hcju/isabelleps huggingface dataset:
split: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'e list \<Rightarrow> 'd"   and list :: "'e list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = ((list = [] \<longrightarrow> P f1) \<and> (\<forall>x21 x22. list = x21 # x22 \<longrightarrow> P (f2 x21 x22)))"
poly_add_in_carrier: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "set (poly_add p1 p2) \<subseteq> carrier R"
assms: shows "mopup_aft_erase_b (2 * n + 3, l, Bk # xs) lm n ires"
step_abs_def: shows "\<up>step \<equiv> map_fun Rep (map_fun id Abs) step"
sat1_refine: shows "satisfiable cnf = (let gr = imp_graph cnf in \<forall>x\<in>vars_of_cnf cnf. \<not> ((Pos x, Neg x) \<in> gr\<^sup>* \<and> (Neg x, Pos x) \<in> gr\<^sup>*))"
prems_1: fixes i :: "nat" assumes "i < length (a # is)" shows "i \<le> (a # is) ! i"
Compositionality_Seq: fixes c1 :: "('exp, 'id) MWLsCom"   and c2 :: "('exp, 'id) MWLsCom" assumes "WHATWHERE_Secure [c1]"   and "WHATWHERE_Secure [c2]"   and "unique_PPc (c1;c2)" shows "WHATWHERE_Secure [c1;c2]"
HighLowUNIV: shows "H \<union> L = UNIV"
negim: fixes T :: "'a set" assumes "negligible T"   and "T \<subseteq> S" shows "negligible (f ` T)"
axioms_1: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and min_index :: "int" assumes "t_covering_design point_set block_collection u_block_size grouping min_index" shows "block_design point_set block_collection u_block_size"
multiplicity_eq_zero_iff: fixes x :: "'a"   and p :: "'a" assumes "x \<noteq> (0::'a)"   and "\<not> is_unit p" shows "(multiplicity p x = 0) = (\<not> p dvd x)"
opt_def: shows "opt = push_forall \<circ> nnf \<circ> unpower 0 \<circ> clearQuantifiers"
simps: fixes a1 :: "nat"   and a2 :: "com" shows "(a1 \<turnstile>' a2) = ((\<exists>l. a1 = l \<and> a2 = SKIP) \<or> (\<exists>x a l. a1 = l \<and> a2 = x ::= a \<and> sec a \<le> sec x \<and> l \<le> sec x) \<or> (\<exists>l c\<^sub>1 c\<^sub>2. a1 = l \<and> a2 = c\<^sub>1;; c\<^sub>2 \<and> l \<turnstile>' c\<^sub>1 \<and> l \<turnstile>' c\<^sub>2) \<or> (\<exists>b l c\<^sub>1 c\<^sub>2. a1 = l \<and> a2 = IF b THEN c\<^sub>1 ELSE c\<^sub>2 \<and> sec b \<le> l \<and> l \<turnstile>' c\<^sub>1 \<and> l \<turnstile>' c\<^sub>2) \<or> (\<exists>b l c. a1 = l \<and> a2 = WHILE b DO c \<and> sec b \<le> l \<and> l \<turnstile>' c) \<or> (\<exists>l c l'. a1 = l' \<and> a2 = c \<and> l \<turnstile>' c \<and> l' \<le> l))"
arrI: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.arr (fst f) \<and> S.arr (snd f) \<and> S.src (fst f) = S.trg (snd f)" shows "S.VV.arr f"
cod_char: fixes f :: "'b \<times> 'b" shows "B.VV.cod f = (if B.VV.arr f then B.VxV.cod f else B.VxV.null)"
assms_3: shows "continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
assms_2: assumes "subgraph H G"   and "arcs H = {e \<in> arcs G. tail G e \<in> verts H \<and> head G e \<in> verts H}" shows "P"
le_less: fixes y :: "('a, 'b) Term.term"   and x :: "('a, 'b) Term.term" shows "y \<unrhd> x = (y \<rhd> x \<or> x = y)"
lang_pow_simps: fixes n :: "nat"   and A :: "'b list set" shows "A ^^ Suc n = A ^^ n @@ A"
SourcesA23_L1: shows "Sources level1 sA23 = {sA11}"
step_Normal_elim_cases_6: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and b :: "'b set"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "'b"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Cond b c1 c2, Normal s) \<rightarrow> u" obtains "u = (c1, Normal s)" and "s \<in> b"   | "u = (c2, Normal s)" and "s \<notin> b"
