hcju/isabelleps huggingface dataset:
mem_vertices: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and v :: "'b \<times> nat list" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "(v |\<in>| Solved_Task.vertices freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions) = (fst v \<in> set conclusions \<and> (snd v = [] \<or> snd v \<in> (#) 0 ` it_paths (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent [fidx (Abstract_Task.conc_forms conclusions) (fst v), 0] (freshenLC v_away) (Abstract_Task.to_it freshenLC renameLCs lconsts subst subst_lconsts anyP antecedent consequent rules assumptions conclusions (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions (fst v))))))"
A: fixes u :: "nat \<Rightarrow> 'a" assumes "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi" shows "limsup (\<lambda>n. ereal (dist x (u n) - dist y (u n))) \<le> ereal (Busemann_function_at xi x' y') + ereal (dist x x' + dist y y')"
triangle: fixes x :: "'c Gromov_completion"   and z :: "'c Gromov_completion"   and y :: "'c Gromov_completion" shows "turn_into_distance extended_predist x z \<le> turn_into_distance extended_predist x y + turn_into_distance extended_predist y z"
eval_subst_trm: fixes e :: "(name, nat) finfun"   and i :: "name"   and t :: "trm"   and u :: "trm" shows "eval_trm e (subst i t u) = eval_trm (finfun_update e i (eval_trm e t)) u"
append_prefixD: fixes xs :: "'e list"   and ys :: "'e list"   and zs :: "'e list" assumes "prefix (xs @ ys) zs" shows "prefix xs zs"
poincare_collinear3_poincare_line_general: fixes a :: "complex_homo"   and a1 :: "complex_homo"   and a2 :: "complex_homo" assumes "poincare_collinear {a, a1, a2}"   and "a1 \<noteq> a2"   and "a1 \<noteq> inversion a2" shows "a \<in> circline_set (poincare_line a1 a2)"
simps_2: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'d prog"   and h :: "'b"   and a :: "'c"   and T :: "ty" shows "heap_base.conf_extRet typeof_addr P h (RetExc a) T = heap_base.conf typeof_addr P h (Addr a) (Class Throwable)"
deadends_in_V: fixes p :: "Player" shows "deadends p \<subseteq> V"
assms_1: shows "otherwith P I PO \<sigma> \<sigma>' a"
Re_complex_div_lt_0: fixes a :: "complex"   and b :: "complex" shows "(Re (a / b) < 0) = (Re (a * cnj b) < 0)"
un_B_cases: fixes y :: "'c" assumes "y \<in> UNIV" obtains x :: "'c dual" where "y = un_B x"
unclocked_single_sound: fixes v :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and bv :: "'b state \<times> (v, v) result" assumes "Big_Step_Unclocked_Single.evaluate v s e bv" shows "Big_Step_Unclocked.evaluate v s e bv"
sub: fixes i :: "name"   and t :: "tm"   and u :: "tm"   and v :: "tm"   and x :: "tm" assumes "atom i \<sharp> tms" shows "(P t u v)(i::=x) = P (subst i x t) (subst i x u) (subst i x v)"
idx_in_elem_inserted: fixes history :: "nat \<Rightarrow> ('c \<times> ('c, 'd) operation) event list"   and i :: "'e"   and e :: "'e \<times> 'f \<times> bool"   and n :: "'e option"   and xs :: "('e \<times> ('e, 'f) operation) event list" assumes "rga history"   and "Deliver (i, Insert e n) \<in> set xs" shows "fst e \<in> set (indices xs)"
pt_eqvt_fun2b: fixes f :: "'a \<Rightarrow> 'c" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "\<forall>pi x. pi \<bullet> f x = f (pi \<bullet> x)" shows "supp f = {}"
images_are_bij: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and g :: "'c" assumes "group_action G \<phi> M"   and "g \<in> carrier G" shows "bij_betw (\<phi> g) M M"
treewidth_lower_bound_1: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a" assumes "Graph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "1 \<le> Graph.treewidth G"
syms: shows "\<forall>i>M. num_syms (g i) = num_syms (g (Suc i))"
AAss1Throw1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and i :: "'addr expr1"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Throw a\<lfloor>i\<rceil> := e,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
filt_equiv_def: fixes F :: "'c \<Rightarrow> 'd measure"   and M :: "'d measure"   and N :: "'d measure" shows "filt_equiv F M N = (sets M = sets N \<and> Filtration.filtration M F \<and> (\<forall>t A. A \<in> sets (F t) \<longrightarrow> (emeasure M A = 0) = (emeasure N A = 0)))"
