hcju/isabelleps huggingface dataset:
cast_document_ptr_not_node_ptr_2: fixes node_ptr :: "(_) node_ptr"   and document_ptr :: "(_) document_ptr" shows "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 node_ptr \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 document_ptr"
sigma_algebra_single_set: fixes X :: "'c set"   and S :: "'c set" assumes "X \<subseteq> S" shows "sigma_algebra S {{}, X, S - X, S}"
mult_right_zero: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<odot> mbot = mbot"
is_unit_mult_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "dvd.dvd times (times a b) one = (dvd.dvd times a one \<and> dvd.dvd times b one)"
comp_associative: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<odot> y \<odot> z = x \<odot> (y \<odot> z)"
inj_comm_ring_hom_axioms: shows "inj_comm_ring_hom to_fract"
upd_set'_aux3: fixes m :: "('c, 'a) mapping"   and k :: "'c"   and v :: "'a"   and a :: "'b" assumes "Mapping.lookup m k = Some v" shows "upd_set' m d f {b. b = k \<or> (a, b) \<in> A} = Mapping.update k (f v) (upd_set' m d f {b. (a, b) \<in> A})"
inverse_def: shows "inverse = (\<lambda>u\<in>G. THE v. v \<in> G \<and> u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>)"
bin_fst_nemp: shows "u\<^sub>0 \<noteq> \<epsilon>"
Cons_1: fixes c1 :: "('a, 'b) conf"   and c2 :: "('a, 'b) conf" assumes "finite_labels ls"   and "finite (pred c1)"   and "finite (pred c2)"   and "\<forall>e\<in>pred c1. finite (Bexp.vars e)"   and "\<forall>e\<in>pred c2. finite (Bexp.vars e)"   and "c2 \<sqsubseteq> c1"   and "feasible c2 ls" shows "feasible c1 ls"
open_bcontfun_def: fixes S :: "('a \<Rightarrow>\<^sub>C 'b) set" shows "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
renamings_apart_is_Var: fixes Cs :: "('a, nat) Term.term literal multiset list" shows "\<forall>\<sigma>\<in>set (renamings_apart Cs). \<forall>x. is_Var (\<sigma> x)"
is_arrow_of_tabulations_in_maps: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg s s.tab (tab\<^sub>0 s) (tab\<^sub>1 s) s s.tab (tab\<^sub>0 s) (tab\<^sub>1 s) s"
Rule_in_nodes: fixes r :: "'rule" shows "(Rule r \<in> i.R nodes) = (r \<in> i.R rules)"
cases: fixes a1 :: "'c"   and a2 :: "'c"   and r :: "('c \<times> 'c) set" assumes "(a1, a2) \<in> r\<^sup>*" obtains a :: "'c" where "a1 = a__" and "a2 = a__"   | a :: "'c" and b :: "'c" and c :: "'c" where "a1 = a" and "a2 = c" and "(a, b) \<in> r\<^sup>*" and "(b, c) \<in> r"
sel_9: fixes x51 :: "Core.term"   and x52 :: "Core.term" shows "un_App2 (x51 $ x52) = x52"
callD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr"   and chain :: "char list"   and rs :: "'c rule list" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m (Call chain)]"   and "s = Undecided"   and "matches \<gamma> m p"   and "\<Gamma> chain = Some rs" obtains "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"   | rs\<^sub>1 :: "'c rule list" and rs\<^sub>2 :: "'c rule list" and m' :: "'c match_expr" where "rs = rs\<^sub>1 @ Rule m' Return # rs\<^sub>2" and "matches \<gamma> m' p" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> Undecided" and "t = Undecided"
cop_F_cases: fixes x :: "'x"   and ds :: "'d set"   and fp :: "'x set" assumes "x \<in> cop_F ds fp" obtains "x \<in> fp"   | "x \<in> CD_on ds (- Rf CH fp) - fp"
cancel_eq: fixes x :: "ordinal"   and y :: "ordinal" shows "(F x = F y) = (x = y)"
b: fixes x :: "'a" assumes "x \<in> S" shows "0 < b x \<and> cball x (b x) \<subseteq> S"
IH: assumes "\<forall>x\<in>set (tl ms). x \<noteq> n" shows "dominates g n m'"
abs_convergent_prod_def: fixes f :: "nat \<Rightarrow> 'e" shows "abs_convergent_prod f = convergent_prod (\<lambda>i. 1 + norm (f i - (1::'e)))"
cSUP_le_iff: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and u :: "'a" assumes "A \<noteq> {}"   and "bdd_above (f ` A)" shows "(Sup (f ` A) \<le> u) = (\<forall>x\<in>A. f x \<le> u)"
