hcju/isabelleps huggingface dataset:
normal_series_subgroups: fixes i :: "nat" assumes "i < length \<HH>" shows "subgroup (\<HH> ! i) G"
ipresIGCons_def: fixes h :: "'e \<Rightarrow> 'f"   and hA :: "'g \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'i, 'e, 'g, 'j) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'k, 'opSym, 'i, 'f, 'h, 'l) model_scheme" shows "ipresIGCons h hA MOD MOD' \<equiv> ipresIGVar h MOD MOD' \<and> ipresIGAbs h hA MOD MOD' \<and> ipresIGOp h hA MOD MOD'"
Rep_resumption_inject: fixes x :: "('h, 'g, 'f, 'i) resumption"   and y :: "('h, 'g, 'f, 'i) resumption" shows "(Rep_resumption x = Rep_resumption y) = (x = y)"
faces_distinct_def: fixes g :: "graph" shows "faces_distinct g \<equiv> distinct (normFaces (faces g))"
error_monad_bind_split: fixes P :: "'a + 'b \<Rightarrow> bool"   and m :: "'a + 'c"   and f :: "'c \<Rightarrow> 'a + 'b" shows "P (m \<bind> f) = ((\<forall>v. m = Inl v \<longrightarrow> P (Inl v)) \<and> (\<forall>v. m = Inr v \<longrightarrow> P (f v)))"
Un_mono: fixes A :: "'d set"   and C :: "'d set"   and B :: "'d set"   and D :: "'d set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<union> B \<subseteq> C \<union> D"
bst_iff_sorted_wrt_less: fixes t :: "'a tree" shows "bst t = Sorted_Less.sorted (inorder t)"
image_mem_set_rel_iff: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c set"   and y :: "'b set"   and R :: "('a \<times> 'b) set" shows "((f ` x, y) \<in> \<langle>R\<rangle>set_rel) = ((x, y) \<in> \<langle>br f top O R\<rangle>set_rel)"
fpower_inductl: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'b"   and i :: "nat" assumes "mono f"   and "mono g"   and "h \<le> g \<sqinter> (f \<circ> h)" shows "h \<le> fpower f i \<circ> g"
subexps_induct: fixes Rs :: "'c exp list \<Rightarrow> bool"   and R :: "'c exp \<Rightarrow> bool"   and es :: "'c exp list" assumes "\<And>es. subexps es = {} \<Longrightarrow> Rs es"   and "\<And>e. (\<And>e'. subexp_of e' e \<Longrightarrow> R e') \<Longrightarrow> R e"   and "\<And>es. (\<And>e'. e' \<in> subexps es \<Longrightarrow> R e') \<Longrightarrow> Rs es" shows "(\<forall>e'. e' \<in> subexps es \<longrightarrow> R e') \<and> Rs es"
exp_is_max_div: fixes m :: "'a"   and p :: "'a" assumes "m \<noteq> (0::'a)"   and "prime p" shows "\<not> p dvd m div p ^ multiplicity p m"
basic_trans_rules_32: fixes a :: "'d"   and b :: "'d"   and A :: "'d set" assumes "a = b"   and "b \<in> A" shows "a \<in> A"
Inv2c_inner_def: fixes s :: "state"   and p :: "Proc" shows "Inv2c_inner s p = ((phase s p = 0 \<longrightarrow> dblock s p = InitDB \<and> disksWritten s p = {} \<and> (\<forall>d. \<forall>br\<in>blocksRead s p d. proc br = p \<and> block br = disk s d p)) \<and> (phase s p \<noteq> 0 \<longrightarrow> mbal (dblock s p) \<in> Ballot p \<and> bal (dblock s p) \<in> Ballot p \<union> {0} \<and> (\<forall>d. \<forall>br\<in>blocksRead s p d. mbal (block br) < mbal (dblock s p))) \<and> (phase s p \<in> {2, 3} \<longrightarrow> bal (dblock s p) = mbal (dblock s p)) \<and> outpt s p = (if phase s p = 3 then inp (dblock s p) else NotAnInput) \<and> chosen s \<in> allInput s \<union> {NotAnInput} \<and> (\<forall>p. inpt s p \<in> allInput s \<and> (chosen s = NotAnInput \<longrightarrow> outpt s p = NotAnInput)))"
