hcju/isabelleps huggingface dataset:
hereditarily_fs_implies_finite_supp: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "finite (supp t\<^sub>\<alpha>)"
map_psubst_new_free: fixes c :: "'a"   and n :: "'a"   and z :: "('a, 'b) form list" assumes "c \<noteq> n" shows "news n (map (psubst (id(n := c))) z)"
is_Bisimulation_def: fixes as :: "'b list"   and ps :: "('b rexp set \<times> 'b rexp set) list" shows "is_Bisimulation as ps = (\<forall>(R, S)\<in>set ps. Atoms R \<union> Atoms S \<subseteq> set as \<and> Nullable R = Nullable S \<and> (\<forall>a\<in>set as. (Pderiv a R, Pderiv a S) \<in> set ps))"
snd3_simp: fixes a :: "'b"   and b :: "'a"   and c :: "'c" shows "snd3 (a, b, c) = b"
assms_7: shows "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |h \<turnstile> get_owner_document ptr'|\<^sub>r)|\<^sub>r"
2: shows "(a, l) \<in> \<langle>R\<rangle>ras_rel"
indmhom_injec: fixes N :: "('e, 'b, 'f) Module_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "R module N"   and "f \<in> mHom R M N" shows "injec\<^bsub>M /\<^sub>m (ker\<^bsub>M,N\<^esub> f),N\<^esub> f\<^sup>\<flat>\<^bsub>R M, N\<^esub>"
generated_ringE: fixes a :: "'a set" assumes "a \<in> G.generated_ring" obtains C :: "'a set set" where "finite C" and "disjoint C" and "C \<subseteq> generated_ring" and "a = \<Union> C"
r_ge_two: shows "2 \<le> \<Lambda> * (\<v> - 1) div (\<k> - 1)"
image_subset_iff_funcset: fixes F :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'c set" shows "(F ` A \<subseteq> B) = (F \<in> A \<rightarrow> B)"
frontier_closures: fixes S :: "'c set" shows "frontier S = closure S \<inter> closure (- S)"
has_channel_def: fixes p :: "'a"   and q :: "'a" shows "has_channel p q = (\<exists>i. channel i = Some (p, q))"
prefixes_hd: fixes s :: "nat \<Rightarrow> nat option"   and t :: "nat"   and j :: "nat"   and b :: "nat" assumes "total_cons s"   and "eval (total_cons.r_prefixes s) [t, j] \<down>= b" shows "e_nth b 0 = j"
x: shows "({#fst \<beta>_step#}, {#fst \<beta>_step, fst \<alpha>_step#}) \<in> mul_eq r"
goodBinp_imp_qGoodBinp_pickBinp: fixes binp :: "'c \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs set option" assumes "goodBinp binp" shows "qGoodBinp (pickBinp binp)"
mem_sphere_0: fixes x :: "'a"   and e :: "real" shows "(x \<in> sphere (0::'a) e) = (norm x = e)"
map_dual_dual: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(\<partial>\<^sub>F f = g) = (\<partial>\<^sub>F g = f)"
Arr_iff_dom_in_domain: fixes f :: "'a" shows "left (trg \<mu>) f = (local.dom f \<in> Left.domains f)"
assms_3: shows "0 \<in>\<^sub>\<circ> \<alpha>"
simps_1: fixes n :: "nat" shows "s'_A n 0 = s_A n"
choose_in_Strategies: fixes v :: "'a" assumes "v \<in> S" shows "choose v \<in> Strategies"
depth_fst: fixes P :: "'b trm" shows "depth P < depth (Fst P)"
axioms: fixes G :: "'a set" assumes "AbGroup G" shows "Group G"
branch_existence': fixes \<rho> :: "'c run"   and \<S>\<^sub>1 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list"   and k :: "nat" assumes "\<rho> \<in> \<lbrakk> \<S>\<^sub>1 \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g" shows "\<exists>\<S>\<^sub>2. \<S>\<^sub>1 \<hookrightarrow>\<^bsup>k\<^esup> \<S>\<^sub>2 \<and> \<rho> \<in> \<lbrakk> \<S>\<^sub>2 \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
butterfly_scaleR_right: fixes \<psi> :: "'c"   and r :: "real"   and \<phi> :: "'b" shows "butterfly \<psi> (r *\<^sub>R \<phi>) = complex_of_real r *\<^sub>C butterfly \<psi> \<phi>"
