hcju/isabelleps huggingface dataset:
axioms_7: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and \<tau>' :: "'d \<Rightarrow> 'e" assumes "inverse_transformations A B F G \<tau> \<tau>'" shows "inverse_transformations_axioms A B \<tau> \<tau>'"
Lp_D_3: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "0 < p"   and "f \<in> space\<^sub>N (\<LL> (ennreal p) M)" shows "Norm (\<LL> (ennreal p) M) f = (LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p)"
ID_ceq: fixes ceq :: "('b \<Rightarrow> 'b \<Rightarrow> bool) option"   and eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.ceq ceq"   and "ID ceq = Some eq" shows "eq = (=)"
simps_19: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x31 :: "poscheckvariant"   and x32 :: "('d, 'e) Term.term"   and x33 :: "('d, 'e) Term.term" shows "(case \<langle>x31: x32 \<doteq> x33\<rangle>\<^sub>s\<^sub>t of send\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f1 x | receive\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f3 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f4 x xa) = f3 x31 x32 x33"
Action_def: fixes L :: "(('f, 'd) expr \<times> ('e data \<Rightarrow> bool) \<times> 'd set \<times> 'e pupdate) set" shows "Action L = action ` L"
inc: shows "supp q \<subseteq> as \<union> q \<bullet> as"
prl: fixes prc :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and prl :: "'a \<Rightarrow> bool"   and g :: "'a"   and f :: "'a" assumes "precedence prc prl"   and "prl g" shows "snd (prc f g) = True"
dbl_simps_5: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus (numeral.numeral one plus k) = numeral.numeral one plus (num.Bit0 k)"
UN_ball_bex_simps_3: fixes A :: "'b set set"   and P :: "'b \<Rightarrow> bool" shows "(\<exists>x\<in>\<Union> A. P x) = (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
nonzero: fixes h :: "'a \<Rightarrow> real" assumes "h \<in> powr_closure g" shows "\<forall>\<^sub>F x in F. h x \<noteq> 0"
uop_ueval: fixes f :: "'e \<Rightarrow> 'c"   and x :: "('e, 'd) uexpr"   and b :: "'d" shows "\<lbrakk>uop f x\<rbrakk>\<^sub>e b = f (\<lbrakk>x\<rbrakk>\<^sub>e b)"
gd_term_def: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict \<equiv> ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"
o_apply: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and x :: "'f" shows "(f \<circ> g) x = f (g x)"
equalityD1: fixes A :: "'b set"   and B :: "'b set" assumes "A = B" shows "A \<subseteq> B"
complete_iff_height: fixes kdt :: "'b kdt" shows "KD_Tree.complete kdt = (min_height kdt = height kdt)"
gen: shows "generate G (\<Union>g\<in>gs - {y}. generate G {g}) = generate G (gs - {y})"
real_cond_exp_bsum: fixes M :: "'c measure"   and F :: "'c measure"   and I :: "'d set"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)" shows "AEeq M (real_cond_exp M F (\<lambda>x. \<Sum>i\<in>I. f i x)) (\<lambda>x. \<Sum>i\<in>I. real_cond_exp M F (f i) x)"
commute_def: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "commute r s = ((r\<inverse>)\<^sup>* O s\<^sup>* \<subseteq> s\<^sup>* O (r\<inverse>)\<^sup>*)"
snap_p_1: shows "\<not> ps (S t j) p \<noteq> None"
int_xor_not: fixes x :: "int"   and y :: "int" shows "NOT x XOR y = NOT (x XOR y) \<and> x XOR NOT y = NOT (x XOR y)"
poly_poly_of_focus: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and a :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "poly (poly_of_focus x p) a = poly_eval (\<lambda>_. a) (focus {x} p)"
str_eq_def: fixes A :: "'a list set" shows "\<approx>A \<equiv> {(x, y). \<forall>z. (x @ z \<in> A) = (y @ z \<in> A)}"
RedStaticDownDynCast: fixes E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and a :: "nat"   and Cs :: "char list list"   and Cs' :: "char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   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" shows "P,E \<turnstile> \<langle>Cast C (ref (a, Cs @ [C] @ Cs')),s\<rangle> \<rightarrow> \<langle>ref (a, Cs @ [C]),s\<rangle>"
i1_ips: shows "ipassmt i1 = Some i1_ips"
