hcju/isabelleps huggingface dataset:
dvd_prime_factors: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "y \<noteq> zero"   and "dvd.dvd times x y" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize x) \<subseteq> set_mset (factorial_semiring.prime_factorization zero times one normalize y)"
that: fixes t :: "'a" assumes "t \<in> T"   and "d s \<le> d t" shows "thesis"
eq_refl: fixes x :: "'a multiset"   and y :: "'a multiset" assumes "x = y" shows "x \<subseteq># y"
outstanding_non_volatile_writes_unshared_nth_update: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and i :: "nat"   and sb :: "'b memref list"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "outstanding_non_volatile_writes_unshared \<S> ts"   and "i < length ts"   and "non_volatile_writes_unshared \<S> sb" shows "outstanding_non_volatile_writes_unshared \<S> (ts[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
same_prob: fixes R' :: "('a \<times> 'a) set" assumes "linorder_on A R'" shows "emeasure M {R'} = emeasure M {R}"
cases: fixes a1 :: "'node SDG_node set"   and a2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a3 :: "'node list"   and a4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a5 :: "'edge list"   and a6 :: "'node list"   and a7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "a1,a2 \<turnstile> (a3,a4) =a5\<Rightarrow>\<^sub>\<tau> (a6,a7)" obtains ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" where "a1 = Sa__" and "a2 = fa__" and "a3 = msa__" and "a4 = sb__" and "a5 = []" and "a6 = msa__" and "a7 = sb__" and "length msa__ = length sb__"   | S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" and ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and a :: "'edge" and ms' :: "'node list" and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as :: "'edge list" and ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "a1 = S" and "a2 = f" and "a3 = ms" and "a4 = s" and "a5 = a # as" and "a6 = ms''" and "a7 = s''" and "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')" and "S,f \<turnstile> (ms',s') =as\<Rightarrow>\<^sub>\<tau> (ms'',s'')"
that: fixes ys :: "'a list"   and zs :: "'a stream" assumes "xs = ys @- x ## zs"   and "x \<notin> set ys" shows "thesis"
exhaust_sel: fixes BsSkewElem :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem" obtains "BsSkewElem = BsSkewBinomialHeapStruc.Element (BsSkewBinomialHeapStruc.un_Element1 BsSkewElem) (BsSkewBinomialHeapStruc.eprio BsSkewElem) (BsSkewBinomialHeapStruc.un_Element3 BsSkewElem)"
subst_apply_stateful_strand_def: fixes S :: "('e, 'f) stateful_strand_step list"   and \<theta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta> \<equiv> map (\<lambda>x. x \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) S"
outputIneqTrans: fixes a :: "name"   and b :: "name"   and P :: "pi"   and c :: "name"   and d :: "name"   and P' :: "pi" assumes "a{b}.P \<longmapsto> c[d] \<prec> P'"   and "a \<noteq> c \<or> b \<noteq> d" shows "False"
newton_coefficients: shows "local.newton_coefficients = rev (map local.c [0..<n])"
SigmaD1: fixes a :: "'d"   and b :: "'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "(a, b) \<in> Sigma A B" shows "a \<in> A"
finsum_Un_disjoint: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> nat \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "g \<in> A \<rightarrow> carrier P"   and "g \<in> B \<rightarrow> carrier P" shows "finsum P g (A \<union> B) = finsum P g A \<oplus>\<^bsub>P\<^esub> finsum P g B"
fds_truncate_plus_remainder: fixes m :: "nat"   and f :: "'a fds" shows "fds_truncate m f + fds_remainder m f = f"