drlex_pm_antisym: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" assumes "drlex_pm s t"   and "drlex_pm t s" shows "s = t"
simps_42: fixes \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula" shows "safe_formula (formula.And \<phi> \<psi>) = (safe_formula \<phi> \<and> (safe_assignment (fv \<phi>) \<psi> \<or> safe_formula \<psi> \<or> fv \<psi> \<subseteq> fv \<phi> \<and> (is_constraint \<psi> \<or> (case \<psi> of formula.Neg x \<Rightarrow> safe_formula x | _ \<Rightarrow> False))))"
rel_option_None2: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e option" shows "rel_option P x None = (x = None)"
lossless_spmf_of_set: fixes A :: "'b set" shows "lossless_spmf (spmf_of_set A) = finite' A"
x_in: shows "x \<in> sum_list ` listset (map cone ps)"
nat_add_offset_less: fixes y :: "nat"   and n :: "nat"   and x :: "nat"   and m :: "nat"   and sz :: "nat" assumes "y < 2 ^ n"   and "x < 2 ^ m"   and "sz = m + n" shows "x * 2 ^ n + y < 2 ^ sz"
surjective: fixes r :: "('c, 'd, 'e) conf_scheme" shows "r = \<lparr>store = store r, pred = pred r, \<dots> = more r\<rparr>"
Un_left_commute: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
radical_sqrt_simultaneous_linear_equation: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real"   and e :: "real"   and f :: "real"   and x :: "real"   and y :: "real" assumes "a \<in> radical_sqrt"   and "b \<in> radical_sqrt"   and "c \<in> radical_sqrt"   and "d \<in> radical_sqrt"   and "e \<in> radical_sqrt"   and "f \<in> radical_sqrt"   and "\<not> (a * e - b * d = 0 \<and> a * f - c * d = 0 \<and> e * c = b * f)"   and "a * x + b * y = c"   and "d * x + e * y = f" shows "x \<in> radical_sqrt \<and> y \<in> radical_sqrt"
greaterThan_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b" shows "ord.greaterThan less l \<equiv> {x. less l x}"
valid_write_sops: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<forall>sop\<in>write_sops sb. valid_sop sop"
assms_2: shows "\<forall>ops\<in>set \<pi>. \<exists>op. ops = [op]"
it_dfsT_def: shows "it_dfsT \<equiv> init \<bind> WHILE\<^sub>T cond step"
count_single: fixes b :: "'e"   and a :: "'e" shows "count {#b#} a = (if b = a then 1 else 0)"
func_mem: fixes x :: "'a" assumes "x \<in> carrier" shows "f x \<in> carrier\<^bsub>T\<^esub>"
cindex_poly_ubd_def: fixes q :: "real poly"   and p :: "real poly" shows "cindex_poly_ubd q p = (THE l. \<forall>\<^sub>F r in at_top. cindexE (- r) r (\<lambda>x. poly q x / poly p x) = real_of_int l)"
set_count_poly_roots_mset: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "set_mset (poly_roots_mset p) = {x. poly p x = (0::'b)}"
deg_fun_zero: shows "deg_fun 0 = (0::'a)"
fract_poly_smult: fixes c :: "'a"   and p :: "'a poly" shows "map_poly to_fract (smult c p) = smult (to_fract c) (map_poly to_fract p)"
assms_3: shows "quicksort_pre R h xs0 lo hi xs"
A: fixes U :: "'a set" shows "\<not> (\<exists>v\<in>sinks I D u ys. \<exists>w\<in>U. (v, w) \<in> I) \<longrightarrow> c_ipurge_aux I D U (xs @ ipurge_tr I D u ys) = c_ipurge_aux I D U (xs @ ys)"
point_eq_iff: fixes M :: "point"   and N :: "point" shows "(M = N) = (abscissa M = abscissa N \<and> ordinate M = ordinate N)"
zero_below_val_2: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "prime p"   and "x \<in> padic_set p"   and "int n \<le> padic_val p x" shows "x n = 0"
PAC_checker_l_step_alt_def: fixes a :: "(char list list \<times> int) list"   and bcd :: "char list code_status \<times> char list set \<times> (nat, (char list list \<times> int) list) fmap"   and e :: "((char list list \<times> int) list, char list, nat) pac_step" shows "PAC_checker_l_step a bcd e = (let (b, c, d) = bcd in PAC_checker_l_step' a b c d e)"