connected_empty: fixes "open" :: "'c set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.connected open {}"
invertible_means_mult_id: fixes A :: "'a mat" assumes "invertible_mat A" shows "matr_option (dim_row A) (mat_inverse A) * A = 1\<^sub>m (dim_row A)"
atypes_inhabited: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and a :: "'c" assumes "typed_model arity public Ana \<Gamma>" shows "\<exists>c. \<Gamma> (Fun c []) = Var a \<and> intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity (Fun c []) \<and> public c \<and> arity c = 0"
bi_unique_iff: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "bi_unique R = ((\<forall>z. \<exists>\<^sub>\<le>\<^sub>1 x. R x z) \<and> (\<forall>z. \<exists>\<^sub>\<le>\<^sub>1 x. R z x))"
RedStaticDownCast: fixes E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and a :: "nat"   and Cs :: "char list list"   and Cs' :: "char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   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" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>ref (a, Cs @ [C] @ Cs'),s\<rangle> \<rightarrow> \<langle>ref (a, Cs @ [C]),s\<rangle>"
invertibleI: fixes u :: "('a set \<times> 'b) set"   and v :: "('a set \<times> 'b) set" assumes "add_rel u v = \<lfloor>U, \<zero>\<^bsub>U\<^esub>\<rfloor>"   and "add_rel v u = \<lfloor>U, \<zero>\<^bsub>U\<^esub>\<rfloor>"   and "u \<in> carrier_direct_lim"   and "v \<in> carrier_direct_lim" shows "invertible u"
take_imp_prefix: fixes n :: "nat"   and xs :: "'b list" shows "prefix (xs \<down> n) xs"
word_add_le_mono1: fixes i :: "'b word"   and j :: "'b word"   and k :: "'b word" assumes "i \<le> j"   and "unat j + unat k < 2 ^ LENGTH('b)" shows "i + k \<le> j + k"
IH: fixes w1 :: "'a list"   and w2 :: "'a list" assumes "w \<in> w1 \<otimes> w2" shows "rev w \<in> rev w1 \<otimes> rev w2"
knows_Says: fixes A :: "agent"   and A' :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" shows "knows A (Says A' B X # evs) = insert X (knows A evs)"
union_left_cancel: fixes K :: "'k multiset"   and M :: "'k multiset"   and N :: "'k multiset" shows "(K + M = K + N) = (M = N)"
set_borel_integral_eq_integral_1: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "set_integrable lborel S f" shows "f integrable_on S"
rep_in_iso_class: fixes F :: "'a set" assumes "is_iso_class F" shows "iso_class_rep F \<in> F"
lr_POR_arg1_rel: shows "POR_arg1_rel \<in> unlr POR.delta"
linear_independent_injective_image: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and s :: "('a, 'b) vec set" assumes "Vector_Spaces.linear (*s) (*s) f"   and "vec.independent s"   and "inj_on f (vec.span s)" shows "vec.independent (f ` s)"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>" obtains "\<phi>.A.arr f" and "\<phi>.A.dom f = a" and "\<phi>.A.cod f = b"
axioms_2: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e" assumes "module_hom s1 s2 f" shows "module s2"
cod_simps_2: shows "H\<mu>H\<nu>\<pi>.cod.leg0 = H\<mu>H\<nu>\<pi>.leg0"
closedin_prod_Times_iff: fixes X :: "'c topology"   and Y :: "'d topology"   and S :: "'c set"   and T :: "'d set" shows "closedin (prod_topology X Y) (S \<times> T) = (S = {} \<or> T = {} \<or> closedin X S \<and> closedin Y T)"
disj_commute: fixes x :: "'a"   and y :: "'a" shows "x OR y = y OR x"
match_4: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'c"   and f :: "'c"   and h :: "'c" assumes "category C"   and "partial_magma.arr C (C g f)"   and "partial_magma.arr C (C h g)" shows "partial_magma.arr C (C h (C g f))"
f: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and e :: "('a \<times> 'a) set"   and m :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and ov :: "('a \<times> 'a) set"   and d :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and f :: "('a \<times> 'a) set"   and p :: "'a"   and q :: "'a" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> f) = (\<exists>k l u. meets k l \<and> meets l p \<and> meets p u \<and> meets k q \<and> meets q u)"
