hcju/isabelleps huggingface dataset:
simps_6: fixes x61 :: "msg"   and x62 :: "msg"   and y61 :: "msg"   and y62 :: "msg" shows "(\<lbrace>x61, x62\<rbrace> = \<lbrace>y61, y62\<rbrace>) = (x61 = y61 \<and> x62 = y62)"
distrib_imp1: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "\<And>x y z. inf_prod x (sup_prod y z) = sup_prod (inf_prod x y) (inf_prod x z)" shows "sup_prod x (inf_prod y z) = inf_prod (sup_prod x y) (sup_prod x z)"
a_Unroll: fixes \<alpha> :: "Syntax.tvar"   and \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty" assumes "atom \<alpha> \<sharp> \<Gamma>"   and "\<Gamma> \<turnstile> e, eP, eV : Mu \<alpha> \<tau>" shows "\<Gamma> \<turnstile> Unroll e, Unroll eP, Unroll eV : subst_type \<tau> (Mu \<alpha> \<tau>) \<alpha>"
not_col_permutation_5: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C" shows "\<not> Col A C B"
consec_un: fixes i :: "nat_int"   and j :: "nat_int"   and n :: "nat" shows "consec i j \<and> n \<notin> Rep_nat_int i \<union> Rep_nat_int j \<longrightarrow> n \<^bold>\<notin> i \<squnion> j"
M_def: shows "M \<equiv> \<lambda>s. MAX j\<in>{s..n}. real_of_int (a j) powr (1 / real_of_int (2 ^ j))"
restrict_bind_spmf: fixes p :: "'c spmf"   and g :: "'c \<Rightarrow> 'b spmf"   and A :: "'b set" shows "(p \<bind> g) \<upharpoonleft> A = p \<bind> (\<lambda>x. g x \<upharpoonleft> A)"
Sard_lemma0: fixes a :: "(real, 'a) vec"   and P :: "(real, 'a) vec set"   and m :: "real"   and e :: "real" assumes "a \<noteq> 0"   and "P \<subseteq> {x. a \<bullet> x = 0}"   and "0 \<le> m"   and "0 \<le> e" obtains S :: "(real, 'a) vec set" where "S \<in> lmeasurable" and "{z. norm z \<le> m \<and> (\<exists>t\<in>P. norm (z - t) \<le> e)} \<subseteq> S" and "Sigma_Algebra.measure lebesgue S \<le> 2 * e * (2 * m) ^ (CARD('a) - 1)"
f'_def: shows "f' = (\<lambda>t. Im (reversepath g t - z) / Re (reversepath g t - z))"
sel: fixes x :: "'d" shows "run_nondet (NondetT x) = x"
transfer_iff: fixes p :: "bool"   and P :: "nat \<Rightarrow> bool"   and q :: "bool"   and Q :: "nat \<Rightarrow> bool" assumes "p \<equiv> eventually P \<U>"   and "q \<equiv> eventually Q \<U>" shows "p = q \<equiv> \<forall>\<^sub>F n in \<U>. P n = Q n"
low_mds_eq_def: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2 \<equiv> \<forall>x. dma x = Low \<and> x \<notin> mds AsmNoRead \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
borel_measurable_lim_metric: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "\<And>i. f i \<in> borel_measurable M" shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
Try: fixes G :: "prog"   and s0 :: "st"   and c1 :: "stmt"   and s1 :: "state"   and s2 :: "state"   and C :: "qtname"   and vn :: "vname"   and c2 :: "stmt"   and s3 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>c1\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2"   and "if G,s2\<turnstile>catch C then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<rightarrow> s3 else s3 = s2" shows "G\<turnstile>Norm s0 \<midarrow>Try c1 Catch(C vn) c2\<rightarrow> s3"
all_less: shows "\<forall>i<r. 0 < rec_exec f (xs @ [i])"
mono_set: fixes S :: "'a set" shows "mono_set S = (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
ffd_ffb_demorgan: fixes f :: "'d \<Rightarrow> 'c set" shows "\<partial> \<circ> fd\<^sub>\<F> f = fb\<^sub>\<F> f \<circ> \<partial>"
conditional_min_upper_bound: fixes C :: "'a set"   and Ca :: "'a set set" assumes "C \<in> carrier fTo D"   and "Chain (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}) Ca" shows "\<exists>X. minimum_elem (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}) (upper_bounds (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}) Ca) X"
