hcju/isabelleps huggingface dataset:
prodsame: shows "prod_list (map ((!) sign) (a # xa)) = sign ! a * prod_list (map ((!) sign) xa)"
hyps_1: fixes A :: "'a set" assumes "xs = sorted_list_of_set A"   and "finite A" shows "sorted_list_of_set {x \<in> A. P x} = filter P (sorted_list_of_set A)"
affine_hull_face_of_disjoint_rel_interior: fixes S :: "'b set"   and F :: "'b set" assumes "convex S"   and "F face_of S"   and "F \<noteq> S" shows "affine hull F \<inter> rel_interior S = {}"
length_1_last_hd: fixes ns :: "'b list" assumes "length ns = 1" shows "last ns = hd ns"
assms_7: shows "q \<cdot> l \<le> l \<cdot> q"
map_poly_hom_smult: fixes c :: "int"   and p :: "int poly" shows "of_int_poly (smult c p) = smult (of_int c) (of_int_poly p)"
20: fixes x :: "'a" shows "- x \<sqinter> x = bot"
restrict_ext: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "restrict f A = restrict g A"
order_def: fixes S :: "('b, 'c) monoid_scheme" shows "order S = card (carrier S)"
le_succ_iff: fixes i :: "hf"   and j :: "hf" assumes "Ord i"   and "Ord j" shows "(succ i \<le> succ j) = (i \<le> j)"
assms: shows "\<not> summable (\<lambda>n. norm (f n * z ^ n))"
Inf_lower2: 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 u :: "'b"   and A :: "'b set"   and v :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "u \<in> A"   and "less_eq u v" shows "less_eq (Inf A) v"
correctCompositionDiffLevelsA22: shows "correctCompositionDiffLevels sA22"
toSet_def: fixes X :: "'b odlist" shows "toSet X = set (toList X)"
assms: fixes xs :: "'a dlist" assumes "\<And>ys. DList.card ys < DList.card xs \<Longrightarrow> P ys" shows "P xs"
ffun_minus_plus_commute: fixes g :: "('a, 'b) ffun"   and h :: "('a, 'b) ffun"   and f :: "('a, 'b) ffun" assumes "fdom g \<inter> fdom h = {}" shows "f - g + h = f + h - g"
long_chain2_card_geq: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set" assumes "long_ch_by_ord2 f X"   and "finite X" shows "3 \<le> card X"
subclsSI: 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 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" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "Shares D \<in> set Bs" shows "subclsSp P C D"
set_disconnected_nodes_get_ancestors_si: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_ancestors_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "l_set_disconnected_nodes_get_ancestors_si set_disconnected_nodes_locs get_ancestors_si_locs" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_ancestors_si_locs. r h h')"
groupF_Union_set: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "\<Union> (set ` set (groupF f xs)) = set xs"
partition_on_empty: fixes P :: "'c set set" shows "partition_on {} P = (P = {})"
hyps_2: fixes \<Gamma>\<^sub>2 :: "binding list"   and t\<^sub>1 :: "trm"   and \<Gamma>\<^sub>1 :: "binding list"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmarkRecord.type"   and ts :: "trm list" assumes "\<Gamma>\<^sub>2 \<turnstile> t\<^sub>1 : T"   and "\<Gamma>\<^sub>1 @ \<Delta>\<^sub>1 @ \<Gamma>\<^sub>2 \<turnstile> t\<^sub>2 : T\<^sub>2"   and "\<turnstile> p \<rhd> t\<^sub>1 \<Rightarrow> ts" shows "\<down>\<^sub>e \<parallel>\<Delta>\<^sub>1\<parallel> 0 \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<turnstile> t\<^sub>2[\<parallel>\<Gamma>\<^sub>1\<parallel> \<mapsto>\<^sub>s ts] : \<down>\<^sub>\<tau> \<parallel>\<Delta>\<^sub>1\<parallel> \<parallel>\<Gamma>\<^sub>1\<parallel> T\<^sub>2"
floorlog_geD: fixes w :: "nat"   and b :: "nat"   and x :: "nat" assumes "w \<le> floorlog b x"   and "0 < w" shows "b ^ (w - 1) \<le> x"
