hcju/isabelleps huggingface dataset:
eval_eq: fixes c :: "com"   and s :: "state"   and t :: "state" shows "<c,s> -c-> t = (\<exists>n. <c,s> -n-> t)"
eq_equiv_class_iff2: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "equiv A r"   and "x \<in> A"   and "y \<in> A" shows "({x} // r = {y} // r) = ((x, y) \<in> r)"
IH_1: fixes x :: "'a" assumes "x \<in> A" shows "random_bst (L x) = map_pmf bst_of_list (pmf_of_set (permutations_of_set (L x)))"
verts_slim_in_G: shows "pverts S.slim \<subseteq> pverts slim"
simps_3: fixes x2 :: "nat" shows "Suc x2 \<noteq> 0"
diff_add_assoc2_ennreal: fixes b :: "ennreal"   and a :: "ennreal"   and c :: "ennreal" assumes "b \<le> a" shows "a - b + c = a + c - b"
ind_zero_str_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and U :: "'c set"   and V :: "'c set" assumes "ind_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str U" shows "ind_sheaf.ind_zero_str zero_str U V \<equiv> zero_str (U \<inter> V)"
scaleR_conv_of_real: fixes r :: "real"   and x :: "'b" shows "r *\<^sub>R x = of_real r * x"
simps_1: shows "lvl \<langle>\<rangle> = 0"
simps_2: fixes v :: "Vars.var"   and e :: "exp"   and "binds" :: "(Vars.var \<times> exp) list" shows "ABinds ((v, e) # binds) = ABind v e \<squnion> ABinds (delete v binds)"
LeadsETo_Union: fixes S :: "'d set set"   and F :: "'d program"   and CC :: "'d set set"   and B :: "'d set" assumes "\<And>A. A \<in> S \<Longrightarrow> F \<in> A LeadsTo[CC] B" shows "F \<in> \<Union> S LeadsTo[CC] B"
lnull_ltlI: fixes xs :: "'b llist" assumes "lnull xs" shows "lnull (ltl xs)"
local_flow_def: fixes f :: "'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and \<phi> :: "real \<Rightarrow> 'b \<Rightarrow> 'b" shows "local_flow f T S \<phi> \<equiv> picard_lindeloef (\<lambda>t. f) T S 0 \<and> local_flow_axioms f T S \<phi>"
prems: fixes xs :: "'a list" assumes "xs \<in> set xss" shows "Suc i \<le> length xs"
nv_def: shows "normal_vectors ws = map (\<lambda>i. normal_vector (set (ws @ us) - {us ! i})) [0..<length us]"
flv_round_def: fixes r :: "nat"   and r_votes :: "process \<Rightarrow> val option"   and r_decisions :: "process \<Rightarrow> val option" shows "flv_round r r_votes r_decisions = {(s, s'). r = opt_v_state.next_round s \<and> opt_no_defection s r_votes \<and> d_guard r_decisions r_votes \<and> s' = s\<lparr>opt_v_state.next_round := Suc r, last_vote := last_vote s ++ r_votes, opt_v_state.decisions := opt_v_state.decisions s ++ r_decisions\<rparr>}"
cases: fixes a1 :: "char list"   and a2 :: "char list"   and 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" assumes "P \<turnstile> a1 \<prec>\<^sup>1 a2" obtains C :: "char list" and Bs :: "base list" and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list" and D :: "char list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(Bs, rest)\<rfloor>" and "D \<in> baseClasses Bs"
valid_SE_def: fixes \<sigma> :: "'c"   and m :: "'c \<Rightarrow> (bool \<times> 'c) option" shows "(\<sigma> \<Turnstile> m) = (m \<sigma> \<noteq> None \<and> fst (the (m \<sigma>)))"
set_2: fixes x21 :: "'a"   and x22 :: "'a list" shows "set (x21 # x22) = x21 \<triangleright> set x22"
