hcju/isabelleps huggingface dataset:
append_Cons: fixes x :: "'e"   and xs :: "'e list"   and ys :: "'e list" shows "append (Cons x xs) ys = Cons x (append xs ys)"
eq\<rho>_adm: fixes f :: "'a \<Rightarrow> Vars.var \<Rightarrow> Value"   and g :: "'a \<Rightarrow> Vars.var \<Rightarrow> Value"   and a :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "cont f"   and "cont g" shows "adm (\<lambda>x. eq\<rho> a (f x) (g x))"
resChainEqvt: fixes perm :: "(name \<times> name) list"   and xvec :: "name list"   and P :: "('a, 'b, 'c) psi" shows "perm \<bullet> \<lparr>\<nu>*xvec\<rparr>P = \<lparr>\<nu>*perm \<bullet> xvec\<rparr>perm \<bullet> P"
bt_augment_v_subset_2: fixes q :: "('c, 'd) pq"   and ts :: "('d, 'c) bintree list" shows "set |q| \<subseteq> set |bts_augment ts q|"
segment_1: shows "\<forall>\<^sub>F x' in at x within {x. \<forall>j\<in>Basis. x \<bullet> j \<in> X j}. x' \<in> {x. \<forall>j\<in>Basis. x \<bullet> j \<in> X j}"
sq_norm_poly_def: fixes p :: "'b poly" shows "\<parallel>p\<parallel>\<^sup>2 \<equiv> sum_list (map sq_norm (coeffs p))"
vector_derivative_part_circlepath01: fixes x :: "real"   and z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "vector_derivative (part_circlepath z r s t) (at x within {0..1}) = \<i> * complex_of_real r * (complex_of_real t - complex_of_real s) * exp (\<i> * complex_of_real (linepath s t x))"
d_IN_alt_def: fixes y :: "'k"   and G :: "('k, 'l) graph_scheme"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "\<And>x. (x, y) \<notin> \<^bold>E\<^bsub>G\<^esub> \<Longrightarrow> g (x, y) = 0" shows "d_IN g y = (\<Sum>\<^sup>+ x\<in>\<^bold>I\<^bold>N\<^bsub>G\<^esub> y. g (x, y))"
leadsTo_Basis: fixes F :: "'d program"   and A :: "'d set"   and B :: "'d set" assumes "F \<in> A ensures B" shows "F \<in> A \<longmapsto> B"
distinct_filter: fixes xs :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "distinct xs" shows "distinct (List.filter P xs)"
axioms: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" assumes "VectorSpace smult V" shows "Module smult V"
restrict_assignment: fixes b :: "'a ifex"   and ass :: "'a \<Rightarrow> bool"   and var :: "'a"   and val :: "bool" shows "val_ifex b (ass(var := val)) = val_ifex (restrict b var val) ass"
ospecD2: fixes x :: "'d option"   and P :: "'d \<Rightarrow> bool" assumes "ospec x P" shows "\<exists>y. x = Some y \<and> P y"
assms_1: shows "(\<sigma>, NodeS ii \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) S U"
T_def_spec: fixes P :: "'a process" shows "T P = {tr. \<exists>a. a \<in> F P \<and> tr = fst a}"
ext: fixes J :: "'a set"   and P :: "'a set set"   and j0 :: "'a"   and I :: "'a set"   and j :: "'a" assumes "J \<in> P"   and "j0 \<in> J"   and "j0 \<in> gen P I"   and "j \<in> J" shows "j \<in> gen P I"
trans_FD: fixes P :: "'d process"   and Q :: "'d process"   and S :: "'d process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D Q"   and "Q \<sqsubseteq>\<^sub>F\<^sub>D S" shows "P \<sqsubseteq>\<^sub>F\<^sub>D S"
sub1: shows "BVO ODE1 \<subseteq> BVO (OProd ODE1 ODE2)"
PROJ_DOM_IDEMPOT: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set" shows "prob_proj PROB (prob_dom PROB) = PROB"
tail_def: fixes S :: "'c list" shows "tail S \<equiv> if S = [] then [] else tl S"
sublinear_intros_4: fixes x :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and y :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "sublinear (wp x)"   and "sublinear (wp y)"   and "healthy (wp x)"   and "healthy (wp y)" shows "sublinear (wp (x ;; y))"
