hcju/isabelleps huggingface dataset:
eval_deterministic: fixes s1 :: "'state"   and s2 :: "'state"   and s3 :: "'state" assumes "\<And>x y z. \<lbrakk>x \<rightarrow> y; x \<rightarrow> z\<rbrakk> \<Longrightarrow> y = z"   and "s1 \<rightarrow>\<^sup>* s2"   and "s1 \<rightarrow>\<^sup>* s3"   and "finished (\<rightarrow>) s2"   and "finished (\<rightarrow>) s3" shows "s2 = s3"
algorithmic_symmetry_2: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> s \<leftrightarrow> t : T" shows "\<Gamma> \<turnstile> t \<leftrightarrow> s : T"
zcount_empty: fixes a :: "'c" shows "zcount {#}\<^sub>z a = 0"
tranclp_trans_induct: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ x y"   and "\<And>x y. r x y \<Longrightarrow> P x y"   and "\<And>x y z. \<lbrakk>r\<^sup>+\<^sup>+ x y; P x y; r\<^sup>+\<^sup>+ y z; P y z\<rbrakk> \<Longrightarrow> P x z" shows "P x y"
fresherI2: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and r' :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "nsqn\<^sub>r r = nsqn\<^sub>r r'"   and "\<pi>\<^sub>5 r' \<le> \<pi>\<^sub>5 r" shows "r \<sqsubseteq> r'"
append: fixes xs :: "('c \<Rightarrow> ('d, 'c) Term.term) list"   and ys :: "('c \<Rightarrow> ('d, 'c) Term.term) list" shows "subst_monoid_mult.prod_list (xs @ ys) = subst_monoid_mult.prod_list xs \<circ>\<^sub>s subst_monoid_mult.prod_list ys"
cf: shows "cod\<^bsub>CatDom F\<^esub> f = Y"
that: fixes i :: "nat"   and s :: "'a"   and R :: "nat set"   and s' :: "'a"   and R' :: "nat set" assumes "ns = NodeS i s R"   and "ns' = NodeS i s' R'" shows "thesis"
nonzero_neg_divide_eq_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "(uminus (divide a b) = c) = (uminus a = times c b)"
locally_mono: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and Q :: "'b set \<Rightarrow> bool" assumes "locally P S"   and "\<And>T. P T \<Longrightarrow> Q T" shows "locally Q S"
del_list_sorted_app: fixes xs :: "'c list"   and x :: "'c"   and ys :: "'c list" assumes "Sorted_Less.sorted (xs @ [x])" shows "List_Ins_Del.del_list x (xs @ ys) = xs @ List_Ins_Del.del_list x ys"
add: fixes b1 :: "'b"   and b2 :: "'b" shows "g (b1 + b2) = g b1 + g b2"
not_less: fixes x :: "'d Word.word"   and y :: "'d Word.word" shows "(\<not> x <s y) = (y \<le>s x)"
op_subset_def: fixes X :: "'c set"   and ivl :: "'c set" shows "op_subset X ivl = ivl_rep ivl \<bind> (\<lambda>(i', s'). op_ivl_rep_of_set X \<bind> (\<lambda>(i, s). RETURN ((i' \<le> i ::: bool_rel) \<and> (s \<le> s' ::: bool_rel))))"
intro: fixes L :: "(nat \<times> int \<times> nat) list" assumes "distinct (map \<alpha> L)"   and "\<forall>u w v. (u, w, v) \<in> set L \<longrightarrow> u \<noteq> v" shows "fromlist L"
min_set_def: fixes X :: "'a set" shows "min_set X = {y \<in> X. \<forall>x\<in>X. x \<le> y \<longrightarrow> x = y}"
tllist_of_llist_eq_lappendt_conv: fixes a :: "'b"   and xs :: "'a llist"   and ys :: "'a llist"   and zs :: "('a, 'b) tllist" shows "(tllist_of_llist a xs = lappendt ys zs) = (\<exists>xs' a'. xs = lappend ys xs' \<and> zs = tllist_of_llist a' xs' \<and> (lfinite ys \<longrightarrow> a = a'))"
prefix_singleton_TT: fixes x :: "'a"   and ys :: "[:'a:]" shows "(prefix\<cdot>[:x:]\<cdot>ys = TT) = (x \<noteq> \<bottom> \<and> (\<exists>zs. zs \<noteq> \<bottom> \<and> ys = x :# zs))"
divide_minus1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide x (uminus one) = uminus x"
get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) document_ptr"   and getter :: "('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> 'w" assumes "l_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
