hcju/isabelleps huggingface dataset:
interchange_rows_i: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'f" shows "interchange_rows A i j $ i = A $ j"
next'_inv: fixes c0 :: "('loc, 't) configuration"   and c1 :: "('loc, 't) configuration"   and P :: "('loc, 't) configuration \<Rightarrow> bool" assumes "next' c0 c1"   and "P c0"   and "\<And>loc t n. \<lbrakk>P c0; next_change_multiplicity' c0 c1 loc t n\<rbrakk> \<Longrightarrow> P c1"   and "\<And>loc t. \<lbrakk>P c0; next_propagate' c0 c1 loc t\<rbrakk> \<Longrightarrow> P c1" shows "P c1"
funion_fempty_right: fixes A :: "'x fset" shows "A |\<union>| {||} = A"
loc_upb_main: fixes n :: "nat"   and x :: "nat" shows "c_assoc_have_key (pr_gr (loc_upb n x)) (c_pair n x) = 0"
disjnt_insert1: fixes a :: "'a"   and X :: "'a set"   and Y :: "'a set" shows "disjnt (insert a X) Y = (a \<notin> Y \<and> disjnt X Y)"
length_lesss_le_sorted: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list"   and i :: "nat" assumes "Linorder_Relations.sorted_wrt R xs"   and "i < length xs" shows "length (lesss R (xs ! i) xs) \<le> i"
to_tree_order_child: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and parent :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "h \<turnstile> Shadow_DOM.get_child_nodes parent \<rightarrow>\<^sub>r children"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<noteq> ptr"   and "child \<in> set children"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set nodes" shows "parent \<in> set nodes"
UNIV_I: fixes x :: "'f" shows "x \<in> UNIV"
impl_of_delete: fixes ks :: "'c list"   and t :: "('c, 'd) trie" shows "impl_of (delete ks t) = delete_trie ks (impl_of t)"
node_local_nos_inc: fixes i :: "nat"   and R\<^sub>i :: "nat set" shows "\<langle>i : optoy i \<langle>\<langle>\<^bsub>i\<^esub> qmsg : R\<^sub>i\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg (\<lambda>_ _. True) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). nos_inc (\<sigma> i) (\<sigma>' i))"
induct: fixes P :: "('c \<times> 'd \<times> bool) list \<Rightarrow> 'c \<times> 'd \<times> bool \<Rightarrow> 'c option \<Rightarrow> bool"   and a0 :: "('c \<times> 'd \<times> bool) list"   and a1 :: "'c \<times> 'd \<times> bool"   and a2 :: "'c option" assumes "\<And>xs e. P xs e None"   and "\<And>e i. P [] e (Some i)"   and "\<And>x xs e i. (fst x \<noteq> i \<Longrightarrow> P xs e (Some i)) \<Longrightarrow> P (x # xs) e (Some i)" shows "P a0 a1 a2"
assms_1: shows "A \<equiv> MS (MSLCons (C, e) xs) f"
simps: fixes a1 :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and a2 :: "'e"   and a3 :: "'f \<Rightarrow> 'g"   and a4 :: "'g"   and a5 :: "'e"   and a6 :: "'f \<Rightarrow> 'g" shows "a1 \<turnstile> \<langle>a2, a3\<rangle> \<rightarrow>\<^bsup>a4\<^esup> \<langle>a5, a6\<rangle> = (\<exists>u A l d. a1 = A \<and> a2 = l \<and> a3 = u \<and> a4 = d \<and> a5 = l \<and> a6 = u \<oplus> d \<and> u \<turnstile> inv_of A l \<and> u \<oplus> d \<turnstile> inv_of A l \<and> (0::'g) \<le> d)"
commute: fixes a :: "'c"   and b :: "'c" shows "lcm a b = lcm b a"
mset_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "mset (map f xs) = image_mset f (mset xs)"
finite_g_out: fixes A :: "'v set" shows "\<integral>\<^sup>+y\<in>A. g (x, y)\<partial>count_space UNIV \<noteq> \<top>"
csqrt_square: fixes b :: "complex" assumes "0 < Re b \<or> Re b = 0 \<and> 0 \<le> Im b" shows "csqrt (b\<^sup>2) = b"
