hcju/isabelleps huggingface dataset:
legal_executionE: fixes is_justification :: "'a prog \<Rightarrow> ('b \<times> ('c, 'b) obs_event action) llist \<times> (nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> ('c, 'b) justifying_execution) \<Rightarrow> bool"   and P :: "'a prog"   and \<E> :: "('b \<times> ('c, 'b) obs_event action) llist set"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" assumes "gen_legal_execution is_justification P \<E> (E, ws)" obtains J :: "nat \<Rightarrow> ('c, 'b) justifying_execution" where "E \<in> \<E>" and "P \<turnstile> (E, ws) \<surd>" and "is_justification P (E, ws) J" and "range (justifying_exec \<circ> J) \<subseteq> \<E>"
cInf_eq_minimum: fixes z :: "'g multiset"   and X :: "'g multiset set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> z \<subseteq># x" shows "Inf X = z"
Festino: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A E B"   and "A I C" shows "B Z C"
elims: fixes x :: "'a mat"   and y :: "'a mat \<times> 'a mat \<times> 'a mat" assumes "Smith_2xn x = y" obtains A :: "'a mat" where "x = A" and "y = (if dim_col A = 0 then (1\<^sub>m (dim_row A), A, 1\<^sub>m 0) else if dim_col A = 1 then let (P, S) = Smith_nx1 A in (P, S, 1\<^sub>m (dim_col A)) else if dim_col A = 2 then Smith_2x2 A else let A1 = mat_of_cols (dim_row A) [col A 0]; A2 = mat_of_cols (dim_row A) (map (col A) [1..<dim_col A]); (P1, D1, Q1) = Smith_2xn A2; C = P1 * A1 @\<^sub>c P1 * A2 * Q1; D = mat_of_cols (dim_row A) [col C 0, col C 1]; E = mat_of_cols (dim_row A) (map (col C) [2..<dim_col A]); (P2, D2, Q2) = Smith_2x2 D; H = P2 * D * Q2 @\<^sub>c P2 * E; k = div_op (H $$ (0, 2)) (H $$ (0, 0)); H2 = addcol (- k) 2 0 H; (uu_, uu_, uu_, H2_DR) = split_block H2 1 1; (H_1xn, Q3) = Smith_1xn H2_DR; S = four_block_mat (Matrix.mat 1 1 (\<lambda>(a, b). H $$ (0, 0))) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m 1 1) H_1xn; Q1' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_col A - 1) 1) Q1; Q2' = four_block_mat Q2 (0\<^sub>m 2 (dim_col A - 2)) (0\<^sub>m (dim_col A - 2) 2) (1\<^sub>m (dim_col A - 2)); Q_div_k = addrow_mat (dim_col A) (- k) 0 2; Q3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_col A - 1) 1) Q3 in (P2 * P1, S, Q1' * Q2' * Q_div_k * Q3'))"
cmp_simps_4: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.dom (\<Phi> (f, g)) = F f \<star>\<^sub>D F g"
AL_update1a: fixes a :: "'a"   and c :: "'a"   and L :: "('a \<times> 'b) list"   and b :: "'b" assumes "a = c" shows "L[a\<mapsto>b]\<down>c = Some b"
initmissing_def: fixes \<sigma> :: "(nat \<Rightarrow> state option) \<times> 'a" shows "op0.initmissing \<sigma> = (\<lambda>i. case fst \<sigma> i of None \<Rightarrow> op0.someinit i | Some s \<Rightarrow> s, snd \<sigma>)"
add_cancel_left_right: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and a :: "'k"   and b :: "'k" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus a b = a) = (b = zero)"
set_cat_cat: fixes U :: "'b set" shows "category (set_cat U)"
and_num_1048576_128: shows "1048576 AND 128 = 0"
attractive_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "attractive A less_eq \<equiv> semiattractive A less_eq \<and> attractive_axioms A less_eq"