execute_trap_def: fixes uu :: "unit" shows "execute_trap uu \<equiv> do select_trap (); err_mode \<leftarrow> gets err_mode_val; if err_mode = True then return () else do psr_val \<leftarrow> gets (cpu_reg_val PSR); s_val \<leftarrow> gets (\<lambda>s. ucast (get_S psr_val)); curr_win \<leftarrow> get_curr_win (); new_cwp \<leftarrow> gets (\<lambda>s. word_of_int ((uint curr_win - 1) mod NWINDOWS)); new_psr_val \<leftarrow> gets (\<lambda>s. update_PSR_exe_trap new_cwp 0 s_val psr_val); write_cpu new_psr_val PSR; exe_trap_st_pc (); exe_trap_wr_pc (); return () od od"
induct: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>list. P (TVar list)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P ty1 \<rightarrow> ty2" shows "P ty"
subst_cls_lists_nth: fixes Cs :: "'a literal multiset list"   and \<sigma>s :: "'s list"   and i :: "nat" assumes "length Cs = length \<sigma>s"   and "i < length Cs" shows "(Cs \<cdot>\<cdot>cl \<sigma>s) ! i = Cs ! i \<cdot> \<sigma>s ! i"
f: fixes x :: "'a" assumes "x \<in> s - t" shows "f x \<le> (0::'b)"
dim_col: fixes A :: "'c mat"   and i :: "nat" shows "dim_vec (col A i) = dim_row A"
unat_mask_word64: fixes n :: "nat" shows "unat (mask n) = 2 ^ min n word_bits - 1"
push_forall_eval: fixes \<phi> :: "atom fm" shows "\<forall>xs. eval \<phi> xs = eval (push_forall \<phi>) xs"
5: shows "\<forall>f\<in>set F'. \<forall>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>) \<longrightarrow> f \<cdot>\<^sub>p \<delta> \<circ>\<^sub>s \<I> \<notin> (set (map snd D) \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>) - (set (map snd (dbproj i D)) \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>)"
assms_1: fixes i :: "nat" shows "A i \<in> sets M"
distinct_7: fixes x1 :: "'a" shows "prot_atom.Atom x1 \<noteq> Bottom"
hyps: shows "(set l' = {a}) = (remdups l' = [a])"
similar_selectlike_mem: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and V :: "variable set"   and X :: "(variable \<Rightarrow> real) set" assumes "Vagree \<nu> \<omega> V" shows "(\<nu> \<in> selectlike X \<omega> V) = (\<nu> \<in> X)"
spref_take: fixes p :: "'b list"   and w :: "'b list" assumes "p <p w" shows "take \<^bold>|p\<^bold>| w = p"
bernoulli_density_def: fixes p :: "real"   and b :: "bool" shows "bernoulli_density p b = ennreal (if p \<in> {0..1} then if b then p else 1 - p else 0)"
sum_tendsto_fds_perzeta: fixes s :: "complex"   and q :: "real" assumes "1 < Re s" shows "(\<lambda>n. \<Sum>k\<in>{0<..n}. exp (complex_of_real (2 * real k * pi * q) * \<i>) * of_nat k powr - s) \<longlonglongrightarrow> eval_fds (fds_perzeta q) s"
try_spmf_bind_out1: fixes f :: "'b \<Rightarrow> 'c spmf"   and p :: "'b spmf"   and q :: "'b spmf" assumes "\<And>x. lossless_spmf (f x)" shows "TRY p ELSE q \<bind> f = TRY p \<bind> f ELSE (q \<bind> f)"
norm_PXtrans2: fixes P :: "pol"   and x :: "nat"   and Q :: "pol"   and Q2 :: "pol"   and n :: "nat" assumes "isnorm (PX P x Q)"   and "isnorm Q2" shows "isnorm (PX P (Suc (n + x)) Q2)"
mono_Det: fixes P :: "'b process"   and Q :: "'b process"   and S :: "'b process" assumes "P \<sqsubseteq> Q" shows "(P [+] S) \<sqsubseteq> (Q [+] S)"
distinct_13: fixes x5 :: "'b" shows "Null \<noteq> Addr x5"