fls_of_fpxs_uminus: fixes f :: "'b fpxs" shows "fls_of_fpxs (- f) = - fls_of_fpxs f"
vars2'_def: shows "st2.vars2' = ind_in_set st2.vars0 ` vars1"
dim_kernel_non_zero_jordan_block_pow: fixes a :: "'a"   and n :: "nat"   and k :: "nat" assumes "a \<noteq> (0::'a)" shows "vectorspace.dim class_ring (module_vec TYPE('a) n\<lparr>carrier := mat_kernel (jordan_block n a ^\<^sub>m k)\<rparr>) = 0"
assms: shows "eucl_of_list (take DIM('a) xs) = x"
mat_both_ordered_semiring_def: fixes n :: "nat"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'c" shows "mat_both_ordered_semiring n gt b \<equiv> ring_mat TYPE('b) n \<lparr>ordered_semiring.geq = (\<ge>\<^sub>m), gt = mat_comp_all gt, max = max\<^sub>m, \<dots> = b\<rparr>"
less_sets_imp_sorted_list_of_set: fixes A :: "'a set"   and B :: "'a set" assumes "A \<lless> B"   and "finite A"   and "finite B" shows "list_of A < list_of B"
sifum_lang_no_dma_axioms: shows "sifum_lang_no_dma ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars"
tendsto_norm: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'d"   and F :: "'c filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
sum_flatten2: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite A"   and "finite B" shows "\<Sum> {\<Sum> {f x y |x. x \<in> A} |y. y \<in> B} = \<Sum> {f x y |x y. x \<in> A \<and> y \<in> B}"
inner_minus_left: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and x :: "'e"   and y :: "'e" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (uminus x) y = - inner x y"
Compositionality_def: fixes ES1 :: "'b ES_rec"   and ES2 :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>1 :: "'b V_rec"   and \<V>2 :: "'b V_rec" shows "Compositionality ES1 ES2 \<V> \<V>1 \<V>2 \<equiv> ((ES_valid ES1 \<and> ES_valid ES2) \<and> composable ES1 ES2 \<and> isViewOn \<V> E\<^bsub>ES1 \<parallel> ES2\<^esub>) \<and> (isViewOn \<V>1 E\<^bsub>ES1\<^esub> \<and> isViewOn \<V>2 E\<^bsub>ES2\<^esub>) \<and> properSeparationOfViews ES1 ES2 \<V> \<V>1 \<V>2 \<and> wellBehavedComposition ES1 ES2 \<V> \<V>1 \<V>2"
NewA: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and T :: "ty"   and Q :: "'b assn"   and e :: "expr"   and R :: "'b assn" assumes "G,A\<turnstile>{Normal P} .init_comp_ty T. {Q}"   and "G,A\<turnstile>{Q} e-\<succ> {\<lambda>Val:i:. abupd (check_neg i) .; Alloc G (Arr T (the_Intg i)) R}" shows "G,A\<turnstile>{Normal P} New T[e]-\<succ> {R}"
shift_bounds_cl_Suc_ivl: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "prod g {Suc m..Suc n} = (\<Prod>i = m..n. g (Suc i))"
lossless_S2: fixes m2 :: "'msg2"   and out2 :: "'out2" shows "lossless_spmf (S2 m2 out2)"
insert_Int_conv_if: fixes A :: "'a set"   and x :: "'a"   and B :: "'a set" shows "A \<inter> insert x B = (if x \<in> A then insert x (A \<inter> B) else A \<inter> B)"
sorted_drop: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and n :: "nat" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (drop n xs)"
1: shows "set (fst (Step A ([0..<l], fst A [0..<l]) (last [0..<l]))) = {0..<l}"
fset_eqI: fixes A :: "'d fset"   and B :: "'d fset" assumes "\<And>x. (x |\<in>| A) = (x |\<in>| B)" shows "A = B"
cil_contains_empty: fixes wa :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and wb :: "'d list" shows "([] \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb) = (wa = [] \<and> wb = [])"