Diff_triv: fixes A :: "'d set"   and B :: "'d set" assumes "A \<inter> B = {}" shows "A - B = A"
C_def: shows "C = (\<Prod>k<N - M. f (k + M))"
compact_chain: fixes \<F> :: "'a set set" assumes "\<And>S. S \<in> \<F> \<Longrightarrow> compact S"   and "{} \<notin> \<F>"   and "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S" shows "\<Inter> \<F> \<noteq> {}"
pfun_minus_zero: fixes f :: "('a, 'b) pfun" shows "{}\<^sub>p - f = {}\<^sub>p"
verit_comp_simplify1_2: fixes a :: "'a" shows "a \<le> a"
inv_rule_basic: fixes T :: "('b, 'c) TS_scheme"   and P :: "'b set" assumes "init T \<subseteq> P"   and "{P} TS.trans T {> P}" shows "reach T \<subseteq> P"
M: fixes x :: "nat" shows "norm (a x * z ^ x) \<le> M"
UN_extend_simps_1: fixes a :: "'c"   and B :: "'d \<Rightarrow> 'c set"   and C :: "'d set" shows "insert a (\<Union> (B ` C)) = (if C = {} then {a} else \<Union>x\<in>C. insert a (B x))"
reduce_column_def: fixes div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and H :: "'a mat" shows "local.reduce_column div_op H = local.reduce_column_aux div_op [2..<dim_row H] H (1\<^sub>m (dim_row H), H)"
these_def: fixes A :: "'c option set" shows "Option.these A = the ` {x \<in> A. x \<noteq> None}"
unitary11_unitary11_gen: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "unitary11 M" shows "unitary11_gen M"
rel_interior_interior: fixes S :: "'a set" assumes "affine hull S = UNIV" shows "rel_interior S = interior S"
character_data_ptr_kinds_eq_h: shows "character_data_ptr_kinds h2 = character_data_ptr_kinds h |\<union>| {|new_character_data_ptr|}"
splus_right_canc: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and t :: "'d"   and v :: "'c"   and s :: "'d" assumes "term_powerprod pair_of_term term_of_pair" shows "(term_powerprod.splus pair_of_term term_of_pair t v = term_powerprod.splus pair_of_term term_of_pair s v) = (t = s)"
strict_inc_induct: fixes i :: "nat"   and j :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "i < j"   and "\<And>i. j = Suc i \<Longrightarrow> P i"   and "\<And>i. \<lbrakk>i < j; P (Suc i)\<rbrakk> \<Longrightarrow> P i" shows "P i"
negated_normalized_folded_ports_nodisc: fixes disc :: "'b common_primitive \<Rightarrow> bool"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and pts :: "ipt_l4_ports list"   and m :: "'b common_primitive match_expr" assumes "\<forall>a. \<not> disc (C a)"   and "(\<forall>a. \<not> disc (Prot a)) \<or> pts = []"   and "m \<in> set (normalize_match (andfold_MatchExp (map (l4_ports_negate_one C) pts)))" shows "\<not> has_disc disc m"
rbt_greater_prop: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'c"   and t :: "('c, 'd) rbt" shows "ord.rbt_greater less k t = (\<forall>x\<in>set (RBT_Impl.keys t). less k x)"
lappend_lnull1: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lnull xs" shows "xs $ ys = ys"
ag_pOp_commute: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier A"   and "y \<in> carrier A" shows "x \<plusminus> y = y \<plusminus> x"
dom_assoc: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "ide a"   and "ide b"   and "ide c" shows "local.dom \<a>[a, b, c] = (a \<otimes> b) \<otimes> c"
order_less_trans: fixes x :: "'e"   and y :: "'e"   and z :: "'e" assumes "x < y"   and "y < z" shows "x < z"
filterlim_tendsto_pos_mult_at_bot: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "0 < c"   and "filterlim g at_bot F" shows "LIM x F. f x * g x :> at_bot"