le_supI1: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x a" shows "less_eq x (sup a b)"
ntrunc_equality: fixes M :: "('f, 'g) Old_Datatype.node set"   and N :: "('f, 'g) Old_Datatype.node set" assumes "\<And>k. ntrunc k M = ntrunc k N" shows "M = N"
MapsToMorDomCod_3: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "cod\<^bsub>C\<^esub> (f ;;\<^bsub>C\<^esub> g) = cod\<^bsub>C\<^esub> g"
dX_Int_Field_Pd: fixes X :: "'x set"   and d :: "'d" assumes "dX X d \<subseteq> Field (Pd d)" shows "X \<inter> Field (Pd d) = dX X d"
vec_to_iarray_nth: fixes i :: "nat"   and A :: "('c, 'b) vec" assumes "i < CARD('b)" shows "vec_to_iarray A !! i = A $ mod_type_class.from_nat i"
prefix_cases: fixes xs :: "'c list"   and ys :: "'c list" obtains "prefix xs ys"   | "strict_prefix ys xs"   | "xs \<parallel> ys"
otrans_to_dterms: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"   and "wellformed \<Gamma>" shows "\<forall>r\<in>sterms \<Gamma> q. r \<in> dterms \<Gamma> p"
move_matrix_add: fixes A :: "'a matrix"   and B :: "'a matrix"   and j :: "int"   and i :: "int" shows "move_matrix (A + B) j i = move_matrix A j i + move_matrix B j i"
restrict_ext: fixes R :: "('c \<times> 'd) set"   and X :: "'c set" shows "R || X = {(x, y) |x y. x \<in> X \<and> (x, y) \<in> R}"
inf_le1: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y \<le> x"
targetsD_3: fixes b :: "'a"   and \<mu> :: "'a" assumes "b \<in> targets \<mu>" shows "b \<star> \<mu> \<noteq> null"
Abs_Pair\<^sub>b\<^sub>a\<^sub>s\<^sub>e_inject: fixes x :: "\<langle>\<langle>'a \<times> 'b\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "\<langle>\<langle>'a \<times> 'b\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "x \<in> {X. X = UML_Types.bot_class.bot \<or> X = null \<or> fst \<lceil>\<lceil>X\<rceil>\<rceil> \<noteq> UML_Types.bot_class.bot \<and> snd \<lceil>\<lceil>X\<rceil>\<rceil> \<noteq> UML_Types.bot_class.bot}"   and "y \<in> {X. X = UML_Types.bot_class.bot \<or> X = null \<or> fst \<lceil>\<lceil>X\<rceil>\<rceil> \<noteq> UML_Types.bot_class.bot \<and> snd \<lceil>\<lceil>X\<rceil>\<rceil> \<noteq> UML_Types.bot_class.bot}" shows "(Abs_Pair\<^sub>b\<^sub>a\<^sub>s\<^sub>e x = Abs_Pair\<^sub>b\<^sub>a\<^sub>s\<^sub>e y) = (x = y)"
size_eq_Suc_imp_elem: fixes M :: "'k multiset"   and n :: "nat" assumes "size M = Suc n" shows "\<exists>a. a \<in># M"
intensional_simps_30: fixes y :: "'b \<Rightarrow> bool" shows "\<forall>x. y \<equiv> y"
bij_betwI': fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and Y :: "'c set" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y)"   and "\<And>x. x \<in> X \<Longrightarrow> f x \<in> Y"   and "\<And>y. y \<in> Y \<Longrightarrow> \<exists>x\<in>X. y = f x" shows "bij_betw f X Y"
if_distrib: fixes f :: "'b \<Rightarrow> 'a"   and c :: "bool"   and x :: "'b"   and y :: "'b" shows "f (if c then x else y) = (if c then f x else f y)"
Id_onI: fixes a :: "'d"   and A :: "'d set" assumes "a \<in> A" shows "(a, a) \<in> Id_on A"
ring_simprules_4: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<ominus>\<^bsub>R\<^esub> y \<in> carrier R"