SN_def: fixes r :: "('b \<times> 'b) set" shows "SN r = (\<forall>x. SN_on r {x})"
length_findnth: fixes n :: "nat" shows "length (findnth n) = 4 * n"
lhs: shows "filter_mset P (add_mset x A \<times># {#b#}) = filter_mset P (A \<times># {#b#}) + filter_mset P {#(x, b)#}"
1: shows "\<forall>ys. length ys < length xs \<longrightarrow> (harry_meas ys = 0) = (True \<notin> set ys)"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vint_mult"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vint_mult" shows "b = c"
hyps: fixes ps :: "'a list" assumes "\<not> prefix ps ys" shows "P ps ys"
assms_2: assumes "ide (g \<cdot> f)"   and "ide (f \<cdot> g)" shows "T"
next_rbt_keys_generator: shows "list.next rbt_keys_generator = rbt_keys_next"
mtf_eq_mtf2: fixes x :: "'b"   and xs :: "'b list" shows "mtf x xs = mtf2 (length xs - 1) x xs"
not_Iic_eq_empty: fixes h :: "'a" shows "{..h} \<noteq> {}"
E\<^sub>i\<^sub>v: fixes x :: "i"   and y :: "i" shows "(dom x \<cong> cod y \<^bold>\<and> E (cod y) \<^bold>\<leftarrow> E (x \<cdot> y)) \<^bold>\<and> (E (x \<cdot> y) \<^bold>\<leftarrow> dom x \<cong> cod y \<^bold>\<and> E (cod y))"
arcs'_def: fixes xs :: "nat list" shows "arcs' xs = set (arcs (hd xs) (last xs) (butlast (tl xs)))"
execn_to_execn_strip_guards': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "t \<notin> Fault ` F" shows "\<Gamma>\<turnstile> \<langle>strip_guards F c,s\<rangle> =n\<Rightarrow> t"
atLeastAtMost_upto: fixes i :: "int"   and j :: "int" shows "{i..j} = set [i..j]"
taylor_eval: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "a \<in> carrier R"   and "f \<in> carrier P"   and "b \<in> carrier R" shows "to_fun (T\<^bsub>a\<^esub> f) b = to_fun f (b \<oplus> a)"
distinct_degree_factorization_def: fixes f :: "'a mod_ring poly" shows "local.distinct_degree_factorization f = (if degree f = 1 then [(1, f)] else local.dist_degree_factorize_main f (monom 1 1) 0 [])"
oalist_inv_raw_singleton: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and k :: "'a"   and v :: "'b" assumes "comparator comp" shows "comparator.oalist_inv_raw comp [(k, v)] = (v \<noteq> (0::'b))"
list_update_length: fixes xs :: "'c list"   and x :: "'c"   and ys :: "'c list"   and y :: "'c" shows "(xs @ x # ys)[length xs := y] = xs @ y # ys"
red': shows "sat_preserving_effective_redundancy_criterion \<Gamma>' Rf (\<lambda>N. Ri N \<union> (\<Gamma>' - \<Gamma>))"
upper_asymptotic_density_zero_lim: fixes A :: "nat set" assumes "upper_asymptotic_density A = 0" shows "(\<lambda>n. real (card (A \<inter> {..<n})) / real n) \<longlonglongrightarrow> 0"
has_laplace_minus: fixes f :: "real \<Rightarrow> complex"   and F :: "complex"   and S :: "complex"   and g :: "real \<Rightarrow> complex"   and G :: "complex" assumes "(f has_laplace F) S"   and "(g has_laplace G) S" shows "((\<lambda>x. f x - g x) has_laplace F - G) S"
that: fixes u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a"   and env\<^sub>1 :: "(name, 'a) fmap"   and env\<^sub>2 :: "(name, 'a) fmap" assumes "u = app u\<^sub>1 u\<^sub>2"   and "match t\<^sub>1 u\<^sub>1 = Some env\<^sub>1"   and "match t\<^sub>2 u\<^sub>2 = Some env\<^sub>2"   and "env = env\<^sub>1 ++\<^sub>f env\<^sub>2" shows "thesis"
digraph_dg_Par: shows "digraph \<alpha> (dg_Par \<alpha>)"
dim_submatrix_1: fixes A :: "'b mat"   and I :: "nat set"   and J :: "nat set" shows "dim_row (submatrix A I J) = card {i. i < dim_row A \<and> i \<in> I}"
