hcju/isabelleps huggingface dataset:
mem_Collect_eq: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
repeat_mset_cancel2: fixes a :: "nat"   and A :: "'k multiset"   and b :: "nat" shows "(repeat_mset a A = repeat_mset b A) = (a = b \<or> A = {#})"
distinct_2: fixes atom_ordering :: "('a \<times> 'a) set"   and x2 :: "'b"   and x1 :: "'b" assumes "propositional_atoms atom_ordering" shows "Neg x2 \<noteq> Pos x1"
simps_1: fixes A :: "'b list set" shows "A ^^ 0 = {[]}"
UNpart01_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list"   and P :: "nat \<Rightarrow> nat set set" shows "UNpart01 cl dl P \<equiv> {BrnFT cl dl} \<union> UNpart1 cl dl P"
ring_times_0_x: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c" assumes "Ring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<cdot>\<^sub>r\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
assms_1: shows "P' \<longmapsto>\<tau> \<prec> P''"
set_intros_1: fixes y :: "'a"   and x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree" assumes "y \<in> set_tree x21" shows "y \<in> set_tree \<langle>x21, x22, x23\<rangle>"
less_eq_character_data_ptr_def: fixes x :: "(_) character_data_ptr"   and y :: "(_) character_data_ptr" shows "x \<le> y \<equiv> case x of character_data_ptr.Ref i \<Rightarrow> case y of character_data_ptr.Ref j \<Rightarrow> i \<le> j | character_data_ptr.Ext x \<Rightarrow> True | character_data_ptr.Ext i \<Rightarrow> case y of character_data_ptr.Ref x \<Rightarrow> False | character_data_ptr.Ext j \<Rightarrow> i \<le> j"
of_int_code_3: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (Int.Pos k) = numeral.numeral one plus k"
sub: shows "mset ds \<subseteq># mset bef + mset aft"
simps: fixes \<Sigma> :: "'d set"   and \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and q :: "'c"   and n :: "nat" shows "semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n = {p. \<exists>x y. semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w p n = Some y \<and> semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w q n = Some x \<and> y < x \<and> \<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 p}"
inj_on_one_iff': fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" assumes "h \<in> hom G H"   and "Group.group G"   and "Group.group H" shows "inj_on h (carrier G) = (\<forall>x. x \<in> carrier G \<longrightarrow> h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>)"
admissibleI: fixes ord :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and P :: "'h \<Rightarrow> bool"   and lub :: "'h set \<Rightarrow> 'h" assumes "\<And>A. \<lbrakk>Complete_Partial_Order.chain ord A; A \<noteq> {}; \<forall>x\<in>A. P x\<rbrakk> \<Longrightarrow> P (lub A)" shows "ccpo.admissible lub ord P"
map_add_comm: fixes m1 :: "'c \<Rightarrow> 'd option"   and m2 :: "'c \<Rightarrow> 'd option" assumes "\<And>x. x \<in> dom m1 \<and> x \<in> dom m2 \<Longrightarrow> m1 x = m2 x" shows "m1 ++ m2 = m2 ++ m1"
ind_ring_morphisms_def: fixes V :: "'b set"   and W :: "'b set" shows "ind_ring_morphisms V W \<equiv> \<rho> (X \<inter> V) (X \<inter> W)"
assms_2: shows "mat (row_length M2) (length M2) M2"
is_lasso_pre_def: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'c"   and m :: "'b"   and L :: "('Q \<times> nat) lasso" shows "graph.is_lasso_pre (b_graph.to_gbg_ext (degeneralize_ext ma) m) L \<equiv> lasso_v0 L \<in> g_V0 (b_graph.to_gbg_ext (degeneralize_ext ma) m) \<and> path (g_E (b_graph.to_gbg_ext (degeneralize_ext ma) m)) (lasso_v0 L) (lasso_reach L) (lasso_va L) \<and> path (g_E (b_graph.to_gbg_ext (degeneralize_ext ma) m)) (lasso_va L) (lasso_cycle L) (lasso_va L)"