simps_26: fixes f1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x11 :: "'b"   and x12 :: "'c"   and x13 :: "'c" shows "(case Trans x11 x12 x13 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f1 x11 x12 x13"
euclidean_size_greater_0_iff: fixes divide :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and times :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and one :: "'f"   and modulo :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and euclidean_size :: "'f \<Rightarrow> nat"   and b :: "'f" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "(0 < euclidean_size b) = (b \<noteq> zero)"
add_point_def: fixes p :: "'a" shows "combine_sys.add_point p \<equiv> insert p \<V>\<^sup>+"
ipoly_poly_add_nonzero: fixes p :: "int poly"   and q :: "int poly"   and x :: "'a"   and y :: "'a" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "ipoly p x = (0::'a)"   and "ipoly q y = (0::'a)"   and "of_int (lead_coeff p) \<noteq> (0::'a)"   and "of_int (lead_coeff q) \<noteq> (0::'a)" shows "poly_add p q \<noteq> 0"
simps: fixes m :: "nat" shows "s2 m = 2 * m + 2"
has_uniq_bottom: fixes P :: "'a set" assumes "domain.has_bottom P" shows "\<exists>!z. z \<in> P \<and> (\<forall>x\<in>P. x \<succeq> z)"
monad_state_stateT': assumes "monad Monad_Overloading.return Monad_Overloading.bind" shows "monad_state Monad_Overloading.return Monad_Overloading.bind get put"
N_ge_0: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool" assumes "grover_state n f" shows "0 < grover_state.N n"
Inf_apply: fixes A :: "('b \<Rightarrow> 'a) set"   and x :: "'b" shows "Inf A x = (INF f\<in>A. f x)"
length_Suc_conv: fixes xs :: "'b list"   and n :: "nat" shows "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
aux_4: fixes t :: "nat" assumes "ne \<noteq> 0"   and "t \<le> \<tau> \<sigma> (ne - 1)"   and "enat (\<tau> \<sigma> (ne - 1) - t) \<le> right I"   and "\<exists>i. \<tau> \<sigma> i = t" shows "\<exists>X. (t, X) \<in> set auxlist"
wary_apps: fixes s :: "('s, 'v) tm"   and ss :: "('s, 'v) tm list" assumes "wary s"   and "\<And>sa. sa \<in> set ss \<Longrightarrow> wary sa"   and "enat (length ss) \<le> arity s" shows "wary (apps s ss)"
max_min_distrib2: fixes a :: "'t"   and b :: "'t"   and c :: "'t" shows "ord_term_lin.max a (ord_term_lin.min b c) = ord_term_lin.min (ord_term_lin.max a b) (ord_term_lin.max a c)"
conseq2: fixes G :: "'a triple set"   and P :: "'a assn"   and c :: "com"   and Q' :: "'a assn"   and Q :: "'a assn" assumes "G|-{P}. c .{Q'}"   and "\<forall>Z s. Q' Z s \<longrightarrow> Q Z s" shows "G|-{P}. c .{Q}"
split_sels_1: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = ((nat = 0 \<longrightarrow> P f1) \<and> (nat = Suc (nat.pred nat) \<longrightarrow> P (f2 (nat.pred nat))))"
listsp_mono: fixes A :: "'g \<Rightarrow> bool"   and B :: "'g \<Rightarrow> bool" assumes "A \<le> B" shows "listsp A \<le> listsp B"
simps_1: fixes i :: "nat"   and v :: "'a" shows "[][i := v] = []"
uminus_inf: shows "~\<^sub>h \<infinity>\<^sub>h = \<infinity>\<^sub>h"
distinct_1: fixes x11 :: "nat"   and x12 :: "nat"   and x21 :: "nat"   and x22 :: "nat" shows "Less x11 x12 \<noteq> Eq x21 x22"
cases: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and a1 :: "'addr expr list"   and a2 :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and a3 :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and a4 :: "'addr expr list"   and a5 :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" assumes "extTA,P,t \<turnstile> \<langle>a1,a2\<rangle> [-a3\<rightarrow>] \<langle>a4,a5\<rangle>" obtains e :: "'addr expr" and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action" and e' :: "'addr expr" and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" and es :: "'addr expr list" where "a1 = e # esa__" and "a2 = sa__" and "a3 = taa__" and "a4 = e' # esa__" and "a5 = s'a__" and "extTA,P,t \<turnstile> \<langle>e,sa__\<rangle> -taa__\<rightarrow> \<langle>e',s'a__\<rangle>"   | es :: "'addr expr list" and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action" and es' :: "'addr expr list" and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" and v :: "'addr val" where "a1 = Val v # es" and "a2 = s" and "a3 = ta" and "a4 = Val v # es'" and "a5 = s'" and "extTA,P,t \<turnstile> \<langle>es,s\<rangle> [-ta\<rightarrow>] \<langle>es',s'\<rangle>"
