hcju/isabelleps huggingface dataset:
add_zero: fixes a :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "a \<in> \<O> U" shows "add_sheaf_spec U (zero_sheaf_spec U) a = a"
tiny_dg_digraph: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "tiny_digraph \<alpha> \<CC>" shows "digraph \<alpha> \<CC>"
well_def1: shows "well_def_signs (length qs1) signs1"
case_3: fixes f1 :: "agent \<Rightarrow> agent \<Rightarrow> msg \<Rightarrow> 'a"   and f2 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and f3 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and x31 :: "agent"   and x32 :: "msg" shows "(case Notes x31 x32 of Says x xa xb \<Rightarrow> f1 x xa xb | Gets x xa \<Rightarrow> f2 x xa | Notes x xa \<Rightarrow> f3 x xa) = f3 x31 x32"
substNext_code_correct: fixes g :: "'g" assumes "redundant g" shows "substNext g = substNext_code g (chooseNext' g)"
Usubset: fixes u :: "ZF" assumes "Elem u U" shows "HOLZF.subset u U"
free1: shows "dfree ($f fid2 (\<lambda>i. if i = vid1 then Var vid1 else Const 0))"
D4_1: fixes F :: "\<Pi>\<^sub>1" shows "d\<^sub>1 (\<^bold>\<lambda>x. \<lparr>F,x\<^sup>P\<rparr>) = d\<^sub>1 F"
Tensor_null: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'a list list" assumes "Matrix_Tensor.mult id f" shows "mult.Tensor f xs [] = []"
rref_Gauss_Jordan: fixes A :: "(('d, 'e) vec, 'f) vec" shows "reduced_row_echelon_form (Gauss_Jordan A)"
commute: fixes a :: "'b"   and b :: "'b" shows "a \<cdot> b = b \<cdot> a"
dependent_def: fixes P :: "'b set" shows "dependent P = (\<exists>a\<in>P. a \<in> span (P - {a}))"
eln_mult_add: fixes a :: "ennreal"   and b :: "ennreal" assumes "\<not> (a = \<infinity> \<and> b = 0 \<or> a = 0 \<and> b = \<infinity>)" shows "eln (a * b) = eln a + eln b"
ex_un_null: shows "\<exists>!n. \<forall>f. hom'.comp n f = n \<and> hom'.comp f n = n"
counit_in_hom_1: shows "\<guillemotleft>\<epsilon> : trg f \<rightarrow> trg f\<guillemotright>"
1: shows "C.commutative_square \<nu>\<pi>.leg0 \<rho>.leg1 \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<^sub>1 \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0"
simps: fixes b :: "bool"   and i :: "real"   and d :: "real"   and x\<^sub>1 :: "real"   and x\<^sub>2 :: "real"   and n :: "nat" shows "\<Psi> b i d x\<^sub>1 x\<^sub>2 n = (if x\<^sub>2 \<le> real n then i * (real n - x\<^sub>2) else if real n \<le> x\<^sub>1 \<and> b then d * (x\<^sub>1 - real n) else 0)"
dagger_refl: fixes x :: "'a" shows "(1::'a) \<le> x\<^sup>\<dagger>"
matrix_right_invertible_independent_rows: fixes A :: "(('a, 'c) vec, 'b) vec" shows "(\<exists>B. A ** B = mat (1::'a)) = (\<forall>c. (\<Sum>i\<in>UNIV. c i *s row i A) = 0 \<longrightarrow> (\<forall>i. c i = (0::'a)))"
length_concat_negGorbit_list: fixes gs :: "'g list"   and T :: "'a \<Rightarrow> 'w"   and as :: "'a list" shows "length (concat (HRep.negGorbit_list gs T as)) = length gs * length as"
element_type_alt_simps: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "element_type \<tau> \<sigma> = (Collection \<sigma> = \<tau> \<or> Set \<sigma> = \<tau> \<or> OrderedSet \<sigma> = \<tau> \<or> Bag \<sigma> = \<tau> \<or> Sequence \<sigma> = \<tau>)"
more_arith_simps_7: fixes a :: "'a"   and b :: "'a" shows "- a \<cdot> b = - (a \<cdot> b)"
assms: shows "ideal F \<div> monomial (1::'a) t = ideal (monomial (1::'a) ` S)"
flow_leaves_compact_ivl_right: fixes x0 :: "'a"   and K :: "'a set" assumes "0 \<in> UNIV"   and "x0 \<in> X"   and "bdd_above (existence_ivl0 x0)"   and "Sup (existence_ivl0 x0) \<in> UNIV"   and "compact K"   and "K \<subseteq> X" obtains t :: "real" where "0 \<le> t" and "t \<in> existence_ivl0 x0" and "flow0 x0 t \<notin> K"
