hcju/isabelleps huggingface dataset:
simps: fixes f :: "'d \<Rightarrow> 'c"   and ps :: "('b, 'd) fm list"   and i :: "'d" shows "sub_block f (ps, i) = (sub_list f ps, f i)"
is_aligned_shiftl: fixes w :: "'a word"   and n :: "nat"   and m :: "nat" assumes "is_aligned w (n - m)" shows "is_aligned (w << m) n"
axioms_1: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "('b, 'c) epsym \<Rightarrow> univ list \<Rightarrow> bool" assumes "ProblemIkTpartG_GEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl eintT eintF eintP" shows "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl"
assms: shows "factorize_rat_poly_monic f = (c, fs)"
map_replicate_trivial: fixes x :: "'g"   and i :: "nat" shows "map (\<lambda>i. x) [0..<i] = replicate i x"
rep_eq: fixes x :: "'c fset" shows "fBall x = Ball (fset x)"
dvd_ideal_generated_singleton: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "ring.ideal_generated plus zero uminus times {a} \<subseteq> ring.ideal_generated plus zero uminus times {b}" shows "dvd.dvd times b a"
rot_circ_right_edge_reparam_polar_circ_split: shows "reparam (rec_join [rot_circle_right_edge]) (rec_join [subcube (3 / 4) 1 (1, circle_polar), subcube 0 (1 / 4) (1, circle_polar)])"
intro: fixes P :: "'c llist"   and G :: "('c, 'd) ParityGame_scheme"   and v0 :: "'c" assumes "\<not> lnull P"   and "Digraph.valid_path G P"   and "Digraph.maximal_path G P"   and "lhd P = v0" shows "vm_path_axioms G P v0"
Inter_UNIV_conv_2: fixes A :: "'b set set" shows "(UNIV = \<Inter> A) = (\<forall>x\<in>A. x = UNIV)"
UnionI: fixes X :: "'e set"   and C :: "'e set set"   and A :: "'e" assumes "X \<in> C"   and "A \<in> X" shows "A \<in> \<Union> C"
hyp2: shows "(\<Sum>s\<leftarrow>remdups (map (consistent_sign_vec qs) (characterize_root_list_p p)). z_R I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s})) = (\<Sum>s\<in>set (remdups (map (consistent_sign_vec qs) (characterize_root_list_p p))). z_R I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))"
lifted_less_eq_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c" shows "ord.lifted_less_eq less_eq f g \<equiv> \<forall>x. less_eq (f x) (g x)"
is_type_def: fixes P :: "'b prog"   and T :: "ty" shows "is_type P T \<equiv> case T of Class C \<Rightarrow> is_class P C | _ \<Rightarrow> True"
sint_less: fixes w :: "'c word" shows "sint w < 2 ^ (LENGTH('c) - Suc 0)"
ContraProve: fixes H :: "fm set"   and B :: "fm"   and A :: "fm" assumes "H \<turnstile> B" shows "insert (Neg B) H \<turnstile> A"
wf_stateful_policy_def: fixes \<T> :: "'b stateful_policy" shows "wf_stateful_policy \<T> \<equiv> (fst ` flows_fix \<T> \<subseteq> hosts \<T> \<and> snd ` flows_fix \<T> \<subseteq> hosts \<T>) \<and> flows_state \<T> \<subseteq> flows_fix \<T> \<and> finite (hosts \<T>)"
formulaEntailsItsClauses: fixes clause :: "Literal list"   and formula :: "Literal list list" assumes "clause el formula" shows "formulaEntailsClause formula clause"
float_eq_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "a \<doteq> b = feq a b"
Abs_euclidean_space_inverse: fixes y :: "'d \<Rightarrow> real" assumes "y \<in> UNIV" shows "Rep_euclidean_space (Abs_euclidean_space y) = y"
set_restriction_def: fixes f :: "'a set \<Rightarrow> 'a set" shows "set_restriction f \<equiv> \<exists>P. \<forall>A. f A = {x \<in> A. P x}"