rel_blinfun_def: shows "rel_blinfun \<equiv> (id ---> id ---> blinfun_apply ---> blinfun_apply ---> id) (===>)"
col_submatrix_UNIV: fixes j :: "nat"   and A :: "'c mat"   and J :: "nat set" assumes "j < card {i. i < dim_col A \<and> i \<in> J}" shows "col (submatrix A UNIV J) j = col A (pick J j)"
\<phi>_def: shows "\<phi> = (SOME \<phi>. bij \<phi>)"
closedI: fixes s :: "'a set" assumes "carrier - s open" shows "s closed"
add_left: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and a :: "'d"   and a' :: "'d"   and b :: "'e" assumes "bounded_sesquilinear prod" shows "prod (a + a') b = prod a b + prod a' b"
igSwapIGAbs_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igSwapIGAbs MOD \<equiv> \<forall>zs z1 z2 xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igSwapAbs MOD zs z1 z2 (igAbs MOD xs x X) = igAbs MOD xs (x @xs[z1 \<and> z2]_zs) (igSwap MOD zs z1 z2 X)"
dense_accessible_frontier_point_pairs: fixes S :: "'a set"   and U :: "'a set"   and V :: "'a set" assumes "open S"   and "connected S"   and "openin (top_of_set (frontier S)) U"   and "openin (top_of_set (frontier S)) V"   and "U \<noteq> {}"   and "V \<noteq> {}"   and "U \<noteq> V" obtains g :: "real \<Rightarrow> 'a" where "arc g" and "pathstart g \<in> U" and "pathfinish g \<in> V" and "g ` {0<..<1} \<subseteq> S"
that: fixes r :: "real" assumes "r \<in> \<rat>"   and "\<bar>r - x\<bar> < e" shows "thesis"
simps_2: fixes s1 :: "'d stream"   and s2 :: "'e stream" shows "stl (szip s1 s2) = szip (stl s1) (stl s2)"
rel_S_mono: fixes L1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and L1' :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and L2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and L2' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and Co1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Co1' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Co2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Co2' :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1' :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Contra1 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Contra2' :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and Contra2 :: "'m \<Rightarrow> 'n \<Rightarrow> bool" assumes "L1 \<le> L1'"   and "L2 \<le> L2'"   and "Co1 \<le> Co1'"   and "Co2 \<le> Co2'"   and "Contra1' \<le> Contra1"   and "Contra2' \<le> Contra2" shows "rel_S L1 L2 Co1 Co2 Contra1 Contra2 \<le> rel_S L1' L2' Co1' Co2' Contra1' Contra2'"
powser_inside: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and z :: "'a" assumes "summable (\<lambda>n. f n * x ^ n)"   and "norm z < norm x" shows "summable (\<lambda>n. f n * z ^ n)"
assms_2: fixes r :: "real" assumes "r \<in> A" shows "f holomorphic_on ball c r"
filter_is_subset: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "set (filter P xs) \<subseteq> set xs"
orbit_fixate_self: fixes a :: "'b"   and f :: "'b perm" shows "orbit (fixate a f) a = {a}"
array_rel_imp_same_length: fixes a :: "'a array"   and a' :: "'b array"   and R :: "('a \<times> 'b) set" assumes "(a, a') \<in> \<langle>R\<rangle>array_rel" shows "array_length a = array_length a'"
deg_pm_single: fixes x :: "'b"   and k :: "'a" shows "deg_pm (monomial k x) = k"
size_mset_btree: fixes t :: "'b tree" assumes "invar_btree t" shows "size (mset_tree t) = 2 ^ rank t"
UN_upper: fixes a :: "'b"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "a \<in> A" shows "B a \<subseteq> \<Union> (B ` A)"
abelian_subgroupI3: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme" assumes "additive_subgroup H G"   and "abelian_group G" shows "abelian_subgroup H G"