exhaust: fixes y :: "Syntax.term" obtains "y = Unit"   | x2 :: "Syntax.var" where "y = Var x2"   | x31 :: "Syntax.var" and x32 :: "Syntax.term" where "y = Lam x31 x32"   | x41 :: "Syntax.var" and x42 :: "Syntax.term" where "y = Rec x41 x42"   | x5 :: "Syntax.term" where "y = Inj1 x5"   | x6 :: "Syntax.term" where "y = Inj2 x6"   | x71 :: "Syntax.term" and x72 :: "Syntax.term" where "y = Syntax.Pair x71 x72"   | x81 :: "Syntax.term" and x82 :: "Syntax.var" and x83 :: "Syntax.term" where "y = Syntax.Let x81 x82 x83"   | x91 :: "Syntax.term" and x92 :: "Syntax.term" where "y = App x91 x92"   | x101 :: "Syntax.term" and x102 :: "Syntax.term" and x103 :: "Syntax.term" where "y = Case x101 x102 x103"   | x11 :: "Syntax.term" where "y = Prj1 x11"   | x12 :: "Syntax.term" where "y = Prj2 x12"   | x13 :: "Syntax.term" where "y = Roll x13"   | x14 :: "Syntax.term" where "y = Unroll x14"   | x15 :: "Syntax.term" where "y = Auth x15"   | x16 :: "Syntax.term" where "y = Unauth x16"   | x17 :: "hash" where "y = Hash x17"   | x181 :: "hash" and x182 :: "Syntax.term" where "y = Hashed x181 x182"
exists: shows "insertion (nth_default 0 (L[var := x])) p = val"
Output: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "a{b}.P \<Longrightarrow>a[b] \<prec> P"
card_lists_distinct_length_eq: fixes A :: "'a set"   and k :: "nat" assumes "finite A"   and "k \<le> card A" shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1..card A}"
index_row_2: fixes A :: "'a mat"   and i :: "nat" shows "dim_vec (Matrix.row A i) = dim_col A"
stake_snth: fixes n :: "nat"   and w :: "'b stream"   and x :: "'b stream"   and i :: "nat" assumes "stake n w = stake n x"   and "Suc i \<le> n" shows "w !! i = x !! i"
conjunctive_def: shows "conjunctive.conjunctive (\<sqinter>) (\<sqinter>) (*) = {x. \<forall>y z. x * (y \<sqinter> z) = x * y \<sqinter> (x * z)}"
binop_i: fixes t :: "t"   and \<C> :: "t_context"   and iop :: "binop_i" assumes "is_int_t t" shows "\<C> \<turnstile> [Binop_i t iop] : [t, t] _> [t]"
mono_Inf: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "order.mono less_eq f" shows "f (Inf A) \<le> Inf (f ` A)"
alpha_ub: shows "\<alpha> \<le> 1 / 2"
inorder_baliR: fixes l :: "('c \<times> color) tree"   and a :: "'c"   and r :: "('c \<times> color) tree" shows "Tree2.inorder (baliR l a r) = Tree2.inorder l @ a # Tree2.inorder r"
fresh_close_Var_id: fixes xT :: "char list \<times> Unary_PCF.type"   and t :: "expr"   and k :: "nat" assumes "fresh xT t" shows "close_Var k xT t = t"
decrease_key_def: fixes k0 :: "'a"   and f :: "'a \<Rightarrow>\<^sub>0 'b" shows "decrease_key k0 f = Abs_poly_mapping (\<lambda>k. if k = k0 \<and> lookup f k \<noteq> (0::'b) then lookup f k - (1::'b) else lookup f k)"
set_of_delete_max: fixes t :: "'b tree" assumes "avl t"   and "is_ord t"   and "t \<noteq> ET" shows "set_of (snd (delete_max t)) = set_of t - {rightmost_item t}"
support_in_outcomes: shows "\<forall>oc. \<forall>p\<in>lotteries_on oc. \<forall>a\<in>set_pmf p. a \<in> oc"
degree1_coeffs: fixes p :: "'b poly" assumes "degree p = 1" shows "\<exists>a b. p = [:b, a:] \<and> a \<noteq> (0::'b)"
induct: fixes S :: "'a set set"   and x :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "generate_topology S x"   and "P UNIV"   and "\<And>a b. \<lbrakk>generate_topology S a; P a; generate_topology S b; P b\<rbrakk> \<Longrightarrow> P (a \<inter> b)"   and "\<And>K. \<lbrakk>\<And>k. k \<in> K \<Longrightarrow> generate_topology S k; \<And>k. k \<in> K \<Longrightarrow> P k\<rbrakk> \<Longrightarrow> P (\<Union> K)"   and "\<And>s. s \<in> S \<Longrightarrow> P s" shows "P x"