other: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and p :: "('b, 'c, 'a, 'd) seqp"   and l :: "'d"   and \<sigma>' :: "nat \<Rightarrow> 'b" assumes "wellformed \<Gamma>"   and "(\<sigma>, p) \<in> oreachable A S U"   and "l \<in> labels \<Gamma> p"   and "P (\<sigma>, l)"   and "U \<sigma> \<sigma>'" shows "P (\<sigma>', l)"
code: fixes sp :: "('c, 'b) sp\<^sub>\<nu>"   and sp' :: "('a, 'c) sp\<^sub>\<nu>" shows "sp o\<^sub>\<nu> sp' = In (map_sp\<^sub>\<mu> id (\<lambda>(sp, sp'). sp o\<^sub>\<nu> sp') (out sp o\<^sub>\<mu> out sp'))"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b \<le> a"
not_irreducibleE: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "\<not> comm_semiring_1.irreducible times one zero x" obtains "x = zero"   | "dvd.dvd times x one"   | a :: "'b" and b :: "'b" where "x = times a b" and "\<not> dvd.dvd times a one" and "\<not> dvd.dvd times b one"
cls: shows "class P Object = \<lfloor>(C', fs, ms)\<rfloor>"
homeomorphic_scaling: fixes c :: "real"   and S :: "'a set" assumes "c \<noteq> 0" shows "S homeomorphic (*\<^sub>R) c ` S"
intros_10: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and p' :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat"   and q :: "('e, 'f, 'g, 'h) seqp" assumes "((\<sigma>, p), a, \<sigma>', p') \<in> oseqp_sos \<Gamma> i" shows "((\<sigma>, p \<oplus> q), a, \<sigma>', p') \<in> oseqp_sos \<Gamma> i"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "F.wtE \<xi> \<equiv> \<forall>x. intTF (tpOfV x) (\<xi> x)"
ratfps_of_poly_code: fixes p :: "'b poly" shows "quot_of_ratfps (ratfps_of_poly p) = (p, 1)"
rel_cases: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d option"   and b :: "'e option" assumes "rel_option R a b" obtains "a = None" and "b = None"   | x :: "'d" and y :: "'e" where "a = Some x" and "b = Some y" and "R x y"
simps: fixes l :: "lit"   and l1 :: "lit"   and l2 :: "lit" shows "succs_of_clause l (Upair l1 l2) = (if l = negate l1 then insert l2 else if l = negate l2 then insert l1 else id)"
adj_dv: fixes v :: "'a"   and w :: "'a" shows "(v \<rightarrow>\<^bsub>del_vert u\<^esub> w) = (v \<rightarrow>\<^bsub>G\<^esub> w)"
dim_subset: fixes S :: "('a, 'b) vec set"   and T :: "('a, 'b) vec set" assumes "S \<subseteq> T" shows "vec.dim S \<le> vec.dim T"
emeasure_eq_measure: fixes A :: "'a set" shows "emeasure (point_measure UNIV (\<lambda>xa. ennreal (R x xa))) A = ennreal (Sigma_Algebra.measure (point_measure UNIV (\<lambda>xa. ennreal (R x xa))) A)"
index_add_vec_2: fixes v\<^sub>1 :: "'a vec"   and v\<^sub>2 :: "'a vec" shows "dim_vec (v\<^sub>1 + v\<^sub>2) = dim_vec v\<^sub>2"
AbGroup_inner_dirsum_el_decomp_ex1: fixes Gs :: "'b set list" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs" shows "\<forall>x\<in>\<Oplus>G\<leftarrow>Gs. G. \<exists>!gs. gs \<in> listset Gs \<and> x = sum_list gs"
induced_modulus_def: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and d :: "nat" assumes "dcharacter n \<chi>" shows "dcharacter.induced_modulus n \<chi> d = (d dvd n \<and> (\<forall>a. coprime a n \<and> [a = 1] (mod d) \<longrightarrow> \<chi> a = 1))"
UN_E: fixes b :: "'k"   and B :: "'l \<Rightarrow> 'k set"   and A :: "'l set" assumes "b \<in> \<Union> (B ` A)" obtains x :: "'l" where "x \<in> A" and "b \<in> B x"
closure_eq_closed: fixes a :: "'a set" assumes "closure a = a" shows "a closed"
a_Lam_inv_I: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and x :: "Syntax.var"   and e' :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> Lam x e', eP, eV : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"   and "atom x \<sharp> \<Gamma>" obtains eP' :: "Syntax.term" and eV' :: "Syntax.term" where "eP = Lam x eP'" and "eV = Lam x eV'" and "\<Gamma>(x $$:= \<tau>\<^sub>1) \<turnstile> e', eP', eV' : \<tau>\<^sub>2"