basis_in_finite: fixes \<E> :: "'a set"   and A :: "'a set" assumes "\<E> \<subseteq> carrier"   and "basis_in \<E> A" shows "finite A"
group_left_neutral: fixes a :: "'e" shows "(1::'e) \<cdot> a = a"
det_smult: fixes a :: "'b"   and A :: "'b mat" shows "Determinant.det (a \<cdot>\<^sub>m A) = a ^ dim_col A * Determinant.det A"
map_eq_Cons_conv: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and y :: "'d"   and ys :: "'d list" shows "(map f xs = y # ys) = (\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys)"
all_zero_below: shows "\<forall>a>0::'rows. bezout_iterate (interchange_rows A (0::'rows) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a))) (nrows A - Suc 0) (0::'rows) (mod_type_class.from_nat k) bezout $ a $ mod_type_class.from_nat k = (0::'a)"
subst_typ'_nil: fixes A :: "Core.term" shows "subst_typ' [] A = A"
col_in_angle_out: fixes B :: "'p"   and A :: "'p"   and P :: "'p"   and C :: "'p" assumes "Col B A P"   and "\<not> Bet A B C"   and "P InAngle A B C" shows "B Out A P"
absolutely_integrable_on_image: fixes S :: "(real, 'a) vec set"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec" assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S" shows "f absolutely_integrable_on g ` S"
distinct_keys: fixes t :: "('d, 'e) RBT.rbt" shows "distinct (RBT.keys t)"
B_trusts_OR3: fixes B :: "agent"   and NA :: "msg"   and A :: "agent"   and X' :: "msg"   and NB :: "msg"   and evs :: "event list"   and X :: "msg"   and K :: "nat" assumes "Says B Server \<lbrace>NA, Agent A, Agent B, X', Crypt (shrK B) \<lbrace>NA, NB, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"   and "Gets B \<lbrace>NA, X, Crypt (shrK B) \<lbrace>NB, Key K\<rbrace>\<rbrace> \<in> set evs"   and "B \<notin> bad"   and "evs \<in> otway" shows "Says Server B \<lbrace>NA, Crypt (shrK A) \<lbrace>NA, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Key K\<rbrace>\<rbrace> \<in> set evs"
size_greater_than_median: fixes xs :: "'b list" shows "size (filter_mset ((<) (median xs)) (mset xs)) \<le> length xs div 2"
coeff_eq_zero_mpoly_to_nested_polyD: fixes mp :: "'b mpoly"   and v :: "nat" assumes "mpoly_to_nested_poly mp v = 0" shows "MPoly_Type.coeff mp 0 = (0::'b)"
r_nth_def: shows "r_nth \<equiv> Cn 2 r_ifz [recf.Id 2 0, r_constn 1 0, Cn 2 r_tuple_nth [Cn 2 r_pdec1 [r_dummy 1 r_dec], Cn 2 r_pdec2 [r_dummy 1 r_dec], recf.Id 2 1]]"
vconverse_vconverse: shows "((r\<inverse>\<^sub>\<circ>)\<inverse>\<^sub>\<circ>)\<inverse>\<^sub>\<circ> = r\<inverse>\<^sub>\<circ>"
neg_mask_is_div: fixes w :: "'b word"   and n :: "nat" shows "w AND NOT (mask n) = w div 2 ^ n * 2 ^ n"
op_set_filter_def: fixes P :: "'g \<Rightarrow> bool"   and s :: "'g set" shows "op_set_filter P s \<equiv> {x \<in> s. P x}"
q: shows "q = p - monom_mult (lookup p (t \<oplus> lt f) / lc f) t f"
mreq_endNone: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option"   and x :: "'e"   and b :: "'d list"   and c :: "'d list"   and a :: "'d" assumes "applied_rule_rev C x b = \<bottom>"   and "applied_rule_rev C x c = \<bottom>" shows "applied_rule_rev C x (a # b) = applied_rule_rev C x (a # c)"
subst_term_eqI: fixes \<sigma> :: "'d \<Rightarrow> ('c, 'e) Term.term"   and \<tau> :: "'d \<Rightarrow> ('c, 'e) Term.term" assumes "\<And>t. t \<cdot> \<sigma> = t \<cdot> \<tau>" shows "\<sigma> = \<tau>"
