hcju/isabelleps huggingface dataset:
exchange_alternating_not_in_alternating: fixes n :: "nat"   and s :: "'w"   and t :: "'w"   and w :: "'w"   and cs :: "'w list"   and xs :: "'w list"   and x :: "'w"   and ys :: "'w list" assumes "order.greater_eq n 2"   and "order.greater (add_order (s + t)) n \<or> add_order (s + t) = 0"   and "S_reduced_for w (alternating_list n s t @ cs)"   and "alternating_list n s t @ cs = xs @ [x] @ ys"   and "S_reduced_for w (t # xs @ ys)" shows "order.greater_eq (length xs) n"
pairwise_imageI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y; f x \<noteq> f y\<rbrakk> \<Longrightarrow> P (f x) (f y)" shows "pairwise P (f ` A)"
typ_arity_lookup_code: fixes \<Sigma> :: "exesignature"   and n :: "String.literal" shows "type_arity (translate_signature \<Sigma>) n = lookup (\<lambda>k. k = n) (exetyp_arity_of \<Sigma>)"
apply_fun_range: fixes R :: "'e \<Rightarrow> 'h \<Rightarrow> 'f"   and P :: "'g \<Rightarrow> 'e \<Rightarrow> 'h" shows "{y. \<exists>x. y = (R .. P x)} = range (\<lambda>x. R .. P x)"
eval_quot_dbtm_ignore: fixes e :: "(name, hf) finfun"   and t :: "dbtm"   and e' :: "(name, hf) finfun" shows "\<lbrakk>quot_dbtm t\<rbrakk>e = \<lbrakk>quot_dbtm t\<rbrakk>e'"
lasso_of_prpl_def: fixes prpl :: "'b list \<times> 'b list" shows "lasso_of_prpl prpl \<equiv> case prpl of (pr, pl) \<Rightarrow> \<lparr>lasso_reach = pr, lasso_va = hd pl, lasso_cysfx = tl pl\<rparr>"
Cons_eq: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "\<not> x < y"   and "\<not> y < x"   and "lexordp_eq xs ys" shows "lexordp_eq (x # xs) (y # ys)"
that: fixes q :: "'b list" assumes "trail w q w"   and "set q = set p"   and "set (awalk_verts w q) = set (awalk_verts u p)" shows "thesis"
insert_body_commutes: fixes xs :: "'a list"   and e1 :: "'a"   and e2 :: "'a" shows "insert_body (insert_body xs e1) e2 = insert_body (insert_body xs e2) e1"
in_set_nthsD: fixes x :: "'c"   and xs :: "'c list"   and I :: "nat set" assumes "x \<in> set (nths xs I)" shows "x \<in> set xs"
vimage_inter_cong: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and y :: "'b set" assumes "\<And>w. w \<in> S \<Longrightarrow> f w = g w" shows "f \<^sup>\<inverse> S y = g \<^sup>\<inverse> S y"
integer_hull_integer_cone: fixes C :: "'b vec set"   and n :: "nat" assumes "C \<subseteq> carrier_vec n"   and "C \<subseteq> \<int>\<^sub>v" shows "gram_schmidt.integer_hull n (gram_schmidt.cone n C) = gram_schmidt.cone n C"
decseq_def: fixes X :: "nat \<Rightarrow> 'd" shows "decseq X = (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
strong_barbed_simulation_impl_weak_preservation_of_barbs: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "strong_reduction_simulation Rel (Calculus CWB) \<and> rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a CWB)" shows "rel_weakly_preserves_barbs Rel CWB"
step: fixes y :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and x :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "y \<in> set_pmf (K_cfg x)"   and "x \<in> {cfg. \<exists>cfg' s. cfg' \<in> R_G.valid_cfg \<and> cfg = repcs s cfg' \<and> abss s = state cfg'}" shows "y \<in> {cfg. \<exists>cfg' s. cfg' \<in> R_G.valid_cfg \<and> cfg = repcs s cfg' \<and> abss s = state cfg'}"