joint_distributed_measurable1: fixes M :: "'c measure"   and S :: "'d measure"   and T :: "'e measure"   and X :: "'c \<Rightarrow> 'd"   and Y :: "'c \<Rightarrow> 'e"   and f :: "'d \<times> 'e \<Rightarrow> ennreal"   and h1 :: "'f \<Rightarrow> 'c"   and N :: "'f measure" assumes "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) f"   and "h1 \<in> N \<rightarrow>\<^sub>M M" shows "(\<lambda>x. X (h1 x)) \<in> N \<rightarrow>\<^sub>M S"
MkElem_in_hom: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and x :: "'e" assumes "arr f"   and "x \<in> Dom f" shows "\<guillemotleft>MkArr {U} (Dom (local.dom f)) (\<lambda>_\<in>{U}. DOWN (UP x)) : unity \<rightarrow> local.dom f\<guillemotright>"
f_times: fixes m :: "nat"   and n :: "nat" shows "f (m * n) = f m * f n"
ex231c: fixes x :: "'a" shows "x \<le> x * x\<^sup>T * x"
negligible_divide_polynomial1: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> real" assumes "polynomial (\<lambda>x. real (f x))"   and "negligible (\<lambda>\<eta>. 1 / g \<eta>)" shows "negligible (\<lambda>\<eta>. real (f \<eta>) / g \<eta>)"
induct: fixes x1 :: "('Var, 'AExp, 'BExp) Stmt"   and x2 :: "Mode \<Rightarrow> 'Var set"   and x3 :: "'Var \<Rightarrow> 'Val"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and P :: "('Var, 'AExp, 'BExp) Stmt \<Rightarrow> (Mode \<Rightarrow> 'Var set) \<Rightarrow> ('Var \<Rightarrow> 'Val) \<Rightarrow> bool" assumes "\<langle>x1, x2, x3\<rangle> \<in> loc_reach lc"   and "P (fst (fst lc)) (snd (fst lc)) (snd lc)"   and "\<And>c' mds' mem' c'' mds'' mem''. \<lbrakk>\<langle>c', mds', mem'\<rangle> \<in> loc_reach lc; P c' mds' mem'; \<langle>c', mds', mem'\<rangle> \<leadsto> \<langle>c'', mds'', mem''\<rangle>\<rbrakk> \<Longrightarrow> P c'' mds'' mem''"   and "\<And>c' mds' mem' mem''. \<lbrakk>\<langle>c', mds', mem'\<rangle> \<in> loc_reach lc; P c' mds' mem'; \<forall>x\<in>mds' AsmNoWrite. mem' x = mem'' x\<rbrakk> \<Longrightarrow> P c' mds' mem''" shows "P x1 x2 x3"
cyclic_on_lists_succ': fixes xss :: "'b list list"   and A :: "'b set" assumes "distincts xss"   and "A \<in> sset xss" shows "cyclic_on (lists_succ xss) A"
assms: shows "eval cid t t' n (\<forall>\<^sub>bp. \<gamma> p)"
urel_integer_1: fixes pp :: "integer" assumes "p = int_of_integer pp" shows "urel_integer 1 1"
set_intros_3: fixes x1 :: "'c" shows "x1 \<in> vars_term (Var x1)"
closure_greaterThanLessThan: fixes a :: "'b"   and b :: "'b" assumes "a < b" shows "closure {a<..<b} = {a..b}"
igFreshIGVar_def: fixes MOD :: "('d, 'e, 'f, 'g, 'h, 'i, 'j, 'k, 'l) model_scheme" shows "igFreshIGVar MOD \<equiv> \<forall>ys y xs x. ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igFresh MOD ys y (igVar MOD xs x)"
stable_no_blocking_onI2: fixes ds :: "'b set"   and X :: "'a set" assumes "\<And>h X''. Singleton_for_d.blocking_on ds X h X'' \<Longrightarrow> False" shows "Singleton_for_d.stable_no_blocking_on ds X"
inter_subset_eq_union: fixes A :: "'a multiset"   and B :: "'a multiset" shows "A \<inter># B \<subseteq># A + B"
nodes_def: fixes g :: "igraph" shows "nodes g \<equiv> {x. \<exists>y. (x, y) \<in> gra g \<or> (y, x) \<in> gra g}"
tranclp_power: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "P\<^sup>+\<^sup>+ x y = (\<exists>n>0. (P ^^ n) x y)"
tranclp_transitive: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "transitive A r\<^sup>+\<^sup>+"
hom_minus: fixes x :: "'b poly"   and y :: "'b poly"   and p :: "'b poly" shows "(x - y) \<circ>\<^sub>p p = x \<circ>\<^sub>p p - y \<circ>\<^sub>p p"
fmrelI: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'d \<rightharpoonup>\<^sub>f 'e"   and n :: "'d \<rightharpoonup>\<^sub>f 'f" assumes "\<And>x. rel_option R (fmlookup m x) (fmlookup n x)" shows "fmrel R m n"