Mapping_RBT'_inverse: fixes y :: "('b, 'c) rbt" assumes "y \<in> {t. ord.is_rbt cless t \<or> ID ccompare = None}" shows "mapping_rbt.impl_of (Mapping_RBT' y) = y"
same_append_eq: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" shows "(xs @ ys = xs @ zs) = (ys = zs)"
compile_to_env_def: fixes C_info :: "(name, dt_def) fmap" shows "compile_to_env C_info = sem_env C_info \<circ> Rewriting_Sterm.compile \<circ> compress_prule_set \<circ> Rewriting_Pterm.compile \<circ> Compiler.transform_irule_set_iter \<circ> compress_irule_set \<circ> Rewriting_Pterm_Elim.compile \<circ> compress_crule_set \<circ> consts_of \<circ> fcompress \<circ> Rewriting_Nterm.compile' C_info \<circ> fcompress"
pure_pure: fixes P :: "('b \<times> 'a) set" shows "is_pure (pure P)"
la: shows "list_all (\<lambda>m. oiface (fst m) = ifaceAny) (lr_of_tran_fbs rt fw ifs)"
vs_tagged: fixes \<sigma>' :: "'b"   and p :: "'c"   and built_in_chain :: "char list"   and default_policy :: "action"   and t :: "state" shows "rs,stateful_matcher' \<sigma>',p\<turnstile> \<langle>[Rule MatchAny (Call built_in_chain), Rule MatchAny default_policy], Undecided\<rangle> \<Rightarrow> t = rs,stateful_matcher_tagged',packet_tagger' \<sigma>' p\<turnstile> \<langle>[Rule MatchAny (Call built_in_chain), Rule MatchAny default_policy], Undecided\<rangle> \<Rightarrow> t"
simps: fixes B :: "ty"   and C :: "ty"   and X :: "ctrm set" shows "ORRIGHT2 (B OR C) X = {<b>:OrR2 <a>.M b |a b M. fic (OrR2 <a>.M b) b \<and> <a>:M \<in> X}"
Attribute: fixes T :: "Javatype"   and S :: "Javatype"   and f :: "AttId"   and U :: "Javatype" assumes "treach T S"   and "S \<le> dtype f"   and "U \<le> rtype f" shows "treach T U"
liftL_ctxt_type: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and E :: "ctxt"   and T :: "Syntax.type"   and U :: "Syntax.type"   and T1 :: "Syntax.type" assumes "\<Gamma> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : T \<Leftarrow> U" shows "\<forall>k. \<Gamma>\<langle>k:T1\<rangle> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t liftL_ctxt E k : T \<Leftarrow> U"
Suc_3: shows "(P', B', Q') = diagonal_to_Smith_aux_PQ [0..<Suc k] bezout (P, B, Q)"
red_dyn_equiv_red_stat: shows "reducedly_dynamically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F = reducedly_statically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F"
OUadmitFO_def: fixes \<sigma> :: "'h \<Rightarrow> ('i, 'j) trm"   and \<theta> :: "('i + 'h, 'j) ODE"   and U :: "('j + 'j) set" shows "OUadmitFO \<sigma> \<theta> U = ((\<Union>i\<in>{i. Inl (Inr i) \<in> SIGO \<theta>}. FVT (\<sigma> i)) \<inter> U = {})"
nondet_fail_exp_def: fixes return :: "int \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and get :: "(('c \<Rightarrow> int option) \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "('c \<Rightarrow> int option) \<Rightarrow> 'b \<Rightarrow> 'b"   and altc :: "int cset \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b" shows "nondet_fail_exp return bind fail get put altc \<equiv> monad_state_altc return bind get put altc \<and> memoization return bind get put \<and> monad_fail return bind fail"