trunc_err_pdevsE: fixes e :: "nat \<Rightarrow> real"   and p :: "nat"   and xs :: "'a pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1..1}" obtains err :: "'a" where "\<bar>err\<bar> \<le> tdev' p (trunc_err_pdevs p xs)" and "pdevs_val e (trunc_pdevs p xs) = pdevs_val e xs + err"
absolutely_integrable_bounded_measurable_product: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'a"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'b" assumes "bilinear h"   and "f \<in> borel_measurable (lebesgue_on S)"   and "S \<in> sets lebesgue"   and "bounded (f ` S)"   and "g absolutely_integrable_on S" shows "(\<lambda>x. h (f x) (g x)) absolutely_integrable_on S"
self_append_conv2: fixes ys :: "'k list"   and xs :: "'k list" shows "(ys = xs @ ys) = (xs = [])"
simps_2: fixes xs :: "('b, 'c) Combinators list" shows "removeShadowRules1_alternative_rev (DenyAll # xs) = [DenyAll]"
absorb1: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" shows "max a b = a"
non_TC_less_0: fixes x :: "V" shows "\<not> x \<sqsubset> 0"
prems_1: shows "\<Gamma>\<turnstile> (r, s) \<rightarrow> (r', s')"
R1_H2_commute2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R1 (H2 P) = H2 (R1 P)"
ord_eq_le_trans: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a = b"   and "subseq b c" shows "subseq a c"
WriteInner_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds"   and l :: "Locs"   and v :: "Vals" shows "WriteInner ch mm rs p l v \<equiv> $WrRequest ch p l v \<and> (GoodWrite mm rs p l v \<or> BadWrite mm rs p l v) \<and> unchanged (rtrner ch!p)"
Exit_no_sum_SDG_edge_source: fixes Vopt :: "'var option"   and popt :: "('pname \<times> bool) option"   and b :: "bool"   and n' :: "'node SDG_node" assumes "sum_SDG_edge (CFG_node (_Exit_)) Vopt popt b n'" shows "False"
polynomial_function_mult: fixes f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> 'd" assumes "polynomial_function f"   and "polynomial_function g" shows "polynomial_function (\<lambda>x. f x *\<^sub>R g x)"
comp_cod_arr: fixes f :: "'a2"   and b :: "'a2" assumes "C2.arr f"   and "C2.cod f = b" shows "b \<cdot>\<^sub>2 f = f"
sorted_sorted_list_of_set: fixes A :: "'t set" shows "ord_term_lin.sorted (ord_term_lin.sorted_list_of_set A)"
h3: shows "isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\<^sup>2 = isolate_variable_sparse p var 2 * (Var var)\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0"
induce_eq_iff_induced: fixes H :: "('a, 'b) pre_digraph" assumes "induced_subgraph H (G \<restriction> {v. u \<rightarrow>\<^sup>* v})" shows "(G \<restriction> {v. u \<rightarrow>\<^sup>* v}) \<restriction> verts H = H"
tendsto_in\<^sub>N_def: fixes N :: "'a quasinorm"   and u :: "nat \<Rightarrow> 'a"   and x :: "'a" shows "tendsto_in\<^sub>N N u x = (\<lambda>n. Norm N (u n - x)) \<longlonglongrightarrow> 0"
approx_from_below_dense_linorder: fixes y :: "'a"   and x :: "'a" assumes "y < x" shows "\<exists>u. (\<forall>n. u n < x) \<and> u \<longlonglongrightarrow> x"
mzeromap_mHom: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'j) Module_scheme" assumes "R module M"   and "R module N" shows "mzeromap M N \<in> mHom R M N"
bind_split: fixes P :: "'f option \<Rightarrow> bool"   and m :: "'g option"   and f :: "'g \<Rightarrow> 'f option" shows "P (m \<bind> f) = ((m = None \<longrightarrow> P None) \<and> (\<forall>v. m = Some v \<longrightarrow> P (f v)))"
cmp_components_are_iso: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.iso (\<Phi>\<^sub>G (f, g))"