that: fixes ns :: "'node SDG_node list" assumes "CFG_node (_Entry_) cd-ns\<rightarrow>\<^sub>d* CFG_node n'"   and "ns \<noteq> []"   and "\<forall>n''\<in>set ns. parent_node n'' \<in> set (sourcenodes as)" shows "thesis"
insert_ops_rem_prefix: fixes pre :: "('a \<times> 'a option) list"   and suf :: "('a \<times> 'a option) list" assumes "insert_ops (pre @ suf)" shows "insert_ops suf"
ads_d_def: fixes x :: "'a" shows "d x = ad (ad x)"
weak_complete_lattice_criterion1: assumes "\<exists>g. greatest L g (carrier L)"   and "\<And>A. \<lbrakk>A \<subseteq> carrier L; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>i. is_glb L i A" shows "weak_complete_lattice L"
elims_1: fixes x :: "nat \<times> cell list \<times> cell list"   and y :: "bool" assumes "is_final x = y" obtains s :: "nat" and l :: "cell list" and r :: "cell list" where "x = (s, l, r)" and "y = (s = 0)"
openin_open_trans: fixes T :: "'a set"   and S :: "'a set" assumes "openin (top_of_set T) S"   and "open T" shows "open S"
Gromov_product_add: fixes e :: "'a"   and x :: "'a"   and y :: "'a" shows "Gromov_product_at e x y + Gromov_product_at x e y = dist e x"
greedy_path_ltl_VVp: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and P :: "'a llist" assumes "greedy_conforming_path p \<sigma> \<sigma>' v0 = LCons v0 P"   and "v0 \<in> VV p"   and "\<not> deadend v0" shows "\<sigma> v0 = lhd P"
a: shows "\<Gamma> \<turnstile> s \<leftrightarrow> t : T"
simps_1: fixes M :: "(('fun, 'b, 'sets) prot_fun, 'c) Term.term list"   and TI :: "('sets set \<times> 'sets set) list"   and x :: "'c" shows "intruder_synth_mod_timpls' M TI (Var x) = List.member M (Var x)"
b: shows "start_of (layout_of ap) as = Suc (length tp1 div 2)"
emeasure_Un': fixes A :: "'a set"   and M :: "'a measure"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M"   and "A \<inter> B \<in> null_sets M" shows "emeasure M (A \<union> B) = emeasure M A + emeasure M B"
dgrad_setI: fixes d :: "'c \<Rightarrow> nat"   and t :: "'c"   and m :: "nat" assumes "d t \<le> m" shows "t \<in> dgrad_set d m"
4: shows "e\<^sup>T * e * F * w\<^sup>T \<le> e\<^sup>T * e * F"
leg0_simps_5: shows "D.cod (F f) = F f"
LocalLexing_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set" shows "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel \<equiv> CFG \<NN> \<TT> \<RR> \<SS> \<and> LocalLexing_axioms \<TT> Lex Sel"
funny_list_def: shows "funny_list = [let b = True in b, False]"
a2: shows "\<exists>y'>r. \<forall>(d, e, f)\<in>set c. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"
all_eq: fixes f :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a" shows "(\<forall>x\<in>set (grouped_out_arcs G_list). cyclic_on f (set x)) = (\<forall>v\<in>pverts (list_digraph G_list). out_arcs (with_proj (list_digraph G_list)) v \<noteq> {} \<longrightarrow> cyclic_on f (out_arcs (with_proj (list_digraph G_list)) v))"
\<delta>_statesI': fixes q :: "'a"   and l :: "'b"   and qs :: "'a list"   and \<delta> :: "('a, 'b) ta_rule set"   and qi :: "'a" assumes "q \<rightarrow> l qs \<in> \<delta>"   and "qi \<in> set qs" shows "qi \<in> \<delta>_states \<delta>"