low_mds_eq_sym: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2" shows "mem\<^sub>2 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>1"
qf3_even_general: fixes A :: "int" assumes "is_qfN A 3"   and "even A" shows "\<exists>B. A = 4 * B \<and> is_qfN B 3"
assms_4: fixes z :: "'a" assumes "z \<in> A"   and "x \<sqsubseteq> z"   and "y \<sqsubseteq> z" shows "thesis"
elims_2: fixes x :: "bval"   and xa :: "obs" assumes "bs_observe x xa" obtains n :: "nat" and n' :: "nat" where "x = BNat n" and "xa = ONat n'" and "n = n'"   | xb :: "nat" and e :: "exp" and \<rho> :: "(nat \<times> bval) list" where "x = BClos xb e \<rho>" and "xa = OFun"
contractible_space_lower_hemisphere: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "contractible_space (subtopology (nsphere n) {x. x k \<le> 0})"
closed_segment_def: fixes a :: "'b"   and b :: "'b" shows "{a--b} = {(1 - u) *\<^sub>R a + u *\<^sub>R b |u. 0 \<le> u \<and> u \<le> 1}"
list_remove1_def: fixes equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "equal_base.list_remove1 equal \<equiv> \<lambda>uu uua. rec_list (\<lambda>x. []) (\<lambda>y xs xsa x. if equal x y then xs else y # xsa x) uua uu"
commit_def: fixes ck :: "'a"   and m :: "nat" shows "commit ck m = sample_uniform (order \<G>) \<bind> (\<lambda>d. let c = \<^bold>g [^] d \<otimes> ck [^] m in return_spmf (c, d))"
Indirect: fixes s :: "Store"   and l :: "Location"   and k :: "Location" assumes "s\<turnstile> l reachable_from (s@@k)"   and "ref k \<noteq> nullV" shows "s\<turnstile> l reachable_from ref k"
bind_returns_error_I2: fixes f :: "('l, 'm, 'n) prog"   and h :: "'l"   and x :: "'n"   and g :: "'n \<Rightarrow> ('l, 'm, 'o) prog"   and e :: "'m" assumes "pure f h"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> g x \<rightarrow>\<^sub>e e" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>e e"
that: fixes q :: "int mpoly"   and r :: "int mpoly" assumes "p = (MPoly_Type.monom (monomial (Suc 0) x') 1 - s) * q + r"   and "x' \<notin> vars r" shows "thesis"
\<phi>R_def: shows "\<phi>R \<equiv> diag (neg P')"
sortedP_selsort: fixes xs :: "(real \<times> real) list"   and X :: "'a \<times> (real \<times> real) pdevs" assumes "set xs \<subseteq> set (inl (snd X))" shows "ccw.sortedP 0 (ccw.selsort 0 xs)"
prv_imp_imp_com: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (imp \<phi> (imp \<chi> \<psi>)) (imp \<chi> (imp \<phi> \<psi>)))"
d\<^sub>\<kappa>_proper: fixes u :: "\<nu>" shows "d\<^sub>\<kappa> (u\<^sup>P) = Some u"
simps: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and a1 :: "'b"   and a2 :: "'a list"   and a3 :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry" shows "CFG.path sourcenode targetnode valid_edge a1 a2 a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> CFG.valid_node sourcenode targetnode valid_edge n) \<or> (\<exists>n'' as n' a n. a1 = n \<and> a2 = a # as \<and> a3 = n' \<and> CFG.path sourcenode targetnode valid_edge n'' as n' \<and> valid_edge a \<and> sourcenode a = n \<and> targetnode a = n''))"
a_star_redu_ImpL_elim: fixes a :: "coname"   and M :: "trm"   and y :: "name"   and N :: "trm"   and z :: "name"   and R :: "trm" assumes "ImpL <a>.M y.N z \<longrightarrow>\<^sub>a* R" shows "\<exists>M' N'. R = ImpL <a>.M' y.N' z \<and> M \<longrightarrow>\<^sub>a* M' \<and> N \<longrightarrow>\<^sub>a* N'"
