hcju/isabelleps huggingface dataset:
steps_preserve_SN_on: fixes a :: "'a"   and b :: "'a"   and A :: "('a \<times> 'a) set" assumes "(a, b) \<in> A\<^sup>*"   and "SN_on A {a}" shows "SN_on A {b}"
disj_matrices_zero1: fixes B :: "'a matrix" shows "disj_matrices 0 B"
recognises_from_1a: fixes d :: "value option" shows "recognises_execution drinks 1 <1 $:= d, 2 $:= Some (Num 50)> [(STR ''coin'', [Num 50]), (STR ''vend'', [])]"
KP: shows "\<not> (infinite KP \<or> KP = {})"
step2: shows "gdpr_scenario' \<rightarrow>\<^sub>n gdpr_scenario''"
maxim_def: fixes A :: "'b set" shows "s.maxim A \<equiv> THE b. s.isMaxim A b"
Sum1_1: fixes P :: "pi"   and Rs :: "residual"   and Q :: "pi" assumes "P \<Longrightarrow>\<^sub>l Rs" shows "P \<oplus> Q \<Longrightarrow>\<^sub>l Rs"
pdevs_domain_nlex_pdevs: fixes x :: "(real \<times> real) pdevs" shows "pdevs_domain (nlex_pdevs x) = pdevs_domain x"
LeastI_ex: fixes less_eq :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and less :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and P :: "'q \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>x. P x" shows "P (ord.Least less_eq P)"
abs_reach_def: fixes \<Phi> :: "'a ltl_prop_equiv_quotient" shows "abs_reach \<Phi> = {foldl f_abs \<Phi> w |w. True}"
set_iterator_product_foldli_conv: fixes as :: "'e list"   and bs :: "'e \<Rightarrow> 'f list" shows "set_iterator_product (foldli as) (\<lambda>a. foldli (bs a)) = foldli (concat (map (\<lambda>a. map (Pair a) (bs a)) as))"
left_tree_initial: fixes s :: "'c"   and t :: "'c" assumes "s \<in> V\<^bsub>T\<^esub>"   and "t \<in> V\<^bsub>T\<^esub>" shows "s \<in> T.left_tree s t"
eq_refl: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.preorder less_eq less"   and "x = y" shows "less_eq x y"
vfinite_vcard_vfinite: fixes A :: "V" shows "vfinite (vcard A) = vfinite A"
bl_of_nth_simps_1: fixes f :: "nat \<Rightarrow> 'b" shows "bl_of_nth 0 f = []"
elims: fixes x :: "nat set"   and xa :: "nat set \<times> 'b option list set"   and y :: "nat set \<times> 'b option list set" assumes "New_max.projectTable x xa = y" obtains V :: "nat set" and s :: "nat set" and t :: "'b option list set" where "x = V" and "xa = (s, t)" and "y = (s \<inter> V, restrict V ` t)"
Diagonalize_red: fixes a :: "'c local.term" assumes "Ide a" shows "\<^bold>\<lfloor>a\<^bold>\<down>\<^bold>\<rfloor> = \<^bold>\<lfloor>a\<^bold>\<rfloor>"
intros: fixes l :: "'e"   and g :: "('f, 'g) cconstraint"   and a :: "'h"   and r :: "'f list"   and l' :: "'e"   and A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and u :: "'f \<Rightarrow> 'g"   and u' :: "'f \<Rightarrow> 'g" assumes "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'"   and "u \<turnstile> g"   and "u' \<turnstile> inv_of A l'"   and "u' = [r\<rightarrow>(0::'g)]u" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsub>a\<^esub> \<langle>l', u'\<rangle>"
deleteAround_subset: fixes g :: "graph"   and a :: "nat"   and ps :: "(nat \<times> nat) list" shows "set (deleteAround g a ps) \<subseteq> set ps"
P_4: fixes n :: "nat" shows "P (Suc n) \<le> P n"
ladder_n_def: fixes L :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" shows "ladder_n L index = deriv_n (L ! index)"
of_int_eq_0_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus z = zero) = (z = 0)"