inverse_unique: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and inverse :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "Groups.group f z inverse"   and "f a b = z" shows "inverse a = b"
under_empty: fixes a :: "'a"   and r :: "('a \<times> 'a) set" assumes "a \<notin> Field r" shows "under r a = {}"
max_min_same_2: fixes x :: "'b"   and y :: "'b" shows "max (min x y) x = x"
sp_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "sp \<pi> P c t \<equiv> \<exists>s. P s \<and> \<pi>: (c, s) \<Rightarrow> t"
sees_method_fun: fixes P :: "'b prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and TS :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list"   and b' :: "staticb"   and TS' :: "ty list"   and T' :: "ty"   and m' :: "'b"   and D' :: "char list" assumes "P \<turnstile> C sees M, b :  TS\<rightarrow>T = m in D"   and "P \<turnstile> C sees M, b' :  TS'\<rightarrow>T' = m' in D'" shows "b = b' \<and> TS' = TS \<and> T' = T \<and> m' = m \<and> D' = D"
subgroupE_3: fixes H :: "nat set"   and a :: "nat" assumes "subgroup H G"   and "a \<in> H" shows "inv\<^bsub>G\<^esub> a \<in> H"
sum_lower_bound_card: fixes P :: "'a set set" assumes "bp P" shows "W U \<le> real (c * card P)"
extended_ord_is_hom_ord: shows "extended_ord.is_hom_ord None"
Polygamma_approx_complex_of_real: fixes x :: "real"   and m :: "nat"   and j :: "nat" assumes "0 < x"   and "0 < m" shows "Polygamma_approx j m (complex_of_real x) = complex_of_real (Polygamma_approx j m x)"
empt: fixes x :: "bool list" assumes "nfa_is_node A x" shows "\<not> subset_memb x subset_empt"
make\<o>_inverse: fixes y :: "j \<Rightarrow> i \<Rightarrow> bool" assumes "y \<in> UNIV" shows "eval\<o> (make\<o> y) = y"
summation_normalized: fixes B :: "real"   and f :: "nat \<Rightarrow> real"   and n :: "nat" assumes "B \<noteq> 0" shows "(\<Sum>i = 0..<n + 1. f i * B ^ (n - i)) = (\<Sum>i = 0..<n + 1. f i / B ^ i) * B ^ n"
IH: assumes "insert_ops xs"   and "(oid, ref) \<in> set xs" shows "\<exists>pre suf. xs = pre @ [(oid, ref)] @ suf \<and> (\<forall>i\<in>set (map fst pre). i < oid) \<and> (\<forall>a\<in>set (map fst suf). oid < a)"
that: fixes y :: "'a" assumes "max a b = a + y" shows "thesis"
set_fmap_inv1: fixes x :: "'g \<times> 'h"   and F :: "'g \<Rightarrow> 'h option" assumes "fst x \<in> dom F"   and "snd x = the (F (fst x))" shows "(F -- x)(fst x \<mapsto> snd x) = F"
new_def: fixes s :: "Store"   and t :: "New" shows "new s t \<equiv> case t of new_instance C \<Rightarrow> objV C (newOID (Rep_Store s) C) | new_array T l \<Rightarrow> arrV T (newAID (Rep_Store s) T)"
chain'': fixes y :: "'a" shows "Complete_Partial_Order.chain (\<le>) ((\<lambda>f x. f (x, y)) ` Y)"
not_distinct_decomp: fixes ws :: "'a list" assumes "\<not> distinct ws" shows "\<exists>xs ys zs y. ws = xs @ [y] @ ys @ [y] @ zs"
integral_on_image_ubound_weak: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> real"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec" assumes "S \<in> sets lebesgue"   and "f \<in> borel_measurable (lebesgue_on (g ` S))"   and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f (g x)"   and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) integrable_on S"   and "\<And>T. \<lbrakk>T \<subseteq> g ` S; T \<in> sets lebesgue\<rbrakk> \<Longrightarrow> {x \<in> S. g x \<in> T} \<in> sets lebesgue" shows "f integrable_on g ` S \<and> integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x))"
