hcju/isabelleps huggingface dataset:
sel_11: fixes x1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x3 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x4 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x5 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x6 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x7 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x8 :: "((nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping) \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and x9 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x10 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "Direction.le_rat (Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) = x11"
Un_Int_distrib2: fixes B :: "'d set"   and C :: "'d set"   and A :: "'d set" shows "B \<inter> C \<union> A = (B \<union> A) \<inter> (C \<union> A)"
is_empty_def: fixes m :: "('d, 'e) mapping" shows "is_empty m = (keys m = {})"
MM5: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<or> \<box>G \<longrightarrow> \<box>(F \<or> G)"
reduce_system_sign_conditions: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "nat list list" assumes "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "satisfy_equation p qs subsets signs"   and "invertible_mat (M_mat signs subsets)" shows "set (get_signs (reduce_system p (qs, M_mat signs subsets, subsets, signs))) = set (characterize_consistent_signs_at_roots_copr p qs)"
Rep_hierauto_cases: fixes y :: "'a data \<times> ('b, 'c, 'a) seqauto set \<times> 'c set \<times> ('b \<Rightarrow> ('b, 'c, 'a) seqauto set option)" assumes "y \<in> hierauto" obtains x :: "('b, 'c, 'a) hierauto" where "y = Rep_hierauto x"
A_ia_y: shows "A $ ia $ y \<noteq> (0::'a)"
invariant: fixes \<sigma> :: "nat \<Rightarrow> 'c list"   and wf_atom :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and project :: "'c \<Rightarrow> 'c"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and init :: "'d rexp \<Rightarrow> 'e"   and delta :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and final :: "'e \<Rightarrow> bool"   and wf_state :: "'e \<Rightarrow> bool"   and post :: "'e \<Rightarrow> 'e"   and L :: "'e \<Rightarrow> 'c list set"   and n :: "nat"   and r :: "'e"   and s :: "'e"   and st :: "('c list \<times> 'e \<times> 'e) list \<times> ('e \<times> 'e) list \<times> ('e \<times> 'e) set" assumes "rexp_DA \<sigma> wf_atom project lookup init delta final wf_state post L n"   and "rexp_DA.invariant \<sigma> delta final wf_state post n r s st"   and "rexp_DA.test_invariant final st" shows "rexp_DA.invariant \<sigma> delta final wf_state post n r s (rexp_DA.step_invariant \<sigma> delta post n st)"
rsquarefree_poly_roots_eq: fixes p :: "'b poly" assumes "rsquarefree p" shows "poly_roots p = mset_set {x. poly p x = (0::'b)}"
same_ipd_stable: fixes k :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat"   and j :: "nat" assumes "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i"   and "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"   and "i < j"   and "ipd (\<pi> i) = ipd (\<pi> k)" shows "ipd (\<pi> j) = ipd (\<pi> k)"
lemma3_5': fixes r :: "('a \<times> 'a) set"   and \<tau> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma>' :: "'b \<times> ('a \<times> 'b) list"   and \<tau>' :: "'b \<times> ('a \<times> 'b) list"   and \<upsilon> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma>'' :: "'b \<times> ('a \<times> 'b) list"   and \<upsilon>' :: "'b \<times> ('a \<times> 'b) list" assumes "trans r"   and "irrefl r"   and "D2 r (\<tau>, \<sigma>, \<sigma>', \<tau>')"   and "D2 r (\<upsilon>, \<sigma>', \<sigma>'', \<upsilon>')" shows "D2 r ((fst \<tau>, snd \<tau> @ snd \<upsilon>), \<sigma>, \<sigma>'', fst \<tau>', snd \<tau>' @ snd \<upsilon>')"
true_cls_mset_def: fixes I :: "'b set"   and CC :: "'b literal multiset multiset" shows "(I \<Turnstile>m CC) = (\<forall>C\<in>#CC. I \<Turnstile> C)"