can_lock_devreserp: fixes s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t' :: "'thread_id"   and ta' :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and x :: "'addr option \<times> 'addr frame list"   and L :: "('addr + 'thread_id + 'thread_id) set" assumes "s \<in> correct_jvm_state \<Phi>"   and "P \<turnstile> s -t'\<triangleright>ta'\<rightarrow>\<^bsub>jvmd\<^esub> s'"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "execd_mthr.can_sync P t x (shr s') L" shows "\<exists>L'\<subseteq>L. execd_mthr.can_sync P t x (shr s) L'"
of_int_2: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "\<guillemotleft>2\<guillemotright>\<^bsub>R\<^esub> = \<one>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>"
sorted_wrt_map_linord: fixes l :: "('a \<times> 'd) list" shows "sorted_wrt (\<lambda>x y. fst x \<le> fst y) l = sorted (map fst l)"
distinct_9: fixes x1 :: "('a, 'b, 'c) val"   and x61 :: "('a, 'b, 'c) expr"   and x62 :: "('a, 'b, 'c) expr" shows "VE x1 \<noteq> Assign x61 x62"
finite_support_one_oc: fixes outcomes :: "'b set" assumes "card outcomes = 1" shows "\<forall>l\<in>lotteries_on outcomes. finite (set_pmf l)"
UP_smult_assoc2: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "p \<in> carrier P"   and "q \<in> carrier P" shows "a \<odot>\<^bsub>P\<^esub> p \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
cobounded2: fixes b :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd" shows "b \<preceq> a \<oplus> b"
Gj_mem: shows "map of_int_hom.vec_hom fs ! j \<in> (!) (map of_int_hom.vec_hom fs) ` {0..<i}"
naturality: fixes f :: "'a" assumes "A.arr f" shows "A.map (A.cod f) \<cdot>\<^sub>A A.map f = A.map f \<cdot>\<^sub>A A.map (A.dom f)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "zero_hom hom"   and "semigroup_add_hom hom" shows "monoid_add_hom hom"
rv2_Cons: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and xa :: "bd_type_F + bd_type_F"   and xs :: "(bd_type_F + bd_type_F) list"   and x :: "'e" shows "rv2 s1 s2 (xa # xs) x = (case xa of Inl k1 \<Rightarrow> rv1 s1 s2 xs (frombd_F22 s2 x k1) | Inr k2 \<Rightarrow> rv2 s1 s2 xs (frombd_F23 s2 x k2))"
genset_order2_add: fixes s :: "'w" assumes "s \<in> S" shows "s + s = (0::'w)"
iwlsFSwSb_iterAll_termFSwSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Iteration.model" assumes "iwlsFSwSb MOD" shows "termFSwSbImorph (iter MOD) (iterAbs MOD) MOD"
min_def: fixes a :: "'a"   and b :: "'a" shows "ord.min (\<lambda>x y. y \<le> x) a b = (if b \<le> a then a else b)"
eq: shows "steps0 (1, [], Oc \<up> x) tcopy_begin n = (s, l, r)"
assms_1: shows "Partitions.partn_lst r B \<alpha> n"
lower_asymptotic_density_in_01_1: fixes A :: "nat set" shows "ereal (lower_asymptotic_density A) = liminf (\<lambda>x. ereal (real (card (A \<inter> {..<x})) / real x))"
comp: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'b" shows "(-`) f \<circ> (-`) g = (-`) (g \<circ> f)"
Abs_ExpList_Cons: fixes X :: "freeExp"   and Xs :: "freeExp list" shows "Abs_ExpList (X # Xs) \<equiv> Abs_Exp (exprel `` {X}) # Abs_ExpList Xs"
section_is_mono: fixes g :: "'a \<times> 'a" assumes "CC.section g" shows "CC.mono g"