simps_4: fixes x2 :: "'d"   and x11 :: "'c"   and x12 :: "'c" shows "Leaf x2 \<noteq> Branch x11 x12"
M1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>F \<longrightarrow> \<box>[F \<and> \<circle>F]_v"
preserves_cod: fixes f :: "'a" assumes "Go\<tau>.A.arr f" shows "Go\<tau>.B.cod (\<tau> f) = G (Go\<tau>.A.cod f)"
left_diff_distrib: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec"   and z :: "(real, 3) vec" shows "cross3 (x - y) z = cross3 x z - cross3 y z"
mult_of_is_Units: fixes R :: "('c, 'd) ring_scheme" assumes "field R" shows "Multiplicative_Group.mult_of R = units_of R"
le_N: fixes n :: "nat"   and a :: "'a" assumes "(n, a) \<in> set (triangular_to_jnf_vector A)"   and "norm a = 1" shows "n \<le> N"
assms_3: shows "E2 = (\<lambda>s. 1 + 3 * (nat (s ''b'') - nat (s ''g'')))"
validn_def: fixes K :: "nat"   and C :: "Class"   and m :: "Method"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "\<Turnstile>\<^sub>K \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I  = (\<forall>M. mbody_is C m M \<longrightarrow> (\<forall>Mspec Minv Anno. MST\<down>(C, m) = Some (Mspec, Minv, Anno) \<longrightarrow> (\<forall>par code l0. M = (par, code, l0) \<longrightarrow> (\<forall>s0 s. MS M l0 (mkState s0) l s \<longrightarrow> A s0 s \<longrightarrow> (\<forall>k\<le>K. (\<forall>h v. (M, l, s, k, h, v) \<in> Exec \<longrightarrow> B s0 s (h, v)) \<and> (\<forall>ll r. ((M, l, s, k, ll, r) \<in> MStep \<longrightarrow> (\<forall>Q. Anno\<down>ll = Some Q \<longrightarrow> Q s0 r)) \<and> ((M, l, s, k, r) \<in> Reachable \<longrightarrow> I s0 s (heap r))))))))"
rc: fixes s :: "'a"   and t :: "'a"   and a :: "'b"   and u :: "'a"   and b :: "'b" assumes "(s, t) \<in> (E a)\<inverse>"   and "(s, u) \<in> R b" shows "(t, u) \<in> local.conversion' (under r a \<inter> under r b) O (R b)\<^sup>= O local.conversion' (under r b) O E a O local.conversion' (under r b) \<or> (t, u) \<in> local.conversion' (under r a) O (R b)\<^sup>= O local.conversion' (under r a \<union> under r b)"
derived_incl: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H"   and "subgroup H G" shows "derived G K \<subseteq> H"
f_shrink_last_eq_f_shrink_last_take: fixes xs :: "'b list"   and k :: "nat" shows "xs \<down> (length xs div k * k) \<div>\<^sub>l k = xs \<div>\<^sub>l k"
le_infI2: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d"   and x :: "'d"   and a :: "'d" assumes "class.semilattice_inf inf less_eq less"   and "less_eq b x" shows "less_eq (inf a b) x"
containsCallcases: fixes procs :: "(char list \<times> char list list \<times> char list list \<times> cmd) list"   and prog :: "cmd"   and ps :: "char list list"   and p :: "char list" assumes "containsCall procs prog ps p" shows "ps = [] \<and> containsCall procs prog ps p \<or> (\<exists>q ins outs c ps'. ps = ps' @ [q] \<and> (q, ins, outs, c) \<in> set procs \<and> containsCall procs c [] p \<and> containsCall procs prog ps' q)"
eq_onp_compp_Grp: fixes P :: "'n \<Rightarrow> bool"   and A :: "'n set"   and f :: "'n \<Rightarrow> 'o" shows "eq_onp P OO Grp A f = Grp (Collect P \<inter> A) f"
s0g_defs_1: shows "s0g \<equiv> \<lparr>init = s0g_init, trans = s0g_trans, obs = id\<rparr>"
set_pmf_closed: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and s :: "'a"   and D :: "'a pmf"   and t :: "'a" assumes "Finite_Markov_Decision_Process K S"   and "s \<in> S"   and "D \<in> K s"   and "t \<in> set_pmf D" shows "t \<in> S"
ht_at_phl_while: fixes p :: "'a"   and s :: "'a"   and x :: "'a"   and w :: "'a" assumes "\<lbrace>t p \<cdot> t s\<rbrace>x\<lbrace>t p\<rbrace>" shows "\<lbrace>t p\<rbrace>(t s \<cdot> x)\<^sup>\<star> \<cdot> t w\<lbrace>t p \<cdot> t w\<rbrace>"