singleCombinatorsStart: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" assumes "singleCombinators (x # xs)" shows "singleCombinators [x]"
combined_size_bound_integer: fixes x :: "rat" assumes "x \<in> {map of_int_hom.vec_hom fs ! i $ j |i j. i < length (map of_int_hom.vec_hom fs) \<and> j < n} \<union> {gs.\<mu>' i j |i j. j \<le> i \<and> i < length (map of_int_hom.vec_hom fs)} \<union> {gs.\<sigma> l i j |i j l. i < length (map of_int_hom.vec_hom fs) \<and> j \<le> i \<and> l \<le> j}"   and "length (map of_int_hom.vec_hom fs) \<noteq> 0" shows "\<bar>x\<bar> \<le> rat_of_nat (length (map of_int_hom.vec_hom fs)) * gs.N ^ (3 * Suc (length (map of_int_hom.vec_hom fs)))"
equality: fixes r :: "('c, 'd) alpha_rp_scheme"   and r' :: "('c, 'd) alpha_rp_scheme" assumes "ok r = ok r'"   and "wait r = wait r'"   and "tr r = tr r'"   and "ref r = ref r'"   and "alpha_rp.more r = alpha_rp.more r'" shows "r = r'"
distinct_27: fixes ccs1 :: "ccs"   and ccs2 :: "ccs"   and ccs' :: "ccs" shows "ccs1 \<parallel> ccs2 \<noteq> !ccs'"
interleave_set: fixes s :: "'a event list"   and t :: "'a event list"   and C :: "'a event set"   and u :: "'a event list" assumes "s setinterleaves ((t, u), C)" shows "set t \<union> set u \<subseteq> set s"
fresh_pgwt_subterm_fresh: fixes S :: "'fun set"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term" assumes "finite S"   and "wf\<^sub>t\<^sub>r\<^sub>m t"   and "wf\<^sub>t\<^sub>r\<^sub>m s"   and "funs_term s \<subseteq> S" shows "s \<notin> subterms (fresh_pgwt S (\<Gamma> t))"
wlog_zero: fixes u :: "complex_homo"   and P :: "complex_homo \<Rightarrow> bool" assumes "u \<in> unit_disc"   and "\<And>a u. \<lbrakk>u \<in> unit_disc; cmod a < 1; P (moebius_pt (blaschke a) u)\<rbrakk> \<Longrightarrow> P u"   and "P 0\<^sub>h" shows "P u"
uminus_in_intervalD: fixes x :: "'b"   and X :: "'b interval" assumes "x \<in>\<^sub>i - X" shows "- x \<in>\<^sub>i X"
childz_same_tree: fixes l :: "'a rose_tree list"   and t :: "'a rose_tree"   and r :: "'a rose_tree list"   and ts :: "'a rose_tree list"   and e :: "'a \<Rightarrow> 'b"   and p :: "('a \<times> 'a rose_tree list \<times> 'a rose_tree list) list"   and d :: "'a" assumes "(l, t, r) \<in> set (splits ts)" shows "tree_of_zipper\<^sub>m (embed_path e p, embed_source_tree e (Tree (d, ts))) = tree_of_zipper\<^sub>m (embed_path e ((d, l, r) # p), embed_source_tree e t)"
prefix_code_1: fixes xs :: "'b list" shows "(\<epsilon> \<le>p xs) = True"
dlex_pm_antisym: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" assumes "dlex_pm s t"   and "dlex_pm t s" shows "s = t"
exec_to_exec_strip_guards': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "t \<notin> Fault ` F" shows "\<Gamma>\<turnstile> \<langle>strip_guards F c,s\<rangle> \<Rightarrow> t"
fstar_comm: fixes f :: "'b \<Rightarrow> 'b" assumes "Inf_pres f" shows "f \<circ> fstar f = fstar f \<circ> f"