vec_to_iarray_column': fixes k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec" assumes "k < ncols A" shows "vec_to_iarray (column (mod_type_class.from_nat k) A) = column_iarray k (matrix_to_iarray A)"
assms_1: shows "list_all (\<lambda>P. index_free P j) Ps"
eq_\<I>_converter_trans: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and conv' :: "('a, 'b, 'c, 'd) converter"   and conv'' :: "('a, 'b, 'c, 'd) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'"   and "\<I>,\<I>' \<turnstile>\<^sub>C conv' \<sim> conv''" shows "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv''"
dvd_finite_3_unfold: fixes x :: "finite_3"   and y :: "finite_3" shows "(x dvd y) = (x = a\<^sub>2 \<or> x = a\<^sub>3 \<or> y = a\<^sub>1)"
LIM_less_bound: fixes b :: "real"   and x :: "real"   and f :: "real \<Rightarrow> real" assumes "b < x"   and "\<forall>x'\<in>{b<..<x}. 0 \<le> f x'"   and "isCont f x" shows "0 \<le> f x"
bind_def: fixes x :: "('d, 'f) state"   and f :: "'f \<Rightarrow> ('d, 'e) state" shows "x \<bind> f = State (\<lambda>s. case run_state x s of (a, s') \<Rightarrow> run_state (f a) s')"
semilattice_axioms: shows "semilattice (\<cdot>)"
Zero_fract_def: shows "0 = Fraction_Field.Fract (0::'c) (1::'c)"
that_3: fixes x :: "'c" assumes "x \<in> S" shows "g x \<noteq> (0::'d)"
elims: fixes x :: "'c list"   and y :: "'c list" assumes "tail x = y" obtains uu_ :: "'c" and xs :: "'c list" where "x = uu_ # xs" and "y = xs"   | "x = []" and "y = undefined"
Diff_Int2: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set" shows "A \<inter> C - B \<inter> C = A \<inter> C - B"
test_shunting: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "test x"   and "test y" shows "(x \<sqinter> y \<le> z) = (x \<le> ! y \<squnion> z)"
left_unique_alt_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "left_unique R = (R OO R\<inverse>\<inverse> \<le> (=))"
embP2_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "embP2 P = (\<lambda>(ps, n). \<forall>s. P (Partial_Evaluation.emb ps s) \<and> n = (0::'a))"
HR_def: fixes x :: "hf"   and x' :: "hf" shows "HR x x' \<equiv> \<exists>s k. SeqHR x x' s k"
FIN_def: shows "FIN \<equiv> {U. \<exists>\<psi> s. learn_fin \<psi> U s}"
C_def: fixes X :: "'a set"   and m :: "nat" shows "C X m = X \<inter> {x. ball x (1 / real (Suc m)) \<subseteq> d x}"
SCF_R3: fixes xs :: "'c option list"   and P :: "'c option process"   and ys :: "'c option list"   and Y :: "'c option set"   and Q :: "'c option process" assumes "xs \<in> sentences P"   and "(ys, Y) \<in> failures Q"   and "ys \<noteq> []" shows "(xs @ ys, Y) \<in> seq_comp_failures P Q"
guha_deterministic2: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and ft :: "('c, 'e) flow_entry_match list"   and p :: "'d"   and x1 :: "'e"   and a :: "'e" assumes "no_overlaps \<gamma> ft"   and "guha_table_semantics \<gamma> ft p (Some x1)"   and "guha_table_semantics \<gamma> ft p (Some a)" shows "x1 = a"
kg: shows "k \<le> degree g \<and> g \<noteq> 0"
assms_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and exp :: "exp" assumes "atom ` domA \<Gamma> \<sharp>* c"   and "y = Let \<Gamma> exp" shows "P"
fst_initmissing_netgmap_default_aodv_init_netlift: fixes s :: "((state \<times> 'a) \<times> 'b) net_state" shows "fst (initmissing (netgmap fst s)) = Lib.default aodv_init (netlift fst s)"
lossless_R2: fixes m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool" shows "lossless_spmf (R2_14_OT m1 m2)"
set_fold_cfc_def: shows "set_fold_cfc \<equiv> map_fun comp_fun_commute_apply id Finite_Set.fold"
strict_domI1: fixes j :: "nat"   and res :: "nat list"   and i :: "nat" assumes "dom_kildall start ! j = res"   and "i \<in> set res" shows "strict_dom i j"