C14_aux: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and x :: "'b" assumes "class.C_algebra plus times one zero less_eq less star"   and "m \<le> n" shows "times (power.power one times x m) (star (power.power one times x n)) = times (star (power.power one times x n)) (power.power one times x m)"
parts_insert_Hash: fixes X :: "msg"   and H :: "msg set" shows "parts (insert (Hash X) H) = insert (Hash X) (parts H)"
singleton_injective: fixes x :: "'a" assumes "singleton x" shows "is_inj x"
iso_arcs_inv_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and a :: "'b" assumes "digraph_isomorphism hom"   and "a \<in> arcs G" shows "iso_arcs (inv_iso hom) (iso_arcs hom a) = a"
cuncurry_def: shows "cuncurry \<equiv> case_prod"
comp_fun_idem_apply: fixes x :: "('b, 'c) comp_fun_idem" shows "comp_fun_idem_apply x \<in> {f. comp_fun_idem f}"
singular_chain_sum: fixes I :: "'c set"   and p :: "nat"   and X :: "'d topology"   and f :: "'c \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'd) \<Rightarrow>\<^sub>0 int" assumes "\<And>i. i \<in> I \<Longrightarrow> singular_chain p X (f i)" shows "singular_chain p X (sum f I)"
two: shows "2\<^sub>\<nat> = set {0, 1\<^sub>\<nat>}"
subst_atm_list_id_subst: fixes As :: "'a list" shows "As \<cdot>al id_subst = As"
REP_simps_3: fixes A :: "('a, 'a set) arr" assumes "arr A" shows "trg (REP A) = local.Cod A"
that: fixes P''' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'''"   and "\<Psi> \<rhd> P' \<sim> P'''" shows "thesis"
stationary_distribution_iterate': fixes N :: "'s spmf"   and y :: "'s option"   and n :: "nat" assumes "K2.stationary_distribution N" shows "measure_pmf.prob N {y} = measure_pmf.expectation N (\<lambda>x. K2.p x y n)"
distinct_1: fixes x1 :: "nat"   and x21 :: "typ"   and x22 :: "typ" shows "TVar x1 \<noteq> x21 -> x22"
lookup_HSem_heap: fixes y :: "'b"   and h :: "('b \<times> 'a) list"   and \<rho> :: "'b \<Rightarrow> 'c" assumes "y \<in> domA h" shows "(\<lbrace> h \<rbrace>\<rho>) y = \<lbrakk> the (map_of h y) \<rbrakk>\<^bsub>\<lbrace> h \<rbrace>\<rho>\<^esub>"
d2: fixes y :: "'a" assumes "dist y x < d2" shows "flow0 y (t y) \<in> ball x e2 \<inter> {x. a \<bullet> (b - a) < x \<bullet> (b - a) \<and> x \<bullet> (b - a) < b \<bullet> (b - a)}"
simps_1: fixes a :: "(('fsym, 'tp) efsym, 'psym) atm" shows "TE.wtL (Pos a) = TE.wtA a"
dim_row_conjugate: fixes m :: "'b mat" shows "dim_row (conjugate m) = dim_row m"
alphaAll_trans: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and A :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)" shows "(FixVars.qGood X \<longrightarrow> (\<forall>X' X''. X #= X' \<and> X' #= X'' \<longrightarrow> X #= X'')) \<and> (FixVars.qGoodAbs A \<longrightarrow> (\<forall>A' A''. A $= A' \<and> A' $= A'' \<longrightarrow> A $= A''))"
inv_equality: fixes y :: "'a vec"   and x :: "'a vec" assumes "y + x = 0\<^sub>v n"   and "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "\<ominus>\<^bsub>V\<^esub> x = y"
subobjs_rel_rev: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C (rev Cs)"   and "wf_prog wf_md P" shows "P,C \<turnstile> [C] \<sqsubseteq> rev Cs"
l_null: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "\<zero>\<^bsub>F\<^esub> \<otimes>\<^bsub>F\<^esub> x = \<zero>\<^bsub>F\<^esub>"
