hcju/isabelleps huggingface dataset:
encode_problem_parallel_complete_iii_a: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and C :: "sat_plan_variable literal set"   and \<A> :: "sat_plan_variable \<Rightarrow> bool" assumes "is_valid_problem_strips \<Pi>"   and "\<Pi>\<^sub>G \<subseteq>\<^sub>m execute_parallel_plan (\<Pi>\<^sub>I) \<pi>"   and "C \<in> cnf (encode_all_operator_preconditions \<Pi> (\<Pi>\<^sub>\<O>) (length \<pi>))"   and "\<forall>k op. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<longrightarrow> \<A> (Operator k (index (\<Pi>\<^sub>\<O>) op)) = (op \<in> set (\<pi> ! k))"   and "\<forall>l op. length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<le> l \<and> l < length \<pi> \<longrightarrow> \<not> \<A> (Operator l (index (\<Pi>\<^sub>\<O>) op))"   and "\<forall>v k. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<longrightarrow> \<A> (State k (index (\<Pi>\<^sub>\<V>) v)) = ((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v = Some True)" shows "clause_semantics \<A> C"
assms_5: fixes s :: "'a"   and t :: "'a"   and k :: "'k" shows "term_of_pair (s + t, k) = splus (term_of_pair (s, k)) (term_of_pair (t, k))"
root_equals_end_points: fixes r :: "'a"   and x :: "'a" assumes "backward_terminating_path_root r (x\<^sup>T)"   and "x \<noteq> 0" shows "r = end_points x"
thetaAtm_def: fixes atm :: "'b" shows "thetaAtm atm \<equiv> {(Atm atm, Atm atm)}"
as_1: shows "simple_linux_router_nol12 rt fw pi = Some a"
sup_assoc: fixes x :: "'k multiset"   and y :: "'k multiset"   and z :: "'k multiset" shows "x \<union># y \<union># z = x \<union># (y \<union># z)"
extremum_uniqueI: fixes a :: "'b" assumes "a \<le> 0" shows "a = 0"
intros_5: fixes x :: "real"   and y :: "real" assumes "x \<in> radical_sqrt"   and "y \<in> radical_sqrt" shows "x * y \<in> radical_sqrt"
R_O_Id: fixes R :: "('e \<times> 'f) set" shows "R O Id = R"
IH_3: assumes "linears (xs @ ys)" shows "fdisjnt (freess xs) (freess ys)"
2: shows "sum t_A {Suc n..<length rs} = int (T (s_A (Suc n)) (drop (Suc n) rs) (drop (Suc n) as))"
fds_converges_scaleR: fixes f :: "'b fds"   and s :: "'b"   and c :: "real" assumes "fds_converges f s" shows "fds_converges (c *\<^sub>R f) s"
c_d: shows "(c, d) \<in> thetaZOParT"
after_summary_empty_summary: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'c zmultiset" assumes "dataflow_topology summary results_in" shows "dataflow_topology.after_summary results_in M {}\<^sub>A = {#}\<^sub>z"
assms_6: shows "c \<turnstile> ev' \<mapsto> d'"
Sublists_altdef: fixes A :: "'a list set" shows "Sublists A = (\<Union>w\<in>A. set (sublists w))"
assms_1: shows "s \<in> reachable (closed (pnet np n)) TT"
intros_11: fixes \<tau> :: "'a OCL_Types.type" shows "OrderedSet \<tau> \<sqsubset> Collection \<tau>"
path_edge: fixes a :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a" shows "CFGExit_wf.path (src a) [a] (trg a)"
simps_1: fixes ref :: "'c"   and uu :: "'d" shows "list_op_deps (Insert (Some ref) uu) = {ref}"
AsWhileExpr: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and i :: "'e set"   and A :: "'e set"   and p :: "('e, 'd, 'f) ann"   and e :: "('e, 'd, 'f) com"   and a :: "'e set"   and Q :: "'e set"   and r :: "'e set"   and b :: "'e set" assumes "assertionsR \<Gamma> \<Theta> i A p e a" shows "assertionsR \<Gamma> \<Theta> Q A (AnnWhile r i p) (While b e) a"
lm04: fixes X :: "'c set"   and x :: "'c" assumes "trivial X"   and "{x} \<subseteq> X" shows "{x} = X"
Abs1_eq_iff_fresh_1: fixes c :: "'a"   and a :: "'a"   and b :: "'a"   and x :: "'b"   and y :: "'b" assumes "atom c \<sharp> (a, b, x, y)" shows "([{atom a}]set. x = [{atom b}]set. y) = ((a \<leftrightarrow> c) \<bullet> x = (b \<leftrightarrow> c) \<bullet> y)"
assms_2: fixes a :: "'a"   and b :: "'b" assumes "a \<in> A"   and "b \<in> B"   and "P a b" shows "f a = g b"
env_get_mal: fixes x :: "char list" shows "Malformed!x = None"
