hcju/isabelleps huggingface dataset:
choice_n: fixes P :: "'b \<Rightarrow> bool"   and x0 :: "'b"   and r :: "('b \<times> 'b) set"   and n :: "nat" assumes "P x0"   and "\<And>x. P x \<Longrightarrow> \<exists>y. P y \<and> (x, y) \<in> r" shows "P (choice P r n)"
simps_6: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and \<phi> :: "Formula.formula" shows "Formula.sat \<sigma> V v i (formula.Neg \<phi>) = (\<not> Formula.sat \<sigma> V v i \<phi>)"
integral_restrict_Int: fixes T :: "'d set"   and S :: "'d set"   and f :: "'d \<Rightarrow> 'c" shows "integral T (\<lambda>x. if x \<in> S then f x else (0::'c)) = integral (S \<inter> T) f"
twiddleHeap_isPbij: fixes \<beta> :: "(Location \<times> Location) set"   and h :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and hh :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" assumes "twiddleHeap \<beta> h hh" shows "\<beta> \<in> Pbij"
trie_keys_update: fixes k :: "bool list"   and v :: "'b"   and t :: "'b trie_raw" shows "trie_keys (trie_update k v t) = insert k (trie_keys t)"
fromArr_def: fixes x :: "'arr" shows "fromArr x \<equiv> if x \<in> Arr then MkArr False True [x] else null"
WT: fixes q :: "'b"   and s :: "'a" assumes "q \<in> outs_\<I> \<I>"   and "I s" shows "\<I>' \<turnstile>g callee s q \<surd>"
simps_1: fixes x :: "'b" shows "pp_first (Node x) = x"
initmissing_def: fixes \<sigma> :: "(nat \<Rightarrow> state option) \<times> 'a" shows "initmissing \<sigma> = (\<lambda>i. case fst \<sigma> i of None \<Rightarrow> toy_openproc.someinit i | Some s \<Rightarrow> s, snd \<sigma>)"
nonzero_H_rcoset_reps_H: shows "set nonzero_H_rcoset_reps \<subseteq> H"
WT_resource_of_oracle: fixes s :: "'s_rest" assumes "IR1 s" shows "\<I>RA \<oplus>\<^sub>\<I> \<I>RU \<turnstile>res RES (callee_of_rest rest1) s \<surd>"
heap_is_wellformed_children_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "child \<in> set children" shows "child |\<in>| node_ptr_kinds h"
A_2: fixes x :: "'b"   and i :: "nat" shows "open (A x i)"
set_insert: fixes x :: "'c"   and A :: "'c set" assumes "x \<in> A" obtains B :: "'c set" where "A = insert x B" and "x \<notin> B"
Rep_formula_inject: fixes x :: "('e, 'f, 'g) formula"   and y :: "('e, 'f, 'g) formula" shows "(Rep_formula x = Rep_formula y) = (x = y)"
cI: shows "\<Union> (S ` I) \<subseteq> carrier G"
distinct_31: fixes x41 :: "('a, 'b, 'c) formula"   and x42 :: "('a, 'b, 'c) formula"   and x51 :: "'c"   and x52 :: "('a, 'b, 'c) formula" shows "(x41 && x42) \<noteq> Exists x51 x52"
ord_eq_le_trans: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a = b"   and "b \<le> c" shows "a \<le> c"
intersect_segment_xline_fst_snd: fixes prec :: "nat"   and seg :: "(real \<times> real) \<times> real \<times> real"   and x :: "real"   and m :: "real"   and M :: "real" assumes "intersect_segment_xline prec seg x = Some (m, M)" shows "closed_segment (fst seg) (snd seg) \<inter> {p. fst p = x} \<subseteq> {(x, m)..(x, M)}"
herbrand_evalt'_1: fixes e :: "nat \<Rightarrow> 'a hterm"   and ht :: "'a hterm" shows "evalt e HApp (term_of_hterm ht) = ht"
card_Diff1_less: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "card (A - {x}) < card A"
lir_fps_numerator_def: fixes m :: "nat"   and cs :: "'b list"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "lir_fps_numerator m cs f g = (let N = length cs - 1 in Poly (map (\<lambda>k. (\<Sum>i\<le>min N k. cs ! (N - i) * f (k - i)) - g k) [0..<N + m]))"