Sup_img_rule_complete: fixes \<Phi> :: "'b \<Rightarrow> bool" shows "(\<forall>x. x \<in> S \<longrightarrow> f x \<le> SPEC \<Phi>) = (Sup (f ` S) \<le> SPEC \<Phi>)"
rep_nat_pp_pp_inj: fixes x :: "('a, 'b) pp"   and y :: "('a, 'b) pp" assumes "rep_nat_pp x = rep_nat_pp y" shows "x = y"
shortest: fixes xs' :: "('a \<times> 'b) list"   and x :: "'a" assumes "strict_prefix xs' xs"   and "set xs' \<subseteq> outs_\<I> \<I> \<times> UNIV"   and "x \<in> outs_\<I> \<I>" shows "trace_callee run_resource (return_spmf res) xs' x = trace_callee run_resource (return_spmf res') xs' x"
vector_sub_rdistrib: fixes a :: "'a"   and b :: "'a"   and x :: "('a, 'b) vec" shows "(a - b) *s x = a *s x - b *s x"
prim_nemp: fixes u :: "'b list" assumes "primitive u" shows "u \<noteq> \<epsilon>"
natural_transformation_axioms: shows "natural_transformation A B F H \<psi>\<phi>.map"
gcd_exp: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat"   and b :: "'b" assumes "class.semiring_gcd_mult_normalize one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd (power.power one times a n) (power.power one times b n) = power.power one times (gcd a b) n"
brec_pref_brw: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule dlist"   and Qi :: "('a, unit) HashMap.hashmap" assumes "rqrm_invar rqrm"   and "rqrm_prop (ls.\<alpha> \<delta>) rqrm"   and "ls.invar \<delta>"   and "hs.invar Qi" shows "wa_precise_refine (det_wa_wa (brec_det_algo rqrm Qi \<delta>)) (brw_algo (hs.\<alpha> Qi) (ls.\<alpha> \<delta>)) brec_\<alpha>"
sturm_seq_def: fixes ps :: "real poly list"   and p :: "real poly" shows "sturm_seq ps p \<equiv> quasi_sturm_seq ps \<and> sturm_seq_axioms ps p"
ex_un_null: shows "\<exists>!n. \<forall>f. D n f = n \<and> D f n = n"
range_strict_mono_ext: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "range f = range g"   and "strict_mono f"   and "strict_mono g" shows "f = g"
reachable1_in_verts_2: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G"   and "u \<rightarrow>\<^sup>+\<^bsub>G\<^esub> v" shows "v \<in> verts G"
goal1: fixes j :: "nat"   and i :: "nat"   and e :: "nat" assumes "\<forall>k<j. eval r_psi [i, k] \<down>" shows "eval (r_consist_upto r_psi) [j, i, e] = (if \<forall>k<j. eval r_psi [i, k] \<down>= e_nth e k then Some 0 else Some 1)"
sup_left_idem: fixes x :: "'b multiset"   and y :: "'b multiset" shows "x \<union># (x \<union># y) = x \<union># y"
semiring_norm_69: fixes m :: "num" shows "(num.Bit0 m \<le> num.One) = False"
wsorted'_def: shows "wsorted' \<equiv> sorted_wrt (\<lambda>x y. weight (\<alpha> x) \<le> weight (\<alpha> y))"
Units_r_cancel: fixes G :: "('c, 'd) monoid_scheme"   and z :: "'c"   and x :: "'c"   and y :: "'c" assumes "Group.monoid G"   and "z \<in> Units G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "(x \<otimes>\<^bsub>G\<^esub> z = y \<otimes>\<^bsub>G\<^esub> z) = (x = y)"
incseq_SucI: fixes X :: "nat \<Rightarrow> 'b" assumes "\<And>n. X n \<le> X (Suc n)" shows "incseq X"
hyps_5: shows "fs\<langle>l\<rangle>\<^sub>? = \<bottom>"
UN_absorb: fixes k :: "'c"   and I :: "'c set"   and A :: "'c \<Rightarrow> 'd set" assumes "k \<in> I" shows "A k \<union> \<Union> (A ` I) = \<Union> (A ` I)"
coprimeE: fixes p :: "'a"   and q :: "'a" assumes "coprime p q" obtains "\<And>r. \<lbrakk>r dvd p; r dvd q\<rbrakk> \<Longrightarrow> r dvd (1::'a)"
length_remdups_card_conv: fixes xs :: "'a list" shows "|remdups xs| = card (set xs)"
Cidm: fixes X :: "'a" assumes "X \<in> obj" shows "id X maps X to X"