homeomorphic_space_refl: fixes X :: "'a topology" shows "X homeomorphic_space X"
finitely_many_partition_on: fixes A :: "'a set" assumes "finite A" shows "finite {P. partition_on A P}"
simps_1: fixes v1 :: "'c \<times> nat"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "SubRel.ces v1 [] v2 subs = (v1 = v2 \<or> (v1, v2) \<in> subs\<^sup>+)"
n_mult_comm: fixes x :: "'a"   and y :: "'a" shows "!x \<cdot> !y = !y \<cdot> !x"
intro: fixes D :: "nat" assumes "\<not> is_square D" shows "pell D"
gpath_subset: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and V :: "(nat \<times> 'd) set"   and r :: "(nat \<times> 'd) list"   and v :: "nat \<times> 'd"   and U :: "(nat \<times> 'd) set" assumes "gpath A w V r v"   and "set (gstates r v) \<subseteq> U" shows "gpath A w U r v"
sep_disj_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.pre_sep_algebra plus zero sep_disj" shows "sep_disj x zero"
ldropn_lzip: fixes n :: "nat"   and xs :: "'c llist"   and ys :: "'d llist" shows "ldropn n (lzip xs ys) = lzip (ldropn n xs) (ldropn n ys)"
simps_16: fixes f1 :: "'a"   and f2 :: "qtname \<Rightarrow> 'a"   and f3 :: "qtname \<Rightarrow> 'a"   and f4 :: "ty \<Rightarrow> 'a" shows "(case NullT of NullT \<Rightarrow> f1 | IfaceT x \<Rightarrow> f2 x | ClassT x \<Rightarrow> f3 x | ArrayT x \<Rightarrow> f4 x) = f1"
IH: fixes x2a :: "('a, 'b) Term.term" assumes "x2a \<in> set T" shows "(t \<sqsubseteq> x2a) = subterm_of t x2a"
pmf_pair: fixes M :: "'c pmf"   and N :: "'d pmf"   and a :: "'c"   and b :: "'d" shows "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
typeSkip: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and T :: "secLevel" shows "\<Gamma>,T \<turnstile> Skip"
simps_3: fixes v :: "'b list"   and vb :: "'b list"   and vc :: "'b list list" shows "C_merge_all (v # vb # vc) = C_merge_adj (v # vb # vc) + C_merge_all (merge_adj (v # vb # vc))"
in_F: fixes g :: "'v \<times> 'v \<Rightarrow> ennreal" shows "(g \<in> F) = (current \<Gamma> g \<and> wave \<Gamma> g \<and> (\<forall>e. g e \<le> f e) \<and> \<E> (TER g) = \<E> (TER f))"
reachable_inv: fixes v :: "'a"   and e :: "'a"   and t :: "'a"   and r :: "'a" assumes "vector v"   and "e \<le> v * - v\<^sup>T"   and "e * t = bot"   and "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>" shows "(v \<squnion> e\<^sup>T * top)\<^sup>T = r\<^sup>T * (t \<squnion> e)\<^sup>\<star>"
one_closed: shows "\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> \<in> carrier (G\<lparr>carrier := H\<rparr>)"
sings_gen_lists: fixes B :: "'b set" shows "\<langle>sings B\<rangle> = lists B"
scalar_product_def: fixes v :: "('a, 'b) vec"   and w :: "('a, 'b) vec" shows "scalar_product v w = (\<Sum>i\<in>UNIV. v $ i * w $ i)"
merkle_transaction: shows "merkle_interface hash_transaction blinding_of_transaction merge_transaction"
vreal_of_real_in_vrealI: fixes a :: "real" shows "a\<^sub>\<real> \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"
distinct_379: fixes x191 :: "char list"   and x192 :: "char list list"   and x193 :: "bool"   and x194 :: "'a exp"   and x201 :: "char list"   and x202 :: "'a exp"   and x203 :: "char list list"   and x204 :: "'a exp" shows "INIT x191 (x192,x193) \<leftarrow> x194 \<noteq> RI (x201,x202) ; x203 \<leftarrow> x204"
uminus_add_conv_diff: fixes plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and uminus :: "'q \<Rightarrow> 'q"   and a :: "'q"   and b :: "'q" assumes "class.ab_group_add plus zero minus uminus" shows "plus (uminus a) b = minus b a"
norm_cA: fixes i :: "'n"   and j :: "'n" shows "cmod (cA $h i $h j) = A $h i $h j"
least_Upper_cong_l: fixes x :: "'a"   and x' :: "'a"   and A :: "'a set" assumes "x .= x'"   and "x \<in> carrier L"   and "x' \<in> carrier L"   and "A \<subseteq> carrier L" shows "is_lub L x A = is_lub L x' A"