simps_20: fixes Vs :: "char list list"   and C :: "char list"   and e :: "char list exp"   and Cs :: "char list list"   and e' :: "char list exp" shows "compE\<^sub>1 Vs (RI (C,e) ; Cs \<leftarrow> e') = RI (C,compE\<^sub>1 Vs e) ; Cs \<leftarrow> compE\<^sub>1 Vs e'"
nz: shows "of_int x = (0::'a) \<longrightarrow> of_int y = (0::'a)"
MultiApp: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat"   and l1 :: "int"   and r :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and l2 :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, k, l1, r) \<in> MStep"   and "(M, l1, r, n, l2, t) \<in> Step" shows "(M, l, s, Suc k + n, l2, t) \<in> MStep"
class_rec: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and t :: "'a"   and f :: "qtname \<Rightarrow> class \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class G C = Some c"   and "ws_prog G" shows "class_rec G C t f = f C c (if C = Object then t else class_rec G (super c) t f)"
div_mult_self1: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a (times c b)) b = plus c (divide a b)"
ffieldI3: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "b \<in>\<^sub>\<circ> \<F>\<^sub>\<bullet> r"
le_add_same_cancel2: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subseteq># b + a) = ({#} \<subseteq># b)"
wprepare_invs_3: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_erase m lm (l, r) = (\<exists>rn. l = Oc \<up> Suc m \<and> tl r = Bk # <lm> @ Bk \<up> rn)"
new_free_vars_on_heap: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : z" shows "fv (\<Delta>, z) - domA \<Delta> \<subseteq> fv (\<Gamma>, e) - domA \<Gamma>"
replicate_FT_hlp: fixes x :: "nat"   and y :: "nat" assumes "x \<le> 16 \<and> y \<le> 16"   and "replicate (16 - x) False @ replicate x True = replicate (16 - y) False @ replicate y True" shows "x = y"
join_no_cols: fixes n :: "nat"   and t1 :: "'b option list set"   and R :: "nat set"   and t2 :: "'b option list set"   and pos :: "bool" assumes "table n {} t1"   and "table n R t2" shows "join t2 pos t1 = (if pos = (replicate n None \<in> t1) then t2 else {})"
independentD_alt: fixes B :: "'b set"   and X :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "independent B"   and "finite {x. X x \<noteq> (0::'a)}"   and "{x. X x \<noteq> (0::'a)} \<subseteq> B"   and "(\<Sum>x | X x \<noteq> (0::'a). X x *s x) = (0::'b)" shows "X x = (0::'a)"
env_bigger: fixes a :: "char list"   and e :: "'b environment"   and x :: "char list"   and X :: "'b" assumes "a \<notin> env_dom e"   and "x \<in> env_dom e" shows "x \<in> env_dom (e\<lparr>a:X\<rparr>)"
counit_simps_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "partial_magma.arr V \<epsilon>"
isDiagonal_diagonal_step: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and j :: "nat"   and d :: "'a"   and v :: "'a" assumes "isDiagonal A"   and "i < min (nrows A) (ncols A)"   and "j < min (nrows A) (ncols A)" shows "isDiagonal (diagonal_step A i j d v)"
intro: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real" assumes "R2 i j"   and "green_typeI_cube_axioms i j twoC F" shows "green_typeI_cube i j twoC F"
not0_implies_Suc: fixes n :: "nat" assumes "n \<noteq> 0" shows "\<exists>m. n = Suc m"
islcm_def: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and a :: "'c"   and b :: "'c" shows "x lcmof\<^bsub>G\<^esub> a b = (a divides\<^bsub>G\<^esub> x \<and> b divides\<^bsub>G\<^esub> x \<and> (\<forall>y\<in>carrier G. a divides\<^bsub>G\<^esub> y \<and> b divides\<^bsub>G\<^esub> y \<longrightarrow> x divides\<^bsub>G\<^esub> y))"
