hcju/isabelleps huggingface dataset:
tau_eq_state: fixes j :: "nat"   and i :: "nat"   and t :: "nat" assumes "j < e_length (block i t)"   and "j \<noteq> gap i t" shows "\<tau> i j \<down>= e_nth (block i t) j"
subst_preserves_good: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "good Y" shows "good (Y #[X / x]_xs)"
thin: fixes H :: "fm set"   and A :: "fm"   and H' :: "fm set" assumes "H \<turnstile> A"   and "H \<subseteq> H'" shows "H' \<turnstile> A"
items_le_fix_D: fixes k :: "nat"   and I :: "item set"   and x :: "item" assumes "items_le k I = I"   and "x \<in> I" shows "item_end x \<le> k"
poly_add_degree: fixes p1 :: "'a list"   and p2 :: "'a list" shows "degree (poly_add p1 p2) \<le> max (degree p1) (degree p2)"
simps_1: fixes x :: "'a" shows "weight (Var x) = w0"
to_poly_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "to_polynomial R (a \<otimes> b) = to_polynomial R a \<otimes>\<^bsub>P\<^esub> to_polynomial R b"
infsetsum_nat: fixes f :: "nat \<Rightarrow> 'd"   and A :: "nat set" assumes "f abs_summable_on A" shows "infsetsum f A = (\<Sum>n. if n \<in> A then f n else (0::'d))"
contg: shows "continuous_map (subtopology X {x \<in> topspace X. q x \<le> p x}) Y g"
prime_factorization_2_gauss_int: shows "prime_factorization 2 = {#1 + \<i>\<^sub>\<int>, 1 + \<i>\<^sub>\<int>#}"
IH_2: fixes r :: "C"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>)|\<^bsub>C\<cdot>r\<^esub> = (\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>|\<^sup>\<circ>\<^bsub>r\<^esub>\<^esub>)|\<^bsub>C\<cdot>r\<^esub>"
exhaust: fixes y :: "mapping_impl" obtains "y = Mapping_IMPL"
termFSwSbImorph_iff: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Iteration.model_scheme" shows "termFSwSbImorph h hA MOD = (termFSbImorph h hA MOD \<and> ipresSwapAll h hA MOD)"
abs_eq: fixes xa :: "hmultiset zmultiset"   and x :: "hmultiset zmultiset" shows "ZHMSet xa - ZHMSet x = ZHMSet (xa - x)"
eqvt_at_perm: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and q :: "perm" assumes "eqvt_at f x" shows "eqvt_at f (q \<bullet> x)"
aff_dim_halfspace_gt: fixes r :: "real"   and a :: "'a" shows "aff_dim {x. r < a \<bullet> x} = (if a = (0::'a) \<and> 0 \<le> r then - 1 else int DIM('a))"
intro: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "\<And>a b. \<lbrakk>a \<in> M; b \<in> M\<rbrakk> \<Longrightarrow> composition a b \<in> M"   and "unit \<in> M"   and "\<And>a b c. \<lbrakk>a \<in> M; b \<in> M; c \<in> M\<rbrakk> \<Longrightarrow> composition (composition a b) c = composition a (composition b c)"   and "\<And>a. a \<in> M \<Longrightarrow> composition unit a = a"   and "\<And>a. a \<in> M \<Longrightarrow> composition a unit = a" shows "Group_Theory.monoid M composition unit"
list_all_rec_def: shows "list_all_rec \<equiv> \<lambda>uu uua. rec_list (\<lambda>P. True) (\<lambda>a l la P. P a \<and> la P) uua uu"
distinct_36: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Stop \<noteq> x41 ;; x42"
assms: shows "run_one_step d i (s, vs, ves, $Binop_f x271 x272) = (s', vs', res)"
dim_openin: fixes T :: "'a set"   and S :: "'a set" assumes "openin (top_of_set T) S"   and "subspace T"   and "S \<noteq> {}" shows "dim S = dim T"
Can_HcompNml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Can t"   and "Can u"   and "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Can (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u)"