Rep_unit_inject: fixes x :: "Linear_Aggregation_Algebras.unit"   and y :: "Linear_Aggregation_Algebras.unit" shows "(Linear_Aggregation_Algebras.unit.Rep_unit x = Linear_Aggregation_Algebras.unit.Rep_unit y) = (x = y)"
s_prod_iff: fixes a :: "'b"   and A :: "'b set"   and R :: "('b \<times> 'b set) set"   and S :: "('b \<times> 'b set) set" shows "((a, A) \<in> R \<cdot> S) = (\<exists>B. (a, B) \<in> R \<and> (\<exists>f. (\<forall>b\<in>B. (b, f b) \<in> S) \<and> A = \<Union> (f ` B)))"
term_variants_pred_inv_const: fixes P :: "'c \<Rightarrow> 'c list"   and c :: "'c"   and t :: "('c, 'd) Term.term" shows "term_variants_pred P (Fun c []) t = ((\<exists>g\<in>set (P c). t = Fun g []) \<or> t = Fun c [])"
a_def: shows "a \<in> H \<inter> J \<and> a \<noteq> \<one>"
callee_in_procs: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "\<exists>ins outs. (p, ins, outs) \<in> set procs"
elims: fixes xs :: "int list"   and x :: "nat"   and y :: "nat list" assumes "der_stack xs x = y" obtains i :: "nat" where "x = i" and "y = (if has_left xs i then the_left xs i # der_stack xs (the_left xs i) else [])"
le_simps_2: fixes m :: "nat"   and n :: "nat" shows "(m < Suc n) = (m \<le> n)"
intros_12: fixes fop :: "binop_f"   and c1 :: "f64"   and c2 :: "f64" assumes "app_binop_f fop c1 c2 = None" shows "\<lparr>[$C ConstFloat64 c1, $C ConstFloat64 c2, $Binop_f T_f64 fop]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
index_def: fixes xs :: "'c list" shows "index xs = (\<lambda>a. find_index (\<lambda>x. x = a) xs)"
sym_preproc_addnew_complete: fixes g :: "'t \<Rightarrow>\<^sub>0 'c"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and v :: "'t"   and vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "g \<in> set gs"   and "lt g adds\<^sub>t v" shows "monom_mult (1::'c) (pp_of_term v - lp g) g \<in> set (snd (sym_preproc_addnew gs vs fs v))"
eq_rel_eval_trm: fixes v :: "event_data option list"   and n :: "nat"   and t1 :: "Formula.trm"   and t2 :: "Formula.trm" assumes "v \<in> eq_rel n t1 t2"   and "is_simple_eq t1 t2"   and "\<forall>x\<in>fv_trm t1 \<union> fv_trm t2. x < n" shows "Formula.eval_trm (map the v) t1 = Formula.eval_trm (map the v) t2"
h_minus_aux_le_h': fixes j :: "nat"   and e :: "'v \<times> 'v" shows "h_minus_aux j e \<le> h' e"
sorted_sorted_wrt: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq = sorted_wrt less_eq"
path_join: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2) = (path g1 \<and> path g2)"
that: fixes e :: "ennreal" assumes "0 < e"   and "\<And>y. eNorm N (y - x') < e \<Longrightarrow> eNorm N (y - x) < a" shows "thesis"
irc_on_def: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" shows "irc_on A f = (\<forall>B\<subseteq>A. \<forall>a\<in>A. a \<notin> f (B \<union> {a}) \<longrightarrow> f (B \<union> {a}) = f B)"
append_eq_append_conv2: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list"   and ts :: "'d list" shows "(xs @ ys = zs @ ts) = (\<exists>us. xs = zs @ us \<and> us @ ys = ts \<or> xs @ us = zs \<and> ys = us @ ts)"
bell00_index_4: shows "|\<beta>\<^sub>0\<^sub>0\<rangle> $$ (3, 0) = complex_of_real (1 / sqrt 2)"
n_less_ten_unat: assumes "n < 10" shows "unat n < 10"