locate_locate_pred: fixes al :: "'b list"   and f :: "'b \<Rightarrow> nat"   and i :: "nat" assumes "\<And>a. a \<in> set al \<Longrightarrow> 0 < f a"   and "i < lsum f al" shows "locate_pred f al i (locate f al i)"
remdups_eq_nil_iff: fixes x :: "'a list" shows "(remdups x = []) = (x = [])"
simps: fixes a1 :: "nat \<Rightarrow> pdf_type"   and a2 :: "cexpr"   and a3 :: "pdf_type" shows "(a1 \<turnstile>\<^sub>c a2 : a3) = ((\<exists>\<Gamma> v. a1 = \<Gamma> \<and> a2 = CVal v \<and> a3 = val_type v) \<or> (\<exists>\<Gamma> x. a1 = \<Gamma> \<and> a2 = CVar x \<and> a3 = \<Gamma> x) \<or> (\<exists>\<Gamma> e1 t1 e2 t2. a1 = \<Gamma> \<and> a2 = <e1, e2>\<^sub>c \<and> a3 = PRODUCT t1 t2 \<and> \<Gamma> \<turnstile>\<^sub>c e1 : t1 \<and> \<Gamma> \<turnstile>\<^sub>c e2 : t2) \<or> (\<exists>\<Gamma> e t oper t'. a1 = \<Gamma> \<and> a2 = oper $$\<^sub>c e \<and> a3 = t' \<and> \<Gamma> \<turnstile>\<^sub>c e : t \<and> op_type oper t = Some t') \<or> (\<exists>\<Gamma> b e1 t e2. a1 = \<Gamma> \<and> a2 = (IF\<^sub>c b THEN e1 ELSE e2) \<and> a3 = t \<and> \<Gamma> \<turnstile>\<^sub>c b : BOOL \<and> \<Gamma> \<turnstile>\<^sub>c e1 : t \<and> \<Gamma> \<turnstile>\<^sub>c e2 : t) \<or> (\<exists>t \<Gamma> e. a1 = \<Gamma> \<and> a2 = \<integral>\<^sub>c e \<partial>t \<and> a3 = REAL \<and> case_nat t \<Gamma> \<turnstile>\<^sub>c e : REAL))"
separating_def: fixes V :: "'a set"   and F :: "'a \<Rightarrow> 'b set" shows "separating V F \<equiv> \<forall>v1\<in>V. \<forall>v2\<in>V. v1 \<noteq> v2 \<longrightarrow> F v1 \<inter> F v2 = {}"
L4_7: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlr" assumes "\<xi> \<Turnstile>\<^sub>r \<phi>" shows "create_graph \<phi> \<le> SPEC (create_graph_rslt_exist \<xi>)"
exhaust: fixes y :: "cmp_val" obtains "y = LT"   | "y = EQ"   | "y = GT"
simps_3: fixes I :: "('e, 'f, 'd) interp"   and x :: "'d"   and t :: "('e, 'd) trm" shows "prog_sem I (DiffAssign x t) = {(\<nu>, \<omega>). \<omega> = repd \<nu> x (dterm_sem I t \<nu>)}"
simps_141: fixes x12 :: "qtname"   and x101 :: "stmt"   and x102 :: "stmt" shows "Init x12 \<noteq> x101 Finally x102"
S_inductive_sup_compare: fixes X1 :: "'a set"   and X2 :: "'a set" assumes "S_inductive_set D"   and "Chain D X1"   and "Chain D X2"   and "X1 \<subseteq> X2" shows "Algebra2.Sup D X1 \<preceq> Algebra2.Sup D X2"
emeasure_eq_measure2: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> fmeasurable M" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
Lev1_0: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and x :: "'d" shows "Lev1 s1 s2 0 x = {[]}"
tso_store_invD_4: fixes s :: "'a process_name \<Rightarrow> ('b, 'a, 'c, 'd) Model.local_state"   and m :: "'a"   and fl :: "bool"   and ws :: "('b, 'c, 'd) mem_store_action list" assumes "tso_store_inv s" shows "sys_mem_store_buffers (mutator m) s \<noteq> mw_fM fl # ws"
atLs_def: fixes p :: "'f"   and labels :: "'g set set"   and s :: "('h, 'g, 'f, 'i, 'j) system_state" shows "atLs p labels s = (AT s p \<in> labels)"
order_iff: fixes f :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and a :: "'a::type"   and b :: "'a::type" assumes "semilattice_order (f::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (a::'a::type) (b::'a::type) = (a = (f::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) a b)"