cons2: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and top0 :: "top0"   and s2 :: "'b state"   and err :: "v error_result"   and tops :: "top0 list" assumes "evaluate_top ck env s1 top0 (s2, Rerr err)" shows "evaluate_prog ck env s1 (top0 # tops) (s2, Rerr err)"
vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_is_fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p: fixes x :: "('d, 'c) stateful_strand_step" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p x = fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p x \<union> set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p x)"
Gcd_rat_def: shows "Gcd = normalization_euclidean_semiring_class.Gcd"
component_refl: fixes F :: "'d program" shows "F \<le> F"
IH: fixes i :: "nat" assumes "i < length T"   and "f \<in> funs_term (S ! i)" shows "f \<in> funs_term (T ! i) \<or> (\<exists>g\<in>funs_term (T ! i). f \<in> set (P g))"
obj_simps_2: fixes a :: "'b" assumes "B.obj a" shows "src\<^sub>B a = a"
assms_3: shows "f -` rel_interior S \<noteq> {}"
induct: fixes rh :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x1 :: "('e, 'f) T\<^sub>m"   and x2 :: "('e, 'f) T\<^sub>m"   and P :: "('e, 'f) T\<^sub>m \<Rightarrow> ('e, 'f) T\<^sub>m \<Rightarrow> bool" assumes "blinding_of_T rh bo x1 x2"   and "\<And>x y. blinding_of_F rh bo (root_hash_T rh) (\<lambda>x1 x2. blinding_of_T rh bo x1 x2 \<and> P x1 x2) x y \<Longrightarrow> P (T\<^sub>m x) (T\<^sub>m y)" shows "P x1 x2"
bounds_2: shows "\<B>\<^sub>l (l ! j) = \<B>\<^sub>l (l ! i)"
Col_dep2: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and c :: "(real, 'a) vec" shows "real_euclid.Col a b c = dep2 (b - a) (c - a)"
distinct_fold_remove1: fixes xs :: "'b list"   and ys :: "'b list" assumes "distinct xs" shows "distinct (fold remove1 ys xs)"
False: shows "\<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)"
co_agree': fixes s :: "real" shows "Vagree (fst (mk_v I ODE (sol 0, b) (sol s)), s *\<^sub>R (\<chi>i. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol s) $ i else 0)) (fst (mk_v I ODE (sol 0, b) (sol s)), s *\<^sub>R snd (mk_v I ODE (sol 0, b) (sol s))) (FVDiff \<theta>)"
col_permutation_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C" shows "Tarski_neutral_dimensionless.Col Bet C A B"
map_family_triangle2: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and S :: "('f, 'g) aGroup_scheme"   and g :: "'c \<Rightarrow> 'f \<Rightarrow> 'd"   and ff :: "('c \<Rightarrow> 'd) \<Rightarrow> 'h"   and B :: "'h set"   and h1 :: "'h \<Rightarrow> 'f"   and h2 :: "'f \<Rightarrow> 'h" assumes "I \<noteq> {}"   and "\<forall>j\<in>I. aGroup (A j)"   and "aGroup S"   and "\<forall>j\<in>I. g j \<in> aHom S (A j)"   and "ff \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A) \<rightarrow> B"   and "bij_to ff (carrier (a\<Pi>\<^bsub>I\<^esub> A)) B"   and "h1 \<in> aHom (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) ff) S"   and "\<forall>j\<in>I. compos (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) ff) (g j) h1 = ProjInd I A ff j"   and "h2 \<in> aHom S (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) ff)"   and "\<forall>j\<in>I. compos S (ProjInd I A ff j) h2 = g j" shows "\<forall>j\<in>I. compos (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) ff) (ProjInd I A ff j) (compos (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) ff) h2 h1) = ProjInd I A ff j"
up_def: shows "up = Grid.lift (\<lambda>d lm p \<alpha>. snd (up' d lm p \<alpha>))"
