hcju/isabelleps huggingface dataset:
fold_val_elem\<^sub>S\<^sub>e\<^sub>q: fixes \<tau> :: "'d state \<times> 'd state"   and f :: "'g \<Rightarrow> 'f \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> 'e"   and p :: "'g"   and s_set :: "'f list" assumes "\<tau> \<Turnstile> \<upsilon> foldl UML_Sequence.OclIncluding Sequence{} (map (f p) s_set)" shows "list_all (\<lambda>x. \<tau> \<Turnstile> \<upsilon> f p x) s_set"
indep_in_subset: fixes \<E> :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "\<E> \<subseteq> carrier"   and "indep_in \<E> X"   and "Y \<subseteq> X" shows "indep_in \<E> Y"
assms_1: shows "w \<in> adopt_node_locs parent owner_document document_ptr"
iMin_le: fixes k :: "'a"   and I :: "'a set" assumes "k \<in> I" shows "iMin I \<le> k"
lists_succ_set_cyc_permutationss: fixes xss :: "'b list list" assumes "distincts xss" shows "lists_succ ` set (cyc_permutationss xss) = {f. f permutes \<Union> (sset xss) \<and> (\<forall>c\<in>sset xss. cyclic_on f c)}"
mono_neutral_cong: fixes T :: "'c set"   and S :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "finite T"   and "finite S"   and "\<And>i. i \<in> T - S \<Longrightarrow> h i = (0::'d)"   and "\<And>i. i \<in> S - T \<Longrightarrow> g i = (0::'d)"   and "\<And>x. x \<in> S \<inter> T \<Longrightarrow> g x = h x" shows "sum g S = sum h T"
exec_1_iff: fixes \<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"   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"   and 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" shows "P \<turnstile> \<sigma> -jvm\<rightarrow>\<^sub>1 \<sigma>' = (exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>)"
is_pr_Hom_def: fixes \<phi> :: "'a \<Rightarrow> 'b"   and Fs\<^sub>1 :: "'a list set"   and Fs\<^sub>2 :: "'b list set" shows "is_pr_Hom \<phi> Fs\<^sub>1 Fs\<^sub>2 \<equiv> map \<phi> ` Fs\<^sub>1 // {\<cong>} = Fs\<^sub>2 // {\<cong>}"
reachable_in_verts_1: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G"   and "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "u \<in> verts G"
poly_neg_inf_def: fixes p :: "'b poly" shows "poly_neg_inf p \<equiv> if even (degree p) then sgn (lead_coeff p) else - sgn (lead_coeff p)"
connected_equivalence_relation_gen: fixes S :: "'b set"   and a :: "'b"   and b :: "'b"   and P :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "connected S"   and "a \<in> S"   and "b \<in> S"   and "P a"   and "P b"   and "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"   and "\<And>T a. \<lbrakk>openin (top_of_set S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"   and "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x\<in>T. \<forall>y\<in>T. P x \<and> P y \<longrightarrow> R x y)" shows "R a b"
inj_Poly_list_of_vec': fixes n :: "nat" shows "inj_on (Poly \<circ> list_of_vec) {v. dim_vec v = n}"
c2: shows "lookup p (t2 \<oplus> lt f2) \<noteq> (0::'b)"
assms: shows "set (Guards t) \<subseteq> set (Guards t')"
basis_reduction_swap_cost_def: fixes i :: "nat"   and state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray" shows "basis_reduction_swap_cost i state = (let di = d_state state i; dsi = d_state state (Suc i); dim1 = d_state state (i - 1); fi = fi_state state; fim1 = fim1_state state; dmu_i_im1 = dmu_ij_state state i (i - 1); fi' = fim1; fim1' = fi; di' = (dsi * dim1 + dmu_i_im1 * dmu_i_im1) div di; local_cost = 4 in case state of (f, dmus, djs) \<Rightarrow> case swap_mu_cost dmus i dmu_i_im1 dim1 di dsi of (swap_res, swap_cost) \<Rightarrow> let res = (False, i - 1, dec_i (update_im1 (update_i f fi') fim1'), swap_res, iarray_update djs i di') in Let (local_cost + swap_cost) (Pair res))"