indep_set_mono: fixes A :: "'a set set"   and B :: "'a set set"   and A' :: "'a set set"   and B' :: "'a set set" assumes "indep_set A B"   and "A' \<subseteq> A"   and "B' \<subseteq> B" shows "indep_set A' B'"
gen_ramanujan_sum_def: fixes f :: "nat \<Rightarrow> complex"   and g :: "nat \<Rightarrow> complex" shows "s f g = (\<lambda>k n. \<Sum>d | d dvd gcd n k. f d * g (k div d))"
iso_def: fixes f :: "'a1" shows "C1.iso f \<equiv> \<exists>g. C1.inverse_arrows f g"
nn_set_integral_space: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "set_nn_integral M (space M) f = integral\<^sup>N M f"
wls_obtain_fresh: fixes V :: "'varSort \<Rightarrow> 'var set"   and XS :: "'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set set"   and AS :: "'varSort \<Rightarrow> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set set"   and Rho :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option) set"   and zs :: "'varSort" assumes "\<forall>xs. |V xs| <o |UNIV| \<or> finite (V xs)"   and "\<forall>s. |XS s| <o |UNIV| \<or> finite (XS s)"   and "\<forall>xs s. |AS xs s| <o |UNIV| \<or> finite (AS xs s)"   and "\<forall>s. \<forall>X\<in>XS s. wls s X"   and "\<forall>xs s. \<forall>A\<in>AS xs s. wlsAbs (xs, s) A"   and "|Rho| <o |UNIV| \<or> finite Rho"   and "Ball Rho wlsEnv" shows "\<exists>z. (\<forall>xs. z \<notin> V xs) \<and> (\<forall>s. \<forall>X\<in>XS s. fresh zs z X) \<and> (\<forall>xs s. \<forall>A\<in>AS xs s. freshAbs zs z A) \<and> (\<forall>rho\<in>Rho. freshEnv zs z rho)"
mult_1_left_octo: fixes x :: "octo" shows "1 * x = x"
preserves_ide: fixes a :: "'a" assumes "A.ide a" shows "B.ide (A_B.Map x a)"
mat_mult_append: fixes A :: "'b mat"   and nr1 :: "nat"   and nc :: "nat"   and B :: "'b mat"   and nr2 :: "nat"   and v :: "'b vec" assumes "A \<in> carrier_mat nr1 nc"   and "B \<in> carrier_mat nr2 nc"   and "v \<in> carrier_vec nc" shows "(A @\<^sub>r B) *\<^sub>v v = A *\<^sub>v v @\<^sub>v B *\<^sub>v v"
IH: fixes E :: "('b \<times> ('c, 'b) obs_event action) list" assumes "ta_hb_consistent P E E'"   and "ta_hb_consistent P (E @ list_of E') E''" shows "ta_hb_consistent P E (lappend E' E'')"
simps_3: fixes oneC :: "int \<times> (real \<Rightarrow> real \<times> real)"   and v :: "int \<times> (real \<Rightarrow> real \<times> real)"   and va :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" shows "rec_join (oneC # v # va) = coeff_cube_to_path oneC +++ rec_join (v # va)"
that: fixes a :: "'c" assumes "a |\<in>| as"   and "fst (f a) = b" shows "thesis"
solution_uminus_right: fixes x :: "'b"   and y :: "'b" shows "solution (x, - y) = solution (x, y)"
nonzero_bl2wc: fixes b :: "cell list" assumes "\<forall>m. b \<noteq> Bk \<up> m" shows "0 < bl2wc b"
gen_cong_eval_rel_fun: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "rel_fun (rel (gen_cong R)) (gen_cong R) eval eval"
inf_seq_distrib: fixes c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" shows "((c\<^sub>0::'b::refinement_lattice) \<sqinter> (c\<^sub>1::'b::refinement_lattice)) ; (d::'b::refinement_lattice) = c\<^sub>0 ; d \<sqinter> c\<^sub>1 ; d"
assms_2: shows "theta\<inverse> \<subseteq> Wretr (theta\<inverse>)"
FWdelay_bisimulation_lift_flip: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves" assumes "FWdelay_bisimulation_lift final1 r1 final2 r2 bisim \<tau>move1 \<tau>move2" shows "FWdelay_bisimulation_lift final2 r2 final1 r1 (\<lambda>t. flip (bisim t)) \<tau>move2 \<tau>move1"
intro: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog) \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_class_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog" assumes "l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order"   and "l_to_tree_order to_tree_order"   and "l_get_attribute type_wf get_attribute get_attribute_locs"   and "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms to_tree_order first_in_tree_order get_attribute get_element_by_id get_elements_by_class_name get_elements_by_tag_name" shows "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order get_attribute get_attribute_locs get_element_by_id get_elements_by_class_name get_elements_by_tag_name type_wf"