out_arcs_add_arc_iff: fixes G :: "('d, 'c) pre_digraph"   and a :: "'c"   and u :: "'d" shows "out_arcs (pre_digraph.add_arc G a) u = (if tail G a = u then insert a (out_arcs G u) else out_arcs G u)"
set_partition: fixes t :: "'b tree"   and p :: "'b"   and l' :: "'b tree"   and r' :: "'b tree" assumes "bst_wrt (\<le>) t"   and "Splay_Heap.partition p t = (l', r')" shows "set_tree t = set_tree l' \<union> set_tree r'"
assms: shows "<e1,Es1> \<mapsto> <e2,Es2>"
case_rews_1: fixes f1 :: "'a"   and f2 :: "'b \<rightarrow> [:'b:] \<rightarrow> 'a" shows "slist_case\<cdot>f1\<cdot>f2\<cdot>\<bottom> = \<bottom>"
aff_dim_subset: fixes S :: "'c set"   and T :: "'c set" assumes "S \<subseteq> T" shows "aff_dim S \<le> aff_dim T"
itrm_bracket_opaque: fixes extra_rule :: "nat itrm \<Rightarrow> nat itrm \<Rightarrow> bool"   and itrm_bracket :: "nat \<Rightarrow> nat itrm \<Rightarrow> nat itrm option"   and i :: "nat"   and x :: "nat itrm"   and x' :: "nat itrm" assumes "itrm_abstraction extra_rule itrm_bracket"   and "itrm_bracket i x = Some x'" shows "set (opaque x') = set (opaque x) - {i}"
sinv: fixes x :: "'a" shows "summable (\<lambda>n. of_real (inverse (fact n)) * x ^ n)"
resr_top: fixes x :: "'a" shows "x \<rightarrow> \<top> = \<top>"
comp_arr_dom: fixes f :: "(bool, 'arr list) concrete_category.arr"   and a :: "(bool, 'arr list) concrete_category.arr" assumes "arr f"   and "local.dom f = a" shows "f \<cdot> a = f"
neg_assert_def: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<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 dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "mbt_algebra.neg_assert one times inf dual bot x = inf (times (dual x) bot) one"
simps_6: fixes \<phi> :: "'b ltlr"   and \<psi> :: "'b ltlr" shows "ltlr_to_ltln (\<phi> or\<^sub>r \<psi>) = ltlr_to_ltln \<phi> or\<^sub>n ltlr_to_ltln \<psi>"
env_delete_add: fixes x :: "'d"   and m1 :: "'d \<Rightarrow> 'e"   and m2 :: "'d \<Rightarrow> 'e"   and S :: "'d set" shows "env_delete x (m1 ++\<^bsub>S\<^esub> m2) = env_delete x m1 ++\<^bsub>S - {x}\<^esub> env_delete x m2"
emeasure_eq_measure: fixes s :: "'s \<times> 's option"   and A :: "('s \<times> 's option) stream set" shows "emeasure (KN.T s) A = ennreal (KN.T.prob s A)"
elims: fixes x :: "'b eclause"   and y :: "'b literal set" assumes "cl_ecl x = y" obtains C :: "'b literal set" and X :: "'b trm set" where "x = Ecl C X" and "y = C"
inject_3: fixes x31 :: "'a"   and x32 :: "('a, 'b) Expression"   and x33 :: "('a, 'b) Expression"   and x34 :: "'a"   and y31 :: "'a"   and y32 :: "('a, 'b) Expression"   and y33 :: "('a, 'b) Expression"   and y34 :: "'a" shows "(Vx : x31 \<turnstile> x32\<equiv>x33 : x34 = Vx : y31 \<turnstile> y32\<equiv>y33 : y34) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33 \<and> x34 = y34)"
startsby_zero_power_nth_same: fixes a :: "'b fps"   and n :: "nat" assumes "a $ 0 = (0::'b)" shows "a ^ n $ n = (a $ 1) ^ n"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "D.ide (F a)"
subst_poly_add: fixes \<sigma> :: "nat \<Rightarrow> linear_poly"   and p :: "linear_poly"   and q :: "linear_poly" shows "subst_poly \<sigma> (p + q) = subst_poly \<sigma> p + subst_poly \<sigma> q"