inf2D1: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c" assumes "(A \<sqinter> B) x y" shows "A x y"
adopt_node_document_in_heap: fixes h :: "(_) heap"   and owner_document :: "(_) document_ptr"   and node :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "known_ptrs h"   and "type_wf h"   and "h \<turnstile> ok adopt_node owner_document node" shows "owner_document |\<in>| document_ptr_kinds h"
simps_2: fixes a :: "'b" shows "allConsec [a] = {}"
topological_basis_closed: fixes U :: "'a set"   and \<tau> :: "'a set \<Rightarrow> bool"   and B :: "'a set set" assumes "on U with \<tau> : \<guillemotleft>topological_basis\<guillemotright> B" shows "B \<subseteq> Pow U"
has_codomain_iff_arr: fixes f :: "'a1 \<times> 'a2" shows "(A1xA2.codomains f \<noteq> {}) = A1xA2.arr f"
1: shows "{i \<in> rows_with_1. i < dim_row Aw'} = {i. i < prod_list (nths (dims Aw) (Collect even)) \<and> (\<forall>i0\<in>set (digit_encode (nths (dims Aw) (Collect even)) i). i0 < r)}"
nprv_ldsjI: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and \<phi>s :: "'fmla list" assumes "nprv F \<phi>"   and "F \<subseteq> fmla"   and "finite F"   and "set \<phi>s \<subseteq> fmla"   and "\<phi> \<in> set \<phi>s" shows "nprv F (ldsj \<phi>s)"
f_aggregate_Cons: fixes k :: "nat"   and xs :: "'b list"   and ys :: "'b list"   and ag :: "'b list \<Rightarrow> 'b" assumes "0 < k"   and "length xs = k" shows "f_aggregate (xs @ ys) k ag = ag xs # f_aggregate ys k ag"
monomial_decomp_monomial_in_cone: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set"   and a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "monomial_decomp ps"   and "hU \<in> set ps"   and "a \<in> cone hU" shows "monomial (lookup a t) t \<in> cone hU"
singleton_insert_inj_eq': fixes a :: "'c"   and A :: "'c set"   and b :: "'c" shows "(Set.insert a A = {b}) = (a = b \<and> A \<subseteq> {b})"
SN_WCR_reaches_NF: fixes r :: "('b \<times> 'b) set"   and x :: "'b" assumes "SN_on r {x}"   and "WCR_on r {x. SN_on r {x}}" shows "\<exists>!y. (x, y) \<in> r\<^sup>* \<and> y \<in> NF r"
assms_2: shows "P,h \<turnstile>\<^sub>s sh \<surd>"
set_diff_diff_left: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A - B - C = A - (B \<union> C)"
h'_def: fixes x :: "'a" shows "h' x = (let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi)"
Signature_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool" shows "U.Signature wtFsym wtPsym \<equiv> countable {f. wtFsym f} \<and> countable {p. wtPsym p}"
expand_init_def: shows "expand_init \<equiv> 0::'d"
frp_invar_add_def: fixes \<delta>1 :: "('d, 'f) ta_rule set"   and \<delta>2 :: "('e, 'f) ta_rule set" shows "frp_invar_add \<delta>1 \<delta>2 \<equiv> {(Q, W, \<delta>d). \<delta>d = {r \<in> \<delta>_prod \<delta>1 \<delta>2. lhs r \<in> Q - set W}}"
representation_bound: fixes B :: "'b set"   and b :: "'b" assumes "finite B"   and "independent B"   and "b \<in> B"   and "pairwise orthogonal B" obtains m :: "real" where "0 < m" and "\<And>x. x \<in> span B \<Longrightarrow> \<bar>representation B x b\<bar> \<le> m * norm x"
PivotAndUpdate_axioms: shows "PivotAndUpdate eq_idx_for_lvar pivot_and_update_code"
instances_are_entailed: fixes I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and C :: "'b literal set"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "validate_clause I C" shows "validate_clause I (subst_cl C \<sigma>)"