abs_conv_abscissa_leI: fixes c :: "ereal"   and f :: "'a fds" assumes "\<And>c'. c < ereal c' \<Longrightarrow> \<exists>s. s \<bullet> (1::'a) = c' \<and> fds_abs_converges f s" shows "abs_conv_abscissa f \<le> c"
valid_equiv_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(equiv_PL p q \<in> Cn {}) = (q \<in> Cn {p} \<and> p \<in> Cn {q})"
finite: fixes A :: "'b set" assumes "class.finite TYPE('b)" shows "finite A"
butlast_append: fixes xs :: "'b list"   and ys :: "'b list" shows "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
coDhalf: shows "continuous_on ({1 / 2..1} - insert (1 / 2) S) (\<lambda>x. vector_derivative (g2 \<circ> (\<lambda>x. 2 * x - 1)) (at x))"
simps_8: fixes n :: "nat"   and \<phi> :: "Formula.formula" shows "minit0 n (formula.Exists \<phi>) = MExists (minit0 (Suc n) \<phi>)"
assms_1: fixes v :: "'a"   and v' :: "'b" assumes "phi v v'" shows "chi v v'"
properties_loop: fixes i :: "nat"   and j :: "nat" assumes "mu \<le> i" shows "seq (i + j * lambda) = seq i"
quasi_isometry_on_def: fixes lambda :: "real"   and C :: "real"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "lambda C -quasi_isometry_on X f = (1 \<le> lambda \<and> 0 \<le> C \<and> (\<forall>x\<in>X. \<forall>y\<in>X. dist (f x) (f y) \<le> lambda * dist x y + C \<and> 1 / lambda * dist x y - C \<le> dist (f x) (f y)))"
fls_eq_iff: fixes f :: "'b fls"   and g :: "'b fls" shows "(f = g) = (\<forall>n. fls_nth f n = fls_nth g n)"
lsu_Sup_closed: shows "Sup_closed_set (Fix \<nu>\<^sup>\<natural>)"
hyps_8: fixes b :: "trm" assumes "e\<^sub>2 \<Down> b" shows "e\<^sub>2' = b"
simps_3: fixes h :: "heap"   and b :: "bool" shows "typeof\<^bsub>h\<^esub> (Bool b) = \<lfloor>Boolean\<rfloor>"
minus: fixes a :: "complex"   and x :: "'b" shows "a *\<^sub>C - x = - (a *\<^sub>C x)"
elims: fixes x :: "auth_query"   and y :: "'a insec_query" assumes "insec_query_of x = y" obtains "x = Forward" and "y = Forward"   | "x = Drop" and "y = Drop"   | "x = Look" and "y = Look"
Bounded_induct: fixes r :: "('c \<times> 'c) set"   and I :: "'c set"   and F :: "'d program"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and B :: "'d set" assumes "wf r"   and "\<forall>m\<in>I. F \<in> A \<inter> f -` {m} \<longmapsto>w A \<inter> f -` r\<inverse> `` {m} \<union> B" shows "F \<in> A \<longmapsto>w A - f -` I \<union> B"
D\<^sub>i: shows "\<^bold>\<forall>x. \<^bold>\<not> (\<^bold>\<forall>xa. \<^bold>\<not> (((\<^bold>\<forall>x. xa \<cdot> x \<cong> x \<^bold>\<leftarrow> E (xa \<cdot> x)) \<^bold>\<and> (\<^bold>\<forall>x. x \<cdot> xa \<cong> x \<^bold>\<leftarrow> E (x \<cdot> xa))) \<^bold>\<and> x \<cdot> xa \<cong> x))"
synth_mpair: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> synth H" shows "\<lbrace>X, Y\<rbrace> \<in> H \<or> X \<in> synth H \<and> Y \<in> synth H"
f: shows "arr f \<and> inj_on (Fun f) (Dom f)"
distinct_3: fixes x3 :: "nat" shows "INIT \<noteq> HOLD x3"
cmp_interpretation: shows "identity_arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg (g \<star> h) Tgh.\<rho>\<sigma>.tab (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) (tab\<^sub>1 g \<star> Tgh.p\<^sub>1) (g \<star> h) Tgh.tab (tab\<^sub>0 (g \<star> h)) (tab\<^sub>1 (g \<star> h)) (g \<star> h)"