subtensor_listsum: fixes As :: "'b tensor list"   and ds :: "nat list"   and i :: "nat" assumes "\<And>A. A \<in> set As \<Longrightarrow> dims A = ds"   and "ds \<noteq> []"   and "i < hd ds" shows "subtensor (listsum ds As) i = listsum (tl ds) (map (\<lambda>A. subtensor A i) As)"
bind_SE_def: fixes f :: "'e \<Rightarrow> ('g \<times> 'e) option"   and g :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'e) option" shows "bind\<^sub>S\<^sub>E f g = (\<lambda>\<sigma>. case f \<sigma> of None \<Rightarrow> None | Some (out, \<sigma>') \<Rightarrow> g out \<sigma>')"
group_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a" shows "group f z inverse \<equiv> semigroup f \<and> group_axioms f z inverse"
real_greaterThanLessThan_infinity_eq: fixes N :: "ereal" shows "real_of_ereal ` {N<..<\<infinity>} = (if N = \<infinity> then {} else if N = - \<infinity> then UNIV else {real_of_ereal N<..})"
hisPrev_prop_1: fixes n' :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" assumes "\<exists>n''<n'. \<exists>nid'. (n'', nid') \<in> his t n nid" shows "hisPred t n nid n' < n'"
is_set_to_sorted_list_def: fixes ordR :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Rk :: "('h \<times> 'g) set"   and Rs :: "('h \<times> 'g) set \<Rightarrow> ('i \<times> 'g set) set"   and tsl :: "'i \<Rightarrow> 'h list" shows "is_set_to_sorted_list ordR Rk Rs tsl \<equiv> \<forall>s s'. (s, s') \<in> \<langle>Rk\<rangle>Rs \<longrightarrow> (\<exists>l'. (tsl s, l') \<in> \<langle>Rk\<rangle>list_rel \<and> RETURN l' \<le> it_to_sorted_list ordR s')"
cprank_max_listsum: fixes As :: "'b tensor list"   and ds :: "nat list"   and n :: "nat" assumes "\<And>A. A \<in> set As \<Longrightarrow> dims A = ds"   and "\<And>A. A \<in> set As \<Longrightarrow> cprank_max n A" shows "cprank_max (n * length As) (listsum ds As)"
smult_spvec_def: fixes y :: "'b" shows "smult_spvec y = map (\<lambda>a. (fst a, y * snd a))"
prime_int_altdef: fixes p :: "int" shows "prime p = (1 < p \<and> (\<forall>m\<ge>0. m dvd p \<longrightarrow> m = 1 \<or> m = p))"
affine_hull_3_imp_collinear: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<in> affine hull {a, b}" shows "collinear {a, b, c}"
deriv2: fixes s :: "real" assumes "vid1 \<in> ODE_vars I (OVar vid1)"   and "s \<in> {0..t}" shows "((\<lambda>t. sterm_sem I (f1 fid1 vid1) (fst (mk_v I (OVar vid1) (sol 0, b) (sol t)))) has_derivative (\<lambda>t'. t' * frechet I (f1 fid1 vid1) (fst (mk_v I (OVar vid1) (sol 0, b) (sol s))) (snd (mk_v I (OVar vid1) (sol 0, b) (sol s))))) (at s within {0..t})"
trms\<^sub>s\<^sub>t_cons: fixes x :: "('a, 'b) strand_step"   and S :: "('a, 'b) strand_step list" shows "trms\<^sub>s\<^sub>t (x # S) = trms\<^sub>s\<^sub>t\<^sub>p x \<union> trms\<^sub>s\<^sub>t S"
simps: fixes n :: "nat"   and i :: "nat"   and v :: "'a" shows "single_nz_val n i v = vec n (\<lambda>j. if i = j then v else (0::'a))"
xor_uni_samp: fixes y :: "bool" shows "map_spmf ((\<oplus>) y) coin_spmf = map_spmf (\<lambda>b. b) coin_spmf"
l2_inv3_def: shows "l2_inv3 \<equiv> {s. extr (bad s) (ik s) (chan s) \<subseteq> synth (analz (generators s)) \<and> chan s \<subseteq> faked_chan_msgs s \<union> chan_generators}"