rep_eq: fixes x :: "integer" shows "bin_last_integer x = odd (int_of_integer x)"
weaken_post: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2 {P} c {Q}"   and "\<forall>s. Q' s \<le> Q s" shows "\<turnstile>\<^sub>2 {P} c {Q'}"
C1: fixes \<Psi>'' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and \<Psi>''' :: "'b" assumes "(\<Psi>'', R, S) \<in> Rel"   and "\<Psi>'' \<simeq> \<Psi>'''" shows "(\<Psi>''', R, S) \<in> Rel'"
simps_7: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'a) cmd list" assumes "PL cval" shows "PL.brn (ParT cl) = lsum PL.brn cl"
cfE_augment_ss: fixes u :: "nat"   and v :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'capacity"   and \<Delta> :: "'capacity" assumes "(u, v) \<in> cfE_of f" shows "cfE_of (augment_edge f (u, v) \<Delta>) \<subseteq> insert (v, u) (cfE_of f)"
CoxeterSystem_def: fixes S :: "'b set" shows "CoxeterSystem S \<equiv> PreCoxeterSystem S \<and> CoxeterSystem_axioms S"
ok_def: fixes F :: "'d program"   and G :: "'d program" shows "F ok G \<equiv> Acts F \<subseteq> AllowedActs G \<and> Acts G \<subseteq> AllowedActs F"
nonces_Crypt: fixes K :: "nat"   and X :: "msg" shows "nonces (Crypt K X) = nonces X"
atLeastAtMost_singleton: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.order less_eq less" shows "ord.atLeastAtMost less_eq a a = {a}"
ordertype_insert: fixes \<alpha> :: "V"   and U :: "V set" assumes "Ord \<alpha>"   and "U \<subseteq> elts \<alpha>" shows "tp (insert \<alpha> U) = ZFC_in_HOL.succ (tp U)"
assms: shows "A \<in> Ob\<^bsub>set_cat U\<^esub>"
rep_eq: fixes x :: "real" shows "blinfun_apply (blinfun_scaleR_right x) = (*\<^sub>R) x"
min_ground_head_antimono: fixes \<zeta> :: "('s, 'v) hd"   and \<xi> :: "('s, 'v) hd" assumes "local.ground_heads \<zeta> \<subseteq> local.ground_heads \<xi>" shows "wt_sym (min_ground_head \<xi>) + \<delta>\<^sub>h * arity_sym\<^sub>h (min_ground_head \<xi>) \<le> wt_sym (min_ground_head \<zeta>) + \<delta>\<^sub>h * arity_sym\<^sub>h (min_ground_head \<zeta>)"
infinite_int_iff_unbounded: fixes S :: "int set" shows "infinite S = (\<forall>m. \<exists>n. m < \<bar>n\<bar> \<and> n \<in> S)"
subset_CollectI: fixes B :: "'b set"   and A :: "'b set"   and Q :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "B \<subseteq> A"   and "\<And>x. \<lbrakk>x \<in> B; Q x\<rbrakk> \<Longrightarrow> P x" shows "{x \<in> B. Q x} \<subseteq> {x \<in> A. P x}"
comm_ring_hom_axioms: shows "comm_ring_hom of_int"
roifex_Falseif_unique: fixes b :: "'a ifex" assumes "ro_ifex b"   and "\<forall>ass. \<not> val_ifex b ass" shows "b = Falseif"
trapC_def: shows "trapC = cball (0, 0) 2 - ball (0, 0) (1 / 2)"
mat_of_cblinfun_def: fixes f :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "mat_of_cblinfun f = mat (length canonical_basis) (length canonical_basis) (\<lambda>(i, j). crepresentation (set canonical_basis) (f *\<^sub>V canonical_basis ! j) (canonical_basis ! i))"
is_unit_mult_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" 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)"
convex_cball: fixes x :: "'c"   and e :: "real" shows "convex (cball x e)"
map_option_eq_Some: fixes f :: "'d \<Rightarrow> 'c"   and xo :: "'d option"   and y :: "'c" shows "(map_option f xo = \<lfloor>y\<rfloor>) = (\<exists>z. xo = \<lfloor>z\<rfloor> \<and> f z = y)"
PIdl_p: shows "I = PIdl\<^bsub>K [X]\<^esub> p"
Theorem_15: fixes ds :: "'b set" shows "gfp_F ds = (- Rf CH (fp_cop_F ds), fp_cop_F ds)"