morphism_ringed_spaces_def: fixes X :: "'d set"   and is_open\<^sub>X :: "'d set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'d set \<Rightarrow> 'e set"   and \<rho>\<^sub>X :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and one_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and Y :: "'f set"   and is_open\<^sub>Y :: "'f set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'f set \<Rightarrow> 'g set"   and \<rho>\<^sub>Y :: "'f set \<Rightarrow> 'f set \<Rightarrow> 'g \<Rightarrow> 'g"   and d :: "'g"   and add_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and one_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and f :: "'d \<Rightarrow> 'f"   and \<phi>\<^sub>f :: "'f set \<Rightarrow> 'g \<Rightarrow> 'e" shows "morphism_ringed_spaces X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<equiv> im_sheaf X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y f \<and> ringed_space Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y \<and> morphism_ringed_spaces_axioms X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f"
R0_def: shows "R0 = {(s1, s2). \<exists>c1 c1' c2 c2'. s1 = [c1;c2] \<and> s2 = [c1';c2'] \<and> ([c1], [c1']) \<in> R1 \<and> ([c2], [c2']) \<in> R2}"
finsum_Un_disjoint: fixes A :: "'e set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'c" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "g \<in> A \<rightarrow> carrier S"   and "g \<in> B \<rightarrow> carrier S" shows "finsum S g (A \<union> B) = finsum S g A \<oplus>\<^bsub>S\<^esub> finsum S g B"
nat_pp_order_of_le_nat_pp: shows "nat_term_order_of_le = Term_Order.LEX"
rx_Max: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "x \<in> X" shows "r x = Sup {\<rho>. le_vec (\<rho> *s x) (A *v x)}"
wary_ss: fixes sa :: "('s, 'v) tm" assumes "sa \<in> set (sa__ # ss)" shows "wary sa"
Cons_eq_filterD: fixes x :: "'a"   and xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and ys :: "'a list" assumes "x # xs = filter P ys" shows "\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
zip_with_option_eq_Some: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x :: "'d option"   and y :: "'e option"   and z :: "'c" shows "(zip_with_option f x y = Some z) = (\<exists>a b. z = f a b \<and> x = Some a \<and> y = Some b)"
has_overlap_at_k_equiv: fixes rects :: "'a rectangle list"   and ops :: "'a operation list"   and k :: "nat" assumes "is_rect_list rects"   and "ops = all_ops rects"   and "k < length ops"   and "has_overlap_at_k rects k" shows "has_rect_overlap rects"
Sup_specs_def: fixes d :: "nat"   and X :: "real list set" shows "Sup_specs d X = SPEC (\<lambda>r. length r = d \<and> (\<forall>x\<in>X. list_all2 (\<le>) x r))"
powr_eq_0_iff: fixes w :: "'a"   and z :: "'a" shows "(w powr z = (0::'a)) = (w = (0::'a))"
isLastAssertedLiteralSubset: fixes l :: "Literal"   and c :: "Literal list"   and M :: "Literal list"   and c' :: "Literal list" assumes "isLastAssertedLiteral l c M"   and "set c' \<subseteq> set c"   and "l el c'" shows "isLastAssertedLiteral l c' M"
map_simp_ide: fixes a :: "'a" assumes "A.ide a" shows "\<tau>'.map a = B.inv (\<tau> a)"
hull_inc: fixes x :: "'b"   and S :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "x \<in> S" shows "x \<in> P hull S"
OAlist_tc_lookup_def: fixes xs :: "('c, 'd) oalist_tc" shows "OAlist_tc_lookup xs = lookup_pair_tc (list_of_oalist_tc xs)"