x1_in_dom: shows "x1 \<in> valid_pub1"
termFSwSbImorph_iff: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" shows "termFSwSbImorph h hA MOD = (termFSbImorph h hA MOD \<and> ipresSwapAll h hA MOD)"
conforms_set_locals: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L' :: "lname \<Rightarrow> ty option"   and l :: "lname \<Rightarrow> val option"   and L :: "lname \<Rightarrow> ty option" assumes "(x, s)\<Colon>\<preceq>(G, L')"   and "G,s\<turnstile>l[\<sim>\<Colon>\<preceq>]L"   and "x = Some (Jump Ret) \<longrightarrow> l Result \<noteq> None" shows "(x, set_locals l s)\<Colon>\<preceq>(G, L)"
enforce_\<I>_gpv_parametric': fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "bi_unique C"   and "bi_unique R" shows "rel_fun (rel_\<I> C R) (rel_fun (rel_gpv'' A C R) (rel_gpv'' A C R)) enforce_\<I>_gpv enforce_\<I>_gpv"
CD_1: shows "C \<in> f \<turnstile> \<C> \<union> g \<turnstile> \<C>"
matrix_vector_mult_def: fixes m :: "(('c, 'e) vec, 'd) vec"   and x :: "('c, 'e) vec" shows "m *v x \<equiv> \<chi>i. \<Sum>j\<in>UNIV. m $ i $ j * x $ j"
simps_15: fixes x1 :: "'b" shows "syms_hd (Var x1) = {}"
AsWhileInv: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and r :: "'e set"   and i :: "'e set"   and p :: "('e, 'd, 'f) ann"   and b :: "'e set"   and e :: "('e, 'd, 'f) com" shows "assertionsR \<Gamma> \<Theta> Q A (AnnWhile r i p) (While b e) i"
finite_vertices: fixes e :: "'edge" assumes "e \<in> E" shows "finite (vertices e)"
simps: fixes a1 :: "'a"   and a2 :: "'a" shows "rel_12_21 a1 a2 = (a1 = \<^bold>1 \<and> a2 = \<^bold>2 \<or> a1 = \<^bold>2 \<and> a2 = \<^bold>1)"
rebase_dvd_axioms: shows "rebase_dvd TYPE('pq) TYPE('p)"
intros_8: fixes \<Gamma> :: "'a formula multiset"   and F :: "'a formula"   and G :: "'a formula"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> F, G, \<Delta>" shows "\<Gamma> \<Rightarrow> F \<^bold>\<or> G, \<Delta>"
signs_are_cols: fixes signs :: "rat list list"   and subsets :: "nat list list" shows "\<forall>i<length signs. col (alt_matrix_A signs subsets) i = vec (length subsets) (\<lambda>j. z (subsets ! j) (signs ! i))"
adv_def: fixes A :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and s :: "'b list"   and rs :: "'b list" shows "adv A s rs = (if rs = [] then [] else let crs = cruel A (Step A (s, fst A s) (last s)) (length rs - 1) in (0, sort_sws (\<lambda>x. length rs - 1 - count_list crs x) s) # replicate (length rs - 1) (0, []))"
car: shows "carrier (residue_ring (int p)) - {\<zero>\<^bsub>residue_ring (int p)\<^esub>} = {1..int p - 1}"
set_append: fixes xs :: "['b]"   and ys :: "['b]" assumes "finite_list xs" shows "set (xs ++ ys) = set xs \<union> set ys"
v_exists_b_e: fixes vs :: "v list" shows "\<exists>ves. $$* vs = $* ves"
simps: fixes r :: "nat" shows "valu r = lg (r + 1) 2 - 1"
abs_eq: fixes x :: "'a set" shows "Inf (dual ` x) = dual (Sup x)"
Derives1_sentence2: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "Derives1 a i r b" shows "is_sentence b"
IH2: assumes "Vagree \<nu> \<nu>' (FVT \<theta>\<^sub>2)"   and "Iagree I J {Inl x |x. x \<in> SIGT \<theta>\<^sub>2}" shows "dterm_sem I \<theta>\<^sub>2 \<nu> = dterm_sem J \<theta>\<^sub>2 \<nu>'"
equivalence_of_partition: shows "partition.Equivalence rel.Partition = {(x, y). (x, y) \<in> (R \<times> S) \<times> R \<times> S \<and> x \<sim> y}"
horizontal_chop1: fixes v :: "view" shows "\<exists>u w. v=u\<parallel>w"