HSem_reorder: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "map_of \<Gamma> = map_of \<Delta>" shows "\<N>\<lbrace>\<Gamma>\<rbrace>\<rho> = \<N>\<lbrace>\<Delta>\<rbrace>\<rho>"
simps_1: fixes t :: "'b tree234" shows "height (T\<^sub>i t) = height t"
strict_implies_order: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subset># b" shows "a \<subseteq># b"
pdevs_apply_pdevs_of_list_Cons: fixes x :: "'b"   and xs :: "'b list"   and i :: "nat" shows "pdevs_apply (pdevs_of_list (x # xs)) i = (if i = 0 then x else pdevs_apply (pdevs_of_list xs) (i - 1))"
finite_gpv_parametric: shows "rel_fun (rel_\<I> (BNF_Def.Grp UNIV id) (=)) (rel_fun (rel_gpv (BNF_Def.Grp UNIV f) (BNF_Def.Grp UNIV id)) (=)) finite_gpv finite_gpv"
mult1I: fixes M :: "'a multiset"   and a :: "'a"   and M0 :: "'a multiset"   and N :: "'a multiset"   and K :: "'a multiset"   and r :: "('a \<times> 'a) set" assumes "M = add_mset a M0"   and "N = M0 + K"   and "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r" shows "(N, M) \<in> mult1 r"
signed_take_bit_int_greater_eq_self_iff: fixes k :: "int"   and n :: "nat" shows "(k \<le> signed_take_bit n k) = (k < 2 ^ n)"
canE_unitor_3: fixes f :: "'a E.term" assumes "E.Ide f" shows "B.can (f \<^bold>\<star> E.Src f) f = B.runit' (EVAL f)"
wfrec: fixes R :: "('b \<times> 'b) set"   and F :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'b" assumes "wf R" shows "wfrec R F a = F (cut (wfrec R F) R a) a"
simps_1: fixes M :: "'d"   and N :: "'d" shows "bn (M\<lparr>N\<rparr>) = []"
denyNMT: shows "D\<^sub>U \<noteq> \<emptyset>"
isCont_Arcsin_lemma: fixes z :: "complex" assumes "Re (\<i> * z + csqrt (1 - z\<^sup>2)) \<le> 0"   and "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "False"
le_iff_sup: fixes y :: "'b"   and x :: "'b" shows "(y \<le> x) = (x \<sqinter> y = y)"
ex2: shows "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> P2 x = P2'"
Init: fixes Env :: "env"   and B :: "lname set"   and C :: "qtname" shows "Env\<turnstile> B \<guillemotright>\<langle>Init C\<rangle>\<guillemotright> \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
canE_associator_2: fixes f :: "'a E.term"   and g :: "'a E.term"   and h :: "'a E.term" assumes "E.Ide f"   and "E.Ide g"   and "E.Ide h"   and "E.Src f = E.Trg g"   and "E.Src g = E.Trg h" shows "B.can ((f \<^bold>\<star> g) \<^bold>\<star> h) (f \<^bold>\<star> g \<^bold>\<star> h) = B.\<a>' (EVAL f) (EVAL g) (EVAL h)"
split_conv: fixes f :: "'i \<Rightarrow> 'j \<Rightarrow> 'h"   and a :: "'i"   and b :: "'j" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
assms_5: shows "h \<turnstile> get_elements_by_tag_name ptr' idd \<rightarrow>\<^sub>r results"
str_init2_def: fixes dummy :: "'d"   and y :: "('d, 'd IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool, 'd IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) F2"   and i :: "'d IIT" shows "str_init2 dummy y i = snd (snd (Rep_IIT i)) (F2map id (\<lambda>f. f i) (\<lambda>f. f i) y)"
llist_of_eq_LCons_conv: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a llist" shows "(llist_of xs = LCons y ys) = (\<exists>xs'. xs = y # xs' \<and> ys = llist_of xs')"
rbt_less_simps_1: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b" shows "ord.rbt_less less k rbt.Empty = True"
min_max_distrib2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linorder less_eq less" shows "ord.min less_eq a (ord.max less_eq b c) = ord.max less_eq (ord.min less_eq a b) (ord.min less_eq a c)"