vsv_value: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" obtains b :: "V" where "r\<lparr>a\<rparr> = b" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
r_one: fixes x :: "nat" assumes "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x"
sdescending_sdrop: fixes w :: "'b stream"   and k :: "nat" assumes "sdescending w" shows "sdescending (sdrop k w)"
post_del_adjR: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and lv :: "nat"   and r :: "('b \<times> nat) tree"   and r' :: "('b \<times> nat) tree" assumes "invar \<langle>l, (a, lv), r\<rangle>"   and "pre_adjust \<langle>l, (a, lv), r'\<rangle>"   and "post_del r r'" shows "post_del \<langle>l, (a, lv), r\<rangle> (adjust \<langle>l, (a, lv), r'\<rangle>)"
f_altdef: fixes s :: "complex" assumes "0 < Re s"   and "s \<notin> \<nat>" shows "f s = eval_fds (fds_perzeta q) s"
obj_simps_3: fixes a :: "'a arrow_of_spans_data" assumes "obj a" shows "trg a = a"
il_sub_inf_right_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<sqinter> x \<le> z \<sqinter> y"
sc: shows "\<forall>S' x. S' \<union> x \<in> C \<longrightarrow> (\<forall>S\<subseteq>S' \<union> x. S \<in> C)"
nabla_coinduction_var: fixes y :: "'a"   and x :: "'a" assumes "d y \<le>  |x\<rangle> y" shows "d y \<le> \<nabla>x"
subst_typ'_comp: fixes inst1 :: "((variable \<times> String.literal set) \<times> typ) list"   and inst2 :: "((variable \<times> String.literal set) \<times> typ) list"   and t :: "Core.term" shows "subst_typ' inst1 (subst_typ' inst2 t) = subst_typ' (map (apsnd (subst_typ inst1)) inst2 @ inst1) t"
coef_apps: fixes s :: "('s, 'v) tm"   and ss :: "('s, 'v) tm list"   and i :: "nat" shows "coef (apps s ss) i = coef s (i + length ss)"
less_eq_top_is_Top: fixes x :: "'a top" assumes "Top \<le> x" shows "x = Top"
respection: shows "rel_weakly_respects_barbs (indRelRSTPO SRel TRel) (STCalWB SWB TWB)"
nth_non_equal_first_eq: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list"   and n :: "nat" assumes "x \<noteq> y" shows "((x # xs) ! n = y) = (xs ! (n - 1) = y \<and> 0 < n)"
finite_imp_foldSetD: fixes A :: "'b set"   and e :: "'a" assumes "finite A"   and "A \<subseteq> B"   and "e \<in> D" shows "\<exists>x. (A, x) \<in> foldSetD D (\<cdot>) e"
eval\<^sub>G_abs_def: shows "\<up>eval\<^sub>G \<equiv> map_fun id (map_fun Rep Abs) eval\<^sub>G"
abc_a'b'c'_neq: shows "a \<noteq> a' \<and> a \<noteq> b' \<and> a \<noteq> c' \<and> b \<noteq> a' \<and> b \<noteq> b' \<and> b \<noteq> c' \<and> c \<noteq> a' \<and> c \<noteq> b' \<and> c \<noteq> c'"
compl_set: fixes xs :: "'a list" shows "- set xs = List.coset xs"
subspace_is_vs: fixes W :: "'c set" assumes "subspace K W V" shows "vectorspace K (vs W)"
C: shows "C = {(I, S, E)} WHILE Less (V a) (N 3) DO a ::= Plus (V a) (N 1)"
Basis: fixes s :: "'b set"   and S :: "'b set set" assumes "s \<in> S" shows "generate_topology S s"
entangled_def: fixes n :: "nat"   and v :: "complex Matrix.mat" shows "entangled n v \<equiv> \<not> prod_state n v"
bdt_Some_One_iff: fixes t :: "dag"   and var :: "ref \<Rightarrow> nat" shows "(bdt t var = Some One) = (\<exists>p. t = Node Tip p Tip \<and> var p = 1)"
ext: fixes g :: "'a"   and f :: "'a" assumes "g \<cdot>\<^sub>S f \<noteq> S.null" shows "S.seq g f"
Lt_le_LeI: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "Lt a \<le> Le b"
right_one_impl: fixes a :: "'a" shows "(1::'a) r\<rightarrow> a = a"