T1: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>\<box>F = \<box>F"
assms: shows "((x1, x2), y1, y2) \<in> (SIGMA x:UNIV. set_pmf (Kp x))\<^sup>*"
exhaust: fixes y :: "vname" obtains x1 :: "nat" where "y = I x1"   | x2 :: "nat" where "y = R x2"
bind_cond_pmf_cancel: fixes p :: "'b pmf"   and q :: "'c pmf"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"   and "\<And>y. y \<in> set_pmf q \<Longrightarrow> set_pmf p \<inter> {x. R x y} \<noteq> {}"   and "\<And>x y. \<lbrakk>x \<in> set_pmf p; y \<in> set_pmf q; R x y\<rbrakk> \<Longrightarrow> measure_pmf.prob q {y. R x y} = measure_pmf.prob p {x. R x y}" shows "p \<bind> (\<lambda>x. cond_pmf q {y. R x y}) = q"
less_by_empty: fixes A :: "('b \<times> 'b) set"   and B :: "('b \<times> 'b) set" assumes "A = {}" shows "A \<subseteq> B"
B_def: shows "B = A \<inter> Inl ` Field r"
bij_betw_ord_ofI: fixes h :: "'b \<Rightarrow> nat"   and A :: "'b set"   and n :: "nat" assumes "bij_betw h A {0..<n}" shows "bij_betw (ord_of \<circ> h) A (hfset (ord_of n))"
A_FactGroup_defs_2: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" shows "G Mod H = \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, mult = (<#>\<^bsub>G\<^esub>), one = H\<rparr>"
axioms_3: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j, 'k) sa_rec_scheme"   and B :: "('i, 'j, 'l) sa_rec_scheme" assumes "lsimulation R A B" shows "simulation R A B"
T\<^sub>c_def: shows "T\<^sub>c \<equiv> {[], [a\<^sub>c], [a\<^sub>c, b\<^sub>c], [a\<^sub>c, b\<^sub>c, c\<^sub>c], [a\<^sub>c, b\<^sub>c, c\<^sub>c, a\<^sub>c], [b\<^sub>c], [b\<^sub>c, a\<^sub>c], [b\<^sub>c, c\<^sub>c], [b\<^sub>c, a\<^sub>c, c\<^sub>c]}"
abs_eq: fixes x :: "'a poly" shows "- x = Abs_poly (\<lambda>n. - poly.coeff x n)"
inject_4: fixes x41 :: "'d"   and x42 :: "'c"   and x43 :: "('a, 'b, 'c, 'd) Tree"   and y41 :: "'d"   and y42 :: "'c"   and y43 :: "('a, 'b, 'c, 'd) Tree" shows "(tAct x41 x42 x43 = tAct y41 y42 y43) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
splitFace_holds_facesAt_distinct: fixes g :: "graph"   and v :: "nat"   and w :: "nat"   and f :: "face"   and n :: "nat" assumes "pre_splitFace g v w f [countVertices g..<countVertices g + n]"   and "minGraphProps g" shows "facesAt_distinct (snd (snd (splitFace g v w f [countVertices g..<countVertices g + n])))"
pos_half_less: fixes x :: "real" assumes "0 < x" shows "x / 2 < x"
ring_seq_smult_eval: fixes s :: "nat \<Rightarrow> 'a"   and a :: "'a"   and k :: "nat" assumes "s \<in> carrier R\<^bsup>\<omega>\<^esup>"   and "a \<in> carrier R" shows "(a \<odot>\<^bsub>R\<^bsup>\<omega>\<^esup>\<^esub> s) k = a \<otimes> s k"
simps_1: fixes \<C> :: "t_context"   and v :: "v"   and ts :: "checker_type" shows "check_single \<C> (C v) ts = type_update ts [] (Type [typeof v])"
pderiv_minus: fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier P"   and "b \<in> carrier P" shows "pderiv (a \<ominus>\<^bsub>P\<^esub> b) = pderiv a \<ominus>\<^bsub>P\<^esub> pderiv b"
ldrop_eSuc_ltl: fixes n :: "nat"   and xs :: "'c llist" shows "ldropn (Suc n) xs = ldropn n (ltl xs)"
Hintikka_alt: fixes S :: "'b formula set" shows "Hintikka (S::'b formula set) = (\<bottom> \<notin> S \<and> (\<forall>k::'b. Atom k \<in> S \<longrightarrow> \<^bold>\<not> (Atom k) \<in> S \<longrightarrow> False) \<and> (\<forall>(F::'b formula) (G::'b formula) H::'b formula. Con F G H \<longrightarrow> F \<in> S \<longrightarrow> G \<in> S \<and> H \<in> S) \<and> (\<forall>(F::'b formula) (G::'b formula) H::'b formula. Dis F G H \<longrightarrow> F \<in> S \<longrightarrow> G \<in> S \<or> H \<in> S))"