list_all2_conv_all_nth: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>i<length xs. P (xs ! i) (ys ! i)))"
verts_set: fixes G :: "nat graph_rec" assumes "cfg_doms G" shows "set (g_V G) = {0..<length (g_V G)}"
measure_UNION_le: fixes I :: "'a set"   and F :: "'a \<Rightarrow> 'b set"   and M :: "'b measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" shows "Sigma_Algebra.measure M (\<Union> (F ` I)) \<le> (\<Sum>i\<in>I. Sigma_Algebra.measure M (F i))"
WT_gpv_iff_outs_gpv: fixes \<I> :: "('i, 'j) \<I>"   and gpv :: "('k, 'i, 'j) gpv" shows "\<I> \<turnstile>g gpv \<surd> = (outs_gpv \<I> gpv \<subseteq> outs_\<I> \<I>)"
simps_2: fixes r :: "'a routing_rule"   and rs :: "'a routing_rule list"   and p :: "'a word" shows "routing_table_semantics (r # rs) p = (if prefix_match_semantics (routing_match r) p then routing_action r else routing_table_semantics rs p)"
ids_list_comb: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a"   and xs :: "'a list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "pre_term.ids frees consts (term.list_comb app f xs) = pre_term.ids frees consts f |\<union>| ffUnion (fset_of_list (map (pre_term.ids frees consts) xs))"
QF_def: fixes k :: "nat" shows "QF k \<equiv> wfrec pair_less (\<lambda>f (j, i). if j = 0 then AF k i else let r = if i = 0 then f (j - 1, m - 1) else f (j, i - 1) in enum (nxt N (Suc (Max r))) ` {..<del k (if j = k then m - Suc i else i) j})"
weakenR: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and F :: "'b formula" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> F, \<Delta>"
d_def: shows "ps_P.d = prod_list ps2_P.dims0"
ds2_components: fixes M1 :: "('e, 'a, 'f) Module_scheme"   and M2 :: "('e, 'a, 'f) Module_scheme"   and M :: "('e, 'a, 'f) Module_scheme"   and a :: "'e" assumes "R module M1"   and "R module M2"   and "R module M"   and "M1 \<Oplus>\<^bsub>R,M\<^esub> M2"   and "a \<in> carrier M" shows "\<exists>a1\<in>carrier M1. \<exists>a2\<in>carrier M2. a = a1 \<plusminus>\<^bsub>M\<^esub> a2"
least_eq: shows "(LEAST n. Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res $ i $ n \<noteq> (0::'a)) = (LEAST n. A $ i $ n \<noteq> (0::'a))"
sylvester_sub_poly: fixes p :: "'a poly"   and m :: "nat"   and q :: "'a poly"   and n :: "nat"   and v :: "'a vec" assumes "degree p \<le> m"   and "degree q \<le> n"   and "v \<in> carrier_vec (m + n)" shows "poly_of_vec ((sylvester_mat_sub m n p q)\<^sup>T *\<^sub>v v) = poly_of_vec (vec_first v n) * p + poly_of_vec (vec_last v m) * q"
fv_occs: fixes x :: "variable"   and \<tau> :: "typ"   and t :: "Core.term" assumes "(x, \<tau>) \<in> fv t" shows "occs (Fv x \<tau>) t"
cinner_simps_3: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "\<langle>x, y - z\<rangle> = \<langle>x, y\<rangle> - \<langle>x, z\<rangle>"
prj0_simps_2: fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "C.dom \<p>\<^sub>0[f, g] = f \<down>\<down> g"
orient2: shows "delta_gt \<delta> (1::'a) (0::'a)"
eP_def: fixes n :: "nat"   and \<omega> :: "nat \<Rightarrow> 'a" shows "eP n \<omega> = distr (K' x n \<omega>) (Pi\<^sub>M {0..<Suc n} (\<lambda>_. M)) (fun_upd \<omega> n)"
ord_le_eq_trans: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'k"   and b :: "'k"   and c :: "'k" assumes "less_eq a b"   and "b = c" shows "less_eq a c"
modalise_characterise: fixes A :: "('a, 'b) form"   and M :: "'b"   and \<Delta> :: "('a, 'b) form multiset" assumes "A \<in># M \<cdot> \<Delta>" shows "\<exists>B. A = Modal M [B]"