assms_2: shows "\<Theta>(f) \<equiv> BIGTHETA_CONST c1 \<Theta>(eval_primfuns fs)"
OSP_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" shows "A B C OSP P Q \<equiv> \<exists>R. A B CTSP P R \<and> A B CTSP Q R"
gt_hd_trans: fixes \<chi> :: "('s, 'v) hd"   and \<xi> :: "('s, 'v) hd"   and \<zeta> :: "('s, 'v) hd" assumes "\<chi> >\<^sub>h\<^sub>d \<xi>"   and "\<xi> >\<^sub>h\<^sub>d \<zeta>" shows "\<chi> >\<^sub>h\<^sub>d \<zeta>"
to_fun_X: fixes a :: "'a" assumes "a \<in> carrier R" shows "to_fun (X_poly R) a = a"
CallRedsParams: fixes es :: "char list exp list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "char list exp list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and v :: "val"   and M :: "char list" assumes "P \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es',s'\<rangle>" shows "P \<turnstile> \<langle>Val v\<bullet>M(es),s\<rangle> \<rightarrow>* \<langle>Val v\<bullet>M(es'),s'\<rangle>"
qtable_project: fixes n :: "nat"   and A :: "nat set"   and R :: "'a list set"   and P :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set" assumes "qtable (Suc n) A (mem_restr (lift_envs R)) P X" shows "qtable n ((\<lambda>x. x - Suc 0) ` (A - {0})) (mem_restr R) (\<lambda>v. \<exists>x. P ((if 0 \<in> A then Some x else None) # v)) (tl ` X)"
w_def: shows "w = (\<lambda>n x. u n x + v n x)"
as0_4: shows "DA0 \<cdot> \<eta>0 = DA"
c1: shows "c1 = map (\<lambda>d. (i, \<langle>check: pair (t, s) \<doteq> pair (snd d)\<rangle>\<^sub>s\<^sub>t)) Di"
extended_Gromov_product_nonneg: fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "0 \<le> extended_Gromov_product_at e x y"
outstanding_non_volatile_write_disj: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'a memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'a memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length (t # ts)"   and "j < length (t # ts)"   and "i \<noteq> j"   and "(t # ts) ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "(t # ts) ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) \<inter> (outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb\<^sub>j \<union> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb\<^sub>j \<union> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j) \<union> (outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j) - read_only_reads \<O>\<^sub>j (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j)) \<union> (\<O>\<^sub>j \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j))) = {}"
index_mat_addcol_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and a :: "'a"   and k :: "nat"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "addcol a k l A $$ (i, j) = (if k = j then a * A $$ (i, l) + A $$ (i, j) else A $$ (i, j))"
interchange: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and \<tau> :: "'a"   and \<sigma> :: "'a" assumes "Left_a.seq \<nu> \<mu>"   and "Left_a.seq \<tau> \<sigma>"   and "\<mu> \<star> \<sigma> \<noteq> null" shows "Left_a.comp \<nu> \<mu> \<star> Left_a.comp \<tau> \<sigma> = Left_a.comp (\<nu> \<star> \<tau>) (\<mu> \<star> \<sigma>)"
shows_prec_char_def: fixes d :: "nat"   and c :: "char" shows "shows_prec d c = (#) c"
assms_4: fixes \<Psi> :: "'b" shows "(\<Psi>, P, \<tau>.Q) \<in> Rel"
semantics_in_vertices_1: fixes A :: "('b, 'c) labeled_graph"   and a :: "'c"   and b :: "'c"   and e :: "'b allegorical_term" assumes "graph A"   and "(a, b) \<in> :A:\<lbrakk>e\<rbrakk>" shows "a \<in> vertices A"
