hcju/isabelleps huggingface dataset:
assms: fixes \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P" shows "\<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel"
dense_sup_p: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus (sup x (uminus x)) = bot"
cr_uint_def: shows "cr_uint \<equiv> \<lambda>x y. x = Rep_uint y"
\<mu>'_spec: shows "\<forall>x\<in>generated_ring. \<exists>C\<subseteq>M. x = \<Union> C \<and> finite C \<and> disjoint C \<and> \<mu>' x = sum \<mu> C"
DomainpE: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c" assumes "Domainp r a" obtains b :: "'d" where "r a b"
make_invar_thm: fixes I :: "'a tarjan_state \<Rightarrow> bool" assumes "is_invar I" shows "I s"
forallM_index_def: fixes P :: "'a \<Rightarrow> nat \<Rightarrow> 'b + unit"   and xs :: "'a list" shows "forallM_index P xs = ??.Error_Monad.forallM_index_aux P 0 xs"
assms_4: shows "S2 \<sqsubseteq> S1"
IH: shows "foldl (+) (0::'a) (map f xs) = foldl (+) (0::'a) (map f (filter P xs)) + foldl (+) (0::'a) (map f (filter (\<lambda>x. \<not> P x) xs))"
ev: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'" shows "Shallow.ev \<phi> \<pi>l = (\<exists>i. \<phi> (map (sdrop i) \<pi>l))"
hyps_3: shows "\<forall>p\<in>set (fst r). p \<in> derivable' R \<and> (\<exists>n. (p, n) \<in> derivable R)"
perm_is_bij: fixes f :: "atom \<Rightarrow> atom" assumes "f \<in> perm" shows "bij f"
cs_length_one_1: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "IFC prob"   and "length (IFC_def.cs prob \<pi> i) = 1" shows "IFC_def.cs prob \<pi> i = [\<pi> i]"
Ks_4: shows "set Ks \<noteq> {\<zero>}"
coeff_replace_coeff: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" assumes "f (0::'c) = (0::'b)" shows "MPoly_Type.coeff (replace_coeff f p) m = f (MPoly_Type.coeff p m)"
cases: fixes x :: "'a poly" obtains a :: "'a" and p :: "'a poly" where "x = pCons a p"
dictatorI: fixes F :: "(indi \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> (alt \<times> alt) set"   and i :: "indi" assumes "F \<in> SWF"   and "\<forall>P\<in>Prof. \<forall>a b. a \<noteq> b \<longrightarrow> a <\<^bsub>P i\<^esub> b \<longrightarrow> a <\<^bsub>F P\<^esub> b" shows "dictator F i"
exp_conj_unitary: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" assumes "unitary P"   and "unitary Q" shows "unitary (P && Q)"
starfun2_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "*f2* f \<equiv> \<lambda>x. Ifun (star_of f \<star> x)"
cases: fixes x :: "'a trm" obtains x :: "Preliminaries.var" where "xa__ = Var x"   | f :: "'a" and Tl :: "'a trm list" where "xa__ = Fn f Tl"
b2i: shows "b2 \<inter> path_image (p +++ linepath a b) = {}"
refl_rtrancl: fixes r :: "('a \<times> 'a) set" shows "refl (r\<^sup>*)"
measure_subadditive: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> sets M"   and "B \<in> sets M"   and "emeasure M A \<noteq> \<infinity>"   and "emeasure M B \<noteq> \<infinity>" shows "Sigma_Algebra.measure M (A \<union> B) \<le> Sigma_Algebra.measure M A + Sigma_Algebra.measure M B"
sscanl_ww_fusion': fixes wrap :: "'b \<rightarrow> 'a"   and unwrap :: "'a \<rightarrow> 'b"   and f :: "'a \<rightarrow> 'c \<rightarrow> 'a"   and f' :: "'b \<rightarrow> 'c \<rightarrow> 'b"   and z :: "'a"   and xs :: "[:'c:]" assumes "wrap oo unwrap = ID"   and "\<And>z x. x \<noteq> \<bottom> \<Longrightarrow> unwrap\<cdot>(f\<cdot>(wrap\<cdot>z)\<cdot>x) = f'\<cdot>(unwrap\<cdot>(wrap\<cdot>z))\<cdot>x" shows "sscanl\<cdot>f\<cdot>z\<cdot>xs = smap\<cdot>wrap\<cdot>(sscanl\<cdot>f'\<cdot>(unwrap\<cdot>z)\<cdot>xs)"
