<notin> vars q"   and "x' \<notin> vars b" shows "A = 0"
hasw_def: fixes xs :: "'b list"   and S :: "'b set" shows "hasw xs S \<equiv> set xs \<subseteq> S \<and> distinct xs"
finprod_singleton_set: fixes f :: "'b \<Rightarrow> 'a vec"   and a :: "'b" assumes "f a \<in> carrier_vec n" shows "finsum V f {a} = f a"
convert_semantics_R: fixes fml :: "real poly fml"   and x :: "real" shows "real_sem fml x = lookup_sem (fst (convert fml)) (map (\<lambda>p. poly p x) (snd (convert fml)))"
f_expand_1: fixes xs :: "'d message_af list" shows "xs \<odot> Suc 0 = xs"
sup2_def: fixes a :: "'a"   and b :: "'a" shows "a \<squnion>2 b = ((a r\<rightarrow> b) l\<rightarrow> b) \<sqinter> ((b r\<rightarrow> a) l\<rightarrow> a)"
sign_changes_def: fixes xs :: "'b list" shows "sign_changes xs = length (remdups_adj (filter (\<lambda>x. x \<noteq> (0::'b)) (map sgn xs))) - 1"
intro: fixes \<delta> :: "real" assumes "0 < \<delta>" shows "safe_distance_normal_axioms \<delta>"
safety_state_prop: fixes P :: "'b \<Rightarrow> bool" shows "safety \<lceil>P\<rceil>"
const_arity_eq_zero: fixes c :: "'fun" assumes "c \<in> \<C>" shows "arity c = 0"
distinctClosed: fixes p :: "(name \<times> name) list"   and xs :: "'a list" shows "distinct (p \<bullet> xs) = distinct xs"
subresultant_prs_impl_def: fixes f :: "'b poly \<times> 'b \<Rightarrow> 'a"   and G1 :: "'b poly"   and G2 :: "'b poly" shows "subresultant_prs_impl f G1 G2 = (let pmod = pseudo_mod G1 G2; n2 = degree G2; delta_1 = degree G1 - n2; g2 = lead_coeff G2; h2 = g2 ^ delta_1 in if pmod = 0 then f (G2, h2) else let G3 = (- 1) ^ (delta_1 + 1) * pmod; g3 = lead_coeff G3; n3 = degree G3; d2 = n2 - n3; pmod = pseudo_mod G2 G3 in if pmod = 0 then f (G3, if d2 = 1 then g3 else dichotomous_Lazard g3 h2 d2) else let divisor = (- (1::'b)) ^ (d2 + 1) * g2 * h2 ^ d2; G4 = sdiv_poly pmod divisor in subresultant_prs_main_impl f G3 G4 n3 d2 h2)"
HCP_adv_neg: shows "Negligible.negligible etp_advantage"
arr_char: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "dual_category C" shows "partial_magma.arr (dual_category.comp C) f = partial_magma.arr C f"
NSDeMoivre: fixes a :: "real star"   and n :: "nat" shows "hcis a ^ n = hcis (hypreal_of_nat n * a)"
assms: shows "\<psi> \<in> \<P>\<^sup>2"
amult_eq_eq_r: fixes z :: "int"   and a :: "ant"   and b :: "ant" assumes "z \<noteq> 0"   and "a * ant z = b * ant z" shows "a = b"
root_int_floor_def: fixes p :: "nat"   and x :: "int" shows "root_int_floor p x = (if 0 \<le> x then root_int_floor_pos p x else - root_int_ceiling_pos p (- x))"
z_3: shows "erf b - erf a = (b - a) * (2 / sqrt pi * exp (- z\<^sup>2))"
hom: shows "homeomorphism (h ` T) (h ` T) f g"
lprefix_code_3: fixes x :: "'c"   and xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" shows "LCons x xs \<sqsubseteq> LCons y ys = (x = y \<and> xs \<sqsubseteq> ys)"
sum_encode: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "(\<Sum>i = 0..<d1. sum (f i) {0..<d2}) = (\<Sum>k = 0..<d. f (encode1 k) (encode2 k))"
nlcons_invlevel: fixes n :: "nat"   and t :: "('a, 'b) FingerTreeStruc"   and nd :: "('a, 'b) Node" assumes "FingerTreeStruc.is_leveln_ftree n t"   and "FingerTreeStruc.is_leveln_node n nd" shows "FingerTreeStruc.is_leveln_ftree n (FingerTreeStruc.nlcons nd t)"
lowner_lub_form: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat" assumes "matrix_seq dim f" shows "matrix_seq.lowner_is_lub f (mat dim dim (\<lambda>(i, j). lim (\<lambda>n. f n $$ (i, j))))"