srev_srev_ident: fixes xs :: "[:'a:]" shows "srev\<cdot>(srev\<cdot>xs) = xs"
padic_add_zero: fixes p :: "int"   and x :: "nat \<Rightarrow> int" assumes "prime p"   and "x \<in> carrier (padic_int p)" shows "\<zero>\<^bsub>padic_int p\<^esub> \<oplus>\<^bsub>padic_int p\<^esub> x = x"
not_le_imp_less: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and y :: "'k"   and x :: "'k" assumes "class.linorder less_eq less"   and "\<not> less_eq y x" shows "less x y"
conjugate_preserve_poincare_between: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc" shows "poincare_between (conjugate u) (conjugate v) (conjugate w) = poincare_between u v w"
c: shows "c \<in> carrier (reduced_homology_group (int p) (nsphere p))"
stereographic_circline_inv_stereographic_circline: shows "stereographic_circline \<circ> inv_stereographic_circline = id"
vwalk_arcs_append: fixes p :: "'a list"   and q :: "'a list" assumes "p \<noteq> []"   and "q \<noteq> []" shows "vwalk_arcs (p @ q) = vwalk_arcs p @ (last p, hd q) # vwalk_arcs q"
f_def: fixes s :: "('c, 'U, 'd) state"   and s' :: "('e, 'U, 'd) state" shows "f s s' \<equiv> \<forall>u c. u \<notin> GH \<longrightarrow> (\<exists>st. s = State st u c) = (\<exists>st'. s' = State st' u c)"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_def: fixes h :: "(_) heap" shows "new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t h = (let new_shadow_root_ptr = shadow_root_ptr.Ref (Suc (fMax (shadow_root_ptr.the_ref |`| shadow_root_ptrs h))) in (new_shadow_root_ptr, put\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t new_shadow_root_ptr (create_shadow_root_obj Open []) h))"
R_def: fixes h :: "'a"   and w :: "'b"   and e :: "'e" shows "\<Sigma>1.R h w e = init1 h w \<bind> (\<lambda>(r, a). response1 r w e \<bind> (\<lambda>z. return_spmf (a, e, z)))"
Minj_pred_correct: fixes i :: "nat"   and l :: "'a list"   and M :: "mon" assumes "0 < i" shows "Imon (drop 1 l) (Minj_pred i M) = Imon l (Minj i M)"
bit_and_iff: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "bit (a AND b) n = (bit a n \<and> bit b n)"
SN: shows "SN {(x, y). (0::'a) \<le> y \<and> delta_gt \<delta> x y}"
dim_singleton: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and Basis :: "'c set"   and x :: "'c" assumes "finite_dimensional_vector_space scale Basis" shows "vector_space.dim scale {x} = (if x = (0::'c) then 0 else 1)"
RBT_lookup_empty: fixes t :: "('a, 'b) RBT_Impl.rbt" shows "(rbt_lookup t = Map.empty) = (t = rbt.Empty)"
l_comb_add1Tr: fixes A :: "'b set"   and H :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b" assumes "ideal R A"   and "H \<subseteq> carrier M" shows "f \<in> {j. j \<le> n} \<rightarrow> H \<and> s \<in> {j. j \<le> n} \<rightarrow> A \<and> t \<in> {j. j \<le> n} \<rightarrow> A \<longrightarrow> l_comb R M n (\<lambda>x\<in>{j. j \<le> n}. s x \<plusminus>\<^bsub>R\<^esub> t x) f = l_comb R M n s f \<plusminus> l_comb R M n t f"
Parallel: fixes i :: "nat"   and cs :: "('a, 'b, 'c) com list"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and s :: "('a, 'c) xstate"   and c' :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate" assumes "i < length cs"   and "\<Gamma>\<turnstile> (cs ! i, s) \<rightarrow> (c', s')" shows "\<Gamma>\<turnstile> (Parallel cs, s) \<rightarrow> (Parallel (cs[i := c']), s')"
lub_spmf_def: fixes Y :: "'a spmf set" shows "lub_spmf Y = embed_spmf (\<lambda>x. enn2real (\<Squnion>p\<in>Y. ennreal (spmf p x)))"
intros_1: fixes p :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" shows "p \<in> greachable A w V p"
dual_less_eq_iff: fixes x :: "'a dual"   and y :: "'a dual" shows "(x \<le> y) = (undual y \<le> undual x)"