assms_1: fixes t :: "real" assumes "t \<in> closed_segment t0 t1" shows "x t = x t0 + ivl_integral t0 t (\<lambda>t. f t (x t))"
emeasure_SUP_chain: fixes A :: "'c set"   and M :: "'c \<Rightarrow> 'd measure"   and N :: "'d measure"   and X :: "'d set" assumes "\<And>i. i \<in> A \<Longrightarrow> sets (M i) = sets N"   and "X \<in> sets N"   and "Complete_Partial_Order.chain (\<le>) (M ` A)"   and "A \<noteq> {}" shows "emeasure (\<Squnion> (M ` A)) X = (\<Squnion>i\<in>A. emeasure (M i) X)"
rebase_rebase: fixes x :: "'pq mod_ring" shows "@@x = @x"
funcsetI: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f \<in> A \<rightarrow> B"
Rep_coreflexive_inject: fixes x :: "'b coreflexive"   and y :: "'b coreflexive" shows "(Rep_coreflexive x = Rep_coreflexive y) = (x = y)"
inject_4: fixes x4 :: "int"   and y4 :: "int" shows "(Smaller' x4 = Smaller' y4) = (x4 = y4)"
\<gamma>_def: shows "\<gamma> = \<delta> @ \<gamma>'"
Compl: fixes a :: "'b set"   and sp :: "'b set"   and A :: "'b set set" assumes "a \<in> sigma_sets sp A" shows "sp - a \<in> sigma_sets sp A"
assms_1: shows "\<forall>w\<in>space M. 0 \<le> X w"
dep_iff_supset_circuit: fixes X :: "'a set" assumes "X \<subseteq> \<E>" shows "(\<not> indep_in \<E> X) = (\<exists>C. \<E>.circuit C \<and> C \<subseteq> X)"
subpath_reversepath: fixes g :: "real \<Rightarrow> 'a" shows "subpath 1 0 g = reversepath g"
null_measure_eq_density: fixes M :: "'c measure" shows "null_measure M = density M (\<lambda>_. 0)"
transfer: shows "(\<lambda>x y. Abs_atoms (Rep_atoms x \<inter> Rep_atoms y)) = (\<sqinter>)"
scaleR_sq_mtx_diag: fixes c :: "real"   and f :: "'a \<Rightarrow> real" shows "c *\<^sub>R sq_mtx_diag f = (\<d>\<i>\<a>\<g> i. c * f i)"
simps_11: fixes i :: "int"   and a :: "MIR.num" shows "zlfm (Dvd i a) = (if i = 0 then zlfm (Eq a) else let (c, r) = zsplit0 a in if c = 0 then Dvd \<bar>i\<bar> r else if 0 < c then And (Eq (Sub (Floor r) r)) (Dvd \<bar>i\<bar> (CN 0 c (Floor r))) else And (Eq (Sub (Floor r) r)) (Dvd \<bar>i\<bar> (CN 0 (- c) (Neg (Floor r)))))"
mult_left_cancel: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.semiring_no_zero_divisors_cancel plus zero times"   and "c \<noteq> zero" shows "(times c a = times c b) = (a = b)"
ereal_truncation_top: fixes x :: "ereal" shows "(\<lambda>x. min x (ereal (real x))) \<longlonglongrightarrow> x"
axioms_1: fixes R :: "('c, 'd) ring_scheme" assumes "field R" shows "domain R"
asymptotics2': fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "(bs ! i * x + (hs ! i) x) powr p * (1 - ln (bs ! i * x + (hs ! i) x) powr (- e / 2)) \<le> (bs ! i * x) powr p * (1 - ln x powr (- e / 2))"
CptnComp: fixes P :: "'b com option"   and s :: "'b"   and Q :: "'b com option"   and t :: "'b"   and xs :: "('b com option \<times> 'b) list" assumes "(P, s) -c\<rightarrow> (Q, t)"   and "(Q, t) # xs \<in> cptn" shows "(P, s) # (Q, t) # xs \<in> cptn"
relevant_vars_def: shows "relevant_vars \<equiv> \<lambda>S n. {x. relevant_varsp S n x}"
lead_coeff_poly_x_minus_y: fixes p :: "'a poly" shows "lead_coeff (poly_x_minus_y p) = [:lead_coeff p * (- (1::'a)) ^ degree p:]"
Ext_foo_def: shows "Ext_foo = (foo_sig, \<lparr>access = Public, static = False, pars = [z], resT = Class Ext, mbody = \<lparr>lcls = [], stmt = Expr ({Ext,Ext,False}Cast (Class Ext) (!! z)..vee:=Lit (Intg 1));; Return (Lit Null)\<rparr>\<rparr>)"
unat_mono: fixes a :: "'b word"   and b :: "'b word" assumes "a < b" shows "unat a < unat b"