segment_construction: fixes x :: "complex_homo"   and y :: "complex_homo"   and a :: "complex_homo"   and b :: "complex_homo" assumes "x \<in> unit_disc"   and "y \<in> unit_disc"   and "a \<in> unit_disc"   and "b \<in> unit_disc" shows "\<exists>z. z \<in> unit_disc \<and> poincare_between x y z \<and> poincare_distance y z = poincare_distance a b"
power_polys_works: fixes i :: "nat"   and n :: "nat"   and curr_p :: "'b"   and u :: "'b"   and mult_p :: "'b" assumes "i < n"   and "curr_p = curr_p mod u" shows "power_polys mult_p u curr_p n ! i = curr_p * mult_p ^ i mod u"
Image_closed_trancl: fixes r :: "('b \<times> 'b) set"   and X :: "'b set" assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"
Var: fixes rs :: "dB list"   and rs' :: "dB list"   and x :: "nat" assumes "rs [\<rightarrow>\<^sub>l] rs'" shows "Var x \<degree>\<degree> rs \<rightarrow>\<^sub>l Var x \<degree>\<degree> rs'"
Synchronized1Throw1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and V :: "nat"   and a :: "'addr"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>sync\<^bsub>V\<^esub> (Throw a) e,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
bind_fv2_idem: fixes x :: "variable"   and \<tau> :: "typ"   and lev1 :: "nat"   and lev2 :: "nat"   and t :: "Core.term" shows "bind_fv2 (x, \<tau>) lev1 (bind_fv2 (x, \<tau>) lev2 t) = bind_fv2 (x, \<tau>) lev2 t"
assms_3: fixes x :: "'a"   and y :: "'a"   and ys :: "'a list" assumes "xs = x # y # ys" shows "P"
inputSuppPresBisim: fixes P :: "pi"   and Q :: "pi"   and x :: "name"   and a :: "name" assumes "\<And>y. y \<in> supp (P, Q, x) \<Longrightarrow> P[x::=y] \<sim> Q[x::=y]" shows "a<x>.P \<sim> a<x>.Q"
homotopic_loops: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" shows "homotopic_loops s p q = (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and> h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and> (\<forall>x\<in>{0..1}. h (0, x) = p x) \<and> (\<forall>x\<in>{0..1}. h (1, x) = q x) \<and> (\<forall>t\<in>{0..1}. pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)))"
hd_Cons_tl: fixes xs :: "'e list" assumes "xs \<noteq> []" shows "hd xs # tl xs = xs"
count_space_plus: fixes c :: "'b" shows "count_space UNIV = distr (count_space UNIV) (count_space UNIV) ((+) c)"
sort_key_const: fixes c :: "'b"   and xs :: "'a list" shows "sort_key (\<lambda>x. c) xs = xs"
tail_arev: fixes G :: "('c, 'd) pre_digraph"   and arev :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "bidirected_digraph G arev"   and "a \<in> arcs G" shows "tail G (arev a) = head G a"
simps: fixes x :: "nat"   and y :: "nat"   and w :: "nat" shows "singly_balanced_arith x y w = (balanced2_arith x y \<and> balanced2_arith (x + y + 1) w)"
l11_3: 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"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F" shows "\<exists>A' C' D' F'. Tarski_neutral_dimensionless.Out Bet B A' A \<and> Tarski_neutral_dimensionless.Out Bet B C C' \<and> Tarski_neutral_dimensionless.Out Bet E D' D \<and> Tarski_neutral_dimensionless.Out Bet E F F' \<and> Tarski_neutral_dimensionless.Cong3 Cong A' B C' D' E F'"
prv_eql_sym: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "prv (imp (eql t1 t2) (eql t2 t1))"
lookup_inject: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and y :: "'a \<Rightarrow>\<^sub>0 'b" shows "(lookup x = lookup y) = (x = y)"
le: fixes l :: "nat" assumes "k < l \<and> l \<le> n + r \<and> (\<pi> @\<^bsup>n\<^esup> \<pi>') l = ipd (\<pi> k)" shows "m \<le> l"