proots_within_iff: fixes x :: "'b"   and p :: "'b poly"   and s :: "'b set" shows "(x \<in> proots_within p s) = (poly p x = (0::'b) \<and> x \<in> s)"
assms_2: fixes s :: "'a" assumes "s \<in> orbit f t" shows "g s = f s"
axioms_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "OrderingSetIso less_eq less less_eq' less' P f" shows "OrderingSetMap less_eq less less_eq' less' P f"
neg_pf_neg_uvp: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and n :: "nat"   and w :: "bool stream" assumes "stock_portfolio Mkt pf"   and "1 + r < u"   and "pf stk (Suc n) (spick w n True) < 0"   and "val_process Mkt pf n (spick w n True) \<le> 0" shows "cls_val_process Mkt pf (Suc n) (spick w n True) < 0"
simps_2: fixes v :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and va :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.meld (v # va) [] = v # va"
even_mask_iff: fixes n :: "nat" shows "even (mask n) = (n = 0)"
cpx_mat_cnj_prod: fixes M :: "complex mat"   and N :: "complex mat" assumes "dim_col M = dim_row N" shows "M * N\<^sup>\<star> = M\<^sup>\<star> * N\<^sup>\<star>"
from_hma_to_hma\<^sub>m: fixes A :: "'d mat" assumes "A \<in> carrier_mat CARD('e) CARD('f)" shows "Mod_Type_Connect.from_hma\<^sub>m (Mod_Type_Connect.to_hma\<^sub>m A) = A"
isolate_def: fixes x :: "'b"   and S :: "'b set" shows "x isolate S = (x \<in> S \<and> (\<exists>T. open T \<and> T \<inter> S = {x}))"
simps: fixes E :: "('e \<times> 'e) set"   and a1 :: "'e"   and a2 :: "'e list"   and a3 :: "'e" shows "path E a1 a2 a3 = ((\<exists>u. a1 = u \<and> a2 = [] \<and> a3 = u) \<or> (\<exists>u v l w. a1 = u \<and> a2 = u # l \<and> a3 = w \<and> (u, v) \<in> E \<and> path E v l w))"
induct: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>tvar. P (TVar tvar)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1\<rightarrow>ty2)" shows "P ty"
lemma_sin_pi: fixes n :: "nat star" assumes "n \<in> HNatInfinite" shows "(*f* sin) (inverse (hypreal_of_hypnat n)) / inverse (hypreal_of_hypnat n) \<approx> 1"
geq1: shows "r1 \<le> r'\<^sub>1"
lcp_same_len: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list"   and w' :: "'b list" assumes "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|"   and "u \<noteq> v" shows "u \<and>\<^sub>p v = u \<cdot> w \<and>\<^sub>p v \<cdot> w'"
to_hma_from_hma\<^sub>m: fixes v :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "to_hma\<^sub>m (from_hma\<^sub>m v) = v"
in_br_conv: fixes c :: "'d"   and a :: "'e"   and \<alpha> :: "'d \<Rightarrow> 'e"   and I :: "'d \<Rightarrow> bool" shows "((c, a) \<in> br \<alpha> I) = (a = \<alpha> c \<and> I c)"
LLL_measure_def: fixes i :: "nat"   and fs :: "int vec list" shows "LLL_measure i fs = 2 * logD fs + m - i"
A: shows "pr_index_enumerator n m1 = pr_index_enumerator n m2"
prime_elem_multiplicity_prod_mset_distrib: fixes p :: "'a"   and A :: "'a multiset" assumes "prime_elem p"   and "(0::'a) \<notin># A" shows "multiplicity p (\<Prod>\<^sub># A) = \<Sum>\<^sub># (image_mset (multiplicity p) A)"
length_map: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "length (map f xs) = length xs"
proper_interval_not_set_less_eq_auxI: fixes ao :: "'a option"   and x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "proper_interval ao (Some x)"   and "\<forall>a\<in>A. x \<le> a"   and "\<forall>b\<in>B. x \<le> b" shows "\<not> - A \<inter> above ao \<sqsubseteq>' B"