infiniteExecution_def: fixes fe :: "nat \<Rightarrow> ('p, 'v, 's) configuration list"   and ft :: "nat \<Rightarrow> ('p, 'v) message list" shows "infiniteExecution fe ft \<equiv> \<forall>n. execution trans sends start (fe n) (ft n) \<and> prefixList (fe n) (fe (n + 1)) \<and> prefixList (ft n) (ft (n + 1))"
snd_proj_mem: fixes B :: "'a"   and M :: "('b \<times> 'a) set" shows "(B \<in> snd_proj M) = (\<exists>A. (A, B) \<in> M)"
hyps_1: fixes f :: "real poly" assumes "df = degree f"   and "normal_poly f" shows "normal_poly (f * g)"
interleave_comm: fixes a :: "'d list"   and b :: "'d list" shows "a \<otimes> b = b \<otimes> a"
treesp_trees_eq: fixes S :: "'d set" shows "treesp (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> trees S)"
full_0_elim: fixes t :: "'b tree234" assumes "full 0 t" obtains "t = \<langle>\<rangle>"
vpairwiseD_2: fixes R :: "V \<Rightarrow> V \<Rightarrow> bool"   and S :: "V"   and x :: "V"   and y :: "V" assumes "vpairwise R S"   and "x \<in>\<^sub>\<circ> S"   and "y \<in>\<^sub>\<circ> S"   and "x \<noteq> y" shows "R y x"
reachable_nodes_def: fixes g :: "'b ugraph"   and r :: "'b" shows "reachable_nodes g r \<equiv> (edges g)\<^sup>* `` {r}"
memFresh: fixes x :: "name"   and l :: "'d list"   and p :: "'d" assumes "x \<sharp> l"   and "p mem l" shows "x \<sharp> p"
dirichlet_prod'_inversion1: fixes g :: "real \<Rightarrow> 'a"   and a :: "nat \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a"   and x :: "real"   and ainv :: "nat \<Rightarrow> 'a" assumes "\<forall>x\<ge>1. g x = dirichlet_prod' a f x"   and "1 \<le> x"   and "dirichlet_prod a ainv = (\<lambda>n. if n = 1 then 1::'a else (0::'a))" shows "f x = dirichlet_prod' ainv g x"
triangle_right: shows "(g \<star> \<epsilon>) \<cdot> (\<eta> \<star> g) = g"
hpq: shows "\<forall>m\<in>{a + b |a b. a \<in> monomials p \<and> b \<in> monomials q}. lookup m var \<le> MPoly_Type.degree p var + MPoly_Type.degree q var"
cpx_vec_length_inner_prod: fixes v :: "complex Matrix.vec" shows "complex_of_real (\<parallel>v\<parallel>\<^sup>2) = \<langle>v|v\<rangle>"
rep_eq: fixes x :: "'b"   and xa :: "'b fs_set" shows "(x \<in>\<^sub>f\<^sub>s xa) = (x \<in> Rep_fs_set xa)"
expectation_indicator: fixes M :: "'a measure"   and A :: "'a set" assumes "prob_space M"   and "A \<in> sets M" shows "integral\<^sup>L M (rind A) = Sigma_Algebra.measure M A"
reduced_def: fixes \<alpha> :: "rat"   and k :: "nat" shows "gs.reduced \<alpha> k = (gs.weakly_reduced \<alpha> k \<and> (\<forall>i j. i < k \<longrightarrow> j < i \<longrightarrow> \<bar>gs.\<mu> i j\<bar> \<le> 1 / 2))"
I_lift_nnf_qe: fixes qe :: "'a fm \<Rightarrow> 'a fm"   and \<phi> :: "'a fm"   and xs :: "'b list" assumes "\<And>\<phi>. nqfree \<phi> \<Longrightarrow> qfree (qe \<phi>)"   and "\<And>xs \<phi>. nqfree \<phi> \<Longrightarrow> I (qe \<phi>) xs = (\<exists>x. I \<phi> (x # xs))" shows "I (lift_nnf_qe qe \<phi>) xs = I \<phi> xs"