hyps_2: shows "\<not> (\<nexists>v w. v ^ 3 + w ^ 3 = x ^ 3 \<and> v * w * x \<noteq> 0 \<and> even x \<and> coprime v w)"
integral_cong: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "integral s f = integral s g"
filter_replicate: fixes P :: "'g \<Rightarrow> bool"   and n :: "nat"   and x :: "'g" shows "filter P (replicate n x) = (if P x then replicate n x else [])"
set_filt_contain: fixes p :: "real"   and M :: "bool stream measure"   and A :: "bool stream set"   and n :: "nat"   and w :: "bool stream" assumes "infinite_coin_toss_space p M"   and "A \<in> sets (infinite_coin_toss_space.nat_filtration M n)"   and "w \<in> A" shows "pseudo_proj_True n -` {pseudo_proj_True n w} \<subseteq> A"
hd_op_discr: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and oper :: "pdf_operator"   and e :: "expr"   and V :: "nat set"   and V' :: "nat set"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "countable_type (the (expr_type \<Gamma> (oper $$ e)))"   and "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d oper $$ e \<Rightarrow> (\<lambda>\<rho> y. \<integral>\<^sup>+ x. (if op_sem oper x = y then 1 else 0) * f \<rho> x \<partial>stock_measure (the (expr_type \<Gamma> e)))"
refl: fixes x :: "'alt" assumes "x \<in> alts" shows "R i x x"
global_assmsI: fixes v :: "'vertex"   and p :: "'form"   and i :: "'var itself" assumes "v |\<in>| vertices"   and "nodeOf v = Assumption p" shows "labelAtOut v (Reg p) |\<in>| global_assms i"
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 taa :: "('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 = (stka__, loca__, pca__, None)" and "a3 = taa" and "a4 = h'" and "a5 = (stk', loc', pc'a__, xcpa__)" and "(taa, xcpa__, h', [(stk', loc', undefined, undefined, pc'a__)]) \<in> exec_instr (ins ! pca__) P t ha__ stka__ loca__ undefined undefined pca__ []" and "pca__ < length ins" and "ci_app ci (ins ! pca__) P ha__ stka__ loca__ undefined undefined pca__ []"   | 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"
isolated_singularity_at_divide: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and g :: "complex \<Rightarrow> complex" assumes "isolated_singularity_at f z"   and "isolated_singularity_at g z"   and "not_essential g z" shows "isolated_singularity_at (\<lambda>w. f w / g w) z"
le_measureD2: fixes A :: "'b measure"   and B :: "'b measure" assumes "A \<le> B"   and "space A = space B" shows "sets A \<subseteq> sets B"
simps_3: fixes m1 :: "real mat convnet"   and m2 :: "real mat convnet"   and inputs :: "real vec list" shows "evaluate_net (Pool m1 m2) inputs = component_mult (evaluate_net m1 (take (length (input_sizes m1)) inputs)) (evaluate_net m2 (drop (length (input_sizes m1)) inputs))"
substs_preserve_ground_lit: fixes C :: "'b literal set"   and y :: "'b literal"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "ground_clause C"   and "y \<in> C" shows "subst_lit y \<sigma> = y"
abs_mult_pos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq zero x" shows "times (abs y) x = abs (times y x)"
jozsa_transform_coeff_1: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "jozsa f n"   and "i < dim_row (jozsa.jozsa_transform f n) \<and> j < dim_col (jozsa.jozsa_transform f n)" shows "i = j \<longrightarrow> jozsa.jozsa_transform f n $$ (i, j) = of_nat (1 - f (i div 2))"