edge'_simps_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and x :: "'a node" assumes "flow_attainability \<Delta>" shows "flow_attainability.edge' \<Delta> SOURCE x = (x = Inner (source \<Delta>))"
list_update_overwrite: fixes xs :: "'g list"   and i :: "nat"   and x :: "'g"   and y :: "'g" shows "xs[i := x, i := y] = xs[i := y]"
plan_action_path_def: fixes P :: "ast_problem"   and M :: "object atom formula set"   and \<pi>s :: "plan_action list"   and M' :: "object atom formula set" shows "ast_problem.plan_action_path P M \<pi>s M' = ((\<forall>\<pi>\<in>set \<pi>s. ast_problem.wf_plan_action P \<pi>) \<and> ground_action_path M (map (ast_problem.resolve_instantiate P) \<pi>s) M')"
incseq_def: fixes X :: "nat \<Rightarrow> 'b" shows "incseq X = (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
emb_prepend_step: fixes t :: "('c, 'd) tm"   and u :: "('c, 'd) tm"   and s :: "('c, 'd) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b u"   and "u \<unrhd>\<^sub>e\<^sub>m\<^sub>b s" shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
LeftDerivation_append1: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and c :: "symbol list" assumes "LeftDerivation a (D @ [(i, r)]) c" shows "\<exists>b. LeftDerivation a D b \<and> LeftDerives1 b i r c"
global_var_approx_def: fixes prog :: "'b program_scheme" shows "global_var_approx prog \<equiv> set (global_vars prog)"
Let_2: shows "atom ` domA \<Delta> \<sharp>* L"
nth_mem: fixes n :: "nat"   and xs :: "'c list" assumes "n < length xs" shows "xs ! n \<in> set xs"
a1: fixes n'' :: "nat" assumes "n \<le> n''"   and "n'' < n'" shows "eval c t t' n'' \<gamma>'"
is_static_fieldm_simp: fixes n :: "vname"   and f :: "qtname \<times> Decl.field" shows "is_static (fieldm n f) = is_static f"
streamValue5: fixes D :: "nat"   and str :: "nat \<Rightarrow> 'a"   and t :: "nat"   and i :: "nat"   and k :: "nat"   and x :: "'a"   and j :: "nat" assumes "\<forall>j\<le>D. str (t + j + (i + k)) = x"   and "j \<le> D" shows "str (t + i + k + j) = x"
ss_nth_inA: shows "\<forall>i<length ss. ss ! i \<in> A"
dependent_insertD: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and S :: "'b set" assumes "vector_space scale"   and "a \<notin> module.span scale S"   and "module.dependent scale (insert a S)" shows "module.dependent scale S"
cEnum_bool_def: shows "CENUM(bool) = Some (enum_class.enum, enum_class.enum_all, enum_class.enum_ex)"
REC_refine: fixes body :: "('b \<Rightarrow> 'c nres) \<Rightarrow> 'b \<Rightarrow> 'c nres"   and x :: "'b"   and x' :: "'d"   and R :: "('b \<times> 'd) set"   and S :: "('c \<times> 'e) set"   and body' :: "('d \<Rightarrow> 'e nres) \<Rightarrow> 'd \<Rightarrow> 'e nres" assumes "trimono body"   and "(x, x') \<in> R"   and "\<And>f f' x x'. \<lbrakk>\<And>x x'. (x, x') \<in> R \<Longrightarrow> f x \<le> \<Down> S (f' x'); (x, x') \<in> R; REC body' = f'\<rbrakk> \<Longrightarrow> body f x \<le> \<Down> S (body' f' x')" shows "REC body x \<le> \<Down> S (REC body' x')"
RepBaseclass_isBaseclass: fixes 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 C :: "char list"   and Bs :: "base list"   and fs :: "(char list \<times> ty) list"   and ms :: "(char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, fs, ms)\<rfloor>"   and "Repeats D \<in> set Bs" shows "D \<in> baseClasses Bs"
hom_imp_img_monoid: fixes h :: "'c \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "h \<in> hom (H\<lparr>carrier := h ` carrier G, one := h \<one>\<rparr>) H" shows "Group.monoid (H\<lparr>carrier := h ` carrier (H\<lparr>carrier := h ` carrier G, one := h \<one>\<rparr>), one := h \<one>\<^bsub>H\<lparr>carrier := h ` carrier G, one := h \<one>\<rparr>\<^esub>\<rparr>)"