Derive_def: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list" shows "Derive a D = (THE b. Derivation a D b)"
mult_cancel_left: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.semiring_no_zero_divisors_cancel plus zero times" shows "(times c a = times c b) = (c = zero \<or> a = b)"
induct: fixes P1 :: "pat \<Rightarrow> bool"   and P3 :: "(char list \<times> pat) list \<Rightarrow> bool"   and P2 :: "char list \<times> pat \<Rightarrow> bool"   and char_list_pat_prod :: "char list \<times> pat" assumes "\<And>x. P1 (PVar x)"   and "\<And>x. P3 x \<Longrightarrow> P1 (PRcd x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P2 (x1, x2)"   and "P3 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 \<Colon> x2)" shows "P2 char_list_pat_prod"
RestrictedP_Mem2: fixes s :: "tm"   and k :: "tm"   and s' :: "tm"   and a :: "tm"   and b :: "tm" shows "{RestrictedP s k s', HPair a b IN s'} \<turnstile> HPair a b IN s AND a IN k"
greaterThanLessThan_empty_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.dense_linorder less_eq less" shows "(ord.greaterThanLessThan less a b = {}) = less_eq b a"
assms_2: shows "mask1 = mask (LENGTH('a) - 1)"
ah_update_mono: fixes h :: "'a \<Rightarrow> 'a set"   and h' :: "'a \<Rightarrow> 'a set"   and F :: "'a set \<times> 'a set"   and F' :: "'a set \<times> 'a set"   and M :: "'a set"   and M' :: "'a set" assumes "h \<le> h'"   and "F = F'"   and "M \<subseteq> M'" shows "ah_update h F M \<le> ah_update h' F' M'"
gtrp2gtr_s: fixes s :: "'f"   and c :: "'f multiset"   and e :: "'g el_step"   and s' :: "'f"   and c' :: "'f multiset"   and T :: "(('f \<times> 'f multiset) \<times> 'g \<times> 'f \<times> 'f multiset) set" assumes "((s, c), e, s', c') \<in> gtrp T" shows "(add_mset s c, le_rem_s e, add_mset s' c') \<in> gtr T"
src_cod: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.src (Span.cod \<mu>) = Span.src \<mu>"
assms_2: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and U :: "'x set" assumes "(h, U) \<in> set ps" shows "cone (h, U) \<subseteq> P[X]"
meet_ac_1: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> y \<sqinter>\<^bsub>L\<^esub> z = x \<sqinter>\<^bsub>L\<^esub> (y \<sqinter>\<^bsub>L\<^esub> z)"
rel_generatD': fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Out :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "('g, 'i, 'k) generat"   and y :: "('h, 'j, 'l) generat" assumes "rel_generat A Out R x y" shows "is_Pure x = is_Pure y \<and> (is_Pure x \<longrightarrow> is_Pure y \<longrightarrow> A (generat.result x) (generat.result y)) \<and> (\<not> is_Pure x \<longrightarrow> \<not> is_Pure y \<longrightarrow> Out (generat.output x) (generat.output y) \<and> R (continuation x) (continuation y))"