prod_diff1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and a :: "'c" assumes "class.semidom_divide divide plus minus zero times one"   and "finite A"   and "f a \<noteq> zero" shows "comm_monoid_mult.prod times one f (A - {a}) = (if a \<in> A then divide (comm_monoid_mult.prod times one f A) (f a) else comm_monoid_mult.prod times one f A)"
gNot_def: fixes g :: "'a gexp" shows "gNot g \<equiv> Nor g g"
narrow_PrimT: fixes G :: "prog"   and pt :: "prim_ty"   and T :: "ty" assumes "G\<turnstile>PrimT pt\<succ>T" shows "\<exists>t. T = PrimT t"
Wrap_def: fixes x :: "'c" shows "Wrap x \<equiv> x"
simps: fixes f :: "nat \<Rightarrow> 'b"   and i :: "'b"   and n :: "nat" shows "dirichlet_inverse f i n = (if n = 0 then 0::'b else if n = 1 then i else - i * (\<Sum>d | d dvd n \<and> d < n. f (n div d) * dirichlet_inverse f i d))"
simps: fixes a1 :: "'c \<Rightarrow> 'c list"   and a2 :: "('c, 'd) Term.term"   and a3 :: "('c, 'd) Term.term" shows "term_variants_pred a1 a2 a3 = ((\<exists>P x. a1 = P \<and> a2 = Var x \<and> a3 = Var x) \<or> (\<exists>T S P g f. a1 = P \<and> a2 = Fun f T \<and> a3 = Fun g S \<and> length T = length S \<and> (\<forall>x<length T. term_variants_pred P (T ! x) (S ! x)) \<and> g \<in> set (P f)) \<or> (\<exists>T S P f. a1 = P \<and> a2 = Fun f T \<and> a3 = Fun f S \<and> length T = length S \<and> (\<forall>x<length T. term_variants_pred P (T ! x) (S ! x))))"
p_max: fixes x :: "real" shows "delayed_safe_distance.other.p x \<le> delayed_safe_distance.other.p_max"
is_linorder_rel_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "is_linorder_rel R \<equiv> (\<forall>x y. R x y \<or> R y x) \<and> (\<forall>x y z. R x y \<longrightarrow> R y z \<longrightarrow> R x z)"
assms_5: shows "h \<turnstile> attach_shadow_root element_ptr shadow_root_mode \<rightarrow>\<^sub>h h'"
vifunionD: fixes a :: "V"   and I :: "V"   and f :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)" shows "\<exists>i\<in>elts I. a \<in>\<^sub>\<circ> f i"
Rep_pupdate: fixes x :: "'b pupdate" shows "Rep_pupdate x \<in> pupdate"
interleaves_all_nil: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list" shows "xs \<simeq> {xs, [], P}"
iterate_size_def: fixes it :: "(nat \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat" shows "iterate_size it = it (\<lambda>_. True) (\<lambda>x. Suc) 0"
ale_neq_less: fixes x :: "ant"   and y :: "ant" assumes "x \<le> y"   and "x \<noteq> y" shows "x < y"
is_strong_nat_string_simps_2: fixes c :: "char"   and s :: "char list" shows "is_strong_nat_string (c # s) = (is_digit_char c \<and> is_nat_string s \<and> (c = CHR ''0'' \<longrightarrow> s = []))"
normal_moment_nz_1: fixes \<mu> :: "real" shows "has_bochner_integral lborel (\<lambda>x. normal_density \<mu> \<sigma> x * x) \<mu>"
comp_fun_idem_insert_clause: shows "comp_fun_idem insert_clause"
this: shows "DA alphabet1 init1 delta1 accept1 wellformed1 Language1 wf1 Lang1"
AnfangTauschS: fixes \<Gamma> :: "'b formula list"   and \<DD> :: "'b formula"   and \<Delta> :: "'b formula list"   and \<Theta> :: "'b formula list" assumes "\<Gamma> \<Rightarrow> \<DD> # \<Delta> @ \<Theta>" shows "\<Gamma> \<Rightarrow> \<Delta> @ \<DD> # \<Theta>"
prems_1: shows "history_consistent \<theta> (hd_prog p (x # xs)) (x # xs)"
sums_logderiv_zeta: fixes s :: "complex" assumes "1 < Re s" shows "(\<lambda>p. if prime p then complex_of_real (ln (real p)) / (of_nat p powr s - 1) else 0) sums - (deriv zeta s / zeta s)"
ins_list_contains_idem: fixes xs :: "'b list"   and x :: "'b" assumes "sorted_less xs"   and "x \<in> set xs" shows "ins_list x xs = xs"
thm_cont_prop_3: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[Contingent F \<^bold>\<equiv> Contingent (F\<^sup>-) in v]"