conforms_gupd: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv"   and obj :: "obj"   and r :: "oref" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r"   and "s\<le>|gupd(r\<mapsto>obj) s" shows "(x, gupd(r\<mapsto>obj) s)\<Colon>\<preceq>(G, L)"
comp_apply_eq: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d"   and h :: "'e \<Rightarrow> 'b"   and k :: "'d \<Rightarrow> 'e" assumes "f (g x) = h (k x)" shows "(f \<circ> g) x = (h \<circ> k) x"
surj_def: fixes f :: "'e \<Rightarrow> 'd" shows "surj f = (\<forall>y. \<exists>x. y = f x)"
nat_int: fixes x :: "nat"   and y :: "nat" shows "nat (int x + int y) = x + y"
prv_tru_imp: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (eqv (imp tru \<phi>) \<phi>)"
list_eq_iff_nth_eq: fixes xs :: "'d list"   and ys :: "'d list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs ! i = ys ! i))"
repeat_zmset_replicate_zmset: fixes n :: "nat"   and a :: "'b" shows "repeat_zmset n {#a#}\<^sub>z = replicate_zmset n a"
apath_rev_path: fixes G :: "'b pair_pre_digraph"   and v :: "'b"   and p :: "('b \<times> 'b) list"   and u :: "'b" assumes "pair_sym_digraph G" shows "pre_digraph.apath (with_proj G) v (rev_path p) u = pre_digraph.apath (with_proj G) u p v"
extended_distance_basepoint: fixes x :: "'a"   and y :: "'a" shows "extended_distance (basepoint x) (basepoint y) = dist x y"
ImageE: fixes b :: "'b"   and r :: "('c \<times> 'b) set"   and A :: "'c set" assumes "b \<in> r `` A" obtains x :: "'c" where "(x, b) \<in> r" and "x \<in> A"
seq_app_first_tail: fixes s :: "nat \<Rightarrow> 'a" shows "first s ## tail s = s"
set_pmf_mix: fixes a :: "real"   and p :: "'b pmf"   and q :: "'b pmf" assumes "a \<in> {0<..<1}" shows "set_pmf (mix_pmf a p q) = set_pmf p \<union> set_pmf q"
simps_4: fixes f1 :: "'d"   and f2 :: "nat \<Rightarrow> 'd" shows "(case 0 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f1"
sorted_less_rev_set_eq: fixes xs :: "'b list" assumes "Sorted_Less.sorted (rev xs)" shows "sorted_list_of_set (set xs) = rev xs"
graph_def: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and k :: "nat" shows "graph A w k \<equiv> alternate (clean A w) (prune A w) k (gunodes A w)"
map_typeof_hext_mono: fixes h :: "'heap"   and vs :: "'addr val list"   and Ts :: "ty list"   and h' :: "'heap" assumes "map typeof\<^bsub>h\<^esub> vs = map Some Ts"   and "h \<unlhd> h'" shows "map typeof\<^bsub>h'\<^esub> vs = map Some Ts"
all_one_implies_inj_on: fixes A :: "'c set"   and B :: "'d set"   and N :: "nat multiset"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B"   and "\<forall>n. n \<in># N \<longrightarrow> n = 1"   and "number_partition (card A) N"   and "size N \<le> card B"   and "f \<in> functions_of A B N" shows "inj_on f A"
not_neutral_contains_not_neutral: fixes g :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "sum g A \<noteq> (0::'a)" obtains a :: "'b" where "a \<in> A" and "g a \<noteq> (0::'a)"
fps_nth: fixes x :: "'a fps" shows "fps_nth x \<in> {f. True}"
strip_acom: fixes f :: "'d \<Rightarrow> 'c"   and c :: "'d acom" shows "strip (ACom.map_acom f c) = strip c"
Entry_target: fixes a :: "'a" assumes "valid_edge a"   and "targetnode a = Entry" shows "False"