completeness: fixes s :: "form list" assumes "infinite (deriv (ns_of_s s))" shows "\<not> Svalid s"
simps_2: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and ta :: "('a, 'd, 'e) new_thread_action"   and tas :: "('a, 'd, 'e) new_thread_action list" shows "FWThread.thread_oks ts (ta # tas) = (FWThread.thread_ok ts ta \<and> FWThread.thread_oks (redT_updT' ts ta) tas)"
lincomb_diff: fixes A :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and b :: "'c \<Rightarrow> 'a" assumes "finite A"   and "A \<subseteq> carrier V"   and "a \<in> A \<rightarrow> carrier K"   and "b \<in> A \<rightarrow> carrier K" shows "lincomb (\<lambda>v. a v \<ominus>\<^bsub>K\<^esub> b v) A = lincomb a A \<ominus>\<^bsub>V\<^esub> lincomb b A"
alist_product_simps_1: fixes ys :: "('c \<times> 'd) list" shows "local.alist_product [] ys = []"
IH: fixes l :: "('a \<times> 'b) list" assumes "l = map (\<lambda>x. (x, a)) xs" shows "\<forall>x\<in>set xs. map_of l x = Some a"
nn_integral_embed_measure': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'d \<Rightarrow> ennreal" assumes "inj_on f (space M)"   and "g \<in> borel_measurable (embed_measure M f)" shows "integral\<^sup>N (embed_measure M f) g = \<integral>\<^sup>+ x. g (f x) \<partial>M"
pderiv_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" assumes "UP_cring R" shows "UP_cring.pderiv R p = poly_shift (UP_cring.n_mult R p)"
is_monic_set_def: fixes B :: "('t \<Rightarrow>\<^sub>0 'd) set" shows "is_monic_set B \<equiv> \<forall>b\<in>B. b \<noteq> 0 \<longrightarrow> lc b = (1::'d)"
qs: fixes q :: "real poly" assumes "q \<in> set qs" shows "degree q \<le> 2"
pconj_rone: fixes b :: "real" assumes "0 \<le> b" shows "b .& 1 = b"
substAbs_vsubstAbs_trans: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A"   and "good Y"   and "freshAbs ys y1 A" shows "A $[y1 // y]_ys $[Y / y1]_ys = A $[Y / y]_ys"
cases: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and a1 :: "'heap"   and a2 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and a3 :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and a4 :: "'heap"   and a5 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P ins xt t a1 a2 a3 a4 a5" obtains ta :: "('addr, 'thread_id, 'heap) jvm_thread_action" and xcp :: "'addr option" and h' :: "'heap" and stk' :: "'addr val list" and loc' :: "'addr val list" and pc' :: "nat" and pc :: "nat" and h :: "'heap" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = ha__" and "a2 = (stk__, loc__, pc__, None)" and "a3 = ta" and "a4 = h'" and "a5 = (stk', loc', pc'__, xcp__)" and "(ta, xcp__, h', [(stk', loc', undefined, undefined, pc'__)]) \<in> exec_instr (ins ! pc__) P t ha__ stk__ loc__ undefined undefined pc__ []" and "pc__ < length ins" and "ci_app ci (ins ! pc__) P ha__ stk__ loc__ undefined undefined pc__ []"   | h :: "'heap" and xcp :: "'addr" and pc :: "nat" and pc' :: "nat" and d :: "nat" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = h" and "a2 = (stk, loc, pc, \<lfloor>xcp\<rfloor>)" and "a3 = \<lbrace>\<rbrace>" and "a4 = h" and "a5 = (Addr xcp # drop (length stk - d) stk, loc, pc', None)" and "match_ex_table P (cname_of h xcp) pc xt = \<lfloor>(pc', d)\<rfloor>" and "d \<le> length stk"
hyps: fixes s :: "'a"   and ss2 :: "'a \<times> ('b \<times> 'a) list" assumes "(s, xs) \<in> seq ars"   and "ss2 \<in> seq ars"   and "lst (s, xs) = fst ss2" shows "(s, xs @ snd ss2) \<in> seq ars \<and> lst (s, xs @ snd ss2) = lst ss2"
norm_le: fixes y :: "'b" shows "norm (h y) \<le> real DIM('b)"