ge_left_comm: 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"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Ge Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Ge Bet Cong B A C D"
graph_homomorphism_def: fixes G\<^sub>1 :: "('a, 'b) labeled_graph"   and G\<^sub>2 :: "('a, 'c) labeled_graph"   and f :: "('b \<times> 'c) set" shows "graph_homomorphism G\<^sub>1 G\<^sub>2 f = (vertices G\<^sub>1 = Domain f \<and> graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> f `` vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2 \<and> univalent f \<and> edge_preserving f (edges G\<^sub>1) (edges G\<^sub>2))"
isOpair_def: fixes q :: "ZF" shows "isOpair q \<equiv> \<exists>x y. q = Opair x y"
dist1: shows "distinct (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))"
image_f_inv_f: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'c set" assumes "surj f" shows "f ` inv f ` A = A"
length_append: fixes xs :: "'f list"   and ys :: "'f list" shows "length (xs @ ys) = length xs + length ys"
critical_set_mono: fixes x :: "ordinal"   and y :: "ordinal"   and A :: "ordinal set" assumes "x \<le> y" shows "critical_set A y \<subseteq> critical_set A x"
bn_finite: fixes \<alpha> :: "'a" shows "finite (bn \<alpha>)"
hterm_of_fterm_fterm_of_hterm: fixes t :: "hterm" shows "hterm_of_fterm (fterm_of_hterm t) = t"
flip_fresh_fresh: fixes a :: "'f"   and x :: "'e"   and b :: "'f" assumes "atom a \<sharp> x"   and "atom b \<sharp> x" shows "(a \<leftrightarrow> b) \<bullet> x = x"
lideal_axioms: shows "lideal I B addB multB zeroB oneB"
config_rand_dist_perm: fixes init :: "'c list"   and A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and qs :: "'c list" assumes "distinct init" shows "\<forall>x\<in>set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). dist_perm (fst x) init"
simps_1: fixes uu :: "(real, 'b) vec" shows "height (Leaf uu) = 0"
INIT\<^sub>2: shows "INIT mem\<^sub>2"
TR_cont_TR1234: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "TR1 R M N P \<union> TR2 R M N P \<union> TR3 R M N P \<union> TR4 R M N P \<subseteq> TR\<^bsub>R\<^esub> M N P"
iT_defs_3: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] \<equiv> {n..n + d}"
prv_dsj_assoc_imp1: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (dsj \<phi> (dsj \<chi> \<psi>)) (dsj (dsj \<phi> \<chi>) \<psi>))"
less_length: fixes x :: "nat list"   and y :: "nat list" assumes "x <\<^sub>v y" shows "length x = length y"
cone_axioms: shows "D.cone (local.dom p0) (\<lambda>i. if i \<in> Collect J0.arr then \<pi>0 i \<cdot> p0 else if i \<in> Collect J1.arr then \<pi>1 i \<cdot> p1 else null)"
cod_simps_3: shows "\<rho>.cod.leg1 = \<rho>.leg1"
quorum_def: fixes Q :: "'a set" shows "quorum Q \<equiv> \<forall>p\<in>Q \<inter> W. \<exists>Sl\<in>slices p. Sl \<subseteq> Q"
ext_inject: fixes init :: "'c set"   and trans :: "('c \<times> 'd \<times> 'c) set"   and more :: "'e"   and init' :: "'c set"   and trans' :: "('c \<times> 'd \<times> 'c) set"   and more' :: "'e" shows "(\<lparr>init = init, trans = trans, \<dots> = more\<rparr> = \<lparr>init = init', trans = trans', \<dots> = more'\<rparr>) = (init = init' \<and> trans = trans' \<and> more = more')"
intro: fixes T :: "'a \<Rightarrow> 'a"   and G :: "'a set" assumes "T ` G \<subseteq> G" shows "GroupEnd_axioms G T"
while_elim: fixes e :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>While e c, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" shows "c' = Stmt.If e (c ;; While e c) Stop \<and> mds' = mds \<and> mem' = mem"