le_Liminf_iff: fixes C :: "'b"   and F :: "'d filter"   and X :: "'d \<Rightarrow> 'b" shows "(C \<le> Liminf F X) = (\<forall>y<C. \<forall>\<^sub>F x in F. y < X x)"
dvd_imp_degree_le: fixes p :: "'b poly"   and q :: "'b poly" assumes "p dvd q"   and "q \<noteq> 0" shows "degree p \<le> degree q"
nonzero_Reals_inverse: fixes a :: "'a" assumes "a \<in> \<real>"   and "a \<noteq> (0::'a)" shows "inverse a \<in> \<real>"
inf_RETURN_SPEC_1: fixes x :: "'a"   and \<Phi> :: "'a \<Rightarrow> bool" shows "inf (RETURN x) (SPEC \<Phi>) = SPEC (\<lambda>y. y = x \<and> \<Phi> x)"
cont2mono: fixes f :: "'b \<Rightarrow> 'c" assumes "cont f" shows "monofun f"
main: fixes k :: "nat" shows "local.norm_bound (A ^\<^sub>m k) (bP * real n * bQ * real n * c1 + bP * real n * bQ * real n * real k ^ (N - 1))"
tor: shows "(\<sigma>, t) \<in> oreachable (opnet onp p\<^sub>2) (otherwith S (net_tree_ips p\<^sub>2) (oarrivemsg I)) (other U (net_tree_ips p\<^sub>2))"
M\<delta>f: shows "(\<lambda>(x, y). \<delta>f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M R)"
simps_12: fixes x51 :: "bexp"   and x52 :: "com" shows "WHILE x51 DO x52 \<noteq> SKIP"
dickson_less_vI: fixes d :: "'a \<Rightarrow> nat"   and v :: "'t"   and m :: "nat"   and u :: "'t" assumes "d (pp_of_term v) \<le> m"   and "d (pp_of_term u) \<le> m"   and "v \<prec>\<^sub>t u" shows "dickson_less_v d m v u"
l10_14: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and P' :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "P \<noteq> P'"   and "A \<noteq> B"   and "Tarski_neutral_dimensionless.Reflect Bet Cong P P' A B" shows "Tarski_neutral_dimensionless.TS Bet A B P P'"
h0: shows "\<forall>i<length (take_indices subsets S). row (alt_matrix_A signs (take_indices subsets S)) i = row (take_rows_from_matrix (alt_matrix_A signs subsets) S) i"
associatedI: fixes a :: "'b"   and b :: "'b" assumes "a dvd b"   and "b dvd a" shows "normalize a = normalize b"
Int_Diff_Un: fixes A :: "'c set"   and B :: "'c set" shows "A \<inter> B \<union> (A - B) = A"
Inv_preserves_Can_3: fixes t :: "'c local.term" assumes "Can t" shows "Cod (Inv t) = Dom t"
exhaust_sel: fixes pr_operation :: "pr_operation" obtains "pr_operation = PUSH"   | "pr_operation = RELABEL"
length_rows: fixes A :: "'a Matrix.mat" shows "length (rows A) = dim_row A"
same_limit_imp_Gromov_product_tendsto_infinity: fixes z :: "'b Gromov_completion"   and u :: "nat \<Rightarrow> 'b"   and v :: "nat \<Rightarrow> 'b"   and C :: "real"   and a :: "'b" assumes "z \<in> Gromov_boundary"   and "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> z"   and "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> z" shows "\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. C \<le> Gromov_product_at a (u n) (v m)"
mult_right: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "(\<lambda>x. f x * h x) \<in> \<Theta>[F](\<lambda>x. g x * h x)"
f0: assumes "infinite (deriv s)" shows "f 0 \<in> deriv s \<and> fst (f 0) = 0 \<and> infinite (deriv (snd (f 0))) \<and> \<not> is_axiom (s_of_ns (snd (f 0)))"
typeI_two_cubes_trace_horizontal_boundaries: shows "two_chain_horizontal_boundary two_chain_typeI \<subseteq> \<gamma>"
fmapU_writer_def: shows "fmapU \<equiv> writer_map\<cdot>ID"