eval_cases_6: fixes P :: "(char list list \<times> char list exp) prog"   and V :: "char list"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>V:=e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains v :: "val" and h :: "heap" and l :: "sfields" and sh :: "sheap" where "e' = unit" and "s' = (h, l(V \<mapsto> v), sh)" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>Val v,(h, l, sh)\<rangle>"   | e'a :: "char list exp" where "e' = throw e'a" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e'a,s'\<rangle>"
simple_tycon_equal: shows "OFCLASS(simple_tycon, equal_class)"
rel_simps_8: fixes m :: "num" shows "(m < num.One) = False"
coeff_monomila_in_varsD: fixes p :: "int mpoly"   and x :: "nat" assumes "MPoly_Type.coeff p (monomial (Suc 0) x) \<noteq> 0" shows "x \<in> vars p"
S: shows "S = (\<lambda>s. if b \<le> s a then s else s(a := b))"
meta_eq_to_obj_eq: fixes A :: "'e"   and B :: "'e" assumes "A \<equiv> B" shows "A = B"
cases: fixes a :: "lam" assumes "val a" obtains x :: "name" and e :: "lam" where "a = LAM [x].e"   | n :: "nat" where "a = NUM n"   | "a = FALSE"   | "a = TRUE"
inf_idem: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.semilattice_inf inf less_eq less" shows "inf x x = x"
nextl_state: fixes q :: "hf"   and xs :: "'a list" assumes "q \<in> states M" shows "M.nextl q xs \<in> states M"
gt_arg: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and s :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "arity.wary arity_sym arity_var s"   and "is_App s" shows "tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym s (arg s)"
not_fn_prm: fixes A :: "'b trm"   and \<pi> :: "'b prm"   and x :: "'b"   and T :: "PreSimplyTyped.type"   and M :: "'b trm" assumes "\<And>x T M. A \<noteq> Fn x T M" shows "\<pi> \<cdot> A \<noteq> Fn x T M"
atom_map_bot_pres: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<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" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "bounded_lattice.atom_map less_eq less bot bot = {}"
that_2: shows "\<I>,\<I>_full \<turnstile>\<^sub>C conv2 \<sim> conv2'"
get_disconnected_document_is_l_get_disconnected_document: shows "l_get_disconnected_document get_disconnected_document get_disconnected_document_locs"
InvariantReasonClausesHoldsForPrefix: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and p :: "(Literal \<times> bool) list" assumes "InvariantReasonClauses F M"   and "InvariantUniq M"   and "isPrefix p M" shows "InvariantReasonClauses F p"
vconverse_vconverse: shows "(r\<inverse>\<^sub>\<circ>)\<inverse>\<^sub>\<circ> = r"
Inf_if_Inf_from: fixes \<iota> :: "('f \<times> 'l) inference"   and N :: "('f \<times> 'l) set" assumes "\<iota> \<in> Inf_from N" shows "\<iota> \<in> Inf_FL"
cofactor_def: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" shows "cofactor A i j = (- (1::'a)) ^ (i + j) * Determinant.det (mat_delete A i j)"
Radj_def: fixes x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "Radj x y \<nu> = (RSadj x y (fst \<nu>), RSadj x y (snd \<nu>))"
globE: fixes t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'BB"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'BB) \<Rightarrow> nat \<Rightarrow> bool" assumes "pb.eval the_bb t t' n (\<box>\<^sub>b\<gamma>)" shows "\<forall>n'\<ge>n. pb.eval the_bb t t' n' \<gamma>"