Oops_parts_knows_Spy: fixes A :: "agent"   and Timestamp :: "msg"   and B :: "msg"   and K :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Says Server A (Crypt (shrK A) \<lbrace>Timestamp, B, K, X\<rbrace>) \<in> set evs" shows "K \<in> parts (knows Spy evs)"
commute: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "ac_operator f" shows "f a b = f b a"
is_dijkstra_step_def: fixes G :: "graph"   and S :: "state"   and S' :: "state" shows "is_dijkstra_step G S S' = (\<exists>m. is_heap_min m (heap S) \<and> S' = dijkstra_step G m S)"
ab_group_add_axioms: shows "class.ab_group_add (+) (0::'a) (-) uminus"
STAR_subset_closed: fixes x :: "'a star"   and A :: "'a set"   and B :: "'a set" assumes "x \<in> *s* A"   and "A \<subseteq> B" shows "x \<in> *s* B"
LNil_eq_lappend_iff: fixes xs :: "'b llist"   and ys :: "'b llist" shows "(<> = xs $ ys) = (xs = <> \<and> ys = <>)"
snoc_2: shows "pgallery (C0 # (Ds @ [D]) @ [C])"
assms_1: fixes a :: "'a" assumes "a \<in> A" shows "\<exists>b\<in>B. a .= b"
foldl_coin: shows "\<not> foldl (\<noteq>) False (map (\<lambda>c. coin dc c \<noteq> coin dc (c + 1)) [0..<n])"
semiring_normalization_rules_25: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d"   and c :: "'d"   and d :: "'d" assumes "class.comm_semiring_1 times one plus zero" shows "plus a (plus c d) = plus (plus a c) d"
simps_2: fixes S :: "'fun set"   and f :: "'fun"   and T :: "('fun, 'atom) Term.term list" shows "fresh_pgwt S (Fun f T) = Fun f (map (fresh_pgwt S) T)"
isFault_simps_3: fixes f :: "'e" shows "isFault (Fault f) = True"
rHom_one: fixes A :: "('d, 'e) Ring_scheme"   and R :: "('f, 'g) Ring_scheme"   and f :: "'d \<Rightarrow> 'f" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "f 1\<^sub>r\<^bsub>A\<^esub> = 1\<^sub>r\<^bsub>R\<^esub>"
simps_2: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "emb_step_at [] dir.Right (App t s) = s"
i: shows "(a, i) \<in> UNIV \<times> Basis"
psubstAbs_cong: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "goodAbs A"   and "goodEnv rho"   and "goodEnv rho'"   and "\<And>ys y. freshAbs ys y A \<or> rho ys y = rho' ys y" shows "A $[rho] = A $[rho']"
rep_eq: fixes x :: "'a upred" shows "`x` = All \<lbrakk>x\<rbrakk>\<^sub>e"
is_expansion_add: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a"   and G :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "is_expansion F basis"   and "is_expansion G basis" shows "is_expansion (plus F G) basis"
length_snoc: fixes xs :: "'b list"   and x :: "'b" shows "length (xs @ [x]) = Suc (length xs)"
label_stab_map_bij_betw_W_chambers: fixes \<phi> :: "'a \<Rightarrow> 'a"   and w0 :: "'a permutation" assumes "label_wrt C0 \<phi>"   and "fixespointwise \<phi> C0"   and "w0 \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>" shows "bij_betw (\<lambda>v. {w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. w \<rightarrow> \<phi> v = v}) (w0 `\<rightarrow> C0) (dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})) {w0})"
psimps: fixes k :: "nat"   and xs :: "'b list" assumes "fast_select_dom (k, xs)" shows "fast_select k xs = (if length xs \<le> 20 then sort xs ! k else let x = fast_select (((length xs + 4) div 5 - 1) div 2) (map median_le_5 (chop 5 xs)); (ls, es, gs) = threeway_partition x xs in if k < length ls then fast_select k ls else if k < length ls + length es then x else fast_select (k - length ls - length es) gs)"
transfer_mem: fixes x :: "'b star"   and X :: "nat \<Rightarrow> 'b"   and a :: "'b star set"   and A :: "nat \<Rightarrow> 'b set" assumes "x \<equiv> star_n X"   and "a \<equiv> Iset (star_n A)" shows "x \<in> a \<equiv> \<forall>\<^sub>F n in \<U>. X n \<in> A n"
A_def: shows "A \<equiv> {x \<in> vars\<^sub>s\<^sub>t S'. \<exists>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S' \<and> x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<and> x \<notin> set X}"