i_last_message_hold_0: fixes f :: "nat \<Rightarrow> 'b message_af" shows "f \<longmapsto> 0 = (\<lambda>n. last_message (f \<Down> Suc n))"
update_until_def: fixes args :: "Monitor.args"   and rel1 :: "event_data option list set"   and rel2 :: "event_data option list set"   and nt :: "nat"   and aux :: "(nat \<times> event_data option list set \<times> event_data option list set) list" shows "update_until args rel1 rel2 nt aux = map (\<lambda>x. case x of (t, a1, a2) \<Rightarrow> (t, if args_pos args then join a1 True rel1 else a1 \<union> rel1, if mem (nt - t) (args_ivl args) then a2 \<union> join rel2 (args_pos args) a1 else a2)) aux @ [(nt, rel1, if left (args_ivl args) = 0 then rel2 else empty_table)]"
subset_trans: fixes A :: "'h set"   and B :: "'h set"   and C :: "'h set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "A \<subseteq> C"
reachable_vwalk_conv: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>* v) = (\<exists>p. vwalk p G \<and> hd p = u \<and> last p = v)"
downset_iso_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.preorder less_eq less" shows "(ord.downset less_eq x \<subseteq> ord.downset less_eq y) = less_eq x y"
poly_y_x_x_y: shows "poly_y_x x_y = y_x"
exhaust: fixes y :: "'a noption" obtains x1 :: "'a" where "y = nSome x1"   | "y = nNone"
deterministicD: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set"   and t :: "'t"   and x :: "'x"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ta' :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and ta'' :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x'' :: "'x"   and m'' :: "'m" assumes "deterministic I"   and "t \<turnstile> \<langle>x, shr s\<rangle> -ta'\<rightarrow> \<langle>x', m'\<rangle>"   and "t \<turnstile> \<langle>x, shr s\<rangle> -ta''\<rightarrow> \<langle>x'', m''\<rangle>"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "actions_ok s t ta'"   and "actions_ok s t ta''"   and "s \<in> I" shows "ta' = ta'' \<and> x' = x'' \<and> m' = m''"
distinct_3: fixes x1 :: "'e"   and x31 :: "nat"   and x32 :: "'e" shows "broadcast x1 \<noteq> unicast x31 x32"
MGT_BodyN: fixes pn :: "pname"   and G :: "state triple set" assumes "insert ({=}.BODY pn.{->}) G|-{=}.the (body pn).{->}" shows "G|-{=}.BODY pn.{->}"
sumAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<oplus> Q) \<oplus> R \<sim>\<^sub>e P \<oplus> Q \<oplus> R"
set_def: fixes a :: "'s" shows "local.set a = img ` hom unity a"
r2p_def: fixes R :: "('b \<times> 'b) set" shows "r2p R = (\<lambda>x. x \<in> r2s R)"
scaleR_left_commute: fixes a :: "real"   and b :: "real"   and x :: "'c" shows "a *\<^sub>R b *\<^sub>R x = b *\<^sub>R a *\<^sub>R x"
gf: fixes x :: "'a" assumes "x \<in> C" shows "g x = f x"
streams_sset: fixes s :: "'b stream"   and A :: "'b set" assumes "s \<in> streams A" shows "sset s \<subseteq> A"
polyadd_head: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "Polynomial_Expression.degree p \<noteq> Polynomial_Expression.degree q" shows "head (p +\<^sub>p q) = (if Polynomial_Expression.degree p < Polynomial_Expression.degree q then head q else head p)"
Rep_JF1: fixes x :: "'d JF1" shows "Rep_JF1 x \<in> car_final1"
SOUNDNESS: fixes C :: "'a eclause"   and S :: "'a eclause set" assumes "derivable_ecl C S"   and "\<forall>x\<in>S. finite (cl_ecl x)" shows "set_entails_clause (cl_ecl ` S) (cl_ecl C)"
prv_exi_congW: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<in> var"   and "prv (imp \<phi> \<chi>)"   and "prv (exi x \<phi>)" shows "prv (exi x \<chi>)"