prems_2: shows "\<forall>a\<in>insert y F. m \<le> a"
geodesic_segment_param_3: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and t :: "real" assumes "geodesic_segment_between G x y"   and "t \<in> {0..dist x y}" shows "geodesic_segment_param G x t \<in> G"
Allreg_def: shows "Allreg \<equiv> \<Uplus>(range Atom)"
se_star_append: fixes c1 :: "('d, 'e) conf"   and ls1 :: "('d, 'e) label list"   and ls2 :: "('d, 'e) label list"   and c2 :: "('d, 'e) conf" shows "se_star c1 (ls1 @ ls2) c2 = (\<exists>c. se_star c1 ls1 c \<and> se_star c ls2 c2)"
fst_grab_subset: fixes N :: "nat set"   and n :: "nat" shows "fst (grab N n) \<subseteq> N"
insert_remove: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "finite A" shows "sum_0 g (insert x A) = g x + sum_0 g (A - {x})"
h12: shows "[(ainv * b') ^ (4 * k) = 1] (mod 4 * k + 3)"
diff_fun_const: fixes c :: "'e" shows "diff_fun k charts (\<lambda>x. c)"
subsumable_def: fixes subsumeseq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "subsumable subsumeseq \<equiv> (\<forall>x. subsumeseq x x) \<and> (\<forall>x y z. subsumeseq x y \<longrightarrow> subsumeseq y z \<longrightarrow> subsumeseq x z)"
filter_list_length: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and n :: "nat"   and p :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b" assumes "length (filter P xs) = n"   and "\<And>i x. \<lbrakk>x \<in> set xs; P x\<rbrakk> \<Longrightarrow> p ((f ^^ i) x)"   and "\<And>x. \<lbrakk>x \<in> set xs; \<not> P x\<rbrakk> \<Longrightarrow> \<exists>i. \<not> p ((f ^^ i) x)"   and "\<And>x. g (f x) = g x"   and "\<And>x. P (f x) = P x" shows "map g (filter_list_length f p n xs) = map g (filter P xs)"
prems: shows "applied_rule_rev C x (x @ [xs]) = \<lfloor>a\<rfloor>"
insert_noteq_member: fixes b :: "'d"   and B :: "'d multiset"   and c :: "'d"   and C :: "'d multiset" assumes "add_mset b B = add_mset c C"   and "b \<noteq> c" shows "c \<in># B"
llist_induct: fixes P :: "'b llist \<Rightarrow> bool"   and xs :: "'b llist" assumes "ccpo.admissible lSup lprefix P"   and "P <>"   and "\<And>x xs. \<lbrakk>lfinite xs; P xs\<rbrakk> \<Longrightarrow> P (x % xs)" shows "P xs"
simps: fixes \<phi> :: "'d stream \<Rightarrow> 'c"   and xs :: "'d stream" shows "nxt \<phi> xs = \<phi> (stl xs)"
sup_left_commute: fixes x :: "'d multiset"   and y :: "'d multiset"   and z :: "'d multiset" shows "x \<union># (y \<union># z) = y \<union># (x \<union># z)"
that: fixes l :: "nat" assumes "k < l"   and "l \<le> n"   and "\<pi> l = ipd (\<pi> k)" shows "thesis"
match_matches: fixes t :: "term"   and u :: "term"   and env :: "(name, term) fmap" assumes "match t u = Some env"   and "linear t" shows "t \<lesssim> u"
atomic_sub: fixes sub :: "'b run"   and r :: "'b run"   and \<phi> :: "'b TESL_atomic" assumes "sub \<lless> r"   and "sub \<in> \<lbrakk> \<phi> \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L" shows "r \<in> \<lbrakk> \<phi> \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
ThrowRedsNull: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s' :: "(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" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>null,s'\<rangle>" shows "P \<turnstile> \<langle>throw e,s\<rangle> \<rightarrow>* \<langle>THROW NullPointer,s'\<rangle>"
