hcju/isabelleps huggingface dataset:
simps_3: fixes i :: "int"   and c :: "int"   and e :: "Cooper.num"   and d :: "int" shows "d_\<delta> (Dvd i (CN 0 c e)) d = (i dvd d)"
simps_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and y :: "'c"   and ys :: "'c list" assumes "class.linorder less_eq less" shows "linorder.insort_key less_eq f x (y # ys) = (if less_eq (f x) (f y) then x # y # ys else y # linorder.insort_key less_eq f x ys)"
simps_7: fixes C :: "'f"   and p :: "('e, 'f, 'd) formula" shows "FVF (InContext C p) = UNIV"
nth_take_lemma: fixes k :: "nat"   and xs :: "'c list"   and ys :: "'c list" assumes "k \<le> length xs"   and "k \<le> length ys"   and "\<And>i. i < k \<longrightarrow> xs ! i = ys ! i" shows "take k xs = take k ys"
FG_consitutents_n0: assumes "of_nat (card G) \<noteq> (0::'d)" shows "0 \<notin> set ActingGroup.FG_constituents"
trss_no_empty_s: fixes c :: "'f list multiset"   and e :: "('g, 'h) label"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" shows "((([], c), e, s', c') \<in> trss fg) = False"
f_def: fixes x :: "'a list set \<times> 'a trm set" shows "f x = (fst x, Subst (snd x) X (Arden X rhs))"
elims: fixes x :: "int list"   and y :: "int list" assumes "Challenge1A.take x = y" obtains "x = []" and "y = []"   | xa :: "int" where "x = [xa__]" and "y = [xa__]"   | xa :: "int" and ya :: "int" and xs :: "int list" where "x = xa # ya # xs" and "y = (if xa < ya then take_incr (xa # ya # xs) else take_decr (xa # ya # xs))"
strict_implies_not_eq: fixes a :: "'c list"   and b :: "'c list" assumes "strict_suffix a b" shows "a \<noteq> b"
interval_tendsto_neq: fixes f :: "nat \<Rightarrow> real interval"   and a :: "real"   and b :: "real" assumes "f \<longlonglongrightarrow>\<^sub>i a"   and "a \<noteq> b" shows "\<exists>n. \<not> b \<in>\<^sub>i f n"
mksynlr_inverse: fixes y :: "('a \<times> 'b) set" assumes "y \<in> {x. x \<in> synlr}" shows "unsynlr (mksynlr y) = y"
map_upds_twist: fixes a :: "'c"   and as :: "'c list"   and m :: "'c \<Rightarrow> 'd option"   and b :: "'d"   and bs :: "'d list" assumes "a \<notin> set as" shows "m(a \<mapsto> b, as [\<mapsto>] bs) = m(as [\<mapsto>] bs, a \<mapsto> b)"
HRP_ORD_OF: fixes i :: "nat" shows "{} \<turnstile> HRP (ORD_OF i) \<guillemotleft>ORD_OF i\<guillemotright>"
simps_2: fixes indis :: "('a \<times> 'a) set"   and aval :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('c, 'b) com"   and c2 :: "('c, 'b) com" assumes "PL_Indis indis" shows "PL_Indis.SC_discr aval indis (c1 ;; c2) = (PL_Indis.SC_discr aval indis c1 \<and> PL_Indis.SC_discr aval indis c2)"
\<dagger>: shows "{y. y = x \<and> (y, x) \<in> r} = {}"
ring_zero: fixes R :: "('c, 'd) Ring_scheme" assumes "Ring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
hyps: shows "FWNormalisationCore.sorted xs l \<longrightarrow> all_in_list (x # xs) l \<longrightarrow> distinct (x # xs) \<longrightarrow> singleCombinators [x] \<longrightarrow> singleCombinators xs \<longrightarrow> FWNormalisationCore.sorted (FWNormalisationCore.insort x xs l) l"
half_gt_zero: 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less zero (divide a (numeral.numeral one plus (num.Bit0 num.One)))"