beukers_integral1_same: fixes s :: "nat"   and r :: "nat" assumes "s \<le> r"   and "r = s" shows "beukers_integral1 r s = 2 * (Re (zeta 3) - (\<Sum>k = 1..r. 1 / real (k ^ 3)))"
updS_dpm: fixes s :: "'a bdd_scheme"   and x :: "nat \<times> nat \<times> nat"   and r :: "nat" shows "dpm (updS s x r) = dpm s"
Rep_bit0: fixes x :: "'b bit0" shows "Rep_bit0 x \<in> {0..<2 * int CARD('b)}"
embedS_def: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "embedS r r' f \<equiv> embed r r' f \<and> \<not> bij_betw f (Field r) (Field r')"
simple_rotations_rotate_id: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations" shows "r \<circ> r \<circ> r = id"
triples_valid_Suc: fixes ts :: "'b triple set"   and n :: "nat" assumes "||=Suc n:ts" shows "||=n:ts"
kn: fixes k :: "nat" shows "(real k / real n \<le> f x) = (k \<le> m)"
diag_mat_mul_diag_ele: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "diagonal_mat A"   and "diagonal_mat B" shows "\<forall>i<n. (A * B) $$ (i, i) = A $$ (i, i) * B $$ (i, i)"
left_partI: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set"   and v :: "'e"   and u :: "'g"   and s :: "'g"   and t :: "'g" assumes "TreeDecomposition G T bag"   and "v \<in> bag u"   and "u \<in> Tree.left_tree T s t" shows "v \<in> TreeDecomposition.left_part T bag s t"
P'_def: fixes n :: "nat" shows "P' n = mapmeasure n (P (J n)) (\<lambda>_. borel)"
good_B: fixes i :: "nat" shows "B i \<subseteq> A \<and> B i \<in> sets M \<and> 0 < emeasure M (B i)"
inj_group_add_hom_axioms: shows "inj_group_add_hom hom"
DONE: fixes x :: "'a" assumes "r = DONE x" shows "thesis"
square_minus_one_cong_one': fixes n :: "nat" assumes "1 < n" shows "[(n - 1) * (n - 1) = 1] (mod n)"
rsp: shows "eq_onp (\<lambda>x. constant_term (denominator x) \<noteq> (0::'b)) 0 0"
simps_3: fixes h :: "'heap" shows "typeof\<^bsub>h\<^esub> Null = typeof Null"
split_list_last: fixes x :: "'g"   and xs :: "'g list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
factorize_rat_poly_generic_def: fixes f :: "rat poly" shows "local.factorize_rat_poly_generic f = (case rat_to_normalized_int_poly f of (c, g) \<Rightarrow> case factorize_int_poly_generic alg g of (d, fs) \<Rightarrow> (c * rat_of_int d, map (\<lambda>(fi, i). (of_int_poly fi, i)) fs))"
fct_gen_subalgebra_def: fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd" shows "fct_gen_subalgebra M N X = gen_subalgebra M (sigma_sets (space M) {X -` B \<inter> space M |B. B \<in> sets N})"
i_f_join_eq_i_join_i_take: fixes I :: "nat set"   and f :: "nat \<Rightarrow> 'b" assumes "finite I" shows "f \<Join>\<^sub> I = f \<Join>\<^sub> I \<Down> card I"
SUP_union: 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 bot :: "'a"   and top :: "'a"   and M :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (M ` (A \<union> B)) = sup (Sup (M ` A)) (Sup (M ` B))"
Zfun_sum: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "finite s"   and "\<And>i. i \<in> s \<Longrightarrow> Zfun (f i) F" shows "Zfun (\<lambda>x. \<Sum>i\<in>s. f i x) F"
cos_minus: fixes x :: "'a" shows "cos (- x) = cos x"
mult_monom: fixes a :: "'b"   and m :: "nat"   and b :: "'b"   and n :: "nat" shows "monom a m * monom b n = monom (a * b) (m + n)"
jipsen1l: fixes x :: "'a"   and y :: "'a" shows "x \<le> y \<leftarrow> (x \<rightarrow> y)"
1: shows "((\<lambda>x. \<Sum>b\<in>basis. f (g1 x) \<bullet> b * (vector_derivative g1 (at x within {0..1}) \<bullet> b)) has_integral i1) {0..1}"
assms_1: shows "diagram ars (\<tau>, \<sigma>, \<sigma>', \<tau>')"
nat_set_cset_bij_def: shows "nat_set_cset_bij = (SOME f. bij f)"
groupF_empty: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c list" shows "(groupF f xs \<noteq> []) = (xs \<noteq> [])"
vimage_algebra_sigma: fixes X :: "'d set set"   and \<Omega>' :: "'d set"   and f :: "'e \<Rightarrow> 'd"   and \<Omega> :: "'e set" assumes "X \<subseteq> Pow \<Omega>'"   and "f \<in> \<Omega> \<rightarrow> \<Omega>'" shows "vimage_algebra \<Omega> f (sigma \<Omega>' X) = sigma \<Omega> {f -` A \<inter> \<Omega> |A. A \<in> X}"