univ_poly_units': fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (K [X])" shows "(p \<in> Units (K [X]\<^bsub>K [X]\<^esub>)) = (p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>) \<and> p \<noteq> [] \<and> degree p = 0)"
n_circ_6: fixes k :: "nat" shows "p \<notin> path_image (n_circ k)"
tagged_fun_relD_lhs: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'd"   and A :: "('a \<times> 'c) set"   and B :: "('b \<times> 'd) set"   and x :: "'a"   and x' :: "'c" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(Let x f, f' x') \<in> B"
collapse_1: fixes generat :: "('d, 'e, 'f) generat" assumes "is_Pure generat" shows "Pure (generat.result generat) = generat"
F'_eq: fixes m :: "nat" shows "F' m = (\<Inter>i. F m (i + m))"
inv_uptoD: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and m :: "nat"   and i :: "nat"   and j :: "nat" assumes "local.inv_upto p A m"   and "i < n"   and "j < n"   and "j < m" shows "p A i j"
Synchronized1Null: fixes V :: "nat"   and xs :: "'addr val list"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap" assumes "V < length xs" shows "uf,P,t \<turnstile>1 \<langle>sync\<^bsub>V\<^esub> (null) e,(h, xs)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),(h, xs[V := Null])\<rangle>"
rep_eq: shows "generator Nil_prod = (\<lambda>_. Done)"
lambda_predicates_3_3: fixes F :: "\<Pi>\<^sub>3" shows "[[\<^bold>\<lambda>\<^sup>3 (\<lambda>x y z. \<lparr>F,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>) \<^bold>= F]]"
of_nat_eq: fixes k :: "nat" shows "of_nat k = Abs_bit0 (int k mod int CARD('b bit0))"
wrap_def: shows "wrap \<equiv> \<Lambda> r e. KS\<cdot>(r\<cdot>e)"
le_x: shows "ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi \<le> x"
prv_suc_imp_cong: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> atrm"   and "t2 \<in> atrm" shows "prv (imp (eql t1 t2) (eql (suc t1) (suc t2)))"
fset_cong: fixes x :: "'a fset"   and y :: "'a fset" shows "(fset x = fset y) = (x = y)"
case1: shows "direction lineA = 0 \<longrightarrow> lineB \<parallel>  lineA"
Times_def: fixes r :: "'a language"   and s :: "'a language" shows "Times r s = Times_Plus (TimesLR r s)"
simps: fixes a :: "'b set" shows "finite a = (a = {} \<or> (\<exists>A a. a = insert a A \<and> finite A))"
arcs_successor': fixes a :: "nat"   and b :: "nat"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "(a, b) \<in> set (arcs i j xs)"   and "b \<noteq> j" shows "\<exists>c. xs = [b] \<and> a = i \<or> (\<exists>ys. xs = b # c # ys \<and> a = i) \<or> (\<exists>ys. xs = ys @ [a, b] \<and> c = j) \<or> (\<exists>ys zs. xs = ys @ a # b # c # zs)"
Eq\<^sub>EI: fixes x :: "\<kappa>"   and y :: "\<kappa>"   and v :: "i" assumes "\<exists>o\<^sub>1 o\<^sub>2. Some (\<omega>\<nu> o\<^sub>1) = d\<^sub>\<kappa> x \<and> Some (\<omega>\<nu> o\<^sub>2) = d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2" shows "[x \<^bold>=\<^sub>E y in v]"
integral_dis: shows "integral s (\<lambda>x. partial_vector_derivative (\<lambda>a. F a \<bullet> j) i x - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x) = integral s (partial_vector_derivative (\<lambda>a. F a \<bullet> j) i) + integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x)"
compass_reflect_compass_6: shows "apply_cltn2 east equator_reflect = east"
DERIV_imp_deriv: fixes f :: "'b \<Rightarrow> 'b"   and f' :: "'b"   and x :: "'b" assumes "(f has_field_derivative f') (at x)" shows "deriv f x = f'"