box_eq_empty_1: fixes a :: "'c"   and b :: "'c" shows "(box a b = {}) = (\<exists>i\<in>Basis. b \<bullet> i \<le> a \<bullet> i)"
Composeo_undef: fixes \<alpha> :: "game option"   and \<beta> :: "game option" shows "(Composeo \<alpha> \<beta> = undefg) = (\<alpha> = undefg \<or> \<beta> = undefg)"
Liminf_upto_llist_def: fixes Xs :: "'b set llist"   and k :: "enat" shows "Liminf_upto_llist Xs k = (\<Union>i\<in>{i. enat i < llength Xs \<and> enat i \<le> k}. \<Inter> (lnth Xs ` {j. i \<le> j \<and> enat j < llength Xs \<and> enat j \<le> k}))"
never_winning_is_losing_even: fixes p :: "'a list"   and \<sigma> :: "'a list \<Rightarrow> 'a" assumes "position p"   and "\<forall>n. \<not> winning_position_Even ((augment_list \<sigma> ^^ n) p)" shows "induced_play \<sigma> p \<notin> A"
Prefixes_subset_\<P>: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and p :: "(symbol \<times> character list) list"   and Doc :: "character list"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "p \<in> LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u" shows "Prefixes p \<subseteq> LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u"
decreasing_closed_nest: fixes S :: "nat \<Rightarrow> 'b set" assumes "\<And>n. closed (S n)"   and "\<And>n. S n \<noteq> {}"   and "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>S n. \<forall>y\<in>S n. dist x y < e" obtains a :: "'b" where "\<And>n. a \<in> S n"
real_vector_eq_affinity: fixes m :: "real"   and y :: "'b"   and x :: "'b"   and c :: "'b" assumes "m \<noteq> 0" shows "(y = m *\<^sub>R x + c) = (y /\<^sub>R m - c /\<^sub>R m = x)"
0: fixes x :: "'n" assumes "x \<in> S" shows "0 \<le> f x"
trace_callee_resource_of_oracle: fixes callee :: "'c \<Rightarrow> 'b \<Rightarrow> ('a \<times> 'c) spmf"   and s :: "'c"   and xs :: "('b \<times> 'a) list"   and x :: "'b" shows "trace_callee run_resource (return_spmf (RES callee s)) xs x = trace_callee callee (return_spmf s) xs x"
diff_zero: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and a :: "'f" assumes "class.cancel_comm_monoid_add plus minus zero" shows "minus a zero = a"
Abs_star_inject: fixes x :: "(nat \<Rightarrow> 'b) set"   and y :: "(nat \<Rightarrow> 'b) set" assumes "x \<in> star"   and "y \<in> star" shows "(Abs_star x = Abs_star y) = (x = y)"
get_action_def: fixes rule :: "'c rule" shows "get_action rule = (case rule of Rule x1 x2 \<Rightarrow> x2)"
shift_lt: fixes i :: "nat"   and j :: "nat"   and e :: "nat \<Rightarrow> 'a"   and T :: "'a" assumes "i < j" shows "(e\<langle>i:T\<rangle>) j = e (j - 1)"
select_convs_1: fixes carrier :: "'h set"   and more :: "'i" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
mem_val_alt_10_mem_equal_2: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 10 ((pa AND 68719476732) + 2) s1 = mem_val_alt 10 ((pa AND 68719476732) + 2) s2"
subdprodr_def: fixes a :: "nat list"   and b :: "nat list"   and y :: "nat list" assumes "hlde a b" shows "hlde.subdprodr a b y = (\<forall>l\<le>length b. take l b \<bullet> take l y \<le> a \<bullet> map (hlde_ops.max_x a b (take l y)) [0..<length a])"
extendNonEmpty: fixes \<Gamma> :: "('a, 'b) form multiset"   and \<Delta> :: "('a, 'b) form multiset"   and C :: "('a, 'b) sequent" assumes "\<not> (\<Gamma> = \<Empt> \<and> \<Delta> = \<Empt>)" shows "extend ( \<Gamma> \<Rightarrow>* \<Delta>) C \<noteq> C"