subst_mono_fv: fixes x :: "'a"   and t :: "('b, 'a) Term.term"   and s :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "x \<in> fv t" shows "s x \<sqsubseteq> t \<cdot> s"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_3: fixes l :: "'a strand_label"   and a :: "poscheckvariant"   and t :: "('b, 'c) Term.term"   and s :: "('b, 'c) Term.term"   and A :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, \<langle>a: t \<doteq> s\<rangle>) # A) = (l, \<langle>a: t \<doteq> s\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
expand_def: shows "expand \<equiv> REC (\<lambda>expand (n, ns). if new n = {} then if \<exists>n'\<in>ns. old n' = old n \<and> next n' = next n then RETURN (name n, upd_incoming n ns) else expand (\<lparr>name = expand_new_name (name n), incoming = {name n}, new = next n, old = {}, next = {}\<rparr>, {n} \<union> ns) else SPEC (\<lambda>x. x \<in> new n) \<bind> (\<lambda>\<phi>. let n = n\<lparr>new := new n - {\<phi>}\<rparr> in if \<exists>q. \<phi> = prop\<^sub>r(q) \<or> \<phi> = nprop\<^sub>r(q) then if not\<^sub>r \<phi> \<in> old n then RETURN (name n, ns) else expand (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) else if \<phi> = true\<^sub>r then expand (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) else if \<phi> = false\<^sub>r then RETURN (name n, ns) else if \<exists>\<nu> \<mu>. \<phi> = \<nu> and\<^sub>r \<mu> \<or> \<phi> = X\<^sub>r \<nu> then expand (n\<lparr>new := new1 \<phi> \<union> new n, old := {\<phi>} \<union> old n, next := next1 \<phi> \<union> next n\<rparr>, ns) else expand (n\<lparr>new := new1 \<phi> \<union> new n, old := {\<phi>} \<union> old n, next := next1 \<phi> \<union> next n\<rparr>, ns) \<bind> (\<lambda>(nm, nds). expand (n\<lparr>name := nm, new := new2 \<phi> \<union> new n, old := {\<phi>} \<union> old n\<rparr>, nds))))"
HSem_bot_eq: fixes ESem :: "'c \<Rightarrow> ('a \<Rightarrow> 'b) \<rightarrow> 'b"   and \<Gamma> :: "('a \<times> 'c) list" shows "has_ESem.HSem ESem \<Gamma>\<cdot>\<bottom> = evalHeap \<Gamma> (\<lambda>e. ESem e\<cdot>(has_ESem.HSem ESem \<Gamma>\<cdot>\<bottom>))"
Extend_def: fixes S :: "'b fm set"   and C :: "'b fm set set"   and f :: "nat \<Rightarrow> 'b fm" shows "Extend S C f \<equiv> \<Union> (range (extend S C f))"
qs: fixes q :: "complex poly" assumes "q \<in> set qs" shows "set (coeffs q) \<subseteq> \<real> \<and> degree q \<le> 2"
assms_1: fixes x :: "'a" assumes "x \<in> s" shows "(g has_field_derivative g' x) (at x)"
assms: fixes a :: "'a"   and b :: "'b" assumes "(a, b) \<in> set cs" shows "f a b = b"
assms_2: shows "(f, h) \<in> [P']\<^sub>f R \<rightarrow> S"
cr_spmf_option_def: shows "option_to_spmf.cr_option \<equiv> \<lambda>x y. x = return_pmf y"
refl: fixes ko :: "'a key_order"   and a :: "'a" shows "le_of_key_order ko a a"
elimation_rules_strictly_decreasing_meas: fixes \<Gamma>\<^sub>1 :: "'b constr list"   and n\<^sub>1 :: "nat"   and \<Psi>\<^sub>1 :: "'b TESL_atomic list"   and \<Phi>\<^sub>1 :: "'b TESL_atomic list"   and \<Gamma>\<^sub>2 :: "'b constr list"   and n\<^sub>2 :: "nat"   and \<Psi>\<^sub>2 :: "'b TESL_atomic list"   and \<Phi>\<^sub>2 :: "'b TESL_atomic list" assumes "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow>\<^sub>e \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2" shows "(\<Psi>\<^sub>2, \<Psi>\<^sub>1) \<in> measure \<mu>"