KeyWithNonce_Gets: fixes K :: "nat"   and NB :: "nat"   and A :: "agent"   and X :: "msg"   and evs :: "event list" shows "KeyWithNonce K NB (Gets A X # evs) = KeyWithNonce K NB evs"
unitBecomesFalse: fixes uClause :: "Literal list"   and uLiteral :: "Literal"   and valuation :: "Literal list" assumes "isUnitClause uClause uLiteral valuation" shows "clauseFalse uClause (valuation @ [opposite uLiteral])"
merge_respects_hashes: fixes a :: "'a\<^sub>m"   and b :: "'a\<^sub>m" shows "(h a = h b) = (\<exists>ab. m a b = Some ab)"
Quotient_set_dlist: shows "Quotient (eq_onp (\<lambda>xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None)) Abs_dlist list_of_dlist cr_set_dlist"
enn2ereal_nonneg: fixes x :: "ennreal" shows "0 \<le> enn2ereal x"
ty_en_defined: fixes \<Gamma> :: "(String.literal \<Rightarrow> unit option) \<times> (String.literal \<Rightarrow> unit option)"   and fs :: "local_state \<times> global_state"   and a :: "action" assumes "ty_fs \<Gamma> fs"   and "ty_la \<Gamma> a" shows "la_en fs a \<noteq> None"
chine_simps_2: shows "C.dom assoc\<mu>\<nu>\<pi>.chine = assoc\<mu>\<nu>\<pi>.dom.apex"
f: shows "(\<lambda>v. a v \<cdot>\<^sub>v v) \<in> X \<rightarrow> carrier_vec n"
inverse_le_sgn: fixes x :: "real"   and y :: "real" assumes "sgn x = sgn y"   and "x \<le> y" shows "inverse y \<le> inverse x"
simps_2: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int" shows "bval (bexp.Not b) s = (\<not> bval b s)"
doesnt_read_or_modify_def: fixes c :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and x :: "var\<^sub>C" shows "conc.doesnt_read_or_modify c x \<equiv> conc.doesnt_read_or_modify_vars c ({x} \<union> \<C>_vars\<^sub>C x)"
source: fixes S :: "'procS" shows "SourceTerm S \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L SourceTerm S"
fresh_star_def: fixes as :: "atom set"   and x :: "'d" shows "as \<sharp>* x \<equiv> \<forall>a\<in>as. a \<sharp> x"
reduction_step1: fixes M :: "bool \<times> bool \<times> bool \<times> bool"   and c0 :: "bool"   and c1 :: "bool"   and D :: "(bool \<times> bool \<times> bool \<times> bool) \<times> (bool \<times> bool \<times> bool \<times> bool \<times> bool \<times> bool) \<times> 'v_OT121 \<times> 'v_OT121 \<times> 'v_OT121 \<Rightarrow> bool spmf" shows "\<exists>A1. \<bar>spmf (R1_14 M (c0, c1) \<bind> D) True - spmf (R1_14_interm1 M (c0, c1) \<bind> D) True\<bar> = \<bar>spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). R1_OT12 (m0, m1) c0 \<bind> (\<lambda>view. A1 view (m0, m1)))) True - spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). S1_OT12 (m0, m1) () \<bind> (\<lambda>view. A1 view (m0, m1)))) True\<bar>"
simps_22: fixes Vs :: "String.literal list"   and e :: "'a expr" shows "compE1 Vs (throw e) = throw (compE1 Vs e)"
bdd_above_norm_f: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "bdd_above {norm (f x) |x. norm x = 1}"
non_bot_arc_in_partition_xor: fixes x :: "'a"   and y :: "'a" assumes "arc x"   and "x \<noteq> bot" shows "x \<le> - y \<and> \<not> x \<le> - - y \<or> \<not> x \<le> - y \<and> x \<le> - - y"
Child_1: shows "p' \<in> grid b ({d} \<union> ds)"
AtCallExpr: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and f :: "'e"   and b :: "('d, 'e, 'f) com"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and as :: "('d, 'e, 'f) ann list"   and n :: "nat"   and a :: "'d set \<times> ('d, 'e, 'f) com"   and r :: "'d set" assumes "\<Gamma> f = Some b"   and "\<Theta> f = Some as"   and "n < length as"   and "atomicsR \<Gamma> \<Theta> (as ! n) b a" shows "atomicsR \<Gamma> \<Theta> (AnnCall r n) (Call f) a"
runit'_simps_3: fixes f :: "'a" assumes "S.ide f" shows "S.trg (S.runit' f) = S.trg f"