induct: fixes x :: "('a, 'b) trm"   and P :: "('a, 'b) trm \<Rightarrow> bool" assumes "dfree x"   and "\<And>i. P (Var i)"   and "\<And>r. P (Const r)"   and "\<And>args i. \<lbrakk>\<And>i. dfree (args i); \<And>i. P (args i)\<rbrakk> \<Longrightarrow> P ($f i args)"   and "\<And>\<theta>\<^sub>1 \<theta>\<^sub>2. \<lbrakk>dfree \<theta>\<^sub>1; P \<theta>\<^sub>1; dfree \<theta>\<^sub>2; P \<theta>\<^sub>2\<rbrakk> \<Longrightarrow> P (Plus \<theta>\<^sub>1 \<theta>\<^sub>2)"   and "\<And>\<theta>\<^sub>1 \<theta>\<^sub>2. \<lbrakk>dfree \<theta>\<^sub>1; P \<theta>\<^sub>1; dfree \<theta>\<^sub>2; P \<theta>\<^sub>2\<rbrakk> \<Longrightarrow> P (Times \<theta>\<^sub>1 \<theta>\<^sub>2)" shows "P x"
rel_mono_strong: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a stream"   and y :: "'b stream"   and Ra :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "stream_all2 R x y"   and "\<And>z yb. \<lbrakk>z \<in> sset x; yb \<in> sset y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "stream_all2 Ra x y"
ii: fixes s :: "char list \<Rightarrow> int" assumes "I 0 s" shows "\<not> bval b s"
f5: fixes a :: "'a" shows "\<not> a \<preceq> hp \<or> a \<preceq> h"
app: fixes t :: "'a"   and u :: "'a" shows "shadows_consts (app t u) = (shadows_consts t \<or> shadows_consts u)"
log_nz: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t" shows "\<Gamma>,\<gamma>,p\<turnstile> Rule m Log # rs \<Rightarrow>\<^sub>z t"
assms_7: shows "x ^ e mod (P * Q) = y ^ e mod (P * Q)"
simps_2: fixes h :: "'b \<times> bool"   and t :: "('b \<times> bool) list" shows "prefixBeforeLastMarked (h # t) = (if isDecision h \<and> markedElements t = [] then [] else h # prefixBeforeLastMarked t)"
no_renegotiation_union: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and h :: "'f"   and X :: "'d set"   and Y :: "'d set" assumes "ContractsWithUnilateralSubstitutesAndIRC Xd Xh Pd Ch"   and "Xd x \<notin> Xd ` Ch h X"   and "x \<in> X" shows "x \<notin> Ch h (X \<union> Y)"
FIX_def: fixes \<phi> :: "((Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "FIX \<phi> = (\<lambda>(s, t). \<forall>\<Phi>. (\<forall>ss tt. \<phi> \<Phi> (ss, tt) \<longrightarrow> \<Phi> (ss, tt)) \<longrightarrow> \<Phi> (s, t))"
plotkin_exists_def: fixes f :: "ValD" shows "plotkin_exists f \<equiv> if appF\<cdot>f\<cdot>\<bottom> = ValFF then ValFF else if \<exists>n. appF\<cdot>f\<cdot>n = ValTT then ValTT else \<bottom>"
1: shows "dom fsubs \<subseteq> tvs f \<union> dom subs"
substBinp_preserves_wls: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var" assumes "wlsBinp delta binp"   and "wls (asSort ys) Y" shows "wlsBinp delta (binp %%[Y / y]_ys)"
simps_1: fixes x :: "'a" shows "primify (Inl x) = {Inl x, Inr x}"
real_of_ereal_positive_mono: fixes x :: "ereal"   and y :: "ereal" assumes "0 \<le> x"   and "x \<le> y"   and "y \<noteq> \<infinity>" shows "real_of_ereal x \<le> real_of_ereal y"
I_def: shows "I = {m. MPoly_Type.coeff p m \<noteq> 0}"
barityOf_lt_var_INNER: fixes delta :: "'opSym" assumes "wlsOpS delta" shows "|{i. barOf delta i \<noteq> None}| <o |UNIV|"
rep_eq: fixes x :: "'a Matrix.vec" shows "dim_vec x = fst (Rep_vec x)"
convolution_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "convolution f g = (\<lambda>x. \<Sum>y | x - y \<in> supp f \<and> y \<in> supp g. f (x - y) * g y)"
rep_eq: fixes x :: "rat"   and xa :: "linear_poly" shows "Rep_linear_poly (x *R xa) = fun_scale x (Rep_linear_poly xa)"
bisim_composeE: fixes bisim12 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and bisim23 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s3 :: "'i" assumes "(bisim12 \<circ>\<^sub>B bisim23) s1 s3" obtains s2 :: "'h" where "bisim12 s1 s2" and "bisim23 s2 s3"