cases: fixes a1 :: "com"   and a2 :: "state" assumes "a1 \<down> a2" obtains f :: "state \<Rightarrow> state set" and s :: "state" where "a1 = Do f__" and "a2 = s__" and "f__ s__ \<noteq> {}"   | c1 :: "com" and s0 :: "state" and c2 :: "com" where "a1 = (c1__; c2__)" and "a2 = s0" and "c1__ \<down> s0" and "\<And>s1. s0 -c1__\<rightarrow> s1 \<Longrightarrow> c2__ \<down> s1"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and c2 :: "com" where "a1 = IF b__ THEN c1a__ ELSE c2a__" and "a2 = sa__" and "b__ sa__" and "c1a__ \<down> sa__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c2 :: "com" and c1 :: "com" where "a1 = IF ba__ THEN c1 ELSE c2" and "a2 = sb__" and "\<not> ba__ sb__" and "c2 \<down> sb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = WHILE bb__ DO c__" and "a2 = sc__" and "\<not> bb__ sc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = WHILE b DO ca__" and "a2 = sd__" and "b sd__" and "ca__ \<down> sd__" and "\<And>t. sd__ -ca__\<rightarrow> t \<Longrightarrow> WHILE b DO ca__ \<down> t"   | p :: "pname" and s :: "state" where "a1 = CALL p" and "a2 = se__" and "body p \<down> se__"   | c :: "com" and f :: "state \<Rightarrow> state" and s :: "state" and g :: "state \<Rightarrow> state \<Rightarrow> state" where "a1 = LOCAL f; c; g" and "a2 = s" and "c \<down> f s"
functions_of: fixes A :: "'c set"   and B :: "'d set"   and N :: "nat multiset" assumes "finite A"   and "finite B"   and "number_partition (card A) N"   and "size N \<le> card B" shows "Twelvefold_Way_Entry10.functions_of A B N \<in> (A \<rightarrow>\<^sub>E B) // domain_and_range_permutation A B"
ntSupp: fixes x :: "name" shows "supp (nameTerm x) = {x}"
trace_plus: fixes A :: "('c, 'd) sq_matrix"   and B :: "('c, 'd) sq_matrix" shows "Square_Matrix.trace (A + B) = Square_Matrix.trace A + Square_Matrix.trace B"
Int_eq: shows "(\<Sum>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * real (Suc n) * x ^ n) = (\<Sum>k. (- 1) ^ k * x ^ (k * 2))"
simplex_likeD_finite: fixes P :: "'a set" assumes "simplex_like P" shows "finite P"
simps_2: fixes a :: "'c"   and t1 :: "'c bt"   and t2 :: "'c bt"   and t :: "'c bt" shows "Binary_Tree.append (Br a t1 t2) t = Br a (Binary_Tree.append t1 t) (Binary_Tree.append t2 t)"
path_component_of_trans: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "path_component_of X x y"   and "path_component_of X y z" shows "path_component_of X x z"
move_matrix_ortho: fixes A :: "'b matrix"   and j :: "int"   and i :: "int" shows "move_matrix A j i = move_matrix (move_matrix A j 0) 0 i"
zero_in_cone: fixes Vs :: "'a vec set" shows "0\<^sub>v n \<in> cone Vs"
2: shows "point r \<and> r \<le> x\<^sup>T ; 1"
troeger: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x\<^sup>\<star> \<cdot> (y \<cdot> ((x + y)\<^sup>\<star> \<cdot> z) + z) = (x + y)\<^sup>\<star> \<cdot> z"
order_refl: fixes x :: "'a Word.word" shows "x \<le>s x"
image_mset_strict_mono: fixes M :: "'b multiset"   and N :: "'b multiset"   and f :: "'b \<Rightarrow> 'c" assumes "\<forall>x\<in>#M. \<forall>y\<in>#N. x < y \<longrightarrow> f x < f y"   and "M < N" shows "image_mset f M < image_mset f N"
CSP1_def: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "CSP1 P \<equiv> P \<or> (\<lambda>(A, A'). \<not> ok A \<and> tr A \<le> tr A')"
RModuleHomI: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'c"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "RModule R smult M"   and "GroupHom M T"   and "\<And>r m. \<lbrakk>r \<in> R; m \<in> M\<rbrakk> \<Longrightarrow> T (smult r m) = smult' r (T m)" shows "RModuleHom R smult M smult' T"