continuous_closed_graph_gen: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d set" assumes "continuous_on S f"   and "f ` S \<subseteq> T" shows "closedin (top_of_set (S \<times> T)) ((\<lambda>x. (x, f x)) ` S)"
distinct_11: fixes x21 :: "'b aexp"   and x22 :: "'b aexp"   and x41 :: "'b"   and x42 :: "value list" shows "Eq x21 x22 \<noteq> In x41 x42"
negS: shows "- adjuster n w us \<in> local.span (insert (adjuster n w us + w) (set us))"
pullback_sums_map_middle3: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and F :: "'b \<Rightarrow> 'c"   and xs :: "'b list"   and ds :: "'c list"   and d :: "'c"   and e :: "'c"   and f :: "'c"   and fs :: "'c list" assumes "class.monoid_add plus zero"   and "map F (monoid_add.sums plus zero xs) = ds @ [d, e, f] @ fs" shows "\<exists>as a b bs. xs = as @ [a, b] @ bs \<and> d = F (monoid_add.sum_list plus zero as) \<and> e = F (monoid_add.sum_list plus zero (as @ [a])) \<and> f = F (monoid_add.sum_list plus zero (as @ [a, b]))"
min_prio_def: fixes es :: "edge list"   and start :: "integer" shows "min_prio es start = Min (prio ` set es \<union> {start})"
intros_11: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and v :: "v"   and s' :: "s"   and vs :: "v list" assumes "supdate_glob s i j v = s'" shows "\<lparr>s;vs;[$C v, $Set_global j]\<rparr> \<leadsto>_ i \<lparr>s';vs;[]\<rparr>"
oth_class_taut_10_a: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<phi> \<^bold>& \<psi>) in v]"
bound_compare_defs_12: fixes c :: "'a"   and b :: "'a option" shows "c <\<^sub>u\<^sub>b b = \<lhd>\<^sub>u\<^sub>b (<) c b"
Cons_3: shows "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"
scalar_left_mono: fixes u :: "'b vec"   and n :: "nat"   and v :: "'b vec"   and w :: "'b vec" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n"   and "w \<in> carrier_vec n"   and "\<And>i. i < n \<Longrightarrow> v $ i \<le> u $ i"   and "\<And>i. i < n \<Longrightarrow> (0::'b) \<le> w $ i" shows "v \<bullet> w \<le> u \<bullet> w"
resolution_and_max_literal: fixes atom_ordering :: "('a \<times> 'a) set"   and R :: "'a Literal set"   and P1 :: "'a Literal set"   and P2 :: "'a Literal set"   and A :: "'a"   and M :: "'a Literal" assumes "propositional_atoms atom_ordering"   and "R = propositional_atoms.resolvent_upon P1 P2 A"   and "propositional_atoms.strictly_maximal_literal atom_ordering P1 (Pos A)"   and "propositional_atoms.strictly_maximal_literal atom_ordering P2 (Neg A)"   and "propositional_atoms.strictly_maximal_literal atom_ordering R M" shows "(atom M, A) \<in> atom_ordering"
runit_def: fixes f :: "'a" shows "PB.runit f \<equiv> right_hom_with_unit.runit (\<cdot>) (\<star>) \<a> (\<i>' (src f)) (src f) f"
bliftM2_strict2: fixes f :: "'c \<rightarrow> 'b \<rightarrow> 'a"   and x :: "'c Box" shows "bliftM2 f\<cdot>x\<cdot>\<bottom> = \<bottom>"
ev_introL: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'"   and "Shallow.ev \<phi> (map stl \<pi>l)" shows "Shallow.ev \<phi> \<pi>l"
separation_by_component_closed_pointwise: fixes S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "closed S"   and "\<not> connected_component (- S) a b" obtains C :: "'a set" where "C \<in> components S" and "\<not> connected_component (- C) a b"
A3: shows "\<forall>TP SQ. TP \<in>T P \<and> SQ \<in>S Q \<longrightarrow> (TP, \<lbrakk>SQ\<rbrakk>) \<in> TRel"
isolated_verts_impl_def: fixes G :: "'b list \<times> ('b \<times> 'b) list" shows "isolated_verts_impl G = filter (\<lambda>v. \<not> (\<exists>e\<in>set (snd G). fst e = v)) (fst G)"
CR: shows "(ci, c) \<in> R \<rightarrow> bool_rel"
sign_num_def: fixes x :: "real" shows "sign_num x \<equiv> if x = 0 then 0 else if 0 < x then 1 else - 1"