Least_equality: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq x y" shows "ord.Least less_eq P = x"
hom_lead_coeff: fixes p :: "int poly" shows "Polynomial.lead_coeff (Ring_Hom_Poly.of_int_poly p) = of_int (Polynomial.lead_coeff p)"
name_list_set_fresh: fixes xvec :: "name list"   and x :: "'c" shows "set xvec \<sharp>* x = xvec \<sharp>* x"
not_top_less: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top :: "'c"   and a :: "'c" assumes "class.order_top less_eq less top" shows "\<not> less top a"
tensor_in_hom: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set"   and g :: "'c local.term set"   and c :: "'c local.term set"   and d :: "'c local.term set" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>"   and "\<guillemotleft>g : c \<rightarrow>\<^sub>S d\<guillemotright>" shows "\<guillemotleft>f \<otimes>\<^sub>S g : a \<otimes>\<^sub>S c \<rightarrow>\<^sub>S b \<otimes>\<^sub>S d\<guillemotright>"
one_eq: shows "\<one>\<^bsub>G\<^esub> = 1"
seqr_left_one_point: fixes x :: "'b \<Longrightarrow> 'c"   and P :: "('d, 'c) urel"   and v :: "'b"   and Q :: "('c, 'e) urel" assumes "vwb_lens x" shows "(P \<and> $x\<acute> =\<^sub>u \<guillemotleft>v\<guillemotright>) ;; Q = P\<lbrakk>\<guillemotleft>v\<guillemotright>/$x\<acute>\<rbrakk> ;; Q\<lbrakk>\<guillemotleft>v\<guillemotright>/$x\<rbrakk>"
strict_trans1: fixes b :: "'g"   and a :: "'g"   and c :: "'g" assumes "b \<le> a"   and "c < b" shows "c < a"
dg_Set_components_1: fixes \<alpha> :: "V" shows "dg_Set \<alpha>\<lparr>Obj\<rparr> = Vset \<alpha>"
exec_meth_take_xt': fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'a instr list"   and ins' :: "'a instr list"   and xt' :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'b"   and h :: "'c"   and stk :: "'a val list"   and loc :: "'a val list"   and pc :: "nat"   and xcp :: "'a option"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and h' :: "'c"   and s' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option" assumes "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P (ins @ ins') (xt' @ xt) t h (stk, loc, pc, xcp) ta h' s'"   and "pc < length ins"   and "pc \<notin> pcs xt" shows "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins xt' t h (stk, loc, pc, xcp) ta h' s'"
bij_betw_singleton_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and g :: "'a \<Rightarrow> 'b"   and a :: "'a" assumes "bij_betw f A B"   and "bij_betw g A B"   and "a \<in> A"   and "\<And>x. \<lbrakk>x \<in> A; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x" shows "f a = g a"
r_cancel_one': fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier S"   and "a \<in> carrier S" shows "(x = a \<oplus>\<^bsub>S\<^esub> x) = (a = \<zero>\<^bsub>S\<^esub>)"
ltakeWhile_cong: fixes xs :: "'c llist"   and ys :: "'c llist"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> lset ys \<Longrightarrow> P x = Q x" shows "ltakeWhile P xs = ltakeWhile Q ys"
lift_imp_trans: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and C :: "'a \<Rightarrow> bool" assumes "\<turnstile> A \<longrightarrow> B"   and "\<turnstile> B \<longrightarrow> C" shows "\<turnstile> A \<longrightarrow> C"
sup_bot_right: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and bot :: "'d"   and x :: "'d" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup x bot = x"