assms_2: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and U :: "'x set" assumes "(h, U) \<in> set ps" shows "h \<noteq> 0"
arctan_upper_42_def: shows "arctan_lower_43 \<equiv> \<lambda>x. (15159 * x ^ 6 + 147455 * x ^ 4 + 345345 * x\<^sup>2 + 225225) * x / (35 * (35 * x ^ 8 + 1260 * x ^ 6 + 6930 * x ^ 4 + 12012 * x\<^sup>2 + 6435))"
zero: fixes U :: "'b set"   and V :: "'b set" assumes "Subspace.subspace U V"   and "vectorspace V" shows "(0::'b) \<in> U"
invKey_shrK: fixes A :: "agent" shows "invKey (shrK A) = shrK A"
P: fixes g :: "'a" assumes "is_glb L g {x, y}" shows "P g"
oLog_def: fixes b :: "ordinal" shows "oLog b = (\<lambda>x. if 1 < b then oInv ((**) b) x else 0)"
SUP_const: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'e set"   and f :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<not> A = {}" shows "Sup ((\<lambda>i. f) ` A) = f"
nccD_4: fixes X :: "'c set" assumes "ncc X" shows "X \<noteq> {}"
cases: fixes a :: "(nat \<times> nat) set" assumes "a \<in> domino" obtains i :: "nat" and j :: "nat" where "a = {(ib__, jb__), (ib__, Suc jb__)}"   | i :: "nat" and j :: "nat" where "a = {(i, j), (Suc i, j)}"
M_def: shows "M \<equiv> sum_upto (\<lambda>n. moebius_mu n / real n)"
intros_1: fixes r :: "('b \<times> 'c) set" shows "([], []) \<in> listrel r"
assms: shows "b_graph.is_acc_run (degeneralize_ext m) r"
B: shows "ipurge_tr_rev_aux I D (D ` (X \<union> set zs)) ys = []"
map_replicate: fixes f :: "'c \<Rightarrow> 'b"   and n :: "nat"   and x :: "'c" shows "map f (replicate n x) = replicate n (f x)"
sincos_total_2pi: fixes x :: "real"   and y :: "real" assumes "x\<^sup>2 + y\<^sup>2 = 1" obtains t :: "real" where "0 \<le> t" and "t < 2 * pi" and "x = cos t" and "y = sin t"
Gamma_series_Gamma_series': fixes z :: "'a" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "(\<lambda>n. Gamma_series' z n / Gamma_series z n) \<longlonglongrightarrow> (1::'a)"
cupcake_single_complete: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env :: "v sem_env"   and s :: "'a state"   and e :: "exp0"   and s' :: "'a state"   and res :: "(v, v) result" assumes "cakeml_static_env static_cenv"   and "Big_Step_Unclocked_Single.evaluate env s e (s', res)"   and "cakeml_static_env.is_cupcake_all_env static_cenv env"   and "is_cupcake_exp e" shows "cupcake_evaluate_single env e res"
pth_6: fixes c :: "real"   and x :: "'b"   and y :: "'b" shows "c *\<^sub>R (x + y) \<equiv> c *\<^sub>R x + c *\<^sub>R y"
t: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> U" shows "t (f \<triangleright> n) \<down>= (if s (f \<triangleright> n) \<down>= 0 then 0 else Suc (the (c (the (s (f \<triangleright> n)) - 1))))"
"0_1": shows "1 < length (remdups (map (the_LabelN \<circ> fst) (filter (Not \<circ> is_LabelS) A)))"
content_image_affinity_cbox: fixes m :: "real"   and c :: "'a"   and a :: "'a"   and b :: "'a" shows "content ((\<lambda>x. m *\<^sub>R x + c) ` cbox a b) = \<bar>m\<bar> ^ DIM('a) * content (cbox a b)"
\<mu>\<nu>: shows "\<guillemotleft>Map \<mu> \<star>\<^sub>B Map \<nu> : EVAL (Dom \<mu> \<^bold>\<star> Dom \<nu>) \<rightarrow>\<^sub>B EVAL (Cod \<mu> \<^bold>\<star> Cod \<nu>)\<guillemotright>"
l1_inv3I: fixes x :: "l1_state" assumes "\<And>A B Rb Ra. \<lbrakk>guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; progress x Rb = Some {xpkE, xsk}; guessed_frame Rb xpkE = Some (epubKF (Ra $ kE)); guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; progress x Ra = Some {xpkE, xskE}; can_signal x A B\<rbrakk> \<Longrightarrow> l1_state.signals x (Commit A B \<langle>epubKF (Ra $ kE), NonceF (Rb $ sk)\<rangle>) < l1_state.signals x (Running A B \<langle>epubKF (Ra $ kE), NonceF (Rb $ sk)\<rangle>)" shows "x \<in> l1_inv3"