Complexs_add: fixes a :: "'b"   and b :: "'b" assumes "a \<in> \<complex>"   and "b \<in> \<complex>" shows "a + b \<in> \<complex>"
distinct_3: fixes x1 :: "'b"   and x3 :: "'d" shows "In1 x1 \<noteq> In3 x3"
that: fixes a :: "V" assumes "A = ZFC_in_HOL.set {\<langle>j, a\<rangle>}"   and "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "thesis"
coprime_nat_abs_left_iff: fixes k :: "int"   and n :: "nat" shows "coprime (nat \<bar>k\<bar>) n = coprime k (int n)"
C2: fixes xvec :: "name list"   and \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<Psi>'"   and "xvec \<sharp>* S" shows "(\<Psi>', \<lparr>\<nu>*xvec\<rparr>S \<parallel> T \<parallel> U, S \<parallel> (\<lparr>\<nu>*xvec\<rparr>T \<parallel> U)) \<in> Rel"
proved_terms_well_formed_2: 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 \<Gamma> :: "Core.term set"   and p :: "Core.term" assumes "\<Theta>,\<Gamma> \<turnstile> p" shows "term_ok \<Theta> p"
preserves_dom: fixes f :: "'a" assumes "B.arr f" shows "B.dom (src\<^sub>B f) = src\<^sub>B (B.dom f)"
generate_field_def: shows "generate_field \<equiv> \<lambda>R H. {x. generate_fieldp R (\<lambda>x. x \<in> H) x}"
span_Basis: shows "module_on.span (*\<^sub>R) basis = tangent_space p"
invmode_Static_eq: fixes m :: "'b member_scheme"   and e :: "expr" shows "(invmode m e = Static) = is_static m"
empty_mrexp_mrexps: fixes r :: "'b rexp" shows "empty_mrexp r \<in> mrexps r"
jvm_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" shows "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>"
Rep_check: fixes x :: "check" shows "Rep_check x \<in> {x. x < 2}"
bprogress: fixes \<Gamma> :: "char list \<Rightarrow> ty"   and b :: "bexp"   and s :: "char list \<Rightarrow> val" assumes "\<Gamma> \<turnstile> b"   and "\<Gamma> \<turnstile> s" shows "\<exists>v. tbval b s v"
trcl_rev_cases: fixes c :: "'f"   and w :: "'g list"   and c' :: "'f"   and S :: "('f \<times> 'g \<times> 'f) set" assumes "(c, w, c') \<in> trcl S" obtains "w = []" and "c = c'"   | ch :: "'f" and e :: "'g" and wh :: "'g list" where "w = wh @ [e]" and "(c, wh, ch) \<in> trcl S" and "(ch, e, c') \<in> S"
FGModuleHom_axioms: shows "FGModuleHom G (\<cdot>) V (\<star>) T"
inj_idom_hom_axioms: shows "inj_idom_hom fpxs_of_fls"
fun_rel_comp_dist: fixes R1 :: "('d \<times> 'h) set"   and R2 :: "('e \<times> 'i) set"   and R3 :: "('h \<times> 'f) set"   and R4 :: "('i \<times> 'g) set" shows "(R1 \<rightarrow> R2) O (R3 \<rightarrow> R4) \<subseteq> R1 O R3 \<rightarrow> R2 O R4"
assms_3: shows "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
poly_subst_monomial: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'d \<Rightarrow>\<^sub>0 nat"   and c :: "'c" shows "poly_subst f (monomial c t) = MPoly_Type_Class.punit.monom_mult c 0 (subst_pp f t)"
take_Suc_Cons: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "(x # xs) \<down> Suc n = x # xs \<down> n"
syncWithTick_imp_NTF1: fixes s :: "'c event list"   and t :: "'c event list"   and A :: "'c set"   and u :: "'c event list"   and P :: "'c process"   and Q :: "'c process"   and X :: "'c event set" assumes "s @ [tick] setinterleaves ((t, u), insert tick (ev ` A))"   and "t \<in> T P"   and "u \<in> T Q" shows "\<exists>t u Xa Y. (t, Xa) \<in> F P \<and> (u, Y) \<in> F Q \<and> s setinterleaves ((t, u), insert tick (ev ` A)) \<and> X - {tick} = (Xa \<union> Y) \<inter> insert tick (ev ` A) \<union> Xa \<inter> Y"