simps: fixes p :: "real poly"   and qs :: "real poly list" shows "calculate_data_R p qs = (let len = length qs in if len = 0 then case reduce_system_R p ([1], base_case_info_R) of (a, b, c) \<Rightarrow> (a, b, map (drop 1) c) else if len \<le> 1 then reduce_system_R p (qs, base_case_info_R) else let q1 = take (len div 2) qs; left = calculate_data_R p q1; q2 = drop (len div 2) qs; right = calculate_data_R p q2 in Let (combine_systems_R p (q1, left) (q2, right)) (reduce_system_R p))"
const_on_def: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and x :: "'c" shows "const_on f S x = (\<forall>y\<in>S. f y = x)"
simps_3: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and ao :: "'a option"   and uu :: "'a list" shows "proper_intrvl.proper_interval_Compl_set_aux less proper_interval ao uu [] = False"
fresh_asTerm_qFresh: fixes qX :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "qGood qX" shows "fresh xs x (asTerm qX) = qFresh xs x qX"
rtrancl3p_converseD: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and bs :: "'b list"   and a' :: "'a" assumes "rtrancl3p r a bs a'" shows "rtrancl3p (converse3p r) a' (rev bs) a"
simps: fixes Ps :: "('d multiset \<Rightarrow> 'a) list"   and i :: "'d" shows "indexed_eval Ps i = indexed_eval_aux (rev Ps) i"
inD_\<psi>: fixes b :: "'d"   and a :: "'d"   and x :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "D.ide b"   and "D.ide a"   and "x \<in> inD ` D.hom b a" shows "inD (HomD.\<psi> (b, a) x) = x"
prj_simps_8: shows "C.cod trg_trg_f.Prj\<^sub>0\<^sub>1 = trg.apex"
match_simplematcher_SrcDst_not_2: fixes \<alpha> :: "action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and X :: "'a ipt_iprange"   and a :: "action"   and p :: "('a, 'b) tagged_packet_scheme" shows "matches (common_matcher, \<alpha>) (MatchNot (Match (Dst X))) a p = (p_dst p \<notin> ipt_iprange_to_set X)"
U_3: shows "emeasure lborel (U - S') < ennreal e"
inj_prod_encode: fixes A :: "(nat \<times> nat) set" shows "inj_on prod_encode A"
wf_INTERSECT: fixes n :: "nat"   and xs :: "'b rexp list" shows "local.wf n (INTERSECT xs) = (\<forall>r\<in>set xs. local.wf n r)"
Rep_formula: fixes x :: "('a, 'b, 'c, 'd) formula" shows "Rep_formula x \<in> {t\<^sub>\<alpha>. hereditarily_fs t\<^sub>\<alpha>}"
is_applicable_then_enabled_1: fixes ast_op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "ast_op \<in> set ast\<delta>" shows "\<exists>ast_op'. lookup_operator ((fst \<circ> the \<circ> lookup_action \<circ> abs_ast_operator) ast_op) = Some ast_op'"
IH: fixes i :: "nat" assumes "\<And>i. coeff p i \<in> R" shows "coeff (p \<circ>\<^sub>p q) i \<in> R"
cases: fixes x :: "('t \<Rightarrow>\<^sub>0 'c) list" obtains x :: "'t \<Rightarrow>\<^sub>0 'c" and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" where "xa__ = x # xs"   | "xa__ = []"
a_Hcarr: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme"   and h :: "'c" assumes "additive_subgroup H G"   and "h \<in> H" shows "h \<in> carrier G"
Int_mono: fixes A :: "'e set"   and C :: "'e set"   and B :: "'e set"   and D :: "'e set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<inter> B \<subseteq> C \<inter> D"
find_first_correct_1: fixes x :: "'a"   and xs :: "'a list"   and i :: "nat" assumes "find_first x xs = Some i" shows "i < length xs"
filt_equiv_space: fixes F :: "'b \<Rightarrow> 'c measure"   and M :: "'c measure"   and N :: "'c measure" assumes "filt_equiv F M N" shows "space M = space N"
is_none_code_2: fixes x :: "'a" shows "Option.is_none \<lfloor>x\<rfloor> = False"