cong: fixes jkbp :: "'c \<Rightarrow> ('c, 'd, 'e) GC list"   and envInit :: "'b list"   and envAction :: "'b \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('c \<Rightarrow> 'e) \<Rightarrow> 'b \<Rightarrow> 'b"   and envVal :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and envObs :: "'c \<Rightarrow> 'b \<Rightarrow> 'g" shows "Environment.clock_sim jkbp envInit envAction envTrans envVal envObs \<equiv> Environment.clock_sim jkbp envInit envAction envTrans envVal envObs"
points_index_upper_bound: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2" shows "int (\<B> index ps) \<le> \<Lambda>"
right_idem: fixes a :: "'b upred"   and b :: "'b upred" shows "((a \<and> b) \<and> b) = (a \<and> b)"
curr_val_satisfies_state_def: fixes s :: "('b, 'c) state" shows "\<Turnstile> s \<equiv> \<langle>\<V> s\<rangle> \<Turnstile>\<^sub>s s"
order_iff_strict: fixes b :: "'b"   and a :: "'b" shows "order.greater_eq a b = (order.greater a b \<or> a = b)"
nr: shows "dim_vec (A *\<^sub>v (k \<cdot>\<^sub>v v)) = nr"
m_rcancel: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int" assumes "a \<noteq> \<zero>\<^bsub>UP R\<^esub>"   and "a \<in> carrier (UP R)"   and "b \<in> carrier (UP R)"   and "c \<in> carrier (UP R)" shows "(b \<otimes>\<^bsub>UP R\<^esub> a = c \<otimes>\<^bsub>UP R\<^esub> a) = (b = c)"
append_append_fetch: fixes tp1 :: "(action \<times> nat) list"   and tp :: "(action \<times> nat) list"   and a :: "nat"   and tp2 :: "(action \<times> nat) list"   and b :: "cell" assumes "is_even (length tp1)"   and "is_even (length tp)"   and "length tp1 div 2 < a \<and> a \<le> length tp1 div 2 + length tp div 2" shows "fetch (tp1 @ tp @ tp2) a b = fetch tp (a - length tp1 div 2) b"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f0 :: "'b"   and f1 :: "'b" assumes "category C"   and "parallel_pair_diagram_axioms C f0 f1" shows "parallel_pair_diagram C f0 f1"
simps_2: fixes q :: "hf"   and x :: "'a"   and xs :: "'a list" shows "nextl q (x # xs) = nextl (nxt M q x) xs"
map_iterator_genord_foldli_conv: fixes iti :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option"   and R :: "'e \<times> 'f \<Rightarrow> 'e \<times> 'f \<Rightarrow> bool" shows "map_iterator_genord iti m R = (\<exists>l0. distinct (map fst l0) \<and> m = map_of l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0)"
index_swap_distinct: fixes xs :: "'a list"   and n :: "nat"   and x :: "'a" assumes "distinct xs"   and "Suc n < length xs" shows "index (swap n xs) x = (if x = xs ! n then Suc n else if x = xs ! Suc n then n else index xs x)"
size_4: fixes x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree" shows "size \<langle>x21, x22, x23\<rangle> = size x21 + size x23 + Suc 0"
find_nu_measures_decreases: fixes \<nu> :: "nat" assumes "seq (\<nu> + \<nu>) \<noteq> seq \<nu>" shows "(Suc \<nu>, \<nu>) \<in> find_nu_measures"
OclIsDeleted_def: fixes X :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "X.oclIsDeleted() \<equiv> \<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> then \<lfloor>\<lfloor>oid_of (X \<tau>) \<in> dom (heap (fst \<tau>)) \<and> oid_of (X \<tau>) \<notin> dom (heap (snd \<tau>))\<rfloor>\<rfloor> else invalid \<tau>"