assms_1: shows "\<I>' \<turnstile>g gpv \<surd>"
substs_preserve_well_constrainedness: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "basic_superposition.well_constrained trm_ord C" shows "basic_superposition.well_constrained trm_ord (subst_ecl C \<sigma>)"
wffp_isState_doo: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list" assumes "GM_sec_model.wffp do sl"   and "list_all isState sl" shows "GM_sec_model.doo do (local.getGMState (hd sl)) (map getGMUserCom (tl sl)) = local.getGMState (last sl)"
mset_reduce_subset: fixes Ds :: "'a literal list list"   and C :: "'a literal list"   and E :: "'a literal list" shows "mset (reduce Ds C E) \<subseteq># mset E"
bigo_sum5: fixes f :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b"   and l :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and k :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and A :: "'c \<Rightarrow> 'd set" assumes "f \<in> O(h)"   and "\<forall>x y. (0::'b) \<le> l x y"   and "\<forall>x. (0::'b) \<le> h x" shows "(\<lambda>x. \<Sum>y\<in>A x. l x y * f (k x y)) \<in> O(\<lambda>x. \<Sum>y\<in>A x. l x y * h (k x y))"
wp_loopI: fixes P :: "'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a nd_fun" assumes "\<lceil>P\<rceil> \<le> \<lceil>I\<rceil>"   and "\<lceil>I\<rceil> \<le> \<lceil>Q\<rceil>"   and "\<lceil>I\<rceil> \<le> wp R \<lceil>I\<rceil>" shows "\<lceil>P\<rceil> \<le> wp (LOOP R INV I ) \<lceil>Q\<rceil>"
step_cpoke: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and e :: "'event" assumes "S p q"   and "e \<in> E" shows "weight_spmf (p \<bind> (\<lambda>s. cpoke core1 s e)) = weight_spmf (q \<bind> (\<lambda>s. cpoke core2 s e))"
lm088: fixes x :: "'c"   and R :: "('c \<times> 'd) set"   and y :: "'d" assumes "x \<notin> Domain R" shows "R +< {(x, y)} = R \<union> {(x, y)}"
abc_sym: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c" shows "betw c b a"
hyps_2: shows "h' [f]\<rightarrow>* h''"
fundamental_theorem_of_calculus_partial_vector: fixes a :: "real"   and b :: "real"   and i :: "'a"   and j :: "'b"   and c :: "'a"   and D :: "'a set"   and F :: "'a \<Rightarrow> 'b"   and F_j_i :: "'a \<Rightarrow> real" assumes "a \<le> b"   and "i \<in> Basis"   and "j \<in> Basis"   and "c \<bullet> i = 0"   and "\<forall>p\<in>D. has_partial_vector_derivative (\<lambda>x. F x \<bullet> j) i (F_j_i p) p"   and "{x *\<^sub>R i + c |x. a \<le> x \<and> x \<le> b} \<subseteq> D" shows "((\<lambda>x. F_j_i (x *\<^sub>R i + c)) has_integral F (b *\<^sub>R i + c) \<bullet> j - F (a *\<^sub>R i + c) \<bullet> j) (cbox a b)"
favorites_nonempty: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a" assumes "pref_profile_wf agents alts R"   and "i \<in> agents" shows "favorites R i \<noteq> {}"
nerr: shows "lr_of_tran rt fw ifs = Inr oft"
det_transpose: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "det A\<^sup>T = det A"
distinct_37: fixes x51 :: "('c, 'd) Term.term"   and x52 :: "('c, 'd) Term.term"   and x61 :: "poscheckvariant"   and x62 :: "('c, 'd) Term.term"   and x63 :: "('c, 'd) Term.term" shows "delete\<langle>x51,x52\<rangle> \<noteq> \<langle>x61: x62 \<in> x63\<rangle>"
pochhammer_Suc_prod_rev: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a (Suc n) = comm_monoid_mult.prod times one (\<lambda>i. plus a (semiring_1.of_nat one plus zero (n - i))) {0..n}"
g_def: fixes i :: "nat" shows "g i = f (i + N)"
feasible_boundedD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "real"   and P :: "'c \<Rightarrow> real" assumes "feasible t"   and "bounded_by b P"   and "nneg P" shows "bounded_by b (t P)"
cut_cut_less: fixes i :: "'b set"   and a :: "'b"   and b :: "'b" shows "i \<down>< a \<down>< b = i \<down>< min a b"