bv_to_nat_emsapss_smaller_pq: fixes p :: "nat"   and q :: "nat"   and m :: "bit list" assumes "prime p"   and "prime q"   and "p \<noteq> q"   and "emsapss_encode m (length (nat_to_bv (p * q)) - Suc 0) \<noteq> []" shows "bv_to_nat (emsapss_encode m (length (nat_to_bv (p * q)) - Suc 0)) < p * q"
get_disconnected_nodes_locs_impl: shows "get_disconnected_nodes_locs = CD.a_get_disconnected_nodes_locs"
valid_line_prepend_B_iff: fixes xs :: "color list" shows "valid_line (B # xs) = valid_line xs"
inv_swap_id: fixes a :: "'a"   and b :: "'a" shows "inv' (Fun.swap a b id) = Fun.swap a b id"
INF_greatest: 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 A :: "'b set"   and u :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq u (f i)" shows "less_eq u (Inf (f ` A))"
BinOpRedsVal: fixes e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(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 E :: "char list \<Rightarrow> ty option"   and e\<^sub>2 :: "expr"   and v\<^sub>2 :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and bop :: "bop"   and v :: "val" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v\<^sub>2,s\<^sub>2\<rangle>"   and "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "P,E \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>2\<rangle>"
Gr_Image_image: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'e"   and B :: "'f set" shows "BNF_Def.Gr A f `` B = f ` (A \<inter> B)"
minimal_in_set: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> nat" assumes "xs \<noteq> []" shows "minimal f xs \<in> set xs"
Max_eq_image_if: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite S"   and "finite T"   and "\<forall>x\<in>S. \<exists>y\<in>T. f x \<le> g y"   and "\<forall>x\<in>T. \<exists>y\<in>S. g x \<le> f y" shows "Max (f ` S) = Max (g ` T)"
init_tau_gap: fixes i :: "nat"   and t :: "nat" shows "\<tau> i \<triangleright> (gap i t - 1) = e_take (gap i t) (block i t)"
HC_contrapos_nn: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> F"   and "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H G \<^bold>\<rightarrow> F" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> G"
simps_1: fixes x :: "'b"   and y :: "'b" shows "extT.Val x + extT.Val y = extT.Val (x + y)"
smooth_on_scaleR_fun: fixes k :: "enat"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and c :: "real" assumes "k-smooth_on S g"   and "open S" shows "k-smooth_on S (c *\<^sub>R g)"
proots_within_smult: fixes a :: "'b"   and p :: "'b poly"   and s :: "'b set" assumes "a \<noteq> (0::'b)" shows "proots_within (smult a p) s = proots_within p s"
IH: fixes y :: "term" assumes "(y, t) \<in> measure size"   and "linear y" shows "P y"
A: shows "((s, c), w, s', c') \<in> trcl (trss fg)"
fresher_def: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and r' :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" shows "r \<sqsubseteq> r' \<equiv> nsqn\<^sub>r r < nsqn\<^sub>r r' \<or> nsqn\<^sub>r r = nsqn\<^sub>r r' \<and> \<pi>\<^sub>5 r' \<le> \<pi>\<^sub>5 r"
edge_preserving_def: fixes h :: "('d \<times> 'e) set"   and e1 :: "('f \<times> 'd \<times> 'd) set"   and e2 :: "('f \<times> 'e \<times> 'e) set" shows "edge_preserving h e1 e2 \<equiv> \<forall>(k, v1, v2)\<in>e1. \<forall>v1' v2'. (v1, v1') \<in> h \<and> (v2, v2') \<in> h \<longrightarrow> (k, v1', v2') \<in> e2"
iso_eq: fixes G :: "('c, 'd) monoid_scheme"   and f :: "'c \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme"   and f' :: "'c \<Rightarrow> 'e" assumes "Group.group G"   and "f \<in> Group.iso G H"   and "\<And>x. x \<in> carrier G \<Longrightarrow> f' x = f x" shows "f' \<in> Group.iso G H"
trms\<^sub>s\<^sub>t_def: fixes S :: "('d, 'e) strand_step list" shows "trms\<^sub>s\<^sub>t S \<equiv> \<Union> (trms\<^sub>s\<^sub>t\<^sub>p ` set S)"
less_pre_n: fixes n :: "nat" assumes "0 < n" shows "n - Suc 0 < n"