is_monoid: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "Group.monoid R"
nu_fusion_equal_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b" assumes "class.order less_eq less"   and "ord.galois less_eq l u"   and "ord.isotone less_eq g"   and "ord.isotone less_eq h"   and "order.has_greatest_postfixpoint less_eq g"   and "order.has_greatest_fixpoint less_eq h"   and "less_eq (h (u (l (order.the_greatest_fixpoint less_eq h)))) (u (g (l (order.the_greatest_fixpoint less_eq h))))"   and "h (u (order.the_greatest_postfixpoint less_eq g)) = u (g (order.the_greatest_postfixpoint less_eq g))" shows "order.the_greatest_fixpoint less_eq h = u (order.the_greatest_postfixpoint less_eq g) \<and> order.the_greatest_fixpoint less_eq h = u (order.the_greatest_fixpoint less_eq g)"
cross3_simps_21: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "(a - b \<le> c) = (a \<le> c + b)"
conf: shows "\<forall>i\<in>{0..<dim}. \<forall>j\<in>{0..<dim}. convergent (\<lambda>n. f n $$ (i, j))"
X_plus_closed: fixes a :: "'a" assumes "a \<in> carrier R" shows "X_poly_plus R a \<in> carrier P"
ord_eq_Suc_0_iff: fixes n :: "nat"   and x :: "nat" shows "(ord n x = Suc 0) = [x = 1] (mod n)"
dim_span_eq_card_independent: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'b set" assumes "vector_space scale"   and "\<not> module.dependent scale B" shows "vector_space.dim scale (module.span scale B) = card B"
comp_ring_morphisms: fixes \<eta> :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and addA :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multA :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zeroA :: "'c"   and oneA :: "'c"   and B :: "'d set"   and addB :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multB :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroB :: "'d"   and oneB :: "'d"   and \<theta> :: "'d \<Rightarrow> 'e"   and C :: "'e set"   and addC :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and multC :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zeroC :: "'e"   and oneC :: "'e" assumes "ring_homomorphism \<eta> A addA multA zeroA oneA B addB multB zeroB oneB"   and "ring_homomorphism \<theta> B addB multB zeroB oneB C addC multC zeroC oneC" shows "ring_homomorphism (\<theta> \<circ> \<eta> \<down> A) A addA multA zeroA oneA C addC multC zeroC oneC"
preserves_modes_mem_\<R>_i: shows "preserves_modes_mem (\<R>_rel (cms ! i))"
simps_1: fixes g :: "graph"   and f :: "face"   and u :: "nat"   and n :: "nat" shows "subdivFace' g f u n [] = makeFaceFinal f g"
inf_finite_5_def: fixes x :: "finite_5"   and y :: "finite_5" shows "x \<sqinter> y = (case (x, y) of (a\<^sub>1, xa) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, a\<^sub>5) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, _) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>4, a\<^sub>5) \<Rightarrow> a\<^sub>4 | (a\<^sub>4, _) \<Rightarrow> a\<^sub>1 | (a\<^sub>5, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>5, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>5, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>5, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>5, a\<^sub>5) \<Rightarrow> a\<^sub>5)"
obtain_comp_block_orig: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and bl1 :: "'b set" assumes "incidence_system point_set block_collection"   and "bl1 \<in># incidence_system.complement_blocks point_set block_collection" obtains bl2 :: "'b set" where "bl2 \<in># block_collection" and "bl1 = incidence_system.block_complement point_set bl2"
eq_elim_ax_def: shows "eq_elim_ax \<equiv> mk_eq Core.A Core.B \<longmapsto> Core.A \<longmapsto> Core.B"
hyps: fixes i :: "nat"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p', is', xs, sb @ sbx, \<D>', \<O>, \<R>')]) m = flush_all_until_volatile_write ts m"
maximal_exists_superset: fixes S :: "'b set"   and P :: "'b set \<Rightarrow> bool"   and B :: "'b set" assumes "finite S"   and "\<And>A. P A \<Longrightarrow> A \<subseteq> S"   and "P B" shows "\<exists>A. finite A \<and> maximal A P \<and> B \<subseteq> A"