5: shows "\<forall>a\<in>A. |F' a| =o |F a|"
in_CompE: fixes h :: "'a"   and G :: "'a set"   and F :: "'a set" assumes "h \<in> Comp G F" obtains f :: "'a" and g :: "'a" where "B.is_iso_class F" and "B.is_iso_class G" and "f \<in> F" and "g \<in> G" and "g \<star> f \<cong>\<^sub>B h"
rf: shows "r \<star> f = \<lparr>Chn = r0 \<down>\<down> r0, Dom = \<lparr>Leg0 = k0, Leg1 = r1 \<cdot> k1\<rparr>, Cod = \<lparr>Leg0 = k0, Leg1 = r1 \<cdot> k1\<rparr>\<rparr>"
BinOpRedsThrow1: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and bop :: "bop"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>throw e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e \<guillemotleft>bop\<guillemotright> e\<^sub>2,s,b\<rangle> \<rightarrow>* \<langle>throw e',s',b'\<rangle>"
iso_def: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" shows "Cop_S.iso f \<equiv> \<exists>g. Cop_S.inverse_arrows f g"
prems_2: shows "elementLevel e (m # M') + i \<le> level"
sup1: fixes A :: "'c"   and k :: "'c"   and a :: "'c"   and b :: "'c" assumes "A \<equiv> sup k a" shows "sup A b \<equiv> sup k (sup a b)"
select_convs_7: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getReason \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getReason"
stk_convert: fixes P :: "'b prog"   and ST :: "ty list"   and ST' :: "ty list" shows "P \<turnstile> ST [\<le>] ST' = Listn.le (subtype P) ST ST'"
resolution_class_def: fixes S :: "'a set set" shows "resolution_class S = (partition_on \<V> S \<and> (\<forall>bl\<in>S. bl \<in># \<B>))"
Gamma_series'_def: fixes z :: "'a"   and n :: "nat" shows "Gamma_series' z n = fact (n - 1) * exp (z * of_real (ln (real n))) / pochhammer z n"
sin_2npi: fixes n :: "nat" shows "sin (2 * real n * pi) = 0"
squarefree_part_ge_Suc_0: fixes n :: "nat" shows "Suc 0 \<le> \<Prod>(squarefree_part n)"
assms: shows "run_step d i (s, vs, es) = (s', vs', RSReturn res)"
pseudo_proj_True_proj: fixes n :: "nat"   and w :: "bool stream" shows "pseudo_proj_True n (pseudo_proj_True n w) = pseudo_proj_True n w"
sets_not_eq: fixes A :: "'e set"   and B :: "'e set" assumes "A \<noteq> B"   and "B \<subseteq> A" shows "\<exists>a\<in>A. a \<notin> B"
assms_4: shows "c + 1 \<le> (1 - b) * real x\<^sub>1"
simps_2: fixes y :: "bool"   and ys :: "bool list"   and x :: "bool list" shows "rbl_mult (y # ys) x = (let ws = False # rbl_mult ys x in if y then rbl_add ws x else ws)"
QCongAnNull_def: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" shows "QCongAnNull a \<equiv> QCongA a \<and> (\<forall>A B C. a A B C \<longrightarrow> \<not> B Out A C)"
Standard_star_of: fixes x :: "'a" shows "star_of x \<in> Standard"
finsum_reindex_bij_betw: fixes h :: "'e \<Rightarrow> 'f"   and S :: "'e set"   and T :: "'f set"   and g :: "'f \<Rightarrow> 'a" assumes "bij_betw h S T"   and "g \<in> T \<rightarrow> carrier R" shows "(\<Oplus>x\<in>S. g (h x)) = finsum R g T"
contra: shows "tl (w \<leadsto> v) \<noteq> u \<leadsto> v"