elims_3: fixes x :: "'a ifex" assumes "\<not> ifex_ordered x" obtains v :: "'a" and t :: "'a ifex" and e :: "'a ifex" where "x = IF v t e" and "\<not> (Ball (ifex_var_set t \<union> ifex_var_set e) ((<) v) \<and> ifex_ordered t \<and> ifex_ordered e)"
simps_6: fixes \<Psi> :: "'b" shows "extractFrame (\<lbrace>\<Psi>\<rbrace>) = \<langle>\<epsilon>, \<Psi>\<rangle>"
sorted_on_Cons: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and xs :: "'a list" assumes "Linear_order r" shows "sorted_on r (x # xs) = (x \<in> Field r \<and> sorted_on r xs \<and> (\<forall>y\<in>set xs. (x, y) \<in> r))"
cmp_rfn1: fixes r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set" assumes "0 < r"   and "0 < s"   and "compseries G r f"   and "compseries G s g" shows "f (Suc 0) \<diamondop> f 0 \<inter> g 0 = carrier G"
next_plane0_finals_incr: fixes g' :: "graph"   and p :: "nat"   and g :: "graph"   and f :: "face" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "f \<in> set (finals g)" shows "f \<in> set (finals g')"
enableds_def: fixes p :: "'state" shows "enableds p \<equiv> {a. en a p}"
cases: fixes a1 :: "('a, 'b, 'c) subst"   and a2 :: "('a, 'c) ODE"   and a3 :: "('c + 'c) set" assumes "Oadmit a1 a2 a3" obtains \<sigma> :: "('a, 'b, 'c) subst" and c :: "'c" and U :: "('c + 'c) set" where "a1 = \<sigma>'__" and "a2 = OVar c" and "a3 = Ua__"   | \<sigma> :: "('a, 'b, 'c) subst" and \<theta> :: "('a, 'c) trm" and U :: "('c + 'c) set" and x :: "'c" where "a1 = \<sigma>''__" and "a2 = OSing x \<theta>" and "a3 = Ub__" and "TUadmit \<sigma>''__ \<theta> Ub__" and "TadmitF \<sigma>''__ \<theta>"   | \<sigma> :: "('a, 'b, 'c) subst" and ODE1 :: "('a, 'c) ODE" and U :: "('c + 'c) set" and ODE2 :: "('a, 'c) ODE" where "a1 = \<sigma>" and "a2 = OProd ODE1 ODE2" and "a3 = U" and "Oadmit \<sigma> ODE1 U" and "Oadmit \<sigma> ODE2 U" and "ODE_dom (Osubst ODE1 \<sigma>) \<inter> ODE_dom (Osubst ODE2 \<sigma>) = {}"
F: shows "P \<turnstile> C' sees F,Static:vT in D"
set_tree_ins_eq: fixes k :: "'a"   and t :: "('a \<times> 'b) tree"   and p :: "'b" assumes "k \<notin> keys t" shows "set_tree (ins k p t) = {(k, p)} \<union> set_tree t"
excess_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and v :: "nat" shows "excess f v \<equiv> sum f (incoming v) - sum f (outgoing v)"
fn_o_inv_fn_is_id: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "bij f" shows "f ^^ n \<circ> inv f ^^ n = (\<lambda>x. x)"
assms_1: shows "f \<in> borel_measurable (lebesgue_on S)"
dual_conjunctive: 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"   and "x \<in> conjunctive.conjunctive inf inf times" shows "dual x \<in> disjunctive.disjunctive sup sup times"
subst_domain_compose: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "subst_domain (\<sigma> \<circ>\<^sub>s \<tau>) \<subseteq> subst_domain \<sigma> \<union> subst_domain \<tau>"
that: fixes x :: "'c" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "thesis"
beukers_integral1_integrable: fixes r :: "nat"   and s :: "nat" shows "set_integrable lborel ({0<..<1} \<times> {0<..<1}) (\<lambda>(x, y). - ln (x * y) / (1 - x * y) * x ^ r * y ^ s)"
induct: fixes P :: "'b hp option \<Rightarrow> 'b hp option \<Rightarrow> bool"   and a0 :: "'b hp option"   and a1 :: "'b hp option" assumes "\<And>h. P h None"   and "\<And>v. P None (Some v)"   and "\<And>h1 h2. P (Some h1) (Some h2)" shows "P a0 a1"
take_set: fixes j :: "nat"   and xs :: "'c list"   and x :: "'c" assumes "j \<le> length xs" shows "x \<in> set (take j xs) \<equiv> \<exists>k<j. xs ! k = x"