inject: fixes x31 :: "comb"   and x32 :: "comb"   and y31 :: "comb"   and y32 :: "comb" shows "(x31 \<bullet> x32 = y31 \<bullet> y32) = (x31 = y31 \<and> x32 = y32)"
zmset_of_inverse: fixes M :: "'b multiset" shows "mset_pos (zmset_of M) = M"
integral_discrete_difference: fixes X :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. \<lbrakk>x \<in> space M; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L M g"
card_rcosets_equal: fixes G :: "('a, 'b) monoid_scheme"   and R :: "'a set"   and H :: "'a set" assumes "Group.group G"   and "R \<in> rcosets\<^bsub>G\<^esub> H"   and "H \<subseteq> carrier G" shows "card H = card R"
lookup_def: fixes xs :: "('b, 'c) oalist_ntm" shows "OAlist_lookup_ntm xs = lookup_ko_ntm (list_of_oalist_ntm xs)"
a_notin: shows "to_nat a \<notin> set [i + 1..<min (nrows A) (ncols A)]"
mod_eq_imp_div_right_inj_on: fixes k :: "nat"   and I :: "nat set" assumes "0 < k"   and "\<forall>x\<in>I. \<forall>y\<in>I. x mod k = y mod k" shows "inj_on (\<lambda>x. x div k) I"
right_unique_rel_converter: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "right_total A"   and "right_unique B"   and "right_unique C"   and "right_total R" shows "right_unique (rel_converter A B C R)"
preserves_inv: fixes f :: "'a" assumes "S.iso f" shows "S.src (S.inv f) = S.inv (S.src f)"
wf_set_of_implies_wfP: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "wf (set_of P)" shows "wfP P"
rep_eq: fixes x :: "'a mod_ring" shows "to_int_mod_ring x = Rep_mod_ring x"
signCert_def: fixes K :: "nat"   and X :: "msg" shows "signCert K X = \<lbrace>X, Crypt K X\<rbrace>"
abs_eq: fixes xa :: "('c, 'd) phi \<Rightarrow> 'd"   and x :: "('c, 'd) phi \<Rightarrow> 'd" assumes "eq_onp (\<lambda>xf. True) xa xa"   and "eq_onp (\<lambda>xf. True) x x" shows "(Abs_lifted_distrib_lattice_top xa \<le> Abs_lifted_distrib_lattice_top x) = (\<forall>f. xa f \<le> x f)"
abs_idempotent: fixes abs :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (abs a) = abs a"
gR_galois_imp1: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "D x z"   and "x \<oplus> z \<preceq>\<^sub>R y" shows "z \<preceq>\<^sub>R rquot y x"
Pair_eqvt: fixes p :: "perm"   and x :: "'e"   and y :: "'f" shows "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
UnopElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and f :: "Val \<Rightarrow> Val" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (unop f)" shows "\<exists>v ops. os = v # ops \<and> n = Suc 0 \<and> t = (f v # ops, S, h) \<and> ll = l + 1"
contI: fixes orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and luba :: "'a set \<Rightarrow> 'a"   and lubb :: "'b set \<Rightarrow> 'b"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>Y. \<lbrakk>Complete_Partial_Order.chain orda Y; Y \<noteq> {}\<rbrakk> \<Longrightarrow> f (luba Y) = lubb (f ` Y)" shows "cont luba orda lubb ordb f"
map_id_on_out: fixes x :: "'c"   and xs :: "'c set" assumes "x \<notin> xs" shows "map_id_on xs x = None"
Cons: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and es :: "char list exp list"   and es' :: "char list exp list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e # es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>Val v # es',s\<^sub>2\<rangle>"
orderE: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" obtains "a = a \<squnion> b"