cp\<^sub>P\<^sub>R\<^sub>E: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "PRE self \<tau> = PRE (\<lambda>_. self \<tau>) \<tau>"
atms_size: fixes x :: "Formula.formula"   and r :: "Formula.formula Regex.regex" assumes "x \<in> atms r" shows "size x < regex.size_regex size r"
inf_sup_absorb: fixes x :: "'a upred"   and y :: "'a upred" shows "(x \<and> (x \<or> y)) = x"
simps_2: fixes p :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (floatarith.Mult a b) vs = approx_floatarith p a vs \<bind> (\<lambda>a1. approx_floatarith p b vs \<bind> (\<lambda>a2. Some (mult_aform' p a1 a2)))"
up: fixes d :: "nat"   and dm :: "nat"   and p :: "(nat \<times> int) list"   and lm :: "nat"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" assumes "d < dm"   and "p \<in> sparsegrid dm lm" shows "up dm lm d \<alpha> p = (\<Sum>p'\<in>lgrid p {d} lm - {p}. \<alpha> p' * l2_\<phi> (p' ! d) (p ! d))"
ringE_4: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y"
\<Gamma>_of_mds_def: fixes mds :: "Mode \<Rightarrow> 'Var set" shows "\<Gamma>_of_mds mds \<equiv> \<lambda>x. if x \<notin> \<C> \<and> x \<in> mds AsmNoWrite \<union> mds AsmNoReadOrWrite then if x \<in> mds AsmNoReadOrWrite then Some {pred_False} else Some (dma_type x) else None"
is_SNF_Smith_2xn: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a mat"   and n :: "nat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op"   and "A \<in> carrier_mat 2 n" shows "is_SNF A (Smith_Impl.Smith_2xn Smith_1x2 Smith_2x2 div_op A)"
classes_above_subset: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog" assumes "classes_above P C \<inter> classes_changed P P' = {}" shows "classes_above P C \<subseteq> classes_above P' C"
WHILET_eq_I_rwof: fixes m :: "'a nres"   and c :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a nres" shows "m \<bind> WHILE\<^sub>T c f = m \<bind> WHILE\<^sub>T\<^bsup>rwof m c f\<^esup> c f"
rl_thm: fixes p :: "fm"   and i :: "int"   and bs :: "real list"   and a :: "real" assumes "iszlfm p (real_of_int i # bs)" shows "(\<exists>x. Ifm (real_of_int x # bs) p) = ((\<exists>j\<in>{1..\<delta> p}. Ifm (real_of_int j # bs) (minusinf p)) \<or> (\<exists>(e, c)\<in>set (\<rho> p). \<exists>j\<in>{1..c * \<delta> p}. Ifm (a # bs) (\<sigma> p c (Add e (C j)))))"
chart_basis_domainP_chart_basis_inv: fixes x :: "'a" shows "chart_basis_domainP b (local.chart_basis_inv x)"
exp_ge_one_minus_x_over_n_power_n: fixes x :: "real"   and n :: "nat" assumes "x \<le> real n"   and "0 < n" shows "(1 - x / real n) ^ n \<le> exp (- x)"
reparam_weak_eq_line_integrals_2: fixes \<gamma>1 :: "real \<Rightarrow> 'a"   and \<gamma>2 :: "real \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'a"   and b :: "'a" assumes "reparam_weak \<gamma>1 \<gamma>2"   and "\<gamma>2 C1_differentiable_on {0..1}"   and "continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)" shows "line_integral_exists F {b} \<gamma>1"
E_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b" shows "Graph.E c \<equiv> {(u, v). c (u, v) \<noteq> (0::'b)}"
module_hom_iff: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e" shows "module_hom s1 s2 f = (module s1 \<and> module s2 \<and> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (s1 c x) = s2 c (f x)))"
antisym_order_consistent_self: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "antisymp r" shows "order_consistent r r"
interchange_rows_preserves: fixes i :: "'d"   and a :: "'d"   and j :: "'d"   and A :: "(('e, 'f) vec, 'd) vec" assumes "i \<noteq> a"   and "j \<noteq> a" shows "interchange_rows A i j $ a = A $ a"