IH: fixes x2a :: "('a, 'b \<times> nat) Term.term" assumes "x2a \<in> set T" shows "x2a \<cdot> var_rename n \<circ>\<^sub>s var_rename_inv n = x2a"
not_the1: fixes P :: "'b \<Rightarrow> bool"   and y :: "'b" assumes "\<exists>!x. P x"   and "y \<noteq> (THE x. P x)" shows "\<not> P y"
h2c: shows "0 < B ^ n * B ^ (n mod 2)"
G_def: fixes x :: "'b" shows "G x = g -` {x} \<inter> space M"
naturality: fixes f :: "'a" assumes "A.arr f" shows "\<sigma> (A.cod f) \<cdot>\<^sub>B F f = G f \<cdot>\<^sub>B \<sigma> (A.dom f)"
finite_Di': fixes i :: "nat"   and y :: "nat list" shows "finite (Di' i y)"
M: fixes x :: "'a"   and n :: "nat" assumes "x \<in> S" shows "norm (\<F> n x) \<le> M"
p_div_set: fixes a :: "'a"   and N :: "'a" shows "{p. p \<in># prime_factorization a \<and> \<not> p dvd N} = {p. p \<in># prime_factorization (a * N)} - {p. p \<in># prime_factorization N}"
3: shows "w r w1 \<and> existsAt b w1 \<and> P b w1 \<and> (\<rightharpoondown>(=) a) b"
sum_Sigma_product: fixes S :: "'c set"   and T :: "'c \<Rightarrow> 'd set"   and x :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "finite S"   and "\<And>i. i \<in> S \<Longrightarrow> finite (T i)" shows "(\<Sum>i\<in>S. sum (x i) (T i)) = (\<Sum>(i, j)\<in>Sigma S T. x i j)"
typ_ok_TFreesT_imp_sort_ok_pre: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and T :: "typ"   and x :: "variable"   and S :: "String.literal set" assumes "is_std_sig \<Sigma>"   and "typ_ok_sig \<Sigma> T"   and "(x, S) \<in> tvsT T" shows "wf_sort (subclass (osig \<Sigma>)) S"
right_diff_distrib_numeral: fixes v :: "num"   and b :: "'a"   and c :: "'a" shows "numeral v * (b - c) = numeral v * b - numeral v * c"
lookup_other_ev_Some: fixes ev :: "'b"   and k :: "nat"   and A :: "'b mat"   and i :: "nat" assumes "lookup_other_ev ev k A = Some i" shows "i < k \<and> A $$ (i, i) \<noteq> ev \<and> (\<forall>j. i < j \<and> j < k \<longrightarrow> A $$ (j, j) = ev)"
one_dvd: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times one a"
su: shows "(Some s, u) = wordinterval_CIDR_split1 rs"
H_2: shows "H = THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H A B)"
inj_one_hom_axioms: shows "inj_one_hom hom"
dist_octo_def: fixes x :: "octo"   and y :: "octo" shows "dist x y = norm (x - y)"
Rep_discrete_market: fixes x :: "('d, 'c) discrete_market" shows "Rep_discrete_market x \<in> {(s, a). stk_strict_subs s}"
3: fixes i :: "nat"   and j :: "nat" assumes "i < length (seq' j)" shows "seq i \<in> set (seq' j)"
simps_95: fixes x51 :: "expr"   and x52 :: "bop"   and x53 :: "expr"   and x4 :: "val" shows "x51 \<guillemotleft>x52\<guillemotright> x53 \<noteq> Val x4"
pp_inf_bot_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y = bot) = (- - x \<sqinter> y = bot)"
numeral_One: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and one :: "'g" assumes "class.numeral plus" shows "numeral.numeral one plus num.One = one"
set_tree_mset: fixes t :: "('b \<times> 'c) tree" shows "Tree2.set_tree t = set_mset (mset_tree t)"
mbisim_wset_eq: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2" shows "wset s1 = wset s2"
assms_2: shows "init \<in> {[x, y], [y, x]}"
elims_2: fixes x :: "('a, 'b) message set"   and xa :: "'c + ('a, 'b option) insert_message list" assumes "is_certified_associated_string x xa" obtains M :: "('a, 'b) message set" and v :: "('a, 'b option) insert_message list" where "x = M" and "xa = Inr v" and "is_associated_string M v"
Supraclassical_logic_axioms: shows "Supraclassical_logic Cn \<top> \<bottom> (.\<longrightarrow>.) .\<not> (.\<and>.) (.\<or>.) (.\<longleftrightarrow>.)"