induct: fixes x1 :: "state"   and x2 :: "com"   and x3 :: "state"   and P :: "state \<Rightarrow> com \<Rightarrow> state \<Rightarrow> bool" assumes "x1 -x2\<rightarrow> x3"   and "\<And>t f s. t \<in> f s \<Longrightarrow> P s (Do f) t"   and "\<And>s0 c1 s1 c2 s2. \<lbrakk>s0 -c1\<rightarrow> s1; P s0 c1 s1; s1 -c2\<rightarrow> s2; P s1 c2 s2\<rbrakk> \<Longrightarrow> P s0 (c1; c2) s2"   and "\<And>b s c1 t c2. \<lbrakk>b s; s -c1\<rightarrow> t; P s c1 t\<rbrakk> \<Longrightarrow> P s (IF b THEN c1 ELSE c2) t"   and "\<And>b s c2 t c1. \<lbrakk>\<not> b s; s -c2\<rightarrow> t; P s c2 t\<rbrakk> \<Longrightarrow> P s (IF b THEN c1 ELSE c2) t"   and "\<And>b s c. \<not> b s \<Longrightarrow> P s (WHILE b DO c) s"   and "\<And>b s c t u. \<lbrakk>b s; s -c\<rightarrow> t; P s c t; t -WHILE b DO c\<rightarrow> u; P t (WHILE b DO c) u\<rbrakk> \<Longrightarrow> P s (WHILE b DO c) u"   and "\<And>s p t. \<lbrakk>s -body p\<rightarrow> t; P s (body p) t\<rbrakk> \<Longrightarrow> P s (CALL p) t"   and "\<And>f s c t g. \<lbrakk>f s -c\<rightarrow> t; P (f s) c t\<rbrakk> \<Longrightarrow> P s (LOCAL f; c; g) (g s t)" shows "P x1 x2 x3"
commute: fixes a :: "'a word"   and b :: "'a word" shows "signed.max a b = signed.max b a"
assms_5: shows "finite X \<longrightarrow> a < card X"
separable_impl_def: fixes p :: "int" shows "separable_impl p = (if p \<le> 65535 then separable_impl_main p (finite_field_ops32 (uint32_of_int p)) else if p \<le> 4294967295 then separable_impl_main p (finite_field_ops64 (uint64_of_int p)) else separable_impl_main p (finite_field_ops_integer (integer_of_int p)))"
K_elem_abss_inj: fixes \<mu> :: "('s \<times> ('c \<Rightarrow> real)) pmf"   and l :: "'s"   and u :: "'c \<Rightarrow> real" assumes "\<mu> \<in> K (l, u)"   and "(l, u) \<in> S" shows "inj_on abss (set_pmf \<mu>)"
SolveLoopTermination: fixes state :: "State"   and F0' :: "Literal list list"   and Vbl :: "nat set"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantUniqQ (getQ state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"   and "InvariantNoDecisionsWhenUnit (getF state) (getM state) (currentLevel (getM state))"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"   and "getSATFlag state = UNDEF \<longrightarrow> InvariantEquivalentZL (getF state) (getM state) F0'"   and "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)"   and "finite Vbl"   and "vars F0' \<subseteq> vars F0"   and "vars F0 \<subseteq> Vbl"   and "InvariantVarsM (getM state) F0 Vbl"   and "InvariantVarsQ (getQ state) F0 Vbl"   and "InvariantVarsF (getF state) F0 Vbl" shows "solve_loop_dom state Vbl"
distinct_1: fixes x21 :: "'b"   and x22 :: "'b llist" shows "LNil \<noteq> x21 ## x22"
option_relE: fixes x :: "'a option"   and x' :: "'b option"   and R :: "('a \<times> 'b) set" assumes "(x, x') \<in> \<langle>R\<rangle>option_rel" obtains "x = None" and "x' = None"   | a :: "'a" and a' :: "'b" where "x = Some a" and "x' = Some a'" and "(a, a') \<in> R"
map_pmf_const: fixes c :: "'c"   and M :: "'d pmf" shows "map_pmf (\<lambda>_. c) M = return_pmf c"
dom_const: fixes f :: "'a \<Rightarrow> 'b" shows "dom (\<lambda>x. Some (f x)) = UNIV"
mrNnt: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "applied_rule_rev Cp x p = \<lfloor>a\<rfloor>" shows "p \<noteq> []"
rel_sel: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a list"   and b :: "'b list" shows "list_all2 R a b = ((a = []) = (b = []) \<and> (a \<noteq> [] \<longrightarrow> b \<noteq> [] \<longrightarrow> R (hd a) (hd b) \<and> list_all2 R (tl a) (tl b)))"