is_element_ptr_kind_def: fixes ptr :: "(_) node_ptr" shows "is_element_ptr_kind ptr = (case cast ptr of None \<Rightarrow> False | Some x \<Rightarrow> True)"
assms_1: shows "[a] \<preceq>\<^sub>F\<^sub>I u @- v"
MapsTo_def: fixes CC :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d" shows "f maps\<^bsub>CC\<^esub> X to Y \<equiv> f \<in> mor\<^bsub>CC\<^esub> \<and> dom\<^bsub>CC\<^esub> f = X \<and> cod\<^bsub>CC\<^esub> f = Y"
sbintrunc_sbintrunc: fixes m :: "nat"   and w :: "int" shows "signed_take_bit m (signed_take_bit m w) = signed_take_bit m w"
dining_cryptographers_space_def: fixes n :: "nat" shows "dining_cryptographers_space n \<equiv> 3 \<le> n"
filter_sup_least_upper_bound: fixes H :: "'a set"   and F :: "'a set"   and G :: "'a set" assumes "ord_class.filter H"   and "F \<subseteq> H"   and "G \<subseteq> H" shows "filter_sup F G \<subseteq> H"
card_permutations_of_set: fixes A :: "'a set" assumes "finite A" shows "card (permutations_of_set A) = fact (card A)"
is_monomial_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'b" shows "is_monomial p = (card (keys p) = 1)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>B.\<rr> f : B.R a \<rightarrow>\<^sub>B B.map b\<guillemotright>"
2: shows "insert q (G - {p}) \<subseteq> dgrad_p_set d n"
snd_pseudo_mod_smult_invar_left: fixes lc :: "'b"   and q :: "'b poly"   and x :: "'b"   and r :: "'b poly"   and d :: "'b poly"   and dr :: "nat"   and n :: "nat"   and q' :: "'b poly" shows "snd (pseudo_divmod_main lc q (Polynomial.smult x r) d dr n) = Polynomial.smult x (snd (pseudo_divmod_main lc q' r d dr n))"
basis_reduction_def: fixes \<alpha> :: "rat"   and n :: "nat"   and fs :: "int vec list" shows "basis_reduction \<alpha> n fs = (let m = length fs in LLL_Impl.basis_reduction_main \<alpha> n m True 0 (initial_state m fs))"
ide_mkIde: fixes f0 :: "'a"   and f1 :: "'a" assumes "C.span f0 f1" shows "ide (mkIde f0 f1)"
simps_1: fixes a :: "real" shows "Re (cis a) = cos a"
diff_sum_integral_has_integral_int': shows "((\<lambda>t. pbernpoly 1 t *\<^sub>R f' t) has_integral S - I - (f (real_of_int b) - f (real_of_int a)) /\<^sub>R 2) {real_of_int a..real_of_int b}"
streams_stake_set: fixes pw :: "'b stream"   and A :: "'b set"   and n :: "nat" assumes "pw \<in> streams A" shows "set (stake n pw) \<subseteq> A"
distincts_lists_fcs: fixes G_list :: "'b list \<times> ('b \<times> 'b) list"   and xss :: "('b \<times> 'b) list list"   and as :: "('b \<times> 'b) list" assumes "lists_digraph_map G_list xss"   and "distinct as" shows "distincts (lists_digraph_map.lists_fcs xss as)"
tendsto_at_botI_sequentially: fixes f :: "real \<Rightarrow> 'a"   and y :: "'a" assumes "\<And>X. filterlim X at_bot sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y" shows "(f \<longlongrightarrow> y) at_bot"
sup_ge1: fixes x :: "'a"   and y :: "'a" shows "x \<le> x \<squnion>1 y"
expectation_gpv_mono: fixes fail :: "ennreal"   and fail' :: "ennreal"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal"   and \<I> :: "('b, 'c) \<I>"   and gpv :: "('a, 'b, 'c) gpv" assumes "fail \<le> fail'"   and "f \<le> g" shows "expectation_gpv fail \<I> f gpv \<le> expectation_gpv fail' \<I> g gpv"