HPhase0Read_HInv5_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase0Read s s' p d" shows "HInv5_inner s' p"
cond_spmf_fst_map_Pair1: fixes x :: "'a"   and p :: "'a spmf"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "x \<in> set_spmf p"   and "inj_on f (set_spmf p)" shows "cond_spmf_fst (map_spmf (\<lambda>x. (f x, g x)) p) (f x) = return_spmf (g (inv_into (set_spmf p) f (f x)))"
Sigma_Union: fixes X :: "'d set set"   and B :: "'d \<Rightarrow> 'e set" shows "Sigma (\<Union> X) B = (\<Union>A\<in>X. Sigma A B)"
types_preserved_set_local: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and v' :: "v"   and i :: "nat"   and ts :: "t list"   and ts' :: "t list"   and vi :: "v list"   and v :: "v"   and vs :: "v list" assumes "\<S>\<bullet>\<C> \<turnstile> [$C v', $Set_local i] : ts _> ts'"   and "length vi = i"   and "local \<C> = map typeof (vi @ [v] @ vs)" shows "\<S>\<bullet>\<C> \<turnstile> [] : ts _> ts' \<and> map typeof (vi @ [v] @ vs) = map typeof (vi @ [v'] @ vs)"
eexp_def: fixes x :: "ereal" shows "eexp x = enn2ereal (ennexp x)"
ordIso_transitive: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set"   and r'' :: "('d \<times> 'd) set" assumes "r =o r'"   and "r' =o r''" shows "r =o r''"
prv_imp_cnjL: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "bprv (imp (cnj \<phi> \<chi>) \<phi>)"
parts_UN: fixes H :: "'a \<Rightarrow> msg set"   and A :: "'a set" shows "parts (\<Union> (H ` A)) = (\<Union>x\<in>A. parts (H x))"
cases: fixes a :: "('Var, 'AExp, 'BExp) Stmt" assumes "no_await a" obtains x :: "'Var" and e :: "'AExp" where "a = x \<leftarrow> e"   | c1 :: "('Var, 'AExp, 'BExp) Stmt" and c2 :: "('Var, 'AExp, 'BExp) Stmt" where "a = c1__ ;; c2__" and "no_await c1__" and "no_await c2__"   | c1 :: "('Var, 'AExp, 'BExp) Stmt" and c2 :: "('Var, 'AExp, 'BExp) Stmt" and b :: "'BExp" where "a = Stmt.If b__ c1 c2" and "no_await c1" and "no_await c2"   | c :: "('Var, 'AExp, 'BExp) Stmt" and b :: "'BExp" where "a = While b cb__" and "no_await cb__"   | "a = Skip"   | "a = Stop"   | c :: "('Var, 'AExp, 'BExp) Stmt" and m :: "'Var ModeUpd" where "a = c@[m]" and "no_await c"
l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs \<equiv> l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs \<and> l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf"
null_eqI: fixes n :: "'a \<times> 'a" assumes "\<And>f. CC.comp n f = n \<and> CC.comp f n = n" shows "n = CC.null"
div_mult_mod_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (times (divide a b) b) (modulo a b) = a"
intruder_deduct_hom_DecomposeH: fixes M :: "('fun, 'var) Term.term set"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and T :: "('fun, 'var) Term.term list"   and t\<^sub>i :: "('fun, 'var) Term.term" assumes "\<langle>M;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m t"   and "Ana t = (K, T)"   and "\<And>k. k \<in> set K \<Longrightarrow> \<langle>M;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m k"   and "t\<^sub>i \<in> set T" shows "\<langle>M;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m t\<^sub>i"
fv_ik_is_fv_rcv: fixes S :: "('d, 'c) strand_step list" shows "fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>t S) = \<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S))"
that: fixes y :: "('idx, 'pred, 'act, 'effect) formula" assumes "y \<in> \<A>[F]"   and "supp y \<subseteq> supp (F, P)"   and "y distinguishes P from Q" shows "thesis"
strict_implies_not_eq: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat" assumes "lex_pm_strict b a" shows "a \<noteq> b"
frechet_derivative_at: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "(f has_derivative f') (at x)" shows "f' = frechet_derivative f (at x)"
subdiv_props_2: shows "chain_subdiv_chain one_chain2 subdiv"
sup_least_classes1: fixes c :: "classes1"   and e :: "classes1"   and d :: "classes1" assumes "c \<le> e"   and "d \<le> e" shows "c \<squnion> d \<le> e"