R_map: fixes r :: "'r"   and m :: "'m" assumes "r \<in> R"   and "m \<in> M" shows "T (r \<cdot> m) = r \<star> T m"
G_def: shows "G = (\<lambda>y. - fact k * (x * z) ^ k / (1 - (1 - x * y) * z) ^ (k + 1))"
prj_char_3: shows "SPN_fgh.Prj\<^sub>0 = \<lbrakk>\<lbrakk>TTfgh.p\<^sub>0\<rbrakk>\<rbrakk>"
supt_not_sym: fixes s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" assumes "s \<rhd> t" shows "(t, s) \<notin> {\<rhd>}"
R0: fixes X :: "'b t_type" shows "X \<in> t_ins_set X"
sigma_finite_density_unique: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "sigma_finite_measure (density M f)" shows "(density M f = density M g) = (AE x in M. f x = g x)"
set_bset_eqvt: fixes p :: "perm"   and B :: "'c set['d]" shows "p \<bullet> set_bset B = set_bset (p \<bullet> B)"
sim_val_def: fixes M :: "('d, 'e, 'f) KripkeStructure"   and M' :: "('d, 'e, 'g) KripkeStructure"   and f :: "'f \<Rightarrow> 'g" shows "sim_val M M' f \<equiv> \<forall>u\<in>worlds M. valuation M u = valuation M' (f u)"
quasi_inverse_unique: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "quasi_inverses f g"   and "quasi_inverses f g'" shows "g \<cong> g'"
mon_loc_empty: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme" shows "mon_loc fg [] = {}"
CD_on_range: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and A :: "'d set" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.CD_on Pd ds A \<subseteq> A \<inter> (\<Union>d\<in>ds. Field (Pd d))"
all_red_crit: shows "\<forall>q\<in>Q. calculus Bot_F Inf_F (\<lambda>N1 N2. entails_q q (\<G>_Fset_q q N1) (\<G>_Fset_q q N2)) (Red_I_\<G>_q q) (Red_F_\<G>_q q)"
red: shows "(if \<mu>1 s1' s1 then trsys2.silent_moves else trsys2.silent_movet) s2 s2'"
nt_def: shows "nt \<equiv> \<tau> \<sigma> (ne + length_muaux args aux)"
sup_state_trans: fixes P :: "'c prog"   and a :: "ty list \<times> ty err list"   and b :: "ty list \<times> ty err list"   and c :: "ty list \<times> ty err list" assumes "P \<turnstile> a \<le>\<^sub>i b"   and "P \<turnstile> b \<le>\<^sub>i c" shows "P \<turnstile> a \<le>\<^sub>i c"
init_obj_def: fixes G :: "prog"   and oi :: "obj_tag"   and r :: "oref" shows "init_obj G oi r = gupd(r\<mapsto>\<lparr>tag = oi, values = init_vals (var_tys G oi r)\<rparr>)"
g0: assumes "degree g = 0"   and "coeff g 0 = (0::'a)" shows "g = 0"
gstep_sim2: fixes d :: "('d, 'ls, 'gs) global_config"   and d' :: "('d, 'ls, 'gs) global_config"   and c :: "('c, 'ls, 'gs) global_config" assumes "(d, d') \<in> s2.gstep"   and "rel_gc c d" shows "\<exists>c'. rel_gc c' d' \<and> (c, c') \<in> s1.gstep"
is_SDG_path_CFG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n'" obtains as :: "'edge list" where "parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n'"
count_of_count_mset_lal: fixes y :: "('a \<times> nat) list"   and x :: "'a" assumes "distinct (map fst y)" shows "count_of y x = count (mset (ass_list_to_single_list y)) x"
SIR: fixes i :: "nat" shows "inres (step (f i)) (f (Suc i))"
m_defs_2: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and I0 :: "nat set"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_ZOC theta c d s t I0 P F \<equiv> mC_ZOC_part c d s t I0 P F \<and> inj_on F P \<and> mC_ZOC_wt c d s t I0 P F \<and> mC_ZOC_eff_cont0 theta c d s t I0 F \<and> mC_ZOC_eff_cont theta c d s t I0 P F"
ord_spmf_eq_leD: fixes p :: "'b spmf"   and q :: "'b spmf"   and x :: "'b" assumes "ord_spmf (=) p q" shows "spmf p x \<le> spmf q x"