absolutely_integrable_abs: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "f absolutely_integrable_on S" shows "(\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) absolutely_integrable_on S"
hyps_1: fixes B :: "'a set"   and C :: "'a set" assumes "i = card B"   and "B \<subseteq> A" shows "local.mk_linear_orders_dom (C, B)"
mult_strict_mono': fixes a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "a < b"   and "c < d"   and "(0::'d) \<le> a"   and "(0::'d) \<le> c" shows "a * c < b * d"
wo_adm_extend_distance: shows "adm_wo Bonk_Schramm_extension_unfolded_wo extend_distance"
set_6: fixes x61 :: "('a, 'b, 'c) expr"   and x62 :: "('a, 'b, 'c) expr" shows "RID\<^sub>E (Assign x61 x62) = RID\<^sub>E x61 \<union> RID\<^sub>E x62"
alpha_Tree_induct: fixes t1 :: "('b, 'c, 'd, 'e) Tree"   and t2 :: "('b, 'c, 'd, 'e) Tree"   and P :: "('b, 'c, 'd, 'e) Tree \<Rightarrow> ('b, 'c, 'd, 'e) Tree \<Rightarrow> bool" assumes "t1 =\<^sub>\<alpha> t2"   and "\<And>tset1 tset2. \<lbrakk>\<And>a b. \<lbrakk>a \<in> set_bset tset1; b \<in> set_bset tset2; a =\<^sub>\<alpha> b\<rbrakk> \<Longrightarrow> P a b; rel_bset (=\<^sub>\<alpha>) tset1 tset2\<rbrakk> \<Longrightarrow> P (tConj tset1) (tConj tset2)"   and "\<And>t1 t2. \<lbrakk>t1 =\<^sub>\<alpha> t2; P t1 t2\<rbrakk> \<Longrightarrow> P (tNot t1) (tNot t2)"   and "\<And>f \<phi>. P (tPred f \<phi>) (tPred f \<phi>)"   and "\<And>f1 \<alpha>1 t1 f2 \<alpha>2 t2. \<lbrakk>\<And>p. p \<bullet> t1 =\<^sub>\<alpha> t2 \<Longrightarrow> P (p \<bullet> t1) t2; f1 = f2; \<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)\<rbrakk> \<Longrightarrow> P (tAct f1 \<alpha>1 t1) (tAct f2 \<alpha>2 t2)" shows "P t1 t2"
sum_of_cont_is_cont: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and g :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_continuous f"   and "is_Zp_continuous g" shows "is_Zp_continuous (f \<oplus>\<^bsub>function_ring (carrier Zp) Zp\<^esub> g)"
col_fst_def: fixes A :: "'a Matrix.mat" shows "col_fst A = Matrix.vec (dim_row A) (\<lambda>i. A $$ (i, 0))"
intros_1: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set" assumes "(a, b) \<in> r" shows "(a, b) \<in> r\<^sup>+"
isometry_def: fixes U :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd" shows "isometry U = (U* o\<^sub>C\<^sub>L U = id_cblinfun)"
chine_simps_2: shows "C.dom \<epsilon>.chine = \<epsilon>.dom.apex"
\<sigma>3'': shows "\<sigma>3'' \<in> seq ars"
fully_unsorted_poly_list_rel_def: fixes R :: "('a \<times> char list multiset) set" shows "fully_unsorted_poly_list_rel R = {(xs, ys). (xs, ys) \<in> \<langle>R \<times>\<^sub>r int_rel\<rangle>list_rel O list_mset_rel}"
cases: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" assumes "(a1, a2) \<in> encoding.indRelRT Enc TRel" obtains S :: "'a" where "a1 = SourceTerm S" and "a2 = TargetTerm (Enc S)"   | T1 :: "'b" and T2 :: "'b" where "a1 = TargetTerm T1" and "a2 = TargetTerm T2" and "(T1, T2) \<in> TRel"
listrel1I: fixes x :: "'g"   and y :: "'g"   and r :: "('g \<times> 'g) set"   and xs :: "'g list"   and us :: "'g list"   and vs :: "'g list"   and ys :: "'g list" assumes "(x, y) \<in> r"   and "xs = us @ x # vs"   and "ys = us @ y # vs" shows "(xs, ys) \<in> listrel1 r"
image_Collect: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool" shows "f ` {x. P x} = {f x |x. P x}"