1: fixes \<tau> :: "'d state \<times> 'd state"   and Y :: "'d state \<times> 'd state \<Rightarrow> 'e"   and X :: "'d state \<times> 'd state \<Rightarrow> 'f" assumes "\<tau> \<Turnstile> \<upsilon> Y" shows "\<tau> \<Turnstile>  Pair{X,Y} .First() \<triangleq> X"
less_list_def: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs < ys) = strict (\<le>) xs ys"
fin_sat_extend: fixes S :: "'b formula set"   and F :: "'b formula" assumes "fin_sat S" shows "fin_sat (F \<triangleright> S) \<or> fin_sat (\<^bold>\<not> F \<triangleright> S)"
lift_envs_mem_restr: fixes A :: "'b list set"   and x :: "'b option list"   and a :: "'b option" assumes "mem_restr A x" shows "mem_restr (lift_envs A) (a # x)"
rep_eq: fixes x :: "('c, 'd) lifted_distrib_lattice_top"   and xa :: "('c, 'd) lifted_distrib_lattice_top" shows "(x \<le> xa) = (\<forall>f. Rep_lifted_distrib_lattice_top x f \<le> Rep_lifted_distrib_lattice_top xa f)"
assms_1: shows "b \<succeq>[\<R>] w \<and> (w, b) \<notin> \<R>"
hmset_pair_decompose_less: fixes m1 :: "hmultiset"   and m2 :: "hmultiset" assumes "m1 < m2" shows "\<exists>k n1 n2. m1 = k + n1 \<and> m2 = k + n2 \<and> head_\<omega> n1 < head_\<omega> n2"
finite_measure_finite_Union: fixes S :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "finite S"   and "A ` S \<subseteq> sets M"   and "disjoint_family_on A S" shows "Sigma_Algebra.measure M (\<Union> (A ` S)) = (\<Sum>i\<in>S. Sigma_Algebra.measure M (A i))"
opp_chambers_surj: shows "f \<turnstile> (\<C> - f \<turnstile> \<C>) = f \<turnstile> \<C>"
simps_1: fixes t :: "'a rose_tree" shows "tree_of_zipper ([], t) = t"
initial_state_def: fixes sys :: "('g, 'h, 'i, 'j, 'k, 'l) pre_system_scheme"   and s :: "'i \<Rightarrow> ('g, 'h, 'j, 'k) com list \<times> 'h option \<times> 'k" shows "initial_state sys s = ((\<forall>p. cPGM (s p) = [PGMs sys p] \<and> cTKN (s p) = None) \<and> INIT sys (cLST \<circ> s))"
arbitrary_union_of_Int_eq: fixes P :: "'c set \<Rightarrow> bool" shows "(\<forall>S T. (arbitrary union_of P) S \<and> (arbitrary union_of P) T \<longrightarrow> (arbitrary union_of P) (S \<inter> T)) = (\<forall>S T. P S \<and> P T \<longrightarrow> (arbitrary union_of P) (S \<inter> T))"
simps_5: fixes red :: "('d \<times> nat) rgraph"   and black :: "('d, 'e, 'f) lts"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf :: "('e, 'f) conf"   and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more :: "'g" shows "confs \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = confs"
assms: shows "(as, x) \<approx>set (=) supp p' (as', x')"
sum_list_def: shows "sum_list \<equiv> monoid_list.F (+) (0::'a)"
pow_le_lgth: fixes tid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "honest tid"   and "\<parallel>tid\<parallel>\<^bsub>t n\<^esub>"   and "\<not> (\<exists>b\<in>pin (\<sigma>\<^bsub>tid\<^esub>t n). length (bc (\<sigma>\<^bsub>tid\<^esub>t n)) < length b)" shows "PoW t n \<le> length (bc (\<sigma>\<^bsub>tid\<^esub>t n))"
arr_SetI: fixes \<alpha> :: "V"   and T :: "V" assumes "\<Z> \<alpha>"   and "vfsequence T"   and "vsv (T\<lparr>ArrVal\<rparr>)"   and "vcard T = 3\<^sub>\<nat>"   and "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = T\<lparr>ArrDom\<rparr>"   and "\<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> T\<lparr>ArrCod\<rparr>"   and "T\<lparr>ArrDom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"   and "T\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>" shows "arr_Set \<alpha> T"