lesub_list_impl_same_size: fixes xs :: "'b list"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" assumes "xs [\<sqsubseteq>\<^bsub>r\<^esub>] ys" shows "length ys = length xs"
between_exchange2: fixes A :: "'p"   and B :: "'p"   and D :: "'p"   and C :: "'p" assumes "Bet A B D"   and "Bet B C D" shows "Bet A C D"
Sup_distl: fixes x :: "('b \<times> 'b) set"   and Y :: "('b \<times> 'b) set set" shows "x O \<Union> Y = \<Union> {z. \<exists>y\<in>Y. z = x O y}"
linear_lower: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and u :: "real"   and x :: "real" assumes "\<And>x. x \<in> {a..b} \<Longrightarrow> (f has_real_derivative f' x) (at x within {a..b})"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> f' x \<le> u"   and "x \<in> {a..b}" shows "f b + u * (x - b) \<le> f x"
Inf_less_iff: fixes S :: "'c set"   and a :: "'c" shows "(Inf S < a) = (\<exists>x\<in>S. x < a)"
simps_5: fixes uu :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uv :: "char list"   and b :: "bexp"   and C :: "acom" shows "strip ({uu/uv} WHILE b DO C) = WHILE b DO strip C"
ASum_zero: fixes n :: "nat"   and f :: "nat \<Rightarrow> ant" shows "(\<forall>j\<le>n. f j \<in> Z\<^sub>\<infinity>) \<and> (\<forall>l\<le>n. f l = 0) \<longrightarrow> ASum f n = 0"
append_self_conv: fixes xs :: "'f list"   and ys :: "'f list" shows "(xs @ ys = xs) = (ys = [])"
"1_1": assumes "(0::'a) < x"   and "\<not> x * x - n < \<epsilon>"   and "(0::'a) < (n / x + x) / (2::'a)" shows "sqrt_approx_main_impl \<epsilon> n ((n / x + x) / (2::'a)) = sqrt_approx_main ((n / x + x) / (2::'a))"
refl: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "ordering less_eq less" shows "less_eq a a"
Derivation_unique_dest: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list"   and c :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.Derivation \<NN> \<TT> \<RR> a D b"   and "CFG.Derivation \<NN> \<TT> \<RR> a D c" shows "b = c"
ind_in_set_mono: fixes a :: "nat"   and A :: "nat set"   and b :: "nat" assumes "a \<in> A"   and "b \<in> A"   and "a < b" shows "ind_in_set A a < ind_in_set A b"
nth_least_strict_mono: fixes l :: "nat"   and A :: "'a set"   and k :: "nat" assumes "enat l < esize A"   and "k < l" shows "nth_least A k < nth_least A l"
simps_1: fixes S :: "('a, 'b) fm set"   and f :: "nat \<Rightarrow> ('a, 'b) fm" shows "extend S f 0 = S"
Abs_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e_inject: fixes x :: "\<langle>\<langle>'a list\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "\<langle>\<langle>'a list\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "x \<in> {X. X = UML_Types.bot_class.bot \<or> X = null \<or> (\<forall>x\<in>set \<lceil>\<lceil>X\<rceil>\<rceil>. x \<noteq> UML_Types.bot_class.bot)}"   and "y \<in> {X. X = UML_Types.bot_class.bot \<or> X = null \<or> (\<forall>x\<in>set \<lceil>\<lceil>X\<rceil>\<rceil>. x \<noteq> UML_Types.bot_class.bot)}" shows "(Abs_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e x = Abs_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e y) = (x = y)"
conjugate_of_complex: fixes z :: "complex" shows "conjugate (of_complex z) = of_complex (cnj z)"
wiringD_1: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and cnv :: "('h, 'i, 'j, 'k) converter"   and f :: "'h \<Rightarrow> 'j"   and g :: "'k \<Rightarrow> 'i" assumes "wiring \<I> \<I>' cnv (f, g)" shows "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C"
ordered_res_with_selection_is_model_res: fixes P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set"   and Sel :: "'at set" assumes "ordered_sel_resolvent UNIV P1 P2 C" shows "ordered_model_resolvent Sel (rename_clause Sel P1) (rename_clause Sel P2) (rename_clause Sel C)"
foldmatrix_transposed_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "foldmatrix_transposed f g A m n \<equiv> foldseq g (\<lambda>j. foldseq_transposed f (A j) n) m"
pos_zmult_pos: fixes a :: "int"   and b :: "int" assumes "0 \<le> a"   and "0 < b" shows "a \<le> a * b"