assms_4: shows "\<not> poincare_collinear {0\<^sub>h, x1, x2}"
indep_var_distribution_eq: fixes S :: "'c measure"   and X :: "'a \<Rightarrow> 'c"   and T :: "'c measure"   and Y :: "'a \<Rightarrow> 'c" shows "indep_var S X T Y = (random_variable S X \<and> random_variable T Y \<and> distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))"
prefix_par_Int2: fixes a :: "'a"   and A :: "'a set"   and P :: "'a process"   and Q :: "'a process" assumes "a \<in> A" shows "(a \<rightarrow> P \<lbrakk>A\<rbrakk> a \<rightarrow> Q) = (a \<rightarrow> (P \<lbrakk>A\<rbrakk> Q))"
set_map_2: fixes f1 :: "'f \<Rightarrow> 'd"   and f2 :: "'g \<Rightarrow> 'c"   and f3 :: "'h \<Rightarrow> 'e"   and v :: "('f, 'g, 'h) F1" shows "F1set2 (F1map f1 f2 f3 v) = f2 ` F1set2 v"
quasi_measure_preservingE_2: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and A :: "'c set" assumes "f \<in> quasi_measure_preserving M N"   and "A \<in> sets N" shows "(f -` A \<inter> space M \<in> null_sets M) = (A \<in> null_sets N)"
ordering_ord_ijk: fixes f :: "nat \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a set"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "TernaryOrdering.ordering f ord X"   and "i < j \<and> j < k \<and> (finite X \<longrightarrow> k < card X)" shows "ord (f i) (f j) (f k)"
is_interval_convex: fixes S :: "'b set" assumes "is_interval S" shows "convex S"
dim_subset_UNIV_cart: fixes S :: "(real, 'a) vec set" shows "dim S \<le> CARD('a)"
simps_1: fixes n :: "nat" shows "bl2nat [] n = 0"
brnL_mono: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and n1 :: "nat"   and n2 :: "nat"   and cl :: "('c, 'd, 'a) cmd list" assumes "PL cval"   and "n1 \<le> n2" shows "PL.brnL cl n1 \<le> PL.brnL cl n2"
NegPos_set_2: fixes a :: "'b"   and x :: "'b negation_type list" shows "(Neg a \<in> set x) = (a \<in> set (getNeg x))"
prob_space_restrict_space: fixes S :: "'c set"   and M :: "'c measure" assumes "S \<in> sets M"   and "emeasure M S = 1" shows "prob_space (restrict_space M S)"
simps_2: shows "domain Resp = {xny, xgnx, xgny, xsk, xEnd}"
A: fixes m :: "nat" shows "euler_approx_aux (Suc m) = 1 + Suc m * euler_approx_aux m"
remdupsRemoveAllCommute: fixes a :: "'b"   and list :: "'b list" shows "remdups (removeAll a list) = removeAll a (remdups list)"
nat_mult_le_one: fixes m :: "nat"   and n :: "nat" assumes "m \<le> 1"   and "n \<le> 1" shows "m * n \<le> 1"
invariant: fixes a :: "'a"   and b :: "'a" assumes "P a"   and "a \<rightarrow> b" shows "Q b"
assms_3: shows "a * d - b * c \<noteq> 0"
fmeasurableI: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>" shows "A \<in> fmeasurable M"
inv_comm: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<oplus>\<^bsub>P\<^esub> y = \<zero>\<^bsub>P\<^esub>"   and "x \<in> carrier P"   and "y \<in> carrier P" shows "y \<oplus>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>"
ASSnot: fixes x :: "char list"   and e :: "aexp"   and s :: "char list \<Rightarrow> int"   and p :: "nat"   and t :: "char list \<Rightarrow> int" shows "(\<not> (x ::= e, s) \<Rightarrow> p \<Down> t) = (p \<noteq> Suc 0 \<or> t \<noteq> s[e/x])"