\<alpha>'_ide_simp: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "\<alpha>' (a, b, c) = \<a>\<^sup>-\<^sup>1[a, b, c]"
generat: shows "generat \<in> set_spmf (the_gpv p)"
total_conv_surjective: fixes x :: "'a" shows "(x \<sqinter> top = top) = (top \<sqinter> id x = top)"
G_pos: fixes t :: "real" assumes "t \<in> {a..b}" shows "0 < G t"
cardinalityOneTheElemIdentity: fixes X :: "'a set" shows "(card X = 1) = (X = {the_elem X})"
Hoare_dgr_def: fixes P :: "'b \<Rightarrow> 'f"   and D :: "'b \<times> 'b \<Rightarrow> 'f \<Rightarrow> 'f"   and Q :: "'b \<Rightarrow> 'f" shows "\<turnstile> P{| D |}Q  \<equiv> \<exists>X. (\<forall>u i j. \<Turnstile> X u i{| D (i, j) |}SUP_L_P X (pair u i) j) \<and> P = \<Squnion>range X \<and> Q = \<Squnion>range X \<sqinter> - grd (step D)"
upclose_sfis: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and f :: "'b \<Rightarrow> real"   and b :: "real"   and g :: "'b \<Rightarrow> real" assumes "measure_space M"   and "a \<in> sfis f M"   and "b \<in> sfis g M" shows "\<exists>c. c \<in> sfis (upclose f g) M"
ladder_n_prev_bound: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and u :: "nat" assumes "is_ladder D L"   and "u < length L - 1" shows "ladder_n L u \<le> ladder_prev_n L (length L - 1)"
a: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<le> x"
size_gen_2: fixes xa :: "'c \<Rightarrow> nat"   and x :: "'d \<Rightarrow> nat"   and x21 :: "char list option"   and x22 :: "'d" shows "Tagged_Prod_Sum.sum.size_sum xa x (sum.Inr x21 x22) = x x22 + Suc 0"
Units_pow_closed: fixes x :: "'a"   and d :: "nat" assumes "x \<in> Units R" shows "x [^] d \<in> Units R"
inject_1: fixes a :: "'g"   and a' :: "'g" shows "(Inl a = Inl a') = (a = a')"
assms_4: shows "M' = M @ [(uLiteral, False)]"
enum_word_def: shows "enum = map word_of_nat [0..<2 ^ LENGTH('b)]"
pmf_of_set_permutations_of_multiset_nonempty: fixes A :: "'a multiset" assumes "A \<noteq> {#}" shows "pmf_of_set (permutations_of_multiset A) = pmf_of_multiset A \<bind> (\<lambda>x. pmf_of_set (permutations_of_multiset (A - {#x#})) \<bind> (\<lambda>xs. return_pmf (x # xs)))"
int_div_same_is_1: fixes a :: "int"   and b :: "int" assumes "0 < a" shows "(a div b = a) = (b = 1)"
factorize_rat_poly_monic_square_free_factorization: fixes f :: "rat poly"   and c :: "rat"   and fs :: "(rat poly \<times> nat) list" assumes "factorize_rat_poly_monic f = (c, fs)" shows "square_free_factorization f (c, fs)"
downward_complete_def: shows "downward_complete = (\<forall>C. chain_on (\<sqsubset>) C A \<and> C \<noteq> {} \<longrightarrow> (\<exists>x\<in>A. glb (\<sqsubset>) C x))"
unique_factorization_m_imp_factorization: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" assumes "q.unique_factorization_m f cfs" shows "q.factorization_m f cfs"
states_target_index: fixes i :: "nat"   and p :: "(('d \<times> 'e) \<times> 'f) list"   and q1 :: "'f" assumes "i < length p" shows "states p q1 ! i = target (take (Suc i) p) q1"
ffunion_mono: fixes A :: "'c fset fset"   and B :: "'c fset fset" assumes "A |\<subseteq>| B" shows "ffUnion A |\<subseteq>| ffUnion B"
dom_closed: fixes f :: "'a" assumes "Arr f" shows "Arr (C.dom f)"
idealprod_whole_r: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set" assumes "Ring R"   and "ideal R I" shows "I \<diamondsuit>\<^sub>r\<^bsub>R\<^esub> carrier R = I"