dom_eqI': fixes a :: "'s"   and f :: "'s" assumes "a \<in> S.domains f" shows "a = S.dom f"
I_qe_dlo\<^sub>1: fixes as :: "atom list"   and xs :: "'b list" assumes "\<forall>a\<in>set as. is_Less a"   and "\<forall>a\<in>set as. depends\<^sub>d\<^sub>l\<^sub>o a" shows "DLO.I (qe_dlo\<^sub>1 as) xs = (\<exists>x. \<forall>a\<in>set as. I\<^sub>d\<^sub>l\<^sub>o a (x # xs))"
coincidence_frechet: fixes \<theta> :: "('d, 'e) trm"   and \<nu> :: "(real, 'e) vec \<times> (real, 'e) vec"   and \<nu>' :: "(real, 'e) vec \<times> (real, 'e) vec"   and I :: "('d, 'f, 'e) interp" assumes "dfree \<theta>"   and "Vagree \<nu> \<nu>' (FVDiff \<theta>)" shows "frechet I \<theta> (fst \<nu>) (snd \<nu>) = frechet I \<theta> (fst \<nu>') (snd \<nu>')"
is_OK_returns_heap_I: fixes h :: "'m"   and f :: "('m, 'n, 'o) prog"   and h' :: "'m" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'" shows "h \<turnstile> ok f"
Ipoly_CRat: fixes bs :: "'b list"   and i :: "int"   and j :: "int" shows "\<lparr>C (i, j)\<rparr>\<^sub>p\<^bsup>bs\<^esup> = of_int i / of_int j"
umformung: fixes A :: "('d list \<Rightarrow> 'c pmf) \<times> ('d list \<times> 'c \<Rightarrow> 'd \<Rightarrow> ((nat \<times> nat list) \<times> 'c) pmf)"   and qs :: "'d list"   and init :: "'d list" assumes "\<And>is s q. \<forall>((free, paid), uu_)\<in>set_pmf (snd A (s, is) q). paid = []"   and "set qs \<subseteq> set init"   and "distinct init"   and "\<And>x. x < length qs \<Longrightarrow> finite (set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) (take x qs)))" shows "T\<^sub>p_on_rand' A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. ALGxy A qs init x y)"
set_shadow_root_def: shows "set_shadow_root \<equiv> l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_shadow_root"
A_1_3_2: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma>2 :: "nat \<Rightarrow> nat option"   and e :: "exp"   and C :: "nat"   and \<Gamma>1 :: "nat \<Rightarrow> nat option" assumes "CT;\<Gamma>2 \<turnstile> e : C" shows "CT;\<Gamma>1 ++ \<Gamma>2 \<turnstile> e : C"
lt_MkI_MkI: fixes x :: "int"   and y :: "int" shows "lt\<cdot>(MkI\<cdot>x)\<cdot>(MkI\<cdot>y) = (if x < y then TT else FF)"
vw: shows "w \<cdot> v = v \<cdot> w"
Limit_vdoubleton_in_VsetI: fixes \<alpha> :: "V"   and a :: "V"   and b :: "V" assumes "Limit \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>"   and "b \<in>\<^sub>\<circ> Vset \<alpha>" shows "set {a, b} \<in>\<^sub>\<circ> Vset \<alpha>"
geodesic_segment_reverse_param: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and t :: "real" assumes "geodesic_segment_between G x y"   and "t \<in> {0..dist x y}" shows "geodesic_segment_param G y (dist x y - t) = geodesic_segment_param G x t"
chine_composite: shows "\<mu>\<nu>.chine = \<mu>.leg0 \<down>\<down> \<nu>.leg1"
intros_2: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>*\<^sup>* a c"
vec_of_poly_rev_shifted_def: fixes p :: "'b poly"   and n :: "nat"   and s :: "nat"   and j :: "nat" shows "Missing_Lemmas.vec_of_poly_rev_shifted p n s j \<equiv> vec n (\<lambda>i. if i \<le> j \<and> j \<le> s + i then coeff p (s + i - j) else (0::'b))"
simps: fixes G :: "('c, 'd) Graph_scheme"   and a :: "'c list" assumes "Graph G" shows "Graph.walk G a = (a = [] \<or> (\<exists>v. a = [v] \<and> v \<in> V\<^bsub>G\<^esub>) \<or> (\<exists>v w vs. a = v # w # vs \<and> v \<rightarrow>\<^bsub>G\<^esub> w \<and> Graph.walk G (w # vs)))"
cong_cong_mod_nat: fixes a :: "nat"   and b :: "nat"   and m :: "nat" shows "[a = b] (mod m) = [a mod m = b mod m] (mod m)"