cases: fixes x :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray" obtains f :: "int vec list \<times> int vec list" and mu :: "int iarray iarray" and d :: "int iarray" where "x = (f, mu, d)"
spl: shows "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF newVs"
xin: shows "x \<in> (\<Inter>a\<in>C. {x \<in> S. f x \<inter> T \<inter> ball a (e / 2) \<noteq> {}})"
length_emsapss_smaller_pq: fixes p :: "nat"   and q :: "nat"   and m :: "bit list" assumes "prime p"   and "prime q"   and "emsapss_encode m (length (nat_to_bv (p * q)) - Suc 0) \<noteq> []"   and "(length (nat_to_bv (p * q)) - Suc 0) mod 8 \<noteq> 0" shows "length (remzero (emsapss_encode m (length (nat_to_bv (p * q)) - Suc 0))) < length (nat_to_bv (p * q))"
rep_eq: fixes x :: "'b interval" shows "bounds_of_interval (- x) = (case bounds_of_interval x of (a, b) \<Rightarrow> (- b, - a))"
smc_SemiCAT_def: fixes \<alpha> :: "V" shows "smc_SemiCAT \<alpha> = [ZFC_in_HOL.set {\<CC>. semicategory \<alpha> \<CC>}, all_smcfs \<alpha>, \<lambda>\<FF>\<in>\<^sub>\<circ>all_smcfs \<alpha>. \<FF>\<lparr>HomDom\<rparr>, \<lambda>\<FF>\<in>\<^sub>\<circ>all_smcfs \<alpha>. \<FF>\<lparr>HomCod\<rparr>, \<lambda>\<GG>\<FF>\<in>\<^sub>\<circ>composable_arrs (dg_SemiCAT \<alpha>). \<GG>\<FF>\<lparr>[]\<^sub>\<circ>\<rparr> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG>\<FF>\<lparr>1\<^sub>\<nat>\<rparr>]\<^sub>\<circ>"
assms: shows "2 \<cdot> g \<cdot> x = 2 \<cdot> g \<cdot> h + v \<cdot> v"
X_def: shows "X = {{i, j} |i j. i < j \<and> P (f i) (f j)}"
strict_prefix_take_eq_strict_prefix_take_ex: fixes xs :: "'b list"   and ys :: "'b list" shows "(ys \<down> length xs = xs \<and> xs \<noteq> ys) = ((\<exists>n. ys \<down> n = xs) \<and> xs \<noteq> ys)"
faithful_functor_axioms_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'e" shows "faithful_functor_axioms A F \<equiv> \<forall>f f'. partial_magma.arr A f \<and> partial_magma.arr A f' \<and> partial_magma.dom A f = partial_magma.dom A f' \<and> partial_magma.cod A f = partial_magma.cod A f' \<longrightarrow> F f = F f' \<longrightarrow> f = f'"
prems_2: shows "\<forall>i\<in>insert x I. finite (S i)"
simps: fixes e :: "'a"   and a :: "'b"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" shows "SkewBinomialHeapStruc.skewlink e a t t' = (if a \<le> SkewBinomialHeapStruc.prio t \<and> a \<le> SkewBinomialHeapStruc.prio t' then SkewBinomialHeapStruc.Node e a (Suc (SkewBinomialHeapStruc.rank t)) [t, t'] else if SkewBinomialHeapStruc.prio t \<le> SkewBinomialHeapStruc.prio t' then SkewBinomialHeapStruc.Node (SkewBinomialHeapStruc.val t) (SkewBinomialHeapStruc.prio t) (Suc (SkewBinomialHeapStruc.rank t)) (SkewBinomialHeapStruc.Node e a 0 [] # t' # SkewBinomialHeapStruc.children t) else SkewBinomialHeapStruc.Node (SkewBinomialHeapStruc.val t') (SkewBinomialHeapStruc.prio t') (Suc (SkewBinomialHeapStruc.rank t')) (SkewBinomialHeapStruc.Node e a 0 [] # t # SkewBinomialHeapStruc.children t'))"
reflect_poly_pCons': fixes p :: "'a poly"   and c :: "'a" assumes "p \<noteq> 0" shows "reflect_poly (pCons c p) = reflect_poly p + monom c (Suc (degree p))"
"case": fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(case OptionT x of OptionT x \<Rightarrow> f x) = f x"
af_respectfulness_1: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and \<nu> :: "'a set" assumes "\<phi> \<longrightarrow>\<^sub>P \<psi>" shows "af_letter \<phi> \<nu> \<longrightarrow>\<^sub>P af_letter \<psi> \<nu>"
inf_sup_aci_4: fixes 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 x :: "'a"   and y :: "'a" assumes "class.lattice inf less_eq less sup" shows "inf x (inf x y) = inf x y"
q_2: shows "I = PIdl\<^bsub>K [X]\<^esub> q"