sum_list_plus: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "class.comm_monoid_add plus zero"   and "length xs = length ys" shows "plus (monoid_add.sum_list plus zero xs) (monoid_add.sum_list plus zero ys) = monoid_add.sum_list plus zero (map2 plus xs ys)"
in_hhom_def: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and a :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and b :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.in_hhom \<mu> a b \<equiv> Span.arr \<mu> \<and> Span.src \<mu> = a \<and> Span.trg \<mu> = b"
subseqs_powset: fixes xs :: "'g list" shows "set ` set (subseqs xs) = Pow (set xs)"
deriv_consistency: assumes "infinite UNIV" shows "consistency {set G |G. \<not> G \<turnstile> \<bottom>}"
antisym_conv2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less"   and "less_eq x y" shows "(\<not> less x y) = (x = y)"
less_append: fixes s :: "'a list"   and t :: "'a list"   and a :: "'a list" assumes "s < t" shows "a @ s < a @ t"
simps_2: fixes c :: "color"   and lc :: "color"   and llc :: "('b, 'c) RBT_Impl.rbt"   and lk :: "'b"   and lv :: "'c"   and lrt :: "('b, 'c) RBT_Impl.rbt"   and k :: "'b"   and v :: "'c"   and rt :: "('b, 'c) RBT_Impl.rbt" shows "rbt_min_opt (Branch c (Branch lc llc lk lv lrt) k v rt) = rbt_min_opt (Branch lc llc lk lv lrt)"
right_unique_OO: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "right_unique A"   and "right_unique B" shows "right_unique (A OO B)"
actn_lemma: fixes m :: "nat"   and q :: "nat"   and r :: "nat" shows "rec_exec rec_actn [m, q, r] = actn m q r"
tickFree_implies_front_tickFree: fixes s :: "'b event list" assumes "tickFree s" shows "front_tickFree s"
pred_fun_def: fixes A :: "'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> bool" shows "pred_fun A B = (\<lambda>f. \<forall>x. A x \<longrightarrow> B (f x))"
ge: fixes numnoabs :: "Ferrack.num \<Rightarrow> bool"   and t :: "Ferrack.num"   and bs :: "real list" assumes "numnoabs t" shows "Ifm bs (case rsplit0 t of (x, xa) \<Rightarrow> ge x xa) = Ifm bs (Ge t) \<and> isrlfm (case rsplit0 t of (x, xa) \<Rightarrow> ge x xa)"
cases: fixes a1 :: "'g"   and a2 :: "'node list"   and a3 :: "'var \<Rightarrow> 'node option" assumes "a1 \<turnstile> a2\<Down>a3" obtains g :: "'g" and ns :: "'node list" where "a1 = g" and "a2 = ns" and "a3 = fold (step g) ns Map.empty" and "g \<turnstile> Entry g-ns\<rightarrow>last ns"
frechet_derivative_works_on: fixes f :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and x :: "'b" assumes "f differentiable_on X"   and "x \<in> X" shows "(f has_derivative frechet_derivative f (at x within X)) (at x within X)"
simps: fixes a1 :: "complex"   and b1 :: "complex"   and c1 :: "complex"   and d1 :: "complex"   and a2 :: "complex"   and b2 :: "complex"   and c2 :: "complex"   and d2 :: "complex" shows "(a1, b1, c1, d1) -\<^sub>m\<^sub>m (a2, b2, c2, d2) = (a1 - a2, b1 - b2, c1 - c2, d1 - d2)"
assms: shows "\<infinity>\<^sub>h \<notin> ocircline_set H"
has_contour_integral_bound_circlepath_strong: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and z :: "complex"   and r :: "real"   and k :: "complex set"   and B :: "real" assumes "(f has_contour_integral i) (circlepath z r)"   and "finite k"   and "0 \<le> B"   and "0 < r"   and "\<And>x. \<lbrakk>cmod (x - z) = r; x \<notin> k\<rbrakk> \<Longrightarrow> cmod (f x) \<le> B" shows "cmod i \<le> B * (2 * pi * r)"