timpl_closure_subst_subset: fixes t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and M :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term set"   and \<delta> :: "(('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat \<Rightarrow> (('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and T :: "('sets set \<times> 'sets set) set" assumes "t \<in> M"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M"   and "\<forall>x\<in>fv\<^sub>s\<^sub>e\<^sub>t M. \<exists>a. \<Gamma>\<^sub>v x = Var (prot_atom.Atom a)"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"   and "ground (subst_range \<delta>)"   and "subst_domain \<delta> \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t M"   and "timpl_closure t T \<subseteq> M" shows "timpl_closure (t \<cdot> \<delta>) T \<subseteq> M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>"
same_proc_call_unique_target: fixes a :: "'a"   and a' :: "'a"   and Q\<^sub>1 :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r\<^sub>1 :: "'e"   and p :: "'f"   and fs\<^sub>1 :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and Q\<^sub>2 :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r\<^sub>2 :: "'e"   and fs\<^sub>2 :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "valid_edge a"   and "valid_edge a'"   and "kind a = Q\<^sub>1:r\<^sub>1\<hookrightarrow>\<^bsub>p\<^esub>fs\<^sub>1"   and "kind a' = Q\<^sub>2:r\<^sub>2\<hookrightarrow>\<^bsub>p\<^esub>fs\<^sub>2" shows "targetnode a = targetnode a'"
disjnt_self_iff_empty: fixes S :: "'a set" shows "disjnt S S = (S = {})"
l_get_owner_document_wf_axioms: shows "l_get_owner_document_wf heap_is_wellformed DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_disconnected_nodes get_owner_document get_parent get_child_nodes"
index_of_suc_def: shows "index_of_suc = sch_to_nat Base_suc"
M1: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> \<box>F \<longrightarrow> \<box>[F \<and> \<circle>F]_v"
monoE: fixes f :: "'a \<Rightarrow> 'd"   and x :: "'a"   and y :: "'a" assumes "mono f"   and "x \<le> y" obtains "f x \<le> f y"
terminates_map_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and f :: "'f \<Rightarrow> 'g" assumes "terminates g" shows "terminates (map_raw f g)"
card_remove_fset_less1: fixes xs :: "'b fset"   and x :: "'b" assumes "x |\<in>| xs" shows "card_fset (remove_fset x xs) < card_fset xs"
distinct_fstD: fixes xs :: "('a \<times> 'b) list"   and x :: "'a"   and y :: "'b"   and z :: "'b" assumes "distinct_fst xs"   and "(x, y) \<in> set xs"   and "(x, z) \<in> set xs" shows "y = z"
run_eval: shows "\<exists>run_eval. \<forall>env e s. evaluate True env s e (run_eval env e s)"
wpt_is_assertion: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "mbt_algebra.wpt one times inf top x \<in> mbt_algebra.assertion one times inf less_eq dual top"
no_call_to_def: fixes c :: "char list"   and rs :: "'b rule list" shows "no_call_to c rs \<equiv> \<forall>r\<in>set rs. case get_action r of Call c' \<Rightarrow> c \<noteq> c' | _ \<Rightarrow> True"
less_multiset\<^sub>D\<^sub>M: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(M < N) = (\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = N - X + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> k < a)))"
commute: fixes a :: "bool"   and b :: "bool" shows "(a \<and> b) = (b \<and> a)"
inner_not_same_Basis: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and u :: "'c"   and v :: "'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis"   and "v \<in> Basis"   and "u \<noteq> v" shows "inner u v = 0"