is_full_pmdlI_lt_finite: fixes B :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "finite B"   and "\<And>k. k \<in> component_of_term ` Keys B \<Longrightarrow> \<exists>b\<in>B. b \<noteq> 0 \<and> component_of_term (lt b) = k \<and> lp b = (0::'a)" shows "is_full_pmdl B"
sup_eq_bot_iff: fixes x :: "'b"   and y :: "'b" shows "(x \<squnion> y = \<bottom>) = (x = \<bottom> \<and> y = \<bottom>)"
xy_3: shows "snd x \<in> \<FF> (fst x)"
add_ac_1: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a + b + c = a + (b + c)"
simps: fixes y :: "nat list"   and i :: "nat"   and u :: "nat list" shows "inc y i u = (if i < length y then if u ! i < y ! i then u[i := u ! i + 1] else inc y (Suc i) u else u)"
ct_suffix_ct_list_eq_exists: fixes x1 :: "ct list"   and x2 :: "ct list" assumes "ct_suffix x1 x2" shows "\<exists>ts_c. ct_suffix x1 (to_ct_list ts_c) \<and> ct_list_eq (to_ct_list ts_c) x2"
seqI': fixes f :: "'a arrow_of_spans_data"   and a :: "'a arrow_of_spans_data"   and b :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and c :: "'a arrow_of_spans_data" assumes "V.in_hom f a b"   and "V.in_hom g b c" shows "V.seq g f"
mem_Times_iff: fixes x :: "'e \<times> 'f"   and A :: "'e set"   and B :: "'f set" shows "(x \<in> A \<times> B) = (fst x \<in> A \<and> snd x \<in> B)"
ord_unique: fixes x :: "'a"   and d :: "nat" assumes "x \<in> carrier G" shows "(ord x = d) = (\<forall>n. (x [^] n = \<one>) = (d dvd n))"
iT_Minus_iEx_conv: fixes k :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "(\<diamond> t k \<ominus> I. P t) = (\<diamond> t I \<down>\<le> k. P (k - t))"
filter_replicate: fixes P :: "'a \<Rightarrow> bool"   and n :: "nat"   and x :: "'a" shows "filter P (replicate n x) = (if P x then replicate n x else [])"
perms_imp_MOST_eq: fixes f :: "'b \<Rightarrow> 'b" assumes "f \<in> perms" shows "\<forall>\<^sub>\<infinity>x. f x = x"
ennreal_mono_minus_cancel: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" assumes "a - b \<le> a - c"   and "a < \<top>"   and "b \<le> a"   and "c \<le> a" shows "c \<le> b"
learn_fin_closed_subseteq: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and V :: "(nat \<Rightarrow> nat option) set" assumes "learn_fin \<psi> U s"   and "V \<subseteq> U" shows "learn_fin \<psi> V s"
Gateway_L2: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat" assumes "Gateway req dt a stop lose d ack i vc"   and "msg (Suc 0) req"   and "msg (Suc 0) a"   and "msg (Suc 0) stop"   and "ts lose"   and "ack t = [connection_ok]"   and "req (Suc t) = [send]"   and "\<forall>k\<le>Suc d. lose (t + k) = [False]" shows "i (Suc (t + d)) = inf_last_ti dt t"
omega_least: 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"   and z :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "less_eq (inf (times x z) y) z" shows "less_eq (times (omega x) y) z"
llist_all2_tllist_of_llistI: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" assumes "tllist_all2 A B xs ys" shows "llist_all2 A (llist_of_tllist xs) (llist_of_tllist ys)"
simps_2: fixes orderA :: "('f, 'g) gorder_scheme"   and orderB :: "('d, 'e) gorder_scheme"   and lower :: "'f \<Rightarrow> 'd"   and upper :: "'d \<Rightarrow> 'f"   and more :: "'h" shows "\<Y>\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = orderB"
pred_intros_disj1': fixes K :: "bool"   and P :: "'c \<Rightarrow> bool"   and M :: "'c measure" assumes "\<not> K \<Longrightarrow> pred M P" shows "pred M (\<lambda>x. K \<or> P x)"