sup: shows "{0..<card ({0..<M} \<inter> v0)} \<subseteq> f ` ({0..<M} \<inter> v0)"
project_set_extend_set_Int: fixes A :: "'a set"   and B :: "'c set" shows "project_set h (extend_set h A \<inter> B) = A \<inter> project_set h B"
unique_append: fixes l' :: "('a \<times> 'b) list"   and l :: "('a \<times> 'b) list" assumes "unique l'"   and "unique l"   and "\<forall>(x, y)\<in>set l. \<forall>(x', y')\<in>set l'. x' \<noteq> x" shows "unique (l @ l')"
simps: fixes g :: "(location \<times> location) set"   and a :: "location \<Rightarrow> char list set"   and c :: "actor \<Rightarrow> char list set \<times> char list set"   and l :: "location \<Rightarrow> char list \<times> ((actor \<times> actor set) \<times> nat) set" shows "cgra (Lgraph g a c l) = c"
arccos_ubound: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "arccos y \<le> pi"
associatedI2: fixes G :: "('c, 'd) monoid_scheme"   and u :: "'c"   and a :: "'c"   and b :: "'c" assumes "Group.monoid G"   and "u \<in> Units G"   and "a = b \<otimes>\<^bsub>G\<^esub> u"   and "b \<in> carrier G" shows "a \<sim>\<^bsub>G\<^esub> b"
ss: shows "set ss = (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
real_sqrt_abs2: fixes x :: "real" shows "sqrt (x * x) = \<bar>x\<bar>"
simps: fixes r :: "'a \<Rightarrow> 'b \<times> 'c"   and a :: "'b \<times> 'c \<Rightarrow> 'a" shows "repr (Tuple_Isomorphism r a) = r"
P0: shows "x = [p] \<cdot> \<one> [^] nat (ord_Zp x) \<otimes> ac_Zp x"
prems_2: fixes i :: "nat" assumes "i < Suc k" shows "finite (A i)"
nop_vdomain: shows "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> 2\<^sub>\<nat>"
thetaIfW_Wbis: shows "thetaIfW \<subseteq> Wbis"
simps_2: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and a :: "'d node_action"   and i :: "nat"   and v :: "nat"   and va :: "nat set"   and s :: "'c net_state"   and s' :: "'c net_state" shows "net_ip_action np a i \<langle>v; va\<rangle> s s' = True"
simps_12: fixes i :: "int"   and a :: "MIR.num" shows "zlfm (NDvd i a) = (if i = 0 then zlfm (NEq a) else let (c, r) = zsplit0 a in if c = 0 then NDvd \<bar>i\<bar> r else if 0 < c then Or (NEq (Sub (Floor r) r)) (NDvd \<bar>i\<bar> (CN 0 c (Floor r))) else Or (NEq (Sub (Floor r) r)) (NDvd \<bar>i\<bar> (CN 0 (- c) (Neg (Floor r)))))"
measure_times: fixes A :: "'i \<Rightarrow> 'a set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)" shows "emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
well_shaped_rchild: fixes t :: "'b tree" assumes "well_shaped t" shows "well_shaped (rchild t)"
that: fixes a :: "'a"   and v :: "'a llist" assumes "w = <a> $ v" shows "thesis"
minus_Mp_2: fixes f :: "int poly"   and g :: "int poly" shows "q.eq_m (f - q.Mp g) (f - g)"
zero_applied_to: fixes x :: "'c" shows "0 x = (0::'b)"
Pair_inject: fixes a :: "'h"   and b :: "'i"   and a' :: "'h"   and b' :: "'i" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
M\<^sub>R_2: shows "\<forall>i\<le>n. \<forall>j\<le>n. M\<^sub>R i 0 = \<infinity> \<and> 0 < j \<and> i \<noteq> j \<longrightarrow> M\<^sub>R i j = \<infinity> \<and> M\<^sub>R j i = \<infinity>"
indRelR_modulo_pred_impl_indRelRT_modulo_pred: fixes Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" shows "(\<forall>(P, Q)\<in>indRelR. Pred (P, Q)) = (\<forall>TRel. (\<forall>(TP, TQ)\<in>TRel. Pred (TargetTerm TP, TargetTerm TQ)) = (\<forall>(P, Q)\<in>indRelRT TRel. Pred (P, Q)))"
ag_l_zero: fixes x :: "'a" assumes "x \<in> carrier M" shows "\<zero> \<plusminus> x = x"