single_valued_Id_on: fixes A :: "'b set" shows "single_valued (Id_on A)"
cut_geI: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I"   and "t \<le> x" shows "x \<in> I \<down>\<ge> t"
wf_dual_pnorm_dual: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp_dual" assumes "wf_dual \<Sigma> n r" shows "wf_dual \<Sigma> n (pnorm_dual r)"
polyhedron_halfspace_ge: fixes b :: "real"   and a :: "'b" shows "polyhedron {x. b \<le> a \<bullet> x}"
word_ao_absorbs_6: fixes x :: "'a word"   and y :: "'a word" shows "x OR x AND y = x"
Guard: fixes s :: "'a"   and g :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and f :: "'c" assumes "s \<in> g"   and "\<Gamma>\<turnstile>c \<down> Normal s" shows "\<Gamma>\<turnstile>Guard f g c \<down> Normal s"
inv2Inv_2: fixes f :: "'b"   and g :: "'b" assumes "cinv f g" shows "Cinv f = g"
leof_use_spec_rule: fixes m :: "'b nres"   and \<Psi> :: "'b \<Rightarrow> bool"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "m \<le>\<^sub>n SPEC \<Psi>"   and "m \<le>\<^sub>n SPEC (\<lambda>s. \<Psi> s \<longrightarrow> \<Phi> s)" shows "m \<le>\<^sub>n SPEC \<Phi>"
is_category_with_finite_products: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "cartesian_category C" shows "category_with_finite_products C"
sup_def: fixes a :: "'a"   and b :: "'a" shows "a \<squnion> b = ((a l\<rightarrow> b) r\<rightarrow> b) \<sqinter> ((b l\<rightarrow> a) r\<rightarrow> a)"
group_isomorphisms_def: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and f :: "'a \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'a" shows "group_isomorphisms G H f g \<equiv> f \<in> hom G H \<and> g \<in> hom H G \<and> (\<forall>x\<in>carrier G. g (f x) = x) \<and> (\<forall>y\<in>carrier H. f (g y) = y)"
is_syz_sig_def: fixes d :: "'a \<Rightarrow> nat"   and u :: "'t" shows "is_syz_sig d u = (\<exists>s\<in>dgrad_sig_set' (length fs) d. s \<noteq> 0 \<and> lt s = u \<and> rep_list s = 0)"
comp_assoc_assoc'_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "B.\<a>' f g h \<cdot>\<^sub>B \<a>\<^sub>B[f, g, h] = (f \<star>\<^sub>B g) \<star>\<^sub>B h"
case_eq_if: fixes f :: "'c \<Rightarrow> 'c hp list \<Rightarrow> 'b"   and hp :: "'c hp" shows "(case hp of Hp x xa \<Rightarrow> f x xa) = f (un_Hp1 hp) (hps hp)"
outside_union_restrict: fixes P :: "('c \<times> 'd) set"   and X :: "'c set" shows "P = P outside X \<union> P || X"
upper_triangularI: fixes A :: "'a mat" assumes "\<And>i j. \<lbrakk>j < i; i < dim_row A\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'a)" shows "Matrix.upper_triangular A"
injective_scaleR: fixes c :: "real" assumes "c \<noteq> 0" shows "inj ((*\<^sub>R) c)"
u': fixes g :: "'a \<Rightarrow>f 'b"   and a :: "'a"   and b :: "'b" shows "f' g(a $:= b) = upd g a b (f' g)"
free_vars_map_vars: fixes f :: "nat \<Rightarrow> nat"   and e :: "cexpr" shows "free_vars (map_vars f e) = f ` free_vars e"
bind_split: fixes P :: "'a option \<Rightarrow> bool"   and m :: "'b option"   and f :: "'b \<Rightarrow> 'a option" shows "P (m \<bind> f) = ((m = None \<longrightarrow> P None) \<and> (\<forall>v. m = Some v \<longrightarrow> P (f v)))"