sum_decomp_f': shows "(\<Sum>ia = 0..<length t. if postrecording_event (swap_events i j t) ia then card {ja \<in> {ia + 1..<length (swap_events i j t)}. prerecording_event (swap_events i j t) ja} else 0) = (\<Sum>ia = 0..<i. if postrecording_event (swap_events i j t) ia then card {ja \<in> {ia + 1..<length (swap_events i j t)}. prerecording_event (swap_events i j t) ja} else 0) + (\<Sum>ia = i..<j + 1. if postrecording_event (swap_events i j t) ia then card {ja \<in> {ia + 1..<length (swap_events i j t)}. prerecording_event (swap_events i j t) ja} else 0) + (\<Sum>ia = j + 1..<length t. if postrecording_event (swap_events i j t) ia then card {ja \<in> {ia + 1..<length (swap_events i j t)}. prerecording_event (swap_events i j t) ja} else 0)"
two_comparisons_into_case_order_5: fixes x :: "'a"   and y :: "'a"   and Q :: "'c"   and P :: "'c"   and R :: "'c" shows "(if lt x y then Q else if le x y then P else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
sign_cases: fixes x :: "'a" obtains "sign x = - 1"   | "sign x = 0"   | "sign x = 1"
monofunI: fixes f :: "'e \<Rightarrow> 'f" assumes "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y" shows "monofun f"
e_dist: shows "\<forall>y\<in>S. dist z y < e \<longrightarrow> g y \<noteq> 0"
pgallery: fixes X :: "'b set set"   and xs :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.pmaxsimpchain X xs" shows "SimplicialComplex.maxsimpchain X xs"
pre_add: shows "pre_subdivFace' g f v' v n (Some u # vol)"
th: shows "fps_const \<i> * fps_sin c + fps_const \<i> * fps_sin c = fps_sin c * fps_const (2 * \<i>)"
state_set_card_iii: fixes a :: "'a"   and X :: "'a list" shows "card (state_set (a # X)) = 1 + card (state_set X)"
unstream_enumFromToS: fixes x :: "int\<^sub>\<bottom>"   and y :: "int\<^sub>\<bottom>" shows "unstream\<cdot>(enumFromToS\<cdot>x\<cdot>y) = enumFromToL\<cdot>x\<cdot>y"
inject: fixes x1 :: "le_rel"   and x2 :: "linear_poly"   and x3 :: "'a"   and y1 :: "le_rel"   and y2 :: "linear_poly"   and y3 :: "'a" shows "(Le_Constraint x1 x2 x3 = Le_Constraint y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
lcf_of_X_plus: fixes a :: "'a" assumes "a \<in> carrier R"   and "\<one> \<noteq> \<zero>" shows "X_poly_plus R a (deg R (X_poly_plus R a)) = \<one>"
finite_char: fixes C :: "'b set set" shows "finite_char (mk_finite_char C)"
simps_2: fixes F :: "'b formula"   and Fs :: "'b formula list" shows "\<^bold>\<Or>F # Fs = F \<^bold>\<or> \<^bold>\<Or>Fs"
effect_updI: fixes i :: "nat"   and h :: "heap"   and a :: "'b array"   and h' :: "heap"   and v :: "'b" assumes "i < Array.length h a"   and "h' = Array.update a i v h" shows "effect (a.(i) \<leftarrow> v) h h' a"
map_poly_hom_smult: fixes c :: "'a"   and p :: "'a poly" shows "map_poly hom (smult c p) = smult (hom c) (map_poly hom p)"
has_integral_le: fixes f :: "'a \<Rightarrow> real"   and i :: "real"   and S :: "'a set"   and g :: "'a \<Rightarrow> real"   and j :: "real" assumes "(f has_integral i) S"   and "(g has_integral j) S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x" shows "i \<le> j"
exists_superset_carrier: fixes S :: "'a set" assumes "finite S"   and "S \<noteq> {}"   and "S \<subseteq> carrier (union_ring C)" shows "\<exists>R\<in>C. S \<subseteq> carrier R"
lea_left_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LeA D E F" shows "C B A LeA D E F"