P: shows "vmc2_path G P v p \<sigma> \<sigma>'"
resumption_ord_simps_2: fixes r :: "('d, 'e, 'f) Resumption.resumption"   and a :: "'d option" shows "resumption_ord r (Resumption.resumption.Done a) = (\<exists>aa. r = Resumption.resumption.Done aa \<and> option_ord aa a)"
fps_divide_nth_0': fixes g :: "'a fps"   and f :: "'a fps" assumes "subdegree g = 0" shows "fps_nth (f / g) 0 = fps_nth f 0 / fps_nth g 0"
alpha_trans_1: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "perm"   and x :: "'e"   and y :: "'e"   and q :: "perm"   and z :: "'e"   and bs :: "atom set"   and f :: "'e \<Rightarrow> atom set"   and cs :: "atom set"   and ds :: "atom set" assumes "\<lbrakk>R (p \<bullet> x) y; R (q \<bullet> y) z\<rbrakk> \<Longrightarrow> R ((q + p) \<bullet> x) z"   and "(bs, x) \<approx>set R f p (cs, y)"   and "(cs, y) \<approx>set R f q (ds, z)" shows "(bs, x) \<approx>set R f (q + p) (ds, z)"
1: shows "\<guillemotleft>\<i>[a] : a \<star> a \<Rightarrow> a\<guillemotright>"
map_ident: fixes t :: "'b pmf" shows "map_pmf (\<lambda>x. x) t = t"
less_eqE: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less_eq a b" obtains c :: "'g" where "b = plus a c"
chine_assoc_inverse: shows "C.inverse_arrows trg_trg_f.chine_assoc trg_trg_f.chine_assoc'"
Transset_inf: fixes x :: "hf"   and y :: "hf" assumes "Transset x"   and "Transset y" shows "Transset (x \<sqinter> y)"
simps_1: fixes all :: "String.literal"   and S :: "typ"   and T :: "typ"   and t :: "Core.term" shows "is_all (Ct all S $ Abs T t) = (all = STR ''Pure.all'' \<and> S = (T \<rightarrow> propT) \<rightarrow> propT)"
mat_gt_def: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sd :: "nat"   and A :: "'a mat"   and B :: "'a mat" shows "mat_gt gt sd A B = (A \<ge>\<^sub>m B \<and> (\<exists>i<sd. \<exists>j<sd. gt (A $$ (i, j)) (B $$ (i, j))))"
case_5: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "nat set \<Rightarrow> 'e \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'd"   and f4 :: "nat \<Rightarrow> 'd"   and f5 :: "'e \<Rightarrow> 'd"   and f6 :: "nat \<Rightarrow> 'd"   and f7 :: "'e \<Rightarrow> 'd"   and f8 :: "'d"   and x5 :: "'e" shows "(case send x5 of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f5 x5"
simps_4: fixes G :: "('b, 'a) labeled_graph"   and l :: "'b" shows ":G:\<lbrakk>A_Lbl l\<rbrakk> = getRel l G"
outstanding_non_volatile_refs_owned_or_read_only_axioms: shows "outstanding_non_volatile_refs_owned_or_read_only \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"
LocalSecretsComposition_neg_s: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and m :: "Secrets" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionLoc PQ"   and "correctCompositionKS PQ"   and "sKS m \<notin> specKeysSecrets P"   and "sKS m \<notin> specKeysSecrets Q"   and "\<not> ine P (sE m)"   and "\<not> ine Q (sE m)"   and "sKS m \<notin> LocalSecrets P \<union> LocalSecrets Q" shows "sKS m \<notin> LocalSecrets PQ"
simps_4: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and ds :: "dir list" shows "position_of (App t s) (dir.Right # ds) = position_of s ds"
invariant_arbitraryI: fixes A :: "('a, 'b) automaton"   and I :: "'b \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "\<And>s. s \<in> reachable A I \<Longrightarrow> P s" shows "A \<TTurnstile> (I \<rightarrow>) P"
complement_conv_sub: fixes y :: "'a"   and x :: "'a" shows "- (y * x) * x\<^sup>T \<le> - y"
join_connection: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "le x y = (dlat.join le x y = y)"
lcn''_def: shows "lcn'' = (cms'' ! n, mem'')"
from_to_view: fixes x :: "(view_metadata \<times> view_data) rose_tree" shows "view_merkle.from_view (view_merkle.to_view x) = x"