if_doesnt_read: fixes x :: "'Var"   and e :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and annos :: "'Var ModeUpd list" assumes "x \<notin> bexp_vars e"   and "\<C>_vars x \<inter> bexp_vars e = {}" shows "doesnt_read_or_modify (Stmt.If e c\<^sub>1 c\<^sub>2 \<otimes> annos) x"
snoc: fixes ys :: "'a list" shows "match (name $$ xs) (name $$ ys) = matchs xs ys"
dropWhile_eq_self_iff: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(dropWhile P xs = xs) = (xs = [] \<or> \<not> P (hd xs))"
neutral: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "g = \<lambda>x. 0::'b on A" shows "sum g A = (0::'b)"
Min_eqI: fixes A :: "'a Word.word set"   and x :: "'a Word.word" assumes "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> x \<le>s y"   and "x \<in> A" shows "signed.Min A = x"
Card_order_trans: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "Card_order r"   and "x \<noteq> y"   and "(x, y) \<in> r"   and "y \<noteq> z"   and "(y, z) \<in> r" shows "x \<noteq> z \<and> (x, z) \<in> r"
I_hd_coeff1_mult_a: fixes m :: "int"   and a :: "atom"   and x :: "int"   and xs :: "int list" assumes "0 < m"   and "hd_coeff a dvd m \<or> hd_coeff a = 0" shows "I\<^sub>Z (hd_coeff1 m a) (m * x # xs) = I\<^sub>Z a (x # xs)"
list_all2_Nil: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and ys :: "'f list" shows "list_all2 P [] ys = (ys = [])"
trg_simps_3: fixes F :: "('a E.term, 'a) arr" assumes "arr F" shows "Map (trg F) = EVAL (E.Trg (Dom F))"
l_new_shadow_root_get_tag_name_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_shadow_root_get_tag_name type_wf get_tag_name get_tag_name_locs \<equiv> l_get_tag_name type_wf get_tag_name get_tag_name_locs \<and> l_new_shadow_root_get_tag_name_axioms get_tag_name_locs"
divide_poly_field: fixes f :: "'b poly"   and g :: "'b poly" shows "f div g = fst (pseudo_divmod (smult (((1::'b) / lead_coeff g) ^ (Suc (degree f) - degree g)) f) g)"
x1_and_roots: shows "sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} \<noteq> [] \<longrightarrow> x1 < sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! 0 \<or> sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! (length (sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}) - 1) < x1 \<or> (\<exists>n<length (sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}) - 1. sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! n < x1 \<and> x1 < sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! (n + 1))"
arr_cod: fixes f :: "'b" assumes "D.arr f" shows "D.arr (D.cod f)"
select_convs_4: fixes NTDom :: "('h, 'i, 'j, 'k, 'l, 'm) Functor"   and NTCod :: "('h, 'i, 'j, 'k, 'l, 'm) Functor"   and NatTransMap :: "'h \<Rightarrow> 'k"   and more :: "'g" shows "NatTrans.more \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = more"
simps_6: fixes cl :: "('d, 'e, 'f) cmd list" shows "proper (Par cl) = (cl \<noteq> [] \<and> Ball (set cl) proper)"
rreq_def: shows "rreq \<equiv> \<lambda>(hops, rreqid, dip, dsn, dsk, oip, osn, sip, handled). Rreq hops rreqid dip dsn dsk oip osn sip handled"
seqSubstSimps_2: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and M :: "'d"   and N :: "'d"   and P :: "('d, 'e, 'f) psi"   and \<sigma> :: "(name list \<times> 'd list) list" assumes "substPsi substTerm substAssert substCond" shows "substPsi.seqSubs substTerm substAssert substCond (M\<langle>N\<rangle>.P) \<sigma> = substType.seqSubst substTerm M \<sigma>\<langle>substType.seqSubst substTerm N \<sigma>\<rangle>.substPsi.seqSubs substTerm substAssert substCond P \<sigma>"