mset_ns_mul_ext: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool" shows "((mset xs, mset ys) \<in> ns_mul_ext {(x, y). snd (f x y)} {(x, y). fst (f x y)}) = snd (mul_ext f xs ys)"
assms: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "p A i j"
closed_in_def: fixes S :: "ty"   and \<Gamma> :: "binding list" shows "S closed_in \<Gamma> \<equiv> supp S \<subseteq> ty_dom \<Gamma>"
sbh_step_preserves_separated_tmps: fixes i :: "nat"   and ts :: "((stmt \<times> nat) \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> (stmt \<times> nat) memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "stmt \<times> nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "(stmt \<times> nat) memref list"   and m' :: "nat \<Rightarrow> nat"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')"   and "separated_tmps ts" shows "separated_tmps (ts[i := (p, is', \<theta>', sb', \<D>', \<O>', \<R>')])"
blinding_of_blindable_mono: fixes bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bo' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" assumes "bo \<le> bo'" shows "blinding_of_blindable h bo \<le> blinding_of_blindable h bo'"
float_divr: fixes x :: "float"   and y :: "float"   and prec :: "nat" shows "real_of_float x / real_of_float y \<le> real_of_float (float_divr prec x y)"
Nil_tl: fixes xs :: "'a list" shows "([] = tl xs) = (xs = [] \<or> (\<exists>x. xs = [x]))"
power_mono: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "power_mono"   and "x \<succ> y"   and "(0::'a) \<le> y"   and "1 \<le> n" shows "x ^ n \<succ> y ^ n"
le_max_iff_disj: fixes z :: "'a word"   and x :: "'a word"   and y :: "'a word" shows "(z \<le>s signed.max x y) = (z \<le>s x \<or> z \<le>s y)"
Xij_def: fixes i :: "nat"   and j :: "nat" shows "local.Xij i j = prod_list (map local.X [i..<j])"
\<psi>\<^sub>3_values: fixes i :: "nat" assumes "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then \<Sum>k<2 ^ n. (- 1) ^ (f k + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else \<Sum>k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))" shows "odd i \<longrightarrow> Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then \<Sum>k<2 ^ n. (- 1) ^ (f k + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else \<Sum>k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))) $$ (i, 0) = complex_of_real (\<Sum>k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))"
intros_7: fixes c' :: "cmd"   and l :: "nat"   and c :: "cmd"   and b :: "expr" assumes "labels c' l c" shows "labels (while (b) c') (l + 2) (c;; while (b) c')"
bisim1TryCatch2: fixes P :: "'a prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e :: "'addr expr1"   and C :: "String.literal"   and V :: "nat" assumes "P,e2,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,try e catch(C V) e2,h \<turnstile> ({V:Class C=None; e'}, xs) \<leftrightarrow> (stk, loc, Suc (Suc (length (compE2 e) + pc)), xcp)"
proper_Par_properL: fixes cl :: "('a, 'b, 'c) cmd list" shows "proper (Par cl) = properL cl"
subset_iff_psubset_eq: fixes A :: "'e set"   and B :: "'e set" shows "(A \<subseteq> B) = (A \<subset> B \<or> A = B)"
grounding_of_cls_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and C :: "'b literal multiset" shows "substitution_ops.grounding_of_cls subst_atm C = {substitution_ops.subst_cls subst_atm C \<sigma> |\<sigma>. substitution_ops.is_ground_subst subst_atm \<sigma>}"