awalk_hd_in_verts: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v" shows "u \<in> verts G"
real_poly_factor: fixes p :: "'a poly"   and q :: "'a poly" assumes "set (coeffs (p * q)) \<subseteq> \<real>"   and "set (coeffs p) \<subseteq> \<real>"   and "p \<noteq> 0" shows "set (coeffs q) \<subseteq> \<real>"
Inf_less_eq: 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 A :: "'b set"   and u :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>v. v \<in> A \<Longrightarrow> less_eq v u"   and "A \<noteq> {}" shows "less_eq (Inf A) u"
dd_closed: fixes n' :: "'node"   and V :: "'var"   and n'' :: "'node"   and S :: "'node set" assumes "n' influences V in n''"   and "n'' \<in> wod_backward_slice S" shows "n' \<in> wod_backward_slice S"
Rats_cases': fixes x :: "'a" assumes "x \<in> \<rat>" obtains a :: "int" and b :: "int" where "0 < b" and "coprime a b" and "x = of_int a / of_int b"
casePres: fixes CsQ :: "('c \<times> ('a, 'b, 'c) psi) list"   and CsP :: "('c \<times> ('a, 'b, 'c) psi) list"   and \<Psi> :: "'b"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<phi> Q. (\<phi>, Q) mem CsQ \<Longrightarrow> \<exists>P. (\<phi>, P) mem CsP \<and> guarded P \<and> (\<Psi>, P, Q) \<in> Rel"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> Rel \<Longrightarrow> \<Psi>' \<rhd> R \<leadsto>[Rel] S"   and "Rel \<subseteq> Rel'" shows "\<Psi> \<rhd> Cases CsP \<leadsto>[Rel'] Cases CsQ"
weak_eq: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp"   and "comp x y = Eq" shows "x = y"
fail_reachable: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and FAIL :: "'f \<times> 'f"   and PM :: "('d, 'e, 'f \<times> 'f) FSM" assumes "\<not> M1 \<preceq> M2"   and "well_formed M1"   and "well_formed M2"   and "productF M2 M1 FAIL PM" shows "FAIL \<in> reachable PM (initial PM)"
h1: shows "\<turnstile>\<^sub>p {adjoint (tensor_P (1\<^sub>m N) (mat_incr K)) * (P' + Q) * tensor_P (1\<^sub>m N) (mat_incr K)} Utrans (tensor_P (1\<^sub>m N) (mat_incr K)) {P' + Q}"
step_2_similar: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat (??.Jordan_Normal_Form_Existence.step_2 A) A"
order_trans: fixes y :: "'d"   and x :: "'d"   and z :: "'d" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
symmetric_subset: fixes B :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "B \<subseteq> UNIV" shows "symmetric B (sympartp r)"
Vfrom: fixes a :: "V"   and i :: "V" shows "Vfrom a i = a \<union>\<^sub>\<circ> \<Union>\<^sub>\<circ> ((\<lambda>j\<in>\<^sub>\<circ>i. VPow (Vfrom a j)) `\<^sub>\<circ> i)"
measure_measure_pmf_finite: fixes S :: "'b set"   and M :: "'b pmf" assumes "finite S" shows "measure_pmf.prob M S = sum (pmf M) S"
Iod_sub_sub: fixes D :: "('c, 'd) Order_scheme"   and S :: "'c set"   and T :: "'c set" assumes "Order D"   and "S \<subseteq> T"   and "T \<subseteq> carrier D" shows "Iod (Iod D T) S = Iod D S"
Rep_IF1_inverse: fixes x :: "'d IF1" shows "Abs_IF1 (Rep_IF1 x) = x"
substitution_lemma: fixes x :: "name"   and y :: "name"   and u :: "lam"   and t :: "lam"   and s :: "lam" assumes "x \<noteq> y"   and "x \<sharp> u" shows "t[x::=s][y::=u] = t[y::=u][x::=s[y::=u]]"
simps_3: fixes uv :: "'b set" shows "(FAILi \<le> RES uv) = False"
freeword_funlift_add: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b freeword"   and g' :: "'b freeword" shows "freeword_funlift f (g + g') = freeword_funlift f g + freeword_funlift f g'"
valid2_last_Cons: fixes tr :: "('state, 'act, 'out) trans list"   and tran :: "('state, 'act, 'out) trans" assumes "valid2 tr" shows "last (tran # tr) = last tr"
ub2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "y \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y"
no_tvars_in_term: fixes x :: "Syntax.tvar"   and t :: "Syntax.term" shows "atom x \<sharp> t"
p2_3: fixes x :: "'a"   and K :: "'a set" assumes "(x, K) \<in> p2" shows "K \<subseteq> s2"