assms: shows "coeff (pCons b c2) (degree q) mod lead_coeff q = 0"
rel_game_def: fixes \<A> :: "'pub_input \<Rightarrow> 'witness spmf" shows "rel_game \<A> = TRY G \<bind> (\<lambda>(h, w). \<A> h \<bind> (\<lambda>w'. return_spmf ((h, w') \<in> Rel))) ELSE return_spmf False"
subdeg_sum'_2: shows "fls_subdegree g = - fls_subdegree f"
adverse_Suc: fixes x :: "nat"   and z :: "nat"   and i :: "nat"   and j :: "nat" assumes "0 < x" shows "local.adverse z i j (Suc x) \<down>= (if s (e_snoc (local.adverse z i j \<triangleright> x) 0) \<noteq> s (local.adverse z i j \<triangleright> x) then 0 else if s (e_snoc (local.adverse z i j \<triangleright> x) 1) \<noteq> s (local.adverse z i j \<triangleright> x) then 1 else z)"
sel_9: fixes x31 :: "nat"   and x32 :: "'d"   and x33 :: "'d"   and x34 :: "'e"   and x35 :: "'e"   and x36 :: "'f" shows "getId (Recv x31 x32 x33 x34 x35 x36) = x31"
nu_s: shows "\<nu> 1\<^sub>\<sigma> = 1\<^sub>\<sigma>"
undef_mat_def: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "undef_mat nr nc f \<equiv> \<lambda>(i, j). map (\<lambda>i. map (\<lambda>j. f (i, j)) [0..<nc]) [0..<nr] ! i ! j"
hyps: assumes "n \<le> length lm" shows "{\<lambda>nl. nl = lm} empty_boxes n {\<lambda>nl. nl = 0 \<up> n @ drop n lm}"
Dirichlet_L_conv_eval_fds: fixes s :: "complex" assumes "\<chi> \<noteq> \<chi>\<^sub>0"   and "0 < Re s" shows "Dirichlet_L n \<chi> s = eval_fds (fds \<chi>) s"
assert_nsc_code_def: shows "assert_nsc_code = assert_nsc assert_all_s"
Ioc_inj: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "({a<..b} = {c<..d}) = (b \<le> a \<and> d \<le> c \<or> a = c \<and> b = d)"
comp_arr_dom: fixes f :: "'b"   and a :: "'b" assumes "D.arr f"   and "D.dom f = a" shows "f \<cdot>\<^sub>D a = f"
dist_les_1: fixes a' :: "'b"   and list' :: "['b]" shows "[] \<notsqsubseteq> a' : list'"
pred_succ: fixes i :: "V" shows "ZFC_in_HOL.pred (ZFC_in_HOL.succ i) = i"
degree_coefficients: fixes p :: "poly" shows "Reflected_Multivariate_Polynomial.degree p = length (coefficients p) - 1"
prime_imp_coprime_int: fixes p :: "int"   and n :: "int" assumes "prime p"   and "\<not> p dvd n" shows "algebraic_semidom_class.coprime p n"
chop_eq_Nil_iff: fixes n :: "nat"   and xs :: "'b list" shows "(chop n xs = []) = (n = 0 \<or> xs = [])"
rep_eq: fixes x :: "(nat, 'd) slist" shows "pdevs_apply (Pdevs x) = Pdevs_raw (Rep_slist x)"
nth_list_update: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a"   and j :: "nat" assumes "i < |xs|" shows "xs[i := x] ! j = (if i = j then x else xs ! j)"
fls_inverse_X': assumes "inverse (1::'a) = (1::'a)" shows "inverse fls_X = fls_X_inv"
finite_traversed_states: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "finite (traversed_states s as)"
properD_1: fixes P :: "heap \<times> nat set \<Rightarrow> bool"   and h :: "heap"   and as :: "nat set" assumes "proper P"   and "P (h, as)" shows "in_range (h, as)"
lcons_def: fixes a :: "'a \<times> 'b"   and t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.lcons a t = FingerTreeStruc.nlcons (Tip (fst a) (snd a)) t"