intro: 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 set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   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"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_disconnected_nodes\<^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 set_disconnected_nodes set_disconnected_nodes_locs"   and "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"   and "l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf" shows "l_set_disconnected_nodes_get_host\<^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 set_disconnected_nodes set_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_host get_host_locs"
8: shows "coreflexive (e * (f \<sqinter> q))"
inline_parametric: fixes S :: "'d \<Rightarrow> 'i \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'j \<Rightarrow> bool"   and C' :: "'g \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'l \<Rightarrow> 'm \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun C (rel_gpv (rel_prod (=) S) C'))) (rel_fun (rel_gpv A C) (rel_fun S (rel_gpv (rel_prod A S) C'))) inline inline"
llength_conv_fixp: shows "llength \<equiv> lfp.fixp_fun (\<lambda>llength xs. case xs of LNil \<Rightarrow> 0 | LCons x xs \<Rightarrow> eSuc (llength xs))"
external_call_hconf: fixes t :: "'thread_id"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap"   and U :: "ty" assumes "P,t \<turnstile> \<langle>a\<bullet>M(vs),h\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>"   and "P,h \<turnstile> a\<bullet>M(vs) : U"   and "hconf h" shows "hconf h'"
Lim_null: fixes f :: "'e \<Rightarrow> 'f"   and l :: "'f"   and net :: "'e filter" shows "(f \<longlongrightarrow> l) net = ((\<lambda>x. f x - l) \<longlongrightarrow> (0::'f)) net"
hyps: shows "monoid.truncate (RDirProd_list Rs) = DirProd_list Rs"
no_\<tau>Move1_\<tau>s_to_no_\<tau>Move2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ls2 :: "('l, 't) locks"   and ts2 :: "'t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option"   and m2 :: "'m2"   and ws2 :: "'t \<Rightarrow> 'w wait_set_status option"   and is2 :: "'t set" assumes "s1 \<approx>m (ls2, (ts2, m2), ws2, is2)" shows "\<exists>ts2'. \<tau>mRed2 (ls2, (ts2, m2), ws2, is2) (ls2, (ts2', m2), ws2, is2) \<and> (\<forall>t. wset s1 t = None \<and> (\<exists>x. thr s1 t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> (\<forall>x' m'. \<not> r1.silent_move t (x, shr s1) (x', m'))) \<longrightarrow> wset (ls2, (ts2', m2), ws2, is2) t = None \<and> (\<exists>x. thr (ls2, (ts2', m2), ws2, is2) t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> (\<forall>x' m'. \<not> r2.silent_move t (x, shr (ls2, (ts2', m2), ws2, is2)) (x', m')))) \<and> s1 \<approx>m (ls2, (ts2', m2), ws2, is2)"
int_div_sub_1: fixes m :: "int"   and n :: "int" assumes "1 \<le> m" shows "(n - 1) div m = (if m dvd n then n div m - 1 else n div m)"
simps: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and a :: "('a, 'b) Term.term"   and M :: "('a, 'b) Term.term set" assumes "typed_model arity public Ana \<Gamma>" shows "(a \<in> typed_model.SMP arity Ana \<Gamma> M) = ((\<exists>t. a = t \<and> t \<in> M) \<or> (\<exists>t t'. a = t' \<and> t \<in> typed_model.SMP arity Ana \<Gamma> M \<and> t' \<sqsubseteq> t) \<or> (\<exists>t \<delta>. a = t \<cdot> \<delta> \<and> t \<in> typed_model.SMP arity Ana \<Gamma> M \<and> typed_model.wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<Gamma> \<delta> \<and> (\<forall>t\<in>subst_range \<delta>. intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t)) \<or> (\<exists>t K T k. a = k \<and> t \<in> typed_model.SMP arity Ana \<Gamma> M \<and> Ana t = (K, T) \<and> k \<in> set K))"
part_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and pivot :: "'a"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.part less f pivot xs = (filter (\<lambda>x. less (f x) pivot) xs, filter (\<lambda>x. f x = pivot) xs, filter (\<lambda>x. less pivot (f x)) xs)"