eventually_principal: fixes P :: "'a \<Rightarrow> bool"   and S :: "'a set" shows "eventually P (principal S) = (\<forall>x\<in>S. P x)"
append_cols_def: fixes A :: "'b mat"   and B :: "'b mat" shows "A @\<^sub>c B = four_block_mat A B (0\<^sub>m 0 (dim_col A)) (0\<^sub>m 0 (dim_col B))"
compatible_def: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" shows "compatible G H \<equiv> tail G = tail H \<and> head G = head H"
sigma_protocol: shows "Schnorr_\<Sigma>.\<Sigma>_protocol"
ratpoly_to_intpoly: fixes p :: "'a poly" assumes "\<forall>i. poly.coeff p i \<in> \<rat>" obtains q :: "int poly" and c :: "nat" where "c \<noteq> 0" and "p = Polynomial.smult (inverse (of_nat c)) (of_int_poly q)"
elims_1: fixes x :: "'at Literal"   and xa :: "'at Literal"   and y :: "bool" assumes "literal_ordering x xa = y" obtains L1 :: "'at Literal" and L2 :: "'at Literal" where "x = L1" and "xa = L2" and "y = ((atom L1, atom L2) \<in> atom_ordering)"
k_def: shows "k = card (rcosets\<^bsub>G\<lparr>carrier := conj.stabilizer P\<rparr>\<^esub> P)"
matrix_mul_lid: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.mat (1::'a) ** A = A"
coprimeE: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and p :: "'a"   and q :: "'a" assumes "class.comm_monoid_mult times one"   and "comm_monoid_mult.coprime times one p q" obtains "\<And>r. \<lbrakk>dvd.dvd times r p; dvd.dvd times r q\<rbrakk> \<Longrightarrow> dvd.dvd times r one"
new_document_ok: fixes h :: "(_) heap" shows "h \<turnstile> ok new_document"
hyps: fixes q :: "('v monom \<times> 'a) list" assumes "check_poly_ge p q" shows "p \<ge>p q"
submonoid_inverse_closed: fixes u :: "'a" assumes "normal.sub.invertible u"   and "u \<in> Normal" shows "inverse u \<in> Normal"
hyps: shows "(0::'a) \<le> v \<bullet>c v"
sel_2: fixes g :: "'a set" assumes "g \<in> G ` I" shows "G (sel g) = g"
le_triangle: fixes m :: "nat"   and n :: "nat" shows "m \<le> triangle (n + m)"
in_cont_mkTree: fixes s :: "'state"   and rs :: "'rule stream"   and t' :: "('state \<times> 'rule) tree" assumes "s \<in> S"   and "fair rs"   and "t' |\<in>| cont (mkTree rs s)" shows "\<exists>sl' s'. s' \<in> S \<and> eff (shd (trim rs s)) s sl' \<and> s' |\<in>| sl' \<and> t' = mkTree (stl (trim rs s)) s'"
poly_eq_iff: fixes p :: "'a poly"   and q :: "'a poly" shows "(p = q) = (\<forall>n. coeff p n = coeff q n)"
infmatrixforward: fixes x :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and y :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "x = y" shows "\<forall>a b. x a b = y a b"
not_basis_inD_aux: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<not> \<E>.basis_in \<E> X" shows "\<not> indep_system.basis \<E> (\<E>.indep_in \<E>) X"
insert_commute: fixes x :: "'k"   and y :: "'k"   and A :: "'k set" shows "insert x (insert y A) = insert y (insert x A)"
elims: fixes x :: "'g list"   and y :: "'g list" assumes "remdups_adj x = y" obtains "x = []" and "y = []"   | xa :: "'g" where "x = [xa__]" and "y = [xa__]"   | xa :: "'g" and ya :: "'g" and xs :: "'g list" where "x = xa # ya # xs" and "y = (if xa = ya then remdups_adj (xa # xs) else xa # remdups_adj (ya # xs))"