brw_algo_def: fixes Qi :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set" shows "brw_algo Qi \<delta> \<equiv> \<lparr>wa_cond = brw_cond Qi, wa_step = brw_step \<delta>, wa_initial = brw_initial \<delta>, wa_invar = brw_invar \<delta>\<rparr>"
schnorr_sigma_protocol_def: fixes \<G> :: "'a cyclic_group" shows "schnorr_sigma_protocol \<G> \<equiv> schnorr_base \<G> \<and> Cyclic_Group.cyclic_group \<G>"
foldl_spmf_bind_spmf: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and p :: "'e spmf"   and g :: "'e \<Rightarrow> 'c spmf"   and xs :: "'d list" shows "foldl_spmf f (p \<bind> g) xs = p \<bind> (\<lambda>a. foldl_spmf f (g a) xs)"
zn_z: shows "hom_boundary (int n) (usphere n) (equator n) zn = z"
rax: shows "r 0 0 \<and> r 0 1 \<and> r 0 2 \<and> r 0 3 \<or> r 1 0 \<and> r 1 1 \<and> r 1 2 \<and> r 1 3 \<or> r 2 0 \<and> r 2 1 \<and> r 2 2 \<and> r 2 3 \<or> r 3 0 \<and> r 3 1 \<and> r 3 2 \<and> r 3 3"
assms: shows "\<turnstile>x1 # x2 \<oplus>\<^sub>\<or>\<^bsup>x\<^esup>"
assms_1: shows "has_subprob_density M (count_space A) f"
surj_hom_subgroup: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and F :: "('e, 'f) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e" assumes "subgroup H G"   and "group_hom G F \<phi>"   and "\<phi> ` carrier G = carrier F" shows "subgroup (\<phi> ` H) F"
Zfun: fixes g :: "'d \<Rightarrow> 'a"   and F :: "'d filter" assumes "Zfun g F" shows "Zfun (\<lambda>x. f (g x)) F"
IH: fixes n :: "nat" assumes "wf_formula n \<phi>" shows "lang n (rexp.Inter (rexp_of_alt n \<phi>) (ENC n (FOV \<phi>))) = lang n (rexp.Inter (rexp_of_alt' n \<phi>) (ENC n (FOV \<phi>)))"
consistent_eq_rank_ge_code: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "consistent A b = (let GJ_P = Gauss_Jordan_PA A; P_mult_b = fst GJ_P *v b in Let (if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (snd GJ_P) \<noteq> 0) + 1) ((\<le>) (if \<exists>a. P_mult_b $ a \<noteq> (0::'b) then mod_type_class.to_nat (GREATEST a. P_mult_b $ a \<noteq> (0::'b)) + 1 else 0)))"
clsupp_g_le_W: fixes i :: "nat" assumes "i \<in> N" shows "closure (support_on carrier (g i)) \<subseteq> W i"
analz_analzD: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> analz (analz H)" shows "X \<in> analz H"
provRuleCharacterise: fixes Ps :: "sequent list"   and C :: "sequent" assumes "(Ps, C) \<in> provRules" shows "\<exists>F x A. (C = ( \<Empt> \<Rightarrow>* \<LM> F \<nabla> [x].A  \<RM>) \<or> C = ( \<LM> F \<nabla> [x].A  \<RM> \<Rightarrow>* \<Empt>)) \<and> x \<sharp> set_of_prem ( Ps - A )"
step_a: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and u :: "'f \<Rightarrow> 'g"   and a :: "'h"   and l' :: "'e"   and u' :: "'f \<Rightarrow> 'g" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsub>a\<^esub> \<langle>l', u'\<rangle>" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"
closure_Times: fixes A :: "'b set"   and B :: "'c set" shows "closure (A \<times> B) = closure A \<times> closure B"
\<R>_def: shows "\<R>\<^sub>\<beta> \<equiv> {Regions_Beta.region X I J r |I J r. Regions_Beta.valid_region X k I J r}"
one_ge_zero: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.ordered_semiring_1 one times plus zero less_eq" shows "less_eq zero one"
disj_CSP: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and Q :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "is_CSP_process P"   and "is_CSP_process Q" shows "is_CSP_process (P \<or> Q)"