non_empty_cycle_root_subset_equals: fixes r :: "'a"   and x :: "'a"   and y :: "'a" assumes "non_empty_cycle_root r x"   and "non_empty_cycle_root r y"   and "x \<le> y" shows "x = y"
supp_1: fixes x1 :: "'d" shows "supp (\<langle>\<epsilon>, x1\<rangle>) = supp x1"
simps_2: fixes s :: "('d, 'e) fmap"   and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" shows "sat_precond_as s (a # as) = (fst a \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s a) as)"
taylor_closed: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R" shows "T\<^bsub>a\<^esub> f \<in> carrier P"
nAct_same: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "dynamic_component.lNAct active c t n \<le> n'"   and "n' \<le> dynamic_component.nxtAct active c t n" shows "the_enat (dynamic_component.nAct tCMP active c (enat n') (inf_llist t)) = the_enat (dynamic_component.nAct tCMP active c (enat n) (inf_llist t))"
assms_3: fixes a :: "int" assumes "b \<le> a" shows "f a = g a"
subdegree_geI: fixes f :: "'b fps"   and n :: "nat" assumes "f \<noteq> 0"   and "\<And>i. i < n \<Longrightarrow> f $ i = (0::'b)" shows "n \<le> subdegree f"
Seq2: fixes c\<^sub>1 :: "com"   and s :: "char list \<Rightarrow> int"   and c\<^sub>1' :: "com"   and s' :: "char list \<Rightarrow> int"   and c\<^sub>2 :: "com" assumes "(c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s')" shows "(c\<^sub>1;; c\<^sub>2, s) \<rightarrow> (c\<^sub>1';; c\<^sub>2, s')"
leadsTo_weaken_L: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set" assumes "F \<in> A \<longmapsto> A'"   and "B \<subseteq> A" shows "F \<in> B \<longmapsto> A'"
D: shows "D = Mod_Type_Connect.from_hma\<^sub>m D_HMA"
Image_outside_Domain: fixes R :: "('d \<times> 'e) set"   and A :: "'d set" assumes "Domain R \<inter> A = {}" shows "R `` A = {}"
preserves_reflects_arr: fixes f :: "'a1 \<times> 'a2" shows "B.arr (uncurry G f) = A1xA2.arr f"
l2p_aux: fixes list :: "('a, 'b) Combinators list"   and a :: "('a, 'b) Combinators" assumes "list \<noteq> []" shows "list2FWpolicy (a # list) = a \<oplus> list2FWpolicy list"
coinduct_set: fixes f :: "'a set \<Rightarrow> 'a set"   and a :: "'a"   and X :: "'a set" assumes "mono f"   and "a \<in> X"   and "X \<subseteq> f (X \<union> gfp f)" shows "a \<in> gfp f"
tl_ne_ex: fixes l :: "'a list" assumes "l \<noteq> []" shows "\<exists>x. l = x # tl l"
compactI: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.ccpo Sup less_eq less"   and "ccpo.admissible Sup less_eq (\<lambda>y. \<not> less_eq x y)" shows "ccpo.compact Sup less_eq x"
ENC_FExists: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n (FExists \<phi>)" shows "lang n (ENC n (FOV (FExists \<phi>))) = SAMEQUOT (any, replicate n False) (map \<pi> ` lang (Suc n) (ENC (Suc n) (FOV \<phi>)))"
filter_empty_conv: fixes P :: "'k \<Rightarrow> bool"   and xs :: "'k list" shows "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)"
equivalence_map_map\<^sub>0: fixes a :: "'a" assumes "C.obj a" shows "D.equivalence_map (map\<^sub>0 a)"
mk_BaseDim_code: fixes d :: "'a" shows "mk_BaseDim d = mk_dimvec ((replicate CARD('a) 0)[enum_ind d := 1])"
UN_empty: fixes B :: "'c \<Rightarrow> 'b set" shows "\<Union> (B ` {}) = {}"
AddSA_SAs: fixes SA :: "('d, 'e, 'f) seqauto"   and HA :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates HA = {}"   and "S \<in> HAStates HA" shows "SAs (HA [++] (S, SA)) = insert SA (SAs HA)"
distinct_disc_17: fixes event :: "('a, 'b, 'c) event" assumes "isRecv event" shows "\<not> isRecvMarker event"
Red: shows "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>es',s',b'\<rangle>"