graph_Entry_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b" shows "graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry \<equiv> graph_path \<alpha>e \<alpha>n invar inEdges' \<and> graph_Entry_axioms \<alpha>n invar inEdges' Entry"
zip_arg_subterm_1: fixes s :: "('a, 'b) Term.term"   and t :: "('c, 'd) Term.term"   and X :: "('a, 'b) Term.term list"   and Y :: "('c, 'd) Term.term list"   and f :: "'a" assumes "(s, t) \<in> set (zip X Y)" shows "s \<sqsubset> Fun f X"
FvarsT_num: 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 num :: "'b set"   and n :: "'b" assumes "Syntax_with_Numerals var trm fmla Var FvarsT substT Fvars subst num"   and "n \<in> num" shows "FvarsT n = {}"
lossless_map_pmf: fixes f :: "'e \<Rightarrow> 'd option"   and p :: "'e pmf" shows "lossless_spmf (map_pmf f p) = (\<forall>x\<in>set_pmf p. f x \<noteq> None)"
decr_length_less_iff: fixes n :: "nat" shows "(LENGTH('b) - Suc 0 < n) = (LENGTH('b) \<le> n)"
locally_sound_mode_use_def: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and lc :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)" assumes "sifum_security_init dma \<C>_vars eval" shows "sifum_security_init.locally_sound_mode_use dma \<C>_vars eval lc = (\<forall>c' mds' mem'. ((c', mds'), mem') \<in> sifum_security_init.loc_reach dma eval lc \<longrightarrow> (\<forall>x. (x \<in> mds' GuarNoReadOrWrite \<longrightarrow> sifum_security_init.doesnt_read_or_modify \<C>_vars eval c' x) \<and> (x \<in> mds' GuarNoWrite \<longrightarrow> sifum_security_init.doesnt_modify dma eval c' x)))"
cobounded2: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b \<preceq> b"
ex_three: shows "\<exists>x y z. x \<noteq> y \<and> x \<noteq> z \<and> y \<noteq> z"
perm_swap_2: fixes a :: "name"   and b :: "name"   and x :: "'d" shows "[(a, b)] \<bullet> [(b, a)] \<bullet> x = x"
Sup_subdistl: fixes x :: "'a"   and Y :: "'a set" shows "\<Squnion>{x \<cdot> y |y. y \<in> Y} \<le> x \<cdot> \<Squnion>Y"
permutations_of_multiset_nonempty: fixes A :: "'a multiset" assumes "A \<noteq> {#}" shows "permutations_of_multiset A = (\<Union>x\<in>set_mset A. (#) x ` permutations_of_multiset (A - {#x#}))"
supp_rel_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "supp_rel R x = {a. infinite {b. \<not> R ((a \<rightleftharpoons> b) \<bullet> x) x}}"
kdvd: shows "\<k> dvd \<v> * (\<Lambda> * (\<v> - 1) div (\<k> - 1) - \<k>)"
select_convs_2: fixes ISignature :: "('c, 'd) Signature"   and ICategory :: "('a, 'b) Category"   and ITypes :: "'c \<Rightarrow> 'a"   and IFunctions :: "'d \<Rightarrow> 'b"   and more :: "'e" shows "iC\<^bsub>\<lparr>ISignature = ISignature, ICategory = ICategory, ITypes = ITypes, IFunctions = IFunctions, \<dots> = more\<rparr>\<^esub> = ICategory"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs" shows "l_new_shadow_root_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
add_leaves_bdd_lookup: fixes n :: "nat"   and b :: "'a bdd"   and x :: "'a"   and xs :: "'a list" assumes "bddh n b" shows "(x \<in> set (add_leaves b xs)) = ((\<exists>bs. x = bdd_lookup b bs \<and> is_alph n bs) \<or> x \<in> set xs)"
hom_mult_eq_zero: fixes x :: "'a"   and y :: "'a" assumes "x * y = (0::'a)" shows "[:x:] * [:y:] = 0"
T'_def: shows "T' = (\<lambda>n x. - of_nat (j + n) * c n * inverse x ^ Suc (j + n))"