case_two_psi_longest_prefix: fixes j :: "nat"   and y :: "nat" assumes "\<exists>t. local.prefixes t j \<up>"   and "y = longest_prefix j" shows "(\<forall>x. 0 < x \<and> x < y \<longrightarrow> \<psi> j x \<down>= 0 \<or> \<psi> j x \<down>= 1) \<and> (\<forall>x\<ge>y. \<psi> j x \<up>)"
prems_1: fixes i :: "nat" shows "dickson_less_v d m (seq (Suc i)) (seq i)"
Dgec_def: shows "Dgec \<equiv> {L \<in> (\<lambda>L. L \<inter> {x. c \<le> x \<bullet> i}) ` \<D>. content L \<noteq> 0}"
simps_2: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and U :: "'d set"   and x :: "'c"   and xs :: "'c list" shows "ipurge_tr_rev_aux I D U (x # xs) = (if \<exists>v\<in>sources_aux I D U xs. (D x, v) \<in> I then x # ipurge_tr_rev_aux I D U xs else ipurge_tr_rev_aux I D U xs)"
disc_22: fixes x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c" shows "\<not> isRecvMarker (Send x21 x22 x23 x24 x25 x26)"
conjSize: shows "g <# (P #> inv g) \<in> subgroups_of_size (p ^ a)"
prems: shows "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T \<cdot> \<delta>)"
1: shows "Img f \<subseteq> Cod f \<and> Cod f \<subseteq> Univ \<and> |Cod f| <o \<AA>"
distinct_4: fixes x31 :: "'d"   and x32 :: "'d ifex"   and x33 :: "'d ifex" shows "IF x31 x32 x33 \<noteq> Trueif"
weak_eq: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "comparator comp"   and "comp x y = Eq" shows "x = y"
linorder_cases: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less" obtains "less x y"   | "x = y"   | "less y x"
map_ticket_addsimps_2: fixes f :: "nat \<Rightarrow> nat"   and t :: "thread"   and tk :: "nat" shows "(map_ticket f t = WAIT tk) = (\<exists>tk'. tk = f tk' \<and> t = WAIT tk')"
list_conj_def: fixes ps :: "fm list" shows "list_conj ps \<equiv> foldr Parametric_Ferrante_Rackoff.conj ps T"
sconf: shows "P,E \<turnstile> s'' \<surd>"
dom_CompFun: fixes HA :: "('d, 'e, 'f) hierauto" shows "dom (CompFun HA) = HAStates HA"
r_cast: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and D :: "nat"   and es :: "exp list" assumes "CT \<turnstile> C <: D" shows "CT \<turnstile> Cast D (New C es) \<rightarrow> New C es"
preserves_ide: fixes a :: "'a" assumes "ide a" shows "ide (if arr a then a else null)"
vector_scaleR_commute: fixes A :: "(('d, 'f) vec, 'e) vec"   and c :: "real"   and x :: "('d, 'f) vec" shows "A *v c *\<^sub>R x = c *\<^sub>R (A *v x)"
conj_bot_left: fixes c :: "'a::refinement_lattice" shows "\<bottom> \<iinter> (c::'a::refinement_lattice) = \<bottom>"
chine_assoc'_in_hom: shows "\<guillemotleft>chine_assoc' : \<mu>\<nu>\<pi>.chine \<rightarrow>\<^sub>C \<mu>\<nu>_\<pi>.chine\<guillemotright>"
mask_shift_and_negate: fixes w :: "'b word"   and n :: "nat"   and m :: "nat" shows "(w AND mask n << m) AND NOT (mask n << m) = 0"
det_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "det A = (\<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. A $h i $h p i))"
mat1b: fixes env :: "v sem_env"   and s1 :: "'a state"   and e :: "exp0"   and s2 :: "'a state"   and v1 :: "v"   and pes :: "(pat \<times> exp0) list"   and err :: "v error_result" assumes "Big_Step_Unclocked_Single.evaluate env s1 e (s2, Rval v1)"   and "match_result env s2 v1 pes Bindv = Rerr err" shows "Big_Step_Unclocked_Single.evaluate env s1 (Mat e pes) (s2, Rerr err)"
design_isomorphism_axioms: shows "design_isomorphism \<V> \<B> \<V>' \<B>' \<pi>"