image_stretch_interval_cart: fixes m :: "'a \<Rightarrow> real"   and a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "(\<lambda>x. \<chi>k. m k * x $ k) ` cbox a b = (if cbox a b = {} then {} else cbox (\<chi>k. min (m k * a $ k) (m k * b $ k)) (\<chi>k. max (m k * a $ k) (m k * b $ k)))"
eq: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (subst_bv (Fv x \<tau>') s) (subst_bv (Fv x \<tau>') t)"
simps_4: fixes \<phi> :: "'a fm"   and \<psi> :: "'a fm" shows "depth (And \<phi> \<psi>) = max (depth \<phi>) (depth \<psi>) + 1"
at_pres: fixes x :: "'a"   and y :: "'a" shows "\<lparr>\<tt>\<tt> x\<rparr>\<tt>\<tt> y\<lparr>\<tt>\<tt> y \<cdot> \<tt>\<tt> x \<cdot> \<tt>\<tt> y\<rparr>"
simps_2: fixes F :: "'effect fs_set"   and P :: "'state"   and \<alpha>P' :: "(('act, 'effect) L_action, ('state, 'effect) L_state) residual" shows "EF (F, P) \<rightarrow>\<^sub>L \<alpha>P' = (\<exists>f. f \<in>\<^sub>f\<^sub>s F \<and> \<alpha>P' = \<langle>Eff f,AC (f, F, \<langle>f\<rangle>P)\<rangle>)"
ivl_diff: fixes i :: "'b"   and n :: "'b"   and m :: "'b" assumes "i \<le> n" shows "{i..<m} - {i..<n} = {n..<m}"
invertible_right_inverse: fixes u :: "'a" assumes "IA.additive.sub.invertible u"   and "u \<in> I" shows "addA u (IA.additive.sub.inverse u) = zeroA"
diff_diff_right: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subseteq># b" shows "c - (b - a) = c + a - b"
in_lists_conv_set: fixes xs :: "'g list"   and A :: "'g set" shows "(xs \<in> lists A) = (\<forall>x\<in>set xs. x \<in> A)"
finite_UNIV_sdim_def: shows "finite_UNIV = Phantom(sdim) True"
OrdSum_second: fixes b :: "'d" shows "(1::'c, b) \<in> OrdSum"
Call_lemma: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list"   and C' :: "char list"   and wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D"   and "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M: Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> subtype P T' T \<and> P \<turnstile> C' \<preceq>\<^sup>* D' \<and> is_type P T' \<and> (\<forall>T\<in>set Ts'. is_type P T) \<and> wf_md P D' (M, Ts', T', m')"
simps_2: fixes y :: "'a" shows "flat_orA (allow (deny y)) = allow y"
upd_rows_single: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and f :: "'d \<Rightarrow> ('c, 'd) vec" shows "upd_rows M {i} f = upd_row M i (f i)"
is_Proj_complement: fixes P :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" assumes "is_Proj P" shows "is_Proj (id_cblinfun - P)"
Rep_af: fixes x :: "'d af" shows "??.Abstract_AF.af.Rep_af x \<in> UNIV"
aux_lemma: fixes p :: "nat \<Rightarrow> nat" assumes "p \<in> three_cycles n" shows "p \<in> derived (alt_group n) (carrier (alt_group n))"
assms_5: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>" shows "(a +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b) *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c"
greaterThanLessThan_iff: fixes i :: "'a"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and u :: "'a" shows "(i \<in> ord.greaterThanLessThan less l u) = (less l i \<and> less i u)"
rces_imp_bces: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv1 :: "'d \<times> nat"   and res :: "('d \<times> nat) edge list"   and rv2 :: "'d \<times> nat" assumes "RedBlack prb"   and "SubRel.ces rv1 res rv2 (subs prb)" shows "Graph.ces (fst rv1) (ui_es res) (fst rv2)"
multisets_def: fixes A :: "'b set" shows "multisets A = {M. set_mset M \<subseteq> A}"
nprv_exiE0: fixes x :: "'var"   and \<phi> :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "exi x \<phi> \<in> F"   and "nprv (insert \<phi> F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "x \<in> var"   and "\<psi> \<in> fmla"   and "\<forall>xa\<in>F. x \<notin> Fvars xa"   and "x \<notin> Fvars \<psi>" shows "nprv F \<psi>"
1: shows "\<guillemotleft>local.inv i \<cdot> f : \<I> \<rightarrow> \<I>\<guillemotright>"
c_redu_subst1': fixes M :: "trm"   and M' :: "trm"   and y :: "name"   and c :: "coname"   and P :: "trm" assumes "M \<longrightarrow>\<^sub>c M'" shows "M{y:=<c>.P} \<longrightarrow>\<^sub>c M'{y:=<c>.P}"