LeadsTo_Trans: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "F \<in> A \<longmapsto>w B"   and "F \<in> B \<longmapsto>w C" shows "F \<in> A \<longmapsto>w C"
matrix_mult_index: fixes m1 :: "'a list list"   and nr :: "nat"   and n :: "nat"   and nc :: "nat"   and m2 :: "'a list list"   and i :: "nat"   and j :: "nat" assumes "m1 \<noteq> []"   and "mat nr n m1"   and "mat n nc m2"   and "i < nr"   and "j < nc" shows "(m1 \<circ> m2) ! j ! i = scalar_product (row m1 i) (col m2 j)"
simps_9: fixes f :: "'l \<Rightarrow> 'k"   and x21 :: "'l"   and x22 :: "'l list" shows "map f (x21 # x22) = f x21 # map f x22"
enq: shows "|~ Enq ic q oc \<longrightarrow> BInv ic q oc \<longrightarrow> \<circle>BInv ic q oc"
gr\<rho>_x: shows "grounding_\<rho> x = apps s (replicate (the_enat (arity s - arity_var x)) s)"
filter_min_append_distinct: fixes rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" assumes "reflp rel"   and "distinct xs"   and "distinct ys" shows "distinct (filter_min_append rel xs ys)"
bubble_minD_mset: fixes xs :: "'a list"   and ys :: "'a list" assumes "bubble_min xs = ys" shows "mset xs = mset ys"
rel_inject_1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'a"   and y1 :: "'b" shows "rel_behaviour R (Terminates x1) (Terminates y1) = R x1 y1"
fold_Cons_rev: fixes xs :: "'f list" shows "fold (#) xs = (@) (rev xs)"
pderiv_singleton: fixes a :: "'a" shows "pderiv [:a:] = 0"
nat_to_bv_non0: fixes n :: "nat" assumes "n \<noteq> 0" shows "nat_to_bv n = nat_to_bv (n div 2) @ [if n mod 2 = 0 then \<zero> else \<one>]"
le_infI: fixes w\<^sub>1 :: "'item stream"   and w\<^sub>2 :: "'item stream" assumes "\<And>v\<^sub>1. v\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1 \<Longrightarrow> v\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2" shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2"
assms_3: fixes S :: "'a set" assumes "S \<in> \<F>" shows "openin (top_of_set (\<Union> \<F>)) S"
transfer_start: fixes P :: "bool"   and Q :: "bool" assumes "P"   and "Transfer.Rel (=) P Q" shows "Q"
cases: fixes a1 :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> Arity \<times> Arity list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "consistent a1 a2" obtains ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" and a :: "Arity" and as :: "Arity list" and \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and S :: "stack_elem list" where "a1 = (ae, a, as)" and "a2 = (\<Gamma>, e, S)" and "a_consistent (ae, a, as) (\<Gamma>, e, S)" and "\<And>x. x \<in> thunks \<Gamma> \<Longrightarrow> ae x = up\<cdot>0"
set_of_binsert: fixes x :: "nat"   and t :: "tree" shows "set_of (binsert x t) = set_of t \<union> {x}"
pcorres_Skip: fixes \<phi> :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> bool" shows "pcorres \<phi> G Skip Skip"
hausdorff_distance_self: fixes A :: "'b set" shows "hausdorff_distance A A = 0"
const_imply4: fixes P :: "'d \<Rightarrow> 'e"   and Q :: "'d \<Rightarrow> 'f"   and R :: "'d \<Rightarrow> 'g"   and S :: "'d \<Rightarrow> 'h" assumes "\<And>\<tau> \<tau>'. \<lbrakk>P \<tau> = P \<tau>'; Q \<tau> = Q \<tau>'; R \<tau> = R \<tau>'\<rbrakk> \<Longrightarrow> S \<tau> = S \<tau>'"   and "const P"   and "const Q"   and "const R" shows "const S"