weak_reduction_contrasimulation_and_closures_2: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_contrasimulation Rel Cal" shows "weak_reduction_contrasimulation (Rel\<^sup>+) Cal"
set_pmf_subset_singleton: fixes p :: "'a pmf"   and x :: "'a" shows "(set_pmf p \<subseteq> {x}) = (p = return_pmf x)"
size_fsubset_elem: fixes x1 :: "'b fset"   and x2 :: "'b fset" assumes "\<exists>e. e |\<in>| x1 \<and> e |\<notin>| x2"   and "\<nexists>e. e |\<in>| x2 \<and> e |\<notin>| x1" shows "size x2 < size x1"
assms: shows "g \<turnstile> n-ns\<rightarrow>m"
put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes element_ptr :: "(_) element_ptr"   and element :: "('c, 'a, 'b, 'd, 'g option) RElement_scheme" shows "put element_ptr element = put (cast element_ptr) (cast element)"
normalized: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" shows "normalized M \<equiv> (\<forall>i j. 0 < i \<and> i \<le> n \<and> 0 < j \<and> j \<le> n \<and> M i j \<noteq> \<infinity> \<longrightarrow> Lt (real_of_int (- int ((k \<circ> v') j))) \<le> M i j \<and> M i j \<le> Le (real ((k \<circ> v') i))) \<and> (\<forall>i\<le>n. 0 < i \<longrightarrow> (M i 0 \<le> Le (real ((k \<circ> v') i)) \<or> M i 0 = \<infinity>) \<and> Lt (real_of_int (- int ((k \<circ> v') i))) \<le> M 0 i)"
pre_order_pair_def: fixes S :: "('b \<times> 'b) set"   and NS :: "('b \<times> 'b) set" shows "pre_order_pair S NS \<equiv> refl NS \<and> trans S \<and> trans NS"
rel_interior_convex_cone: fixes S :: "'a set" assumes "convex S" shows "rel_interior (cone hull {1} \<times> S) = {(c, c *\<^sub>R x) |c x. 0 < c \<and> x \<in> rel_interior S}"
p_def: fixes s :: "'a"   and t :: "'a"   and n :: "nat" shows "K1.p s t n = K1.T.prob s {\<omega> \<in> space (K1.T s). (s ## \<omega>) !! n = t}"
one_final_def: fixes g :: "graph" shows "one_final g \<equiv> one_final_but g {}"
int_pow_neg: fixes x :: "nat \<Rightarrow> int"   and i :: "int" assumes "x \<in> carrier Zp" shows "[(- i)] \<cdot> x = \<ominus> ([i] \<cdot> x)"
map_id: fixes t :: "'a + 'b" shows "map_sum id id t = t"
l2_inv4_trans: shows "{l2_inv4 \<inter> l2_inv3 \<inter> l2_inv1} TS.trans l2 {> l2_inv4}"
Sup_apply: fixes A :: "('d \<Rightarrow> 'c) set"   and x :: "'d" shows "(\<Squnion>A) x = (SUP f\<in>A. f x)"
simps_8: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_gexp_aux (Eq e1 e2) (Gt v va) = True"
phis_addN_conv: fixes n :: "'node \<times> 'val"   and m :: "('val, ('node \<times> 'val) set) mapping"   and va :: "'val" shows "Mapping.lookup (phis_addN g n m) va = (case Mapping.lookup m va of None \<Rightarrow> if va \<in> set (case phis g n of None \<Rightarrow> [] | Some x \<Rightarrow> id x) then Some {n} else None | Some N \<Rightarrow> if va \<in> set (case phis g n of None \<Rightarrow> [] | Some x \<Rightarrow> id x) then Some (insert n N) else Some N)"
fsafe_And: fixes p :: "('a, 'b, 'c) formula"   and q :: "('a, 'b, 'c) formula" assumes "fsafe p"   and "fsafe q" shows "fsafe (p && q)"
assms_5: shows "strict_border (take (i - 1) s) (take (j - 1) s)"
is_UP_monomI: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "a \<in> carrier R"   and "p = monom P a n" shows "is_UP_monom p"
assms_3: fixes K :: "'a list set"   and L :: "'a list set"   and x :: "'a" assumes "K \<sim> L" shows "Deriv x K \<sim> Deriv x L"
head_apps: fixes s :: "('c, 'd) tm"   and ss :: "('c, 'd) tm list" shows "head (apps s ss) = head s"
simps_7: fixes X :: "WS1S_Nameful.so"   and \<phi> :: "WS1S_Nameful.ws1s" shows "sos (Ex2 X \<phi>) = remove1 X (sos \<phi>)"