reachable_mode_states_refl: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val" shows "map snd cms \<in> reachable_mode_states (cms, mem)"
default_secure: fixes G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b"   and F :: "('a \<times> 'a) set"   and i :: "'a" assumes "wf_graph G"   and "\<not> sinvar G nP"   and "F \<in> set_offending_flows G nP" shows "(\<not> True \<longrightarrow> i \<in> fst ` F \<longrightarrow> \<not> sinvar G (nP(i := \<bottom>))) \<and> (True \<longrightarrow> i \<in> snd ` F \<longrightarrow> \<not> sinvar G (nP(i := \<bottom>)))"
simps: fixes xs :: "('a \<times> 'c) list"   and ko :: "'o" shows "sort_oalist_raw (xs, ko) = (sort_oalist ko xs, ko)"
non_trivial_field_hom_imp_inj: assumes "field R"   and "h ` carrier R \<noteq> {\<zero>\<^bsub>S\<^esub>}" shows "inj_on h (carrier R)"
mem_carrier: fixes x :: "'c" assumes "x \<in> H" shows "x \<in> carrier (add_monoid M)"
substPB_o: fixes \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<Phi> :: "('c, 'd) lit list set" shows "substPB (subst \<pi>1 \<circ> \<pi>2) \<Phi> = substPB \<pi>1 (substPB \<pi>2 \<Phi>)"
S'_disj: fixes i :: "'a"   and j :: "'a" assumes "i \<noteq> j" shows "S' i \<inter> S' j = {}"
map_eq_Cons_conv: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list"   and y :: "'e"   and ys :: "'e list" shows "(map f xs = y # ys) = (\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys)"
hyps: shows "\<Z> k u \<subseteq> \<W> (\<P> k u) k"
bot_eq_sup_iff: fixes x :: "'a"   and y :: "'a" shows "((zero_class.zero::'a) = x + y) = (x = (zero_class.zero::'a) \<and> y = (zero_class.zero::'a))"
nat_pow_mult: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier P" shows "x [^]\<^bsub>P\<^esub> n \<otimes>\<^bsub>P\<^esub> x [^]\<^bsub>P\<^esub> m = x [^]\<^bsub>P\<^esub> (n + m)"
pre_str_hoare_r: fixes p\<^sub>1 :: "'b upred"   and p\<^sub>2 :: "'b upred"   and C :: "'b hrel"   and q :: "'b upred" assumes "`p\<^sub>1 \<Rightarrow> p\<^sub>2`"   and "\<lbrace>p\<^sub>2\<rbrace> C \<lbrace>q\<rbrace>\<^sub>u" shows "\<lbrace>p\<^sub>1\<rbrace> C \<lbrace>q\<rbrace>\<^sub>u"
gauge_integral_Fubini_universe_y_2: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "integrable lborel f"   and "(\<lambda>x. integral UNIV (\<lambda>y. f (y, x))) \<in> borel_measurable lborel" shows "(\<lambda>x. integral UNIV (\<lambda>y. f (y, x))) integrable_on UNIV"
assms_5: fixes zs :: "('a \<times> 'b) list" assumes "oalist_inv_raw zs" shows "g zs = map_val_pair (\<lambda>k v. f k v (0::'c)) zs"
assms_3: shows "FSbImorph h' hA' MOD' MOD''"
assms: shows "P \<leadsto>\<^sup>^<Rel> Q"
doms_1: shows "subst_domain (rm_vars X \<theta>) \<subseteq> subst_domain \<theta>"
replicate_append_same: fixes i :: "nat"   and x :: "'g" shows "replicate i x @ [x] = x # replicate i x"
mon_env_uncons: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and e :: "('c, 'f) label list el_step"   and w :: "('c, 'f) label list el_step list" shows "mon_env fg (e # w) = (case e of LOC a \<Rightarrow> {} | ENV a \<Rightarrow> mon_w fg a) \<union> mon_env fg w"
dist: shows "\<forall>x\<in>S. \<forall>y\<in>S. dist x y = c * dist (f x) (f y)"
prod_splitE: fixes p :: "'b \<times> 'd"   and P :: "'b \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<forall>a b. p = (a, b) \<longrightarrow> P a b" obtains a :: "'b" and b :: "'d" where "p = (a, b)" and "P a b"