beforex_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and I :: "nat set"   and d :: "nat" shows "(P,C,M \<rhd> xt / I,d) = (\<exists>xt\<^sub>0 xt\<^sub>1. ex_table_of P C M = xt\<^sub>0 @ xt @ xt\<^sub>1 \<and> pcs xt\<^sub>0 \<inter> I = {} \<and> pcs xt \<subseteq> I \<and> (\<forall>pc\<in>I. \<forall>C pc' d'. match_ex_table P C pc xt\<^sub>1 = \<lfloor>(pc', d')\<rfloor> \<longrightarrow> d' \<le> d))"
in_degree_same: fixes z :: "'a" assumes "z \<in> pverts G" shows "in_degree (with_proj (subdivide G e w)) z = in_degree (with_proj G) z"
elementsAppend: fixes a :: "('a \<times> bool) list"   and b :: "('a \<times> bool) list" shows "elements (a @ b) = elements a @ elements b"
alpha_refl_1: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and bs :: "atom set"   and f :: "'e \<Rightarrow> atom set" assumes "R x x" shows "(bs, x) \<approx>set R f 0 (bs, x)"
WTrts_shext_mono: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and es :: "char list exp list"   and Ts :: "ty list"   and sh' :: "sheap" assumes "P,E,h,sh \<turnstile> es [:] Ts"   and "sh \<unlhd>\<^sub>s sh'"   and "\<not> sub_RIs es" shows "P,E,h,sh' \<turnstile> es [:] Ts"
indRelTEQ_impl_TRel_is_weak_barbed_contrasimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_barbed_contrasimulation (indRelTEQ TRel) (STCalWB SWB TWB)" shows "weak_barbed_contrasimulation (TRel\<^sup>*) TWB"
sub_2: shows "vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2"
Struct_axioms: shows "SStruct Ik_intT Ik_intF Ik_intP"
insertCI: fixes a :: "'b"   and B :: "'b set"   and b :: "'b" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> insert b B"
encode_all_frame_axioms_def: fixes \<Pi> :: "'a strips_problem"   and t :: "nat" shows "encode_all_frame_axioms \<Pi> t \<equiv> let l = List.product [0..<t] (\<Pi>\<^sub>\<V>) in \<^bold>\<And>map (\<lambda>(k, v). encode_negative_transition_frame_axiom \<Pi> k v) l @ map (\<lambda>(k, v). encode_positive_transition_frame_axiom \<Pi> k v) l"
simps_1: fixes s :: "'a list" shows "\<ff> s 0 = 0"
Inf_superset_mono: fixes B :: "'a set"   and A :: "'a set" assumes "B \<subseteq> A" shows "\<Sqinter> A \<le> \<Sqinter> B"
fun_relD2: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'e \<Rightarrow> 'f"   and Ra :: "('c \<times> 'e) set"   and Rr :: "('d \<times> 'f) set"   and x' :: "'e"   and r' :: "'f" assumes "(f, f') \<in> Ra \<rightarrow> Rr"   and "f' x' = r'" shows "\<forall>x. (x, x') \<in> Ra \<longrightarrow> (f x, r') \<in> Rr"
singleton_fset_holds: fixes M :: "'b fset"   and f :: "'b \<Rightarrow> 'c"   and m :: "'b" assumes "fBall M (\<lambda>x. fBall M (\<lambda>y. f x = f y))"   and "m |\<in>| M" shows "f m = fthe_elem (f |`| M)"
LI_in_SMP_subset_single: fixes S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S' :: "('fun, 'var) strand_step list"   and \<theta>' :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "(S, \<theta>) \<leadsto> (S', \<theta>')"   and "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta>"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t S)"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t S)"   and "list_all tfr\<^sub>s\<^sub>t\<^sub>p S"   and "trms\<^sub>s\<^sub>t S \<subseteq> SMP M" shows "trms\<^sub>s\<^sub>t S' \<subseteq> SMP M"
infinite_descent0: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "\<And>n. \<lbrakk>0 < n; \<not> P n\<rbrakk> \<Longrightarrow> \<exists>m<n. \<not> P m" shows "P n"
commute: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "abel_semigroup f" shows "f a b = f b a"