cr_fset_def: shows "cr_fset \<equiv> \<lambda>x. (=) (abs_fset x)"
slice1_cdep_slice1: fixes nx :: "'node SDG_node"   and n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "nx \<in> sum_SDG_slice1 n"   and "n s\<longrightarrow>\<^bsub>cd\<^esub> n'" shows "nx \<in> sum_SDG_slice1 n'"
diamond_cube_boundary_to_subdiv_def: fixes gamma :: "int \<times> (real \<Rightarrow> real \<times> real)" shows "diamond_cube_boundary_to_subdiv gamma \<equiv> if gamma = diamond_top_edges then {diamond_top_left_edge, diamond_top_right_edge} else if gamma = diamond_bot_edges then {diamond_bot_left_edge, diamond_bot_right_edge} else {}"
forget_eval_fm: fixes k :: "name"   and A :: "fm"   and e :: "(name, hf) finfun"   and x :: "hf" assumes "atom k \<sharp> A" shows "eval_fm (finfun_update e k x) A = eval_fm e A"
weakSimIFresh: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and x :: "name"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "eqvt Rel"   and "x \<sharp> \<Psi>"   and "x \<sharp> P"   and "x \<sharp> Q"   and "\<And>\<alpha> Q' \<Psi>''. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; \<alpha> \<noteq> \<tau>; bn \<alpha> \<sharp>* subject \<alpha>; bn \<alpha> \<sharp>* C; x \<sharp> \<alpha>; x \<sharp> Q'; x \<sharp> \<Psi>''\<rbrakk> \<Longrightarrow> \<exists>P''. \<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'' \<and> (\<exists>P'. \<Psi> \<otimes> \<Psi>'' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi> \<otimes> \<Psi>'', P', Q') \<in> Rel)"   and "\<And>Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q'; x \<sharp> Q'\<rbrakk> \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto><Rel> Q"
cases: fixes a1 :: "sequent"   and a2 :: "nat"   and R :: "(sequent list \<times> sequent) set" assumes "(a1, a2) \<in> derivable R" obtains C :: "sequent" where "a1 = C" and "a2 = 0" and "([], C) \<in> R"   | r :: "sequent list \<times> sequent" and m :: "nat" where "a1 = snd r" and "a2 = m + 1" and "r \<in> R" and "fst r \<noteq> []" and "\<forall>p\<in>set (fst r). \<exists>n\<le>m. (p, n) \<in> derivable R"
Bv: fixes x :: "'a" shows "(\<lambda>n. blinfun_apply (X n) x) \<longlonglongrightarrow> blinfun_apply (Blinfun v) x"
e: fixes s :: "real" assumes "s \<in> cball t e" shows "s \<in> existence_ivl t0 x0"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_on_left_moving_2 ires rs (l, r) = (wcode_on_left_moving_2_B ires rs (l, r) \<or> wcode_on_left_moving_2_O ires rs (l, r))"
fixedaction_reorder: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and a :: "action"   and m1 :: "'c match_expr list"   and m2 :: "'c match_expr list"   and m3 :: "'c match_expr list"   and s :: "state" shows "approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m a) (m1 @ m2 @ m3)) s = approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m a) (m2 @ m1 @ m3)) s"
simps_2: fixes j :: "nat"   and e :: "'v \<times> 'v" shows "h_minus_aux (Suc j) e = (if e \<in> set (cycle_edges (enum_cycle j)) then h_minus_aux j e + Min {h' e' - h_minus_aux j e' |e'. e' \<in> set (cycle_edges (enum_cycle j))} else h_minus_aux j e)"
scale_right_diff_distrib: fixes a :: "'a"   and x :: "'c"   and y :: "'c" shows "a *b (x - y) = a *b x - a *b y"