rtlI: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option" assumes "ranked_tree_automaton TA A" shows "ta_lang TA \<in> regular_languages A"
acyclicI_order: fixes r :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'loc \<times> 't" assumes "\<And>a b. (a, b) \<in> r \<Longrightarrow> f b<\<^sub>pf a" shows "acyclic r"
PSimQ: fixes \<Psi>' :: "'b" shows "\<Psi>' \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"
map\<^sub>0_simps_3: fixes a :: "'d" assumes "D.obj a" shows "trg\<^sub>D (\<eta>\<^sub>0 a) = GF.map\<^sub>0 a"
usc_at_mem: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> 'c"   and x :: "nat \<Rightarrow> 'b"   and A :: "'c" assumes "usc_at x0 f"   and "x \<longlonglongrightarrow> x0"   and "(f \<circ> x) \<longlonglongrightarrow> A" shows "A \<le> f x0"
CallRedsStatic: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and vs :: "val list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and C :: "char list"   and fs :: "Objects.fields"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> char list exp"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>addr a,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1,b\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,s\<^sub>2,b\<^sub>2\<rangle>"   and "hp s\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>e\<bullet>M(es),s,b\<rangle> \<rightarrow>* \<langle>THROW IncompatibleClassChangeError,s\<^sub>2,b\<^sub>2\<rangle>"
En_def: fixes \<rho> :: "'e V_rec \<Rightarrow> 'e set"   and s :: "'s"   and e :: "'e" shows "En \<rho> s e \<equiv> \<exists>\<beta> \<gamma>. \<exists>s'\<in>S\<^bsub>SES\<^esub>. \<exists>s''\<in>S\<^bsub>SES\<^esub>. s0\<^bsub>SES\<^esub> \<beta>\<Longrightarrow>\<^bsub>SES\<^esub> s \<and> \<gamma> \<upharpoonleft> \<rho> \<V> = \<beta> \<upharpoonleft> \<rho> \<V> \<and> s0\<^bsub>SES\<^esub> \<gamma>\<Longrightarrow>\<^bsub>SES\<^esub> s' \<and> s' e\<longrightarrow>\<^bsub>SES\<^esub> s''"
indep_var_rv2: fixes S :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and T :: "'b measure"   and Y :: "'a \<Rightarrow> 'b" assumes "indep_var S X T Y" shows "random_variable T Y"
bot_acom: fixes c' :: "'a option acom"   and c :: "com" assumes "strip c' = c" shows "\<bottom>\<^sub>c c \<sqsubseteq> c'"
in_set_cyc_permutations_obtain: fixes xs :: "'b list"   and ys :: "'b list" assumes "distinct xs"   and "distinct ys"   and "set xs = set ys" obtains n :: "nat" where "rotate n ys \<in> set (cyc_permutations xs)"
set_add_atoms: fixes r :: "nat rexp"   and as :: "nat list" shows "set (add_atoms r as) = atoms r \<union> set as"
pvalid_next_terminal_admissible: fixes p :: "(symbol \<times> character list) list"   and x :: "item"   and t :: "symbol"   and c :: "character list" assumes "pvalid p x"   and "next_symbol x = Some t"   and "is_terminal t" shows "admissible (p @ [(t, c)])"
induct: fixes x1 :: "(nat \<times> ty) list"   and x2 :: "exp"   and x3 :: "ty"   and P :: "(nat \<times> ty) list \<Rightarrow> exp \<Rightarrow> ty \<Rightarrow> bool" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> x T. lookup \<Gamma> x = Some T \<Longrightarrow> P \<Gamma> (EVar x) T"   and "\<And>\<Gamma> n. P \<Gamma> (ENat n) (TNat n)"   and "\<And>x A \<Gamma> e B. \<lbrakk>(x, A) # \<Gamma> \<turnstile> e : B; P ((x, A) # \<Gamma>) e B\<rbrakk> \<Longrightarrow> P \<Gamma> (ELam x e) (TFun (A \<rightarrow> B))"   and "\<And>\<Gamma> e1 A B