mat_t_mult_sm: fixes k :: "complex"   and A :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_transpose (k *\<^sub>s\<^sub>m A) = k *\<^sub>s\<^sub>m mat_transpose A"
abs_trieP_insertP: fixes ks :: "bool list"   and t :: "trieP" shows "abs_trieP (insertP ks t) = insert ks (abs_trieP t)"
mono_C: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "C i \<subseteq> C j"
interval_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" shows "interval r a b \<equiv> {x. (a, x) \<in> r \<and> (x, b) \<in> r}"
\<P>_k: fixes k :: "'key" assumes "k \<in> keys" shows "prob (fst -` {k} \<inter> msgs) = K k"
mem_box_2: fixes x :: "'c"   and a :: "'c"   and b :: "'c" shows "(x \<in> cbox a b) = (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
card_order_on_well_order_on: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "card_order_on A r" shows "well_order_on A r"
PreLoebian_def: fixes x :: "'a" shows "PreLoebian x = (\<forall>y. d y \<le>  |x\<^sup>\<star>\<rangle> \<Omega> x y)"
cp0: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'a"   and Y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'b"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "f X Y \<tau> = f (\<lambda>_. X \<tau>) (\<lambda>_. Y \<tau>) \<tau>"
rGamma_real_def: fixes x :: "real" shows "rGamma x = Re (rGamma (complex_of_real x))"
uminus_carrier_iff_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" shows "(- A \<in> carrier_mat nr nc) = (A \<in> carrier_mat nr nc)"
suffix_imp_subseq: fixes xs :: "'b list"   and ys :: "'b list" assumes "suffix xs ys" shows "subseq xs ys"
absch: shows "\<forall>x\<in>set_pmf (config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) (take n qs)). (case x of (s, is) \<Rightarrow> real (Suc (index s (qs ! n)))) \<le> real (case x of (is, s) \<Rightarrow> Suc (length init))"
right_total_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "right_total R = (\<forall>y. \<exists>x. R x y)"
butlast_append_Cons: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "butlast (xs @ y # ys) = xs @ butlast (y # ys)"
nth_map_upt: fixes i :: "nat"   and n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "i < n - m" shows "list.map f [m..<n] ! i = f (m + i)"
fv_def: fixes e :: "'c" shows "fv e = {v. atom v \<in> supp e}"
brouwer_surjective_cball: fixes a :: "'a"   and e :: "real"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and S :: "'a set" assumes "continuous_on (cball a e) f"   and "0 < e"   and "x \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> cball a e\<rbrakk> \<Longrightarrow> x + (y - f y) \<in> cball a e" shows "\<exists>y\<in>cball a e. f y = x"
WT_converter_id: fixes \<I> :: "('a, 'b) \<I>" shows "\<I>, \<I> \<turnstile>\<^sub>C 1\<^sub>C \<surd>"
fmlookup_restrict_set: fixes A :: "'b set"   and m :: "('b, 'a) fmap"   and x :: "'b" shows "fmlookup (fmrestrict_set A m) x = (if x \<in> A then fmlookup m x else None)"
matrix_mul_lid: fixes A :: "(('c, 'd) vec, 'e) vec" shows "mat (1::'c) ** A = A"
fdia_rel_im1: fixes R :: "('b \<times> 'b) set"   and P :: "'b set" shows "s2r (R\<inverse> `` P) = relfdia R (s2r P)"
surj_on_respects_range_permutation: fixes A :: "'c set"   and B :: "'d set" shows "(\<lambda>f. f ` A = B) respects range_permutation A B"
image_Un: fixes f :: "'q \<Rightarrow> 'p"   and A :: "'q set"   and B :: "'q set" shows "f ` (A \<union> B) = f ` A \<union> f ` B"
assms_2: fixes K :: "'a list set"   and L :: "'a list set" assumes "K \<sim> L" shows "([] \<in> K) = ([] \<in> L)"
P_ends_on_deadend: fixes n :: "nat" assumes "enat n < llength P"   and "deadend (P $ n)" shows "enat (Suc n) = llength P"