Max_singleton: fixes x :: "'t" shows "ord_term_lin.Max {x} = x"
sumf_nonneg: fixes x :: "'a" shows "0 \<le> (\<Sum>i\<in>{j \<in> N. x \<in> W j}. f i x)"
preserves_hseq: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu> \<and> C.arr \<nu> \<and> src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.arr (local.map \<mu>) \<and> D.arr (local.map \<nu>) \<and> src\<^sub>D (local.map \<mu>) = trg\<^sub>D (local.map \<nu>)"
integral_trans: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and f :: "('test, 'atom, 'choice) cmd \<times> 'state \<Rightarrow> real" assumes "proper (fst cf)" shows "measure_pmf.expectation (local.trans cf) f = (\<Sum>i<brn (fst cf). wt (fst cf) (snd cf) i * f (cont_eff cf i))"
assoc: fixes a :: "'f"   and b :: "'f"   and c :: "'f" shows "a + b + c = a + (b + c)"
subst_forget_ty: fixes X :: "W.tvar"   and T :: "ty"   and S :: "ty" assumes "X \<sharp> T" shows "T[X::=S] = T"
FAccRed: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e\<bullet>F{Cs},s\<rangle> \<rightarrow> \<langle>e'\<bullet>F{Cs},s'\<rangle>"
CastRedsThrow: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "char list exp"   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 C :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>throw a,s'\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>throw a,s'\<rangle>"
term_struct_rel_strong_axioms: shows "term_struct_rel_strong P"
pathsCard_def: fixes ps :: "Vars.var list set"   and x :: "Vars.var" shows "pathsCard ps x = (if \<forall>p\<in>ps. no_call_in_path x p then none else if \<forall>p\<in>ps. one_call_in_path x p then once else many)"
simps: fixes t1 :: "'b option list"   and s :: "nat set"   and t2 :: "'b option list" shows "New_max.isSameIntersection t1 s t2 = (\<forall>i\<in>s. t1 ! i = t2 ! i)"
combine_common_factor: fixes a :: "'a"   and e :: "'a"   and b :: "'a"   and c :: "'a" shows "a * e \<squnion> (b * e \<squnion> c) = (a \<squnion> b) * e \<squnion> c"
MGF_def: fixes P :: "state assn"   and t :: "Term.term"   and G :: "prog" shows "{P} t\<succ> {G\<rightarrow>} = {P} t\<succ> {\<lambda>Y s' s. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y, s')}"
nd_fun_leq: fixes x :: "'a nd_fun"   and y :: "'a nd_fun" shows "(x \<le> y) = (x + y = y)"
simps_7: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and \<phi> :: "'b formula" assumes "formula \<Sigma>" shows "formula.satisfies \<Sigma> (w, I) (FExists \<phi>) = (\<exists>p. formula.satisfies \<Sigma> (w, Inl p # I) \<phi>)"
\<tau>exec_move_conv_\<tau>exec_meth: fixes addr2thread_id :: "'b \<Rightarrow> 'c"   and thread_id2addr :: "'c \<Rightarrow> 'b"   and spurious_wakeups :: "bool"   and empty_heap :: "'a"   and allocate :: "'a \<Rightarrow> htype \<Rightarrow> ('a \<times> 'b) set"   and typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and heap_read :: "'a \<Rightarrow> 'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> bool"   and heap_write :: "'a \<Rightarrow> 'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> 'a \<Rightarrow> bool"   and ci :: "('b, 'a) check_instr"   and P :: "'b J1_prog"   and t :: "'c"   and e :: "'b expr1" shows "JVM_heap_base.\<tau>exec_move addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t e = JVM_heap_base.\<tau>exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci (compP2 P) (compE2 e) (compxE2 e 0 0) t"
simps_2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and ys :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (x # ys) = (Ball (set ys) (less_eq x) \<and> linorder.sorted less_eq ys)"
vbrelation_Limit_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<AA>\<lparr>Arr\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<FF>\<lparr>ArrMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
cs_def: fixes a :: "'a"   and b :: "'a" shows "a \<down>\<^sup>* b \<equiv> \<exists>s. a \<rightarrow>\<^sup>* s \<and> b \<rightarrow>\<^sup>* s"