sparse_row_vector_nprt: fixes v :: "(nat \<times> 'a) list" assumes "sorted_spvec v" shows "sparse_row_vector (nprt_spvec v) = nprt (sparse_row_vector v)"
no_state_change_if_no_event: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and p :: "'p" assumes "c \<turnstile> ev \<mapsto> c'"   and "occurs_on ev \<noteq> p" shows "states c p = states c' p \<and> ps c p = ps c' p"
N_top: shows "- (- top \<sqinter> top) \<sqinter> top = top"
attract: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "((\<lambda>x y. y \<sqsubseteq> x) \<restriction> A) x y"   and "((\<lambda>x y. y \<sqsubseteq> x) \<restriction> A) y x"   and "((\<lambda>x y. y \<sqsubseteq> x) \<restriction> A) y z"   and "x \<in> UNIV"   and "y \<in> UNIV"   and "z \<in> UNIV" shows "((\<lambda>x y. y \<sqsubseteq> x) \<restriction> A) x z"
sint_of_nat_le: fixes b :: "nat"   and a :: "nat" assumes "b < 2 ^ (LENGTH('b) - 1)"   and "a \<le> b" shows "sint (word_of_nat a) \<le> sint (word_of_nat b)"
simps_2: fixes x :: "'b"   and hs :: "'b heap list" shows "no_Empty (Hp x hs) = no_Emptys hs"
simps: fixes a1 :: "nat \<Rightarrow> pdf_type"   and a2 :: "expr"   and a3 :: "pdf_type" shows "(a1 \<turnstile> a2 : a3) = ((\<exists>\<Gamma> x. a1 = \<Gamma> \<and> a2 = Var x \<and> a3 = \<Gamma> x) \<or> (\<exists>\<Gamma> v. a1 = \<Gamma> \<and> a2 = Val v \<and> a3 = val_type v) \<or> (\<exists>\<Gamma> e1 t1 e2 t2. a1 = \<Gamma> \<and> a2 = LET e1 IN e2 \<and> a3 = t2 \<and> \<Gamma> \<turnstile> e1 : t1 \<and> case_nat t1 \<Gamma> \<turnstile> e2 : t2) \<or> (\<exists>\<Gamma> e t oper t'. a1 = \<Gamma> \<and> a2 = oper $$ e \<and> a3 = t' \<and> \<Gamma> \<turnstile> e : t \<and> op_type oper t = Some t') \<or> (\<exists>\<Gamma> e1 t1 e2 t2. a1 = \<Gamma> \<and> a2 = <e1 ,  e2> \<and> a3 = PRODUCT t1 t2 \<and> \<Gamma> \<turnstile> e1 : t1 \<and> \<Gamma> \<turnstile> e2 : t2) \<or> (\<exists>\<Gamma> e dst. a1 = \<Gamma> \<and> a2 = Random dst e \<and> a3 = dist_result_type dst \<and> \<Gamma> \<turnstile> e : dist_param_type dst) \<or> (\<exists>\<Gamma> b e1 t e2. a1 = \<Gamma> \<and> a2 = IF b THEN e1 ELSE e2 \<and> a3 = t \<and> \<Gamma> \<turnstile> b : BOOL \<and> \<Gamma> \<turnstile> e1 : t \<and> \<Gamma> \<turnstile> e2 : t) \<or> (\<exists>\<Gamma> t. a1 = \<Gamma> \<and> a2 = Fail t \<and> a3 = t))"
hyps_1: fixes rs :: "nat list"   and "is" :: "nat list"   and y :: "nat" assumes "rs' = butlast (butlast (butlast rs))"   and "is \<lhd> dims (tensors_from_net (witness_l rs) $ y)"   and "3 \<le> length rs"   and "\<And>r. r \<in> set rs \<Longrightarrow> 0 < r"   and "y < rs ! 0" shows "lookup (tensors_from_net (witness_l rs) $ y) is = (if nths is (Collect even) = nths is {n. odd n} \<and> (\<forall>i\<in>set is. i < last (butlast rs)) then 1 else 0)"
content_pos_le: fixes X :: "'c set" shows "0 \<le> content X"
z: shows "z \<notin> \<real>\<^sub>\<le>\<^sub>0"
sequence_to_failure_reaches_FAIL_ob: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and io :: "('d \<times> 'e) list"   and FAIL :: "'f \<times> 'f"   and PM :: "('d, 'e, 'f \<times> 'f) FSM" assumes "sequence_to_failure M1 M2 io"   and "OFSM M1"   and "OFSM M2"   and "productF M2 M1 FAIL PM" shows "io_targets PM (initial PM) io = {FAIL}"
Sign_synth: fixes X :: "msg"   and K :: "msg"   and H :: "msg set" assumes "Sign X K \<in> synth H" obtains "Sign X K \<in> H"   | "X \<in> synth H" and "K \<in> synth H"