ipurge_ref_subset_union: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'b list"   and X :: "'b set"   and Y :: "'b set" shows "ipurge_ref I D u xs (X \<union> Y) \<subseteq> X \<union> ipurge_ref I D u xs Y"
satisfies_nFEx: fixes \<phi> :: "('a, 'k) aformula"   and \<AA> :: "'i"   and b :: "'k" assumes "lformula \<phi>" shows "(\<AA> \<Turnstile> nFEx b \<phi>) = (\<AA> \<Turnstile> FEx b \<phi>)"
guard_subset_eq_outputs_updates_subsumption: fixes t1 :: "transition"   and t2 :: "transition"   and c :: "nat \<Rightarrow>f value option" assumes "Label t1 = Label t2"   and "Arity t1 = Arity t2"   and "Outputs t1 = Outputs t2"   and "Updates t1 = Updates t2"   and "set (Guards t2) \<subseteq> set (Guards t1)" shows "subsumes t2 c t1"
brokenI: fixes A :: "agent"   and K :: "msg set"   and B :: "agent" assumes "priK A \<in> K \<or> priK B \<in> K \<or> shrK A B \<in> K \<or> shrK B A \<in> K" shows "(A, B) \<in> broken K"
unfold: shows "supto = (\<Lambda> i j. If le\<cdot>i\<cdot>j then i :# supto\<cdot>(i + 1)\<cdot>j else [::])"
relcomp_UNION_distrib2: fixes r :: "'d \<Rightarrow> ('a \<times> 'c) set"   and I :: "'d set"   and s :: "('c \<times> 'b) set" shows "\<Union> (r ` I) O s = (\<Union>i\<in>I. r i O s)"
one_hmultiset_def: shows "1 = \<omega>\<^sub>h^ 0"
contractible_space_subtopology_euclideanreal: fixes S :: "real set" shows "contractible_space (top_of_set S) = is_interval S"
ID1: fixes x :: "'r" shows "ID\<cdot>x = x"
FAssRedsThrow2: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>2 :: "char list exp"   and e :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2\<rangle>"
le_inf_iff: fixes y :: "'c"   and z :: "'c"   and x :: "'c" shows "(y \<squnion> z \<le> x) = (y \<le> x \<and> z \<le> x)"
dvd_normalize_div: fixes b :: "'a"   and a :: "'a" assumes "b dvd a" shows "normalize (a div b) = normalize a div normalize b"
mset_le_single_cases: fixes M :: "'a multiset"   and a :: "'a" assumes "M \<subseteq># {#a#}" obtains "M = {#}"   | "M = {#a#}"
induct: fixes \<phi>s :: "Formula.formula list"   and x1 :: "mregex \<Rightarrow> mregex"   and x2 :: "Formula.formula Regex.regex \<Rightarrow> Formula.formula Regex.regex"   and P :: "(mregex \<Rightarrow> mregex) \<Rightarrow> (Formula.formula Regex.regex \<Rightarrow> Formula.formula Regex.regex) \<Rightarrow> bool" assumes "ok_lctxt \<phi>s x1 x2"   and "P id id"   and "\<And>\<kappa> \<kappa>' mr. \<lbrakk>ok_lctxt \<phi>s \<kappa> \<kappa>'; P \<kappa> \<kappa>'\<rbrakk> \<Longrightarrow> P (\<lambda>t. \<kappa> (MTimes t mr)) (\<lambda>t. \<kappa>' (regex.Times t (from_mregex mr \<phi>s)))" shows "P x1 x2"
assms_2: shows "b \<in>\<^sub>\<circ> dg_prod (I -\<^sub>\<circ> J) \<AA>\<lparr>Arr\<rparr>"
starset_starset_n_eq: fixes X :: "'a set" shows "*s* X = *sn* (\<lambda>n. X)"
seq'_cons: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule"   and s :: "state"   and t :: "state"   and rs :: "'c rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[r], s\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, t\<rangle> \<Rightarrow> t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r # rs, s\<rangle> \<Rightarrow> t'"
spmf_of_gpv_def: fixes gpv :: "('d, 'e, 'f) gpv" shows "spmf_of_gpv gpv = (THE p. gpv = lift_spmf p)"
P1_advantages_eq: fixes x :: "'a \<times> 'a"   and y :: "bool"   and z :: "'e"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'e \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'f \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'g spmf)"   and D :: "'g \<times> 'a \<Rightarrow> bool spmf" shows "mal_def.adv_P1 x y z (P1_S1, P1_S2) \<A> D = P1_adv_real_ideal_model D x y \<A> z"