point_in_one_group: fixes x :: "'a"   and G1 :: "'a set"   and G2 :: "'a set" assumes "x \<in> G1"   and "G1 \<in> \<G>"   and "G2 \<in> \<G>"   and "G1 \<noteq> G2" shows "x \<notin> G2"
less_fun_trans: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "less_fun f g"   and "less_fun g h" shows "less_fun f h"
bad1_bad2: shows "spmf (map_spmf (snd \<circ> snd) game_semi_forgetful) True = spmf (map_spmf snd game_A) True"
unitary_exmat_incr: shows "unitary (tensor_P (1\<^sub>m N) (mat_incr K))"
product_category_def: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "product_category C1 C2 \<equiv> category C1 \<and> category C2"
wf_syscls_is_class_Thread: fixes P :: "'a prog" assumes "wf_syscls P" shows "is_class P Thread"
perfect_hiding: fixes \<A> :: "(('g \<times> 'a) \<times> 'h \<times> 'b \<Rightarrow> (('e \<times> 'e) \<times> 'm) spmf) \<times> ('j \<times> 'd \<Rightarrow> 'm \<Rightarrow> bool spmf)" shows "AND_\<Sigma>_commit.abstract_com.perfect_hiding_ind_cpa \<A>"
subcls_implmt: fixes A :: "qtname"   and B :: "qtname"   and G :: "prog"   and K :: "qtname" assumes "G\<turnstile>A\<preceq>\<^sub>C B"   and "G\<turnstile>B\<leadsto>K" shows "G\<turnstile>A\<leadsto>K"
reach_equiv_beh_states: fixes T :: "('e, 'f) TS_scheme" shows "reach T = \<Union> (set ` beh T)"
cp_eq_cs: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and k' :: "nat" assumes "((\<sigma>, k), \<sigma>', k') \<in> cp" shows "cs\<^bsup>path \<sigma>\<^esup> k = cs\<^bsup>path \<sigma>'\<^esup> k'"
lcoset_join2: fixes H :: "'a set set"   and g :: "'a set" assumes "subgroup H (G Mod N)"   and "g \<in> H" shows "g <#\<^bsub>G Mod N\<^esub> H = H"
fst_eq_Domain: fixes R :: "('c \<times> 'd) set" shows "fst ` R = Domain R"
B: shows "\<forall>x\<in>X. (v, D x) \<in> I"
is_ft_dghmI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<And>a b. \<lbrakk>a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>; b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>\<rbrakk> \<Longrightarrow> v11 (\<FF>\<lparr>ArrMap\<rparr> \<restriction>\<^sup>l\<^sub>\<circ> Hom \<AA> a b)" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>"
routing_ipassmt_wi_def: fixes tbl :: "'a routing_rule list" shows "routing_ipassmt_wi tbl \<equiv> reduce_range_destination (routing_port_ranges tbl wordinterval_UNIV)"
permutation_swap_id: fixes a :: "'a"   and b :: "'a" shows "permutation (Fun.swap a b id)"
rbl_add_take2: fixes bla :: "bool list"   and blb :: "bool list" assumes "length bla \<le> length blb" shows "rbl_add bla (take (length bla) blb) = rbl_add bla blb"
valuate_def: shows "valuate \<equiv> map_fun Rep_linear_poly id fun_valuate"
prj_simps_4: shows "C.arr trg_trg.prj\<^sub>0"
imp_elim: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "imp \<phi> \<psi> \<pi>l"   and "\<phi> \<pi>l" obtains "\<psi> \<pi>l"
phiDefs_def: fixes g :: "'g"   and n :: "'node" shows "phiDefs g n \<equiv> {v. (n, v) \<in> dom (Mapping.lookup (phis g))}"
encode_all_operator_preconditions_def: fixes \<Pi> :: "'a strips_problem"   and ops :: "'a strips_operator list"   and t :: "nat" shows "encode_all_operator_preconditions \<Pi> ops t \<equiv> let l = List.product [0..<t] ops in foldr (\<^bold>\<and>) (map (\<lambda>(t, op). encode_operator_precondition \<Pi> t op) l) (\<^bold>\<not> \<bottom>)"
mono_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and gt' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt' y x"   and "lexext gt ys xs" shows "lexext gt' ys xs"