e2. \<lbrakk>\<Gamma> \<turnstile> e1 : TFun (A \<rightarrow> B); P \<Gamma> e1 (TFun (A \<rightarrow> B)); \<Gamma> \<turnstile> e2 : A; P \<Gamma> e2 A\<rbrakk> \<Longrightarrow> P \<Gamma> (EApp e1 e2) B"   and "\<And>\<Gamma> x e. P \<Gamma> (ELam x e) (TFun \<top>)"   and "\<And>\<Gamma> x e A B. \<lbrakk>\<Gamma> \<turnstile> ELam x e : TFun A; P \<Gamma> (ELam x e) (TFun A); \<Gamma> \<turnstile> ELam x e : TFun B; P \<Gamma> (ELam x e) (TFun B)\<rbrakk> \<Longrightarrow> P \<Gamma> (ELam x e) (TFun (A \<sqinter> B))"   and "\<And>\<Gamma> e A B. \<lbrakk>\<Gamma> \<turnstile> e : A; P \<Gamma> e A; A <: B\<rbrakk> \<Longrightarrow> P \<Gamma> e B"   and "\<And>\<Gamma> e1 n1 e2 n2 f. \<lbrakk>\<Gamma> \<turnstile> e1 : TNat n1; P \<Gamma> e1 (TNat n1); \<Gamma> \<turnstile> e2 : TNat n2; P \<Gamma> e2 (TNat n2)\<rbrakk> \<Longrightarrow> P \<Gamma> (EPrim f e1 e2) (TNat (f n1 n2))"   and "\<And>\<Gamma> e1 e3 B e2. \<lbrakk>\<Gamma> \<turnstile> e1 : TNat 0; P \<Gamma> e1 (TNat 0); \<Gamma> \<turnstile> e3 : B; P \<Gamma> e3 B\<rbrakk> \<Longrightarrow> P \<Gamma> (EIf e1 e2 e3) B"   and "\<And>\<Gamma> e1 n e2 B e3. \<lbrakk>\<Gamma> \<turnstile> e1 : TNat n; P \<Gamma> e1 (TNat n); n \<noteq> 0; \<Gamma> \<turnstile> e2 : B; P \<Gamma> e2 B\<rbrakk> \<Longrightarrow> P \<Gamma> (EIf e1 e2 e3) B" shows "P x1 x2 x3"
bres_anti: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "less_eq_prod x y" shows "less_eq_prod (prod.bres y z) (prod.bres x z)"
csols_def: fixes t0 :: "real"   and x0 :: "'a" shows "csols t0 x0 = {(x, t1). {t0--t1} \<subseteq> T \<and> x t0 = x0 \<and> (x solves_ode f) {t0--t1} X}"
le_cSUP_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq (f ` A)" shows "less_eq x (Sup (f ` A)) = (\<forall>y. less y x \<longrightarrow> (\<exists>i\<in>A. less y (f i)))"
mksynlr_Inf: fixes A :: "('a, 'b) synlr set" shows "unsynlr (Inf A) = \<Inter> (unsynlr ` A)"
substT_Var: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and x :: "'a"   and y :: "'a"   and t :: "'b" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"   and "x \<in> var"   and "y \<in> var"   and "t \<in> trm" shows "substT (Var x) t y = (if x = y then t else Var x)"
Pd_linear': fixes d :: "'b" shows "Linear_order (Pd d)"
lexordp_conv_lexord: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder less_eq less" shows "ord.lexordp less xs ys = ((xs, ys) \<in> lexord {(x, y). less x y})"
pick_proj: fixes P :: "'a \<Rightarrow> bool"   and \<phi> :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "equivP P \<phi>"   and "P x" shows "\<phi> (pick (proj \<phi> x)) x"
set_borel_integral_eq_integral_2: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "set_integrable lborel S f" shows "set_lebesgue_integral lborel S f = integral S f"
known_ptrs_known_ptr: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "known_ptr ptr"
Null_not_in_Obj: shows "Null \<notin> Obj"
simps_2: fixes i :: "nat"   and l :: "nat" shows "gso_int i (Suc l) = (let v = \<mu>' l l \<cdot>\<^sub>v gso_int i l - \<mu>' i l \<cdot>\<^sub>v gso_int l l in if l = 0 then v else map_vec (\<lambda>k. k div \<mu>' (l - 1) (l - 1)) v)"
that: fixes k :: "nat" assumes "u = v \<^sup>@ k" shows "thesis"
a_evalF_decomp: shows "\<aF> = fst (sum_to_tup\<cdot>(\<mu> f. \<Lambda> x. \<Union> (Rep_cfun f ` abs_R\<cdot>x) \<union> abs_g\<cdot>x))"
le_diff_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a (minus c b) = less_eq (plus a b) c"