inner_nth_eucl: fixes x :: "'b"   and y :: "'b" shows "x \<bullet> y = (\<Sum>i<DIM('b). x $\<^sub>e i * y $\<^sub>e i)"
trilean_distributivity_1: fixes a :: "trilean"   and b :: "trilean"   and c :: "trilean" shows "(a \<or>? b) \<and>? c = a \<and>? c \<or>? b \<and>? c"
vimage_eq: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" shows "(a \<in> f -` B) = (f a \<in> B)"
circ_slide: fixes x :: "'a"   and y :: "'a" shows "x * (y * x)\<^sup>\<circ> = (x * y)\<^sup>\<circ> * x"
root_p: shows "x \<in> {x. poly p x = 0}"
ivl_disj_un_one_6: fixes l :: "'a"   and u :: "'a" assumes "l < u" shows "{l<..<u} \<union> {u..} = {l<..}"
coprime_divisors: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "a dvd c"   and "b dvd d"   and "coprime c d" shows "coprime a b"
ground_cs: shows "\<forall>i<n. is_ground_cls (Cs ! i)"
is_mgu_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "is_mgu \<sigma> E = (\<sigma> \<in> unifiers E \<and> (\<forall>\<tau>\<in>unifiers E. \<exists>\<gamma>. \<tau> = \<sigma> \<circ>\<^sub>s \<gamma>))"
ub_exp: fixes y :: "real"   and x :: "float"   and prec :: "nat" assumes "y \<le> exp (real_of_float x)" shows "y \<le> real_of_float (ub_exp prec x)"
homotopic_paths_imp_pathfinish: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c"   and q :: "real \<Rightarrow> 'c" assumes "homotopic_paths s p q" shows "pathfinish p = pathfinish q"
checker_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real" shows "checker s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<equiv> let distance = s\<^sub>o - s\<^sub>e; precond = check_precond s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o; safe_dist1 = first_safe_dist v\<^sub>e a\<^sub>e; safe_dist2 = second_safe_dist v\<^sub>e a\<^sub>e v\<^sub>o a\<^sub>o; cond2 = suff_cond_safe_dist2 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o in precond \<and> (safe_dist1 < distance \<or> safe_dist2 < distance \<and> distance \<le> safe_dist1 \<and> cond2)"
simps_57: fixes x51 :: "expr"   and x52 :: "stmt"   and x53 :: "stmt"   and x31 :: "jump"   and x32 :: "stmt" shows "If(x51) x52 Else x53 \<noteq> x31\<bullet> x32"
vec_lambda_unique: fixes f :: "('b, 'a) vec"   and g :: "'a \<Rightarrow> 'b" shows "(\<forall>i. f $ i = g i) = (vec_lambda g = f)"
simps_36: fixes x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'d" shows "\<tau>\<^sub>n \<noteq> x31\<not>x32:arrive(x33)"
set_of_seq_def: shows "set_of_seq = set_of_pred \<circ> Predicate.pred_of_seq"
jozsa_transform_dim_1: shows "dim_row U\<^sub>f = 2 ^ (n + 1)"
is_unit_poly_iff: fixes p :: "'b poly" shows "(p dvd 1) = (\<exists>c. p = [:c:] \<and> c dvd (1::'b))"
assms_3: shows "\<forall>X\<in>Pow C. f X = X \<longrightarrow> X = C"
erlang_distributedI: fixes X :: "'b \<Rightarrow> real"   and M :: "'b measure"   and l :: "real"   and k :: "nat" assumes "X \<in> borel_measurable M"   and "0 < l"   and "\<And>a. 0 \<le> a \<Longrightarrow> emeasure M {x \<in> space M. X x \<le> a} = ennreal (erlang_CDF k l a)" shows "distributed M lborel X (\<lambda>x. ennreal (erlang_density k l x))"
assms: shows "- (- (x \<squnion> - y) \<squnion> - y) = x"
iMOD_subset: fixes r :: "nat"   and r' :: "nat"   and m :: "nat"   and m' :: "nat" assumes "r \<le> r'"   and "r' mod m = r mod m"   and "m' mod m = 0" shows "[ r', mod m' ] \<subseteq> [ r, mod m ]"
atLeastatMost_empty_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq a b = {}) = (\<not> less_eq a b)"
least_def: fixes L :: "('a, 'b) gorder_scheme"   and l :: "'a"   and A :: "'a set" shows "least L l A = (A \<subseteq> carrier L \<and> l \<in> A \<and> (\<forall>x\<in>A. l \<sqsubseteq>\<^bsub>L\<^esub> x))"
list2FWpolicys_eq: fixes Filter :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "Filter \<noteq> []" shows "Cp (list2policyR Filter) = Cp (list2FWpolicy (rev Filter))"
fresh_atm_3: fixes a :: "name"   and b :: "coname" shows "a \<sharp> b"
IH: fixes gpv :: "('b, 'a, 'c) gpv" assumes "interaction_bounded_by' (\<lambda>_. True) gpv n" shows "finite_gpv \<I>_full gpv"