assms: shows "no_smallest_element_skipped (E \<union> {e}) F"
DK_fresh_not_KeyCryptNonce: fixes DK :: "nat"   and evs :: "event list"   and K :: "nat" assumes "DK \<in> symKeys"   and "Key DK \<notin> used evs"   and "evs \<in> set_cr" shows "\<not> KeyCryptNonce DK K evs"
disc_iff_1: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lnull (ltakeWhile P xs) = (lnull xs \<or> \<not> P (lhd xs))"
hyps_1: shows "p\<^sub>1 = Point x\<^sub>1 y\<^sub>1"
valid_path_edges: fixes G :: "('c, 'd) Graph_scheme"   and P :: "'c llist"   and n :: "nat" assumes "Digraph G"   and "Digraph.valid_path G P"   and "enat (Suc n) < llength P" shows "P $ n \<rightarrow>\<^bsub>G\<^esub> P $ Suc n"
above_diagonal_in_residues: fixes associates :: "'a set"   and residues :: "'a \<Rightarrow> 'a set"   and H :: "(('a, 'b) vec, 'b) vec"   and j :: "'b"   and i :: "'b" assumes "Hermite associates residues H"   and "invertible H"   and "upper_triangular H"   and "j < i" shows "H $ j $ (LEAST n. H $ i $ n \<noteq> (0::'a)) \<in> residues (H $ i $ (LEAST n. H $ i $ n \<noteq> (0::'a)))"
homologous_rel_eq: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "(homologous_rel p X S a = homologous_rel p X S b) = homologous_rel p X S a b"
simps: fixes a1 :: "'s"   and a2 :: "'s cfg" shows "cfg_onp a1 a2 = (\<exists>cfg s. a1 = s \<and> a2 = cfg \<and> state cfg = s \<and> action cfg \<in> K s \<and> (\<forall>x. x \<in> set_pmf (action cfg) \<longrightarrow> cfg_onp x (cont cfg x)))"
fxpnt\<Gamma>: shows "\<Inter> (symbolic_run_interpretation_primitive ` set \<Gamma>) = \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
cd: shows "c / d = rat_of_int a / rat_of_int b"
oghoare_step_Stuck: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and x :: "'e"   and c' :: "('e, 'd, 'f) com"   and P :: "('e, 'd, 'f) ann"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and Q :: "'e set"   and A :: "'e set"   and P' :: "bool" assumes "\<Gamma>\<turnstile> (c, Normal x) \<rightarrow> (c', Stuck)"   and "x \<in> pre P"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P c Q, A" shows "P'"
trace_core_aux_append: fixes core :: "('a, 'b, 'c, 'd, 'e, 'f) core"   and p :: "'a spmf"   and tr :: "('b + 'c \<times> 'e + 'd \<times> 'f) list"   and tr' :: "('b + 'c \<times> 'e + 'd \<times> 'f) list" shows "trace_core_aux core p (tr @ tr') = trace_core_aux core (trace_core_aux core p tr) tr'"
power_numeral_odd: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and w :: "num" assumes "class.monoid_mult one times" shows "power.power one times z (numeral (num.Bit1 w)) = (let w = power.power one times z (numeral w) in times (times z w) w)"
Acc_Trans_Fun_Step_def: fixes k :: "nat"   and trans_fun :: "'e message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and pointwise_shrink :: "'d list \<Rightarrow> 'd"   and x :: "'e message_af"   and c :: "'d" shows "Acc_Trans_Fun_Step k trans_fun pointwise_shrink x c \<equiv> pointwise_shrink (f_Exec_Comp_Stream trans_fun (x # NoMsg\<^bsup>k - Suc 0\<^esup>) c)"
ipset_from_cidr_0: fixes ip :: "'c word" shows "ipset_from_cidr ip 0 = UNIV"
meet_top: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<sqinter> \<top> = x"
simps_1: fixes E :: "char list \<Rightarrow> 'a"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and x :: "char list" shows "eval\<^sub>t E F (\<epsilon> x) = E x"
iff: fixes S :: "'a set" shows "P (f ` S) = Q S"
vertical_composite_axioms: shows "vertical_composite B D (K \<circ> F) (L \<circ> G) (M \<circ> H) (\<sigma> \<circ> \<tau>) (\<mu> \<circ> \<nu>)"