substT_type: fixes D :: "binding list"   and X :: "tyvrs"   and Q :: "ty"   and G :: "binding list"   and t :: "trm"   and T :: "ty"   and P :: "ty" assumes "D @ TVarB X Q # G \<turnstile> t : T"   and "G\<turnstile>P<:Q" shows "D[X \<mapsto> P]\<^sub>e @ G \<turnstile> t[X \<mapsto>\<^sub>\<tau> P] : T[X \<mapsto> P]\<^sub>\<tau>"
strict_implies_not_eq: fixes b :: "'b list"   and a :: "'b list" assumes "strict_prefix b a" shows "a \<noteq> b"
distinct_1: fixes x11 :: "nat set"   and x12 :: "'d"   and x21 :: "nat"   and x22 :: "nat" shows "x11:*cast(x12) \<noteq> x21:deliver(x22)"
sub: fixes a :: "'a" assumes "a \<in> Normal" shows "a \<in> G"
distinct_replace_norm: fixes fs :: "face list"   and newFs :: "face list"   and oldF :: "face" assumes "distinct (normFaces fs)"   and "distinct (normFaces newFs)"   and "set (normFaces fs) \<inter> set (normFaces newFs) \<subseteq> {}" shows "distinct (normFaces (replace oldF newFs fs))"
Pair: shows "chophelper (to_list v p) [] = (a, b)"
pderiv_eval_deriv: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)"   and "a \<in> carrier Zp" shows "deriv f a = pderiv f \<bullet> a"
msg6_cardSK_disj: fixes A :: "agent"   and K :: "nat"   and c :: "msg"   and n :: "msg"   and k' :: "msg"   and cardSK :: "nat"   and X :: "msg"   and Y :: "msg"   and evs :: "event list" assumes "Gets A \<lbrace>Crypt K \<lbrace>c, n, k', Key cardSK, X\<rbrace>, Y\<rbrace> \<in> set evs"   and "evs \<in> set_cr" shows "cardSK \<notin> range (invKey \<circ> pubEK \<circ> CA) \<or> Key cardSK \<in> knows Spy evs"
module_on_def: fixes U\<^sub>M :: "'c set"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "module_on U\<^sub>M scale \<equiv> ((\<forall>x y a. x \<in> U\<^sub>M \<longrightarrow> y \<in> U\<^sub>M \<longrightarrow> scale a (x + y) = scale a x + scale a y) \<and> (\<forall>x a b. x \<in> U\<^sub>M \<longrightarrow> scale (a + b) x = scale a x + scale b x) \<and> (\<forall>x a b. x \<in> U\<^sub>M \<longrightarrow> scale a (scale b x) = scale (a * b) x)) \<and> ((\<forall>x. x \<in> U\<^sub>M \<longrightarrow> scale (1::'d) x = x) \<and> (\<forall>x y. x \<in> U\<^sub>M \<longrightarrow> y \<in> U\<^sub>M \<longrightarrow> x + y \<in> U\<^sub>M)) \<and> (0::'c) \<in> U\<^sub>M \<and> (\<forall>x a. x \<in> U\<^sub>M \<longrightarrow> scale a x \<in> U\<^sub>M)"
dfs_app: fixes g :: "(node \<times> node) list"   and xs :: "node list"   and ys :: "node list"   and zs :: "node list" shows "dfs g (xs @ ys) zs = dfs g ys (dfs g xs zs)"
single_valued_subcls1: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and G :: "'a prog" assumes "wf_prog wf_md G" shows "single_valuedp (subcls1 G)"
set_lebesgue_integral_cong_AE: fixes A :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "A \<in> sets M"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x\<in>A in M. f x = g x" shows "set_lebesgue_integral M A f = set_lebesgue_integral M A g"
list_nth_split_technical: fixes m :: "nat"   and cs :: "'b list" assumes "m < length cs"   and "cs \<noteq> []" shows "\<exists>xs ys. cs = xs @ cs ! m # ys"
intro: fixes return :: "int \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "(('c \<Rightarrow> int option) \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "('c \<Rightarrow> int option) \<Rightarrow> 'b \<Rightarrow> 'b"   and sample :: "int pmf \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b" assumes "memoization return bind get put"   and "monad_state_prob return bind get put sample"   and "monad_fail return bind fail" shows "prob_exp return bind fail get put sample"