wt_elim_cases_7: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and T' :: "ref_ty"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (e InstOf T')\<Colon>T" obtains T :: "ref_ty" where "Ta__ = Inl (PrimT Boolean)" and "E,dt\<Turnstile>e\<Colon>-RefT T" and "is_acc_type (prg E) (pkg E) (RefT T')" and "prg E\<turnstile>RefT T\<preceq>? RefT T'"
simps_1: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and p :: "'d"   and c' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel" shows "distributed_system.next channel trans send recv c (Snapshot p) c' = (distributed_system.can_occur channel trans send recv (Snapshot p) c \<and> ps c' p = Some (states c p) \<and> (\<forall>p'. states c p' = states c' p') \<and> (\<forall>p'. p' \<noteq> p \<longrightarrow> ps c' p' = ps c p') \<and> (\<forall>i. (\<exists>q. channel i = Some (p, q)) \<longrightarrow> msgs c' i = msgs c i @ [Marker]) \<and> (\<forall>i. (\<exists>q. channel i = Some (q, p)) \<longrightarrow> cs c' i = (fst (cs c i), Recording)) \<and> (\<forall>i. (\<nexists>q. channel i = Some (p, q)) \<longrightarrow> msgs c' i = msgs c i) \<and> (\<forall>i. (\<nexists>q. channel i = Some (q, p)) \<longrightarrow> cs c' i = cs c i))"
wellorder_Least_lemma_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and k :: "'b" assumes "class.wellorder less_eq less"   and "P k" shows "less_eq (ord.Least less_eq P) k"
contraction_with_completion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<div> \<phi> = K \<div>\<^bsub>* \<gamma>\<^sub>A\<^sub>G\<^sub>M\<^esub> \<phi>"
inject_9: fixes x1 :: "pi"   and y1 :: "pi" shows "(!x1 = !y1) = (x1 = y1)"
hausdorff_distance_subset: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "A \<noteq> {}"   and "bounded B" shows "hausdorff_distance A B = (SUP x\<in>B. infdist x A)"
square_eq_iff: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "domain R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<otimes>\<^bsub>R\<^esub> x = y \<otimes>\<^bsub>R\<^esub> y) = (x = y \<or> x = \<ominus>\<^bsub>R\<^esub> y)"
is_singleton_altdef: fixes A :: "'e set" shows "is_singleton A = (card A = 1)"
map_2: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and x2 :: "'d error_result" shows "map_result f1 f2 (Rerr x2) = Rerr (map_error_result f2 x2)"
le_zero_iff_zero_pprt: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq a zero = (lattice_ab_group_add.pprt zero sup a = zero)"
typing_elims_2: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and u :: "dB"   and T :: "LambdaType.type" assumes "e \<turnstile> t \<degree> u : T" obtains T :: "LambdaType.type" where "e \<turnstile> t : T \<Rightarrow> Ta__" and "e \<turnstile> u : T"
finiteI_independent: fixes B :: "('b, 'c) vec set" assumes "vec.independent B" shows "finite B"
hyp: shows "cl_ecl C \<noteq> subst_cl C' \<sigma>"
distinct_3: fixes x1 :: "real \<times> real \<times> real"   and x3 :: "real \<times> real \<times> real" shows "LessUni x1 \<noteq> LeqUni x3"
Exit_no_sourcenode: fixes as :: "'edge list"   and n :: "'node"   and n' :: "'node" assumes "(_Exit_) \<in> set (sourcenodes as)"   and "n -as\<rightarrow>* n'" shows "False"