ap_spmf_id: fixes x :: "'d spmf" shows "return_spmf (\<lambda>x. x) \<diamondop> x = x"
endset_final: fixes \<sigma> :: "'state" assumes "\<sigma> \<in> endset" shows "\<forall>P. small P \<sigma> = {}"
UNIV_typedef_to_Quotient: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and T :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "type_definition Rep Abs UNIV"   and "T \<equiv> \<lambda>x y. x = Rep y" shows "Quotient (=) Abs Rep T"
product_coneI: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c"   and a :: "'c"   and \<pi> :: "'b \<Rightarrow> 'c" assumes "discrete_diagram J C D"   and "limit_cone J C D a \<pi>" shows "product_cone J C D a \<pi>"
powr_ms_aux: fixes basis :: "(real \<Rightarrow> real) list"   and F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and abort :: "bool"   and p :: "real" assumes "basis_wf basis"   and "is_expansion_aux F f basis"   and "trimmed_ms_aux F"   and "0 < fst (dominant_term_ms_aux F)" shows "is_expansion_aux (powr_ms_aux abort F p) (\<lambda>x. f x powr p) basis"
assms_2: shows "(c, s) \<rightarrow>c (c', s')"
cases: 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 a1 :: "char list"   and a2 :: "char list" assumes "subcls1p P a1 a2" obtains C :: "char list" and Bs :: "base list" and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list" and D :: "char list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(Bs, rest)\<rfloor>" and "D \<in> baseClasses Bs"
game2_a_def: shows "game2_a = spmf_of_set (nlists UNIV len) \<bind> (\<lambda>r. exec_gpv rf_encrypt \<A>1 Map.empty \<bind> (\<lambda>(((p0, p1), \<sigma>), s1). let bad = r \<in> dom s1 in if valid_plain p0 \<and> valid_plain p1 then coin_spmf \<bind> (\<lambda>b. let pb = if b then p0 else p1 in prf.random_oracle s1 r \<bind> (\<lambda>(pad, s2). let cipher = (r, pb [\<oplus>] pad) in exec_gpv rf_encrypt (\<A>2 cipher \<sigma>) s2 \<bind> (\<lambda>(b', s3). return_spmf (b' = b, bad)))) else coin_spmf \<bind> (\<lambda>b. return_spmf (b, bad))))"
id5: shows "[l..<l + k] = l # [Suc l..<Suc l + kk]"
intros_4: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and f :: "'f"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>c,Fault f\<rangle> =n\<Rightarrow> Fault f"
HR_Ord_D: fixes x :: "hf"   and y :: "hf" assumes "HR x y"   and "Ord x" shows "WR x y"
step: fixes c :: "'a" assumes "(a, c) \<in> r"   and "(c, b) \<in> rtrancl_on F r" shows "thesis"
det_Q: shows "Determinant.det (Matrix.mat 2 2 (\<lambda>(i, j). if i = 0 \<and> j = 0 then s else if i = 0 \<and> j = 1 then - b1 else if i = 1 \<and> j = 0 then t else a1)) = (1::'a)"
impI2: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and q :: "'b"   and A :: "'b set"   and p :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "q \<in> Cn A" shows "imp_PL p q \<in> Cn A"
part_product_in_whole_product: fixes x :: "i"   and y :: "i"   and v :: "i"   and z :: "i" assumes "O x y"   and "P x v \<and> P y z" shows "P (x \<otimes> y) (v \<otimes> z)"
simps_5: fixes N :: "nat"   and n :: "float" shows "plain_floatarith N (Num n) = True"
assms_1: fixes s :: "'a" shows "root (f s) = ROOT s"
paths_second_in_V: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and xs :: "'c list" assumes "DisjointPaths G v0 v1 paths"   and "xs \<in> paths" shows "hd (tl xs) \<in> V\<^bsub>G\<^esub>"
X: shows "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
\<Delta>_simps_1: shows "seq (r \<star> tab\<^sub>0 r) r.tab"
not_None_eq: fixes x :: "'a option" shows "(x \<noteq> None) = (\<exists>y. x = \<lfloor>y\<rfloor>)"