discover_impl_def: fixes u :: "'f"   and v :: "'a"   and s :: "('a, 'e) simple_state_scheme" shows "a.c.discover_impl u v s \<equiv> ASSERT (v \<notin> on_stack s \<and> v \<notin> visited s) \<bind> (\<lambda>_. let s = s\<lparr>ss_stack := (v, a.a.E `` {v}) # ss_stack s\<rparr>; s = s\<lparr>on_stack := insert v (on_stack s)\<rparr> in Let (s\<lparr>visited := insert v (visited s)\<rparr>) RETURN)"
flatten_focus: fixes X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "flatten (focus X p) = p"
zero_less_HMSet_iff: fixes M :: "hmultiset multiset" shows "(0 < HMSet M) = (M \<noteq> {#})"
add_arc_commute: fixes b :: "'b"   and a :: "'b" shows "pre_digraph.add_arc (G.add_arc b) a = pre_digraph.add_arc (G.add_arc a) b"
case_2: fixes f1 :: "'c"   and f2 :: "'c"   and f3 :: "'d \<Rightarrow> 'c" shows "(case dFAILi of dSUCCEEDi \<Rightarrow> f1 | dFAILi \<Rightarrow> f2 | dRETURN x \<Rightarrow> f3 x) = f2"
elims_3: fixes x :: "constraint" assumes "\<not> nonstrict_constr x" obtains v :: "linear_poly" and va :: "rat" where "x = LT v__ va__"   | v :: "linear_poly" and va :: "rat" where "x = GT vb__ vaa__"   | v :: "linear_poly" and va :: "linear_poly" where "x = LTPP vc__ vab__"   | v :: "linear_poly" and va :: "linear_poly" where "x = GTPP v va"
less_eq_Sup: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and u :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>v. v \<in> A \<Longrightarrow> less_eq u v"   and "A \<noteq> {}" shows "less_eq u (Sup A)"
map_le_implies_dom_le: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "f \<subseteq>\<^sub>m g" shows "dom f \<subseteq> dom g"
rwp_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "rwp_on A f = {(x, y). \<exists>B\<subseteq>A. x \<in> B \<and> y \<in> f B}"
splitset_fun_nonempty: fixes xs :: "'c list" shows "splitset_fun xs \<noteq> {}"
network_def: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd" shows "network history msg_id \<equiv> node_histories history \<and> network_axioms history msg_id"
dependent_def: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and P :: "'d set" assumes "vector_space scale" shows "module.dependent scale P = (\<exists>a\<in>P. a \<in> module.span scale (P - {a}))"
ir_runs2sigs_upd_resp_some_raw: fixes runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Ts :: "nat"   and Ta :: "nat"   and s :: "(key \<times> nat \<times> nat) signal" assumes "runz Rb = Some (Resp, [A, B], [])"   and "finite (dom runz)" shows "ir_runs2sigs (runz(Rb \<mapsto> (Resp, [A, B], [aKey Kab, aNum Ts, aNum Ta]))) s = ((ir_runs2sigs runz)(Running [A, B] (Kab, Ts, Ta) := Suc (card (ir_running runz A B Kab Ts Ta)))) s"
lex_convex2: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real"   and u :: "real" assumes "Counterclockwise_2D_Arbitrary.lex p q"   and "Counterclockwise_2D_Arbitrary.lex p r"   and "0 \<le> u"   and "u \<le> 1" shows "Counterclockwise_2D_Arbitrary.lex p (u *\<^sub>R q + (1 - u) *\<^sub>R r)"
map_scale_assoc: fixes c :: "'d"   and d :: "'d"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "c \<cdot> d \<cdot> t = (c * d) \<cdot> t"
map_entry_code: fixes k :: "'b"   and f :: "'c \<Rightarrow> 'c"   and m :: "('b, 'c) mapping" shows "map_entry k f m = (case lookup m k of None \<Rightarrow> m | Some v \<Rightarrow> update k (f v) m)"
single_valued_def: fixes r :: "('d \<times> 'e) set" shows "single_valued r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
shift_left_inj: fixes xs :: "'d list"   and s1 :: "'d stream"   and s2 :: "'d stream" shows "(xs @- s1 = xs @- s2) = (s1 = s2)"