case_cong_weak: fixes generat :: "('b, 'd, 'e) generat"   and generat' :: "('b, 'd, 'e) generat"   and f1 :: "'b \<Rightarrow> 'f"   and f2 :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "generat = generat'" shows "(case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = (case generat' of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa)"
supp_fun_app: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "supp (f x) \<subseteq> supp f \<union> supp x"
span_in_category_axioms: shows "span_in_category (\<cdot>) S"
not_eq_order_implies_strict: fixes a :: "('x, nat) pp"   and b :: "('x, nat) pp" assumes "a \<noteq> b"   and "ord a b" shows "ord_strict a b"
sum_two_elements: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'b" assumes "class.comm_monoid_add plus zero"   and "a \<noteq> b" shows "comm_monoid_add.sum plus zero f {a, b} = plus (f a) (f b)"
n2_ch: shows "|~ \<langle>n2\<rangle>_vars = n2"
Emp_int_def: shows "\<epsilon> \<equiv> abs_interface \<epsilon>\<^sub>c"
assms_1: shows "\<forall>x\<in>X. x \<le> c"
assms_1: fixes z :: "nat" shows "\<lbrace>p \<and> b \<and> v =\<^sub>u \<guillemotleft>z\<guillemotright>\<rbrace> S \<lbrace>p \<and> v <\<^sub>u \<guillemotleft>z\<guillemotright>\<rbrace>\<^sub>u"
\<section>: fixes z :: "'a"   and a :: "'b"   and b :: "'b" assumes "\<And>x. x \<in> T \<and> x \<notin> S \<Longrightarrow> g x = f x"   and "((\<lambda>x. if x \<in> T then f x else (0::'a)) has_integral z) (cbox a b)" shows "((\<lambda>x. if x \<in> T then g x else (0::'a)) has_integral z) (cbox a b)"
uminus_cont: fixes f :: "'c \<Rightarrow> 'd" assumes "f \<in> bcontfun" shows "(\<lambda>x. - f x) \<in> bcontfun"
diff_diff_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (minus a b) c = minus a (plus b c)"
BetS_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.BetS Bet A B C \<equiv> Bet A B C \<and> A \<noteq> B \<and> B \<noteq> C"
eq_on_def: fixes A :: "'c set" shows "eq_on A = (\<lambda>x y. x \<in> A \<and> x = y)"
hash_blindable_def: fixes h :: "'c \<Rightarrow> 'd" shows "hash_blindable h = hash_blindable' \<circ> map_blindable\<^sub>m h id"
\<i>_simps_1: fixes a :: "'c" assumes "C.obj a" shows "D.seq (F \<i>\<^sub>C[a]) (\<Phi> (a, a))"
cone_decomp_map_times: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and s :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "cone_decomp T ps" shows "cone_decomp ((*) s ` T) (map (apfst ((*) s)) ps)"
translation_apply: fixes a :: "'a"   and b :: "'a" assumes "a \<in> M"   and "b \<in> M" shows "(a)\<^sub>L b = a \<cdot> b"
disc_10: fixes x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a" shows "\<not> isSend (RecvMarker x51 x52 x53)"
msubsteq2: fixes vs :: "'b list"   and c :: "poly"   and a :: "poly"   and b :: "tm"   and x :: "'b"   and bs :: "'b list"   and t :: "tm" assumes "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> (0::'b)"   and "islin (Eq (CNP 0 a b))" shows "Ifm vs (x # bs) (msubsteq2 c t a b) = Ifm vs (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> # bs) (Eq (CNP 0 a b))"
Inf_pres_Sup_pres: fixes f :: "'c \<Rightarrow> 'd" shows "Inf_pres f = Sup_pres (\<partial>\<^sub>F f)"
allow_all_id_def: shows "A\<^sub>I \<equiv> A\<^sub>p (id \<circ> Some)"
closed_orbit_def: fixes x :: "'a" shows "rev.closed_orbit x = (\<exists>T\<in>rev.existence_ivl0 x. T \<noteq> 0 \<and> rev.flow0 x T = x)"
sum_closed: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'v" assumes "finite A"   and "f ` A \<subseteq> V" shows "sum f A \<in> V"
assms_3: shows "(c, s) \<rightarrow>*t s'"