sharing_consis: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts'"   and "ts' ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "sharing_consistent (share_all_until_volatile_write ts \<S>) \<O> sb"
Cauchy_uniformly_convergent: fixes X :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "uniformly_Cauchy_on X f" shows "uniformly_convergent_on X f"
lunit_commutes_with_L: fixes f :: "'a" assumes "ide f" shows "PB.lunit (L f) = L (PB.lunit f)"
assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "opr (opr x y) z = opr x (opr y z)"
Dg1_is_gamma2_pathimg: shows "path_image gamma2 = {p. \<exists>x. x \<in> cbox a b \<and> p = (g1 x, x)}"
vec_to_iarray_basis_null_space: fixes A :: "(('b, 'c) vec, 'd) vec" shows "vec_to_iarray ` basis_null_space A = basis_null_space_iarrays (matrix_to_iarray A)"
subset_add_mset: fixes ys :: "'b multiset"   and x :: "'b"   and zs :: "'b multiset" shows "(ys \<subseteq># add_mset x zs) = (ys \<subseteq># zs \<or> (\<exists>xs. xs \<subseteq># zs \<and> ys = add_mset x xs))"
HT'set_r_def: fixes r :: "((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> 'd) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and \<pi> :: "char list \<Rightarrow> com option"   and \<Theta> :: "(((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> 'd) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> com) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool)) set" shows "HT'set_r r \<pi> \<Theta> \<equiv> \<forall>(n, P, c, Q)\<in>\<Theta>. HT' \<pi> (\<lambda>s. r n s \<and> P s) c Q"
v4: shows "atom v4 \<sharp> (t, t', X1, v2, v3)"
close_sym: fixes g :: "graph"   and u :: "nat"   and v :: "nat" assumes "minGraphProps g"   and "u \<in> \<V> g"   and "close g u v" shows "close g v u"
GF_advice_inter: fixes X :: "'b ltln set"   and \<phi> :: "'b ltln"   and S :: "'b ltln set" assumes "X \<inter> subformulas\<^sub>\<mu> \<phi> \<subseteq> S" shows "\<phi>[X \<inter> S]\<^sub>\<Pi>\<^sub>1 = \<phi>[X]\<^sub>\<Pi>\<^sub>1"
wf_trm_subst_compose: fixes \<theta> :: "'b \<Rightarrow> ('fun, 'a) Term.term"   and x :: "'b"   and \<delta> :: "'a \<Rightarrow> ('fun, 'a) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m (\<theta> x)"   and "\<And>x. wf\<^sub>t\<^sub>r\<^sub>m (\<delta> x)" shows "wf\<^sub>t\<^sub>r\<^sub>m ((\<theta> \<circ>\<^sub>s \<delta>) x)"
card_le_dim_spanning: fixes B :: "'a set"   and V :: "'a set" assumes "B \<subseteq> V"   and "V \<subseteq> span B"   and "finite B"   and "card B \<le> dim V" shows "independent B"
simps_1: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "sorted_wrt P [] = True"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "sup b (sup a c) = sup a (sup b c)"
SN_rel_on_imp_SN_rel_on_alt: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and T :: "'a set" assumes "SN_rel_on R S T" shows "SN_rel_on_alt R S T"
sorted_map_remove1: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list"   and x :: "'c" assumes "sorted (map f xs)" shows "sorted (map f (remove1 x xs))"
monomial_sum: fixes a :: "'c"   and f :: "'e \<Rightarrow> 'd"   and C :: "'e set" shows "monomial (sum f C) a = (\<Sum>c\<in>C. monomial (f c) a)"
IH: assumes "\<forall>a\<in>set fs. is_cnf a" shows "is_cnf (foldr (\<^bold>\<and>) fs (\<^bold>\<not> \<bottom>))"
rf_encrypt_def: shows "rf_encrypt = (\<lambda>s plain. spmf_of_set (nlists UNIV len) \<bind> (\<lambda>r. prf.random_oracle s r \<bind> (\<lambda>(pad, s'). return_spmf ((r, plain [\<oplus>] pad), s'))))"
EntryPath_suffix: fixes g :: "'g"   and ns :: "'node list"   and ns' :: "'node list" assumes "EntryPath g ns"   and "suffix ns' ns"   and "ns' \<noteq> []" shows "EntryPath g ns'"
length_suffixes: fixes xs :: "'a list" shows "length (suffixes xs) = Suc (length xs)"
latestAct_def: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>c \<leftarrow> t\<rangle>\<^bsub>n\<^esub> = (GREATEST n'. latestAct_cond c t n n')"
inject: fixes \<Gamma> :: "('k, 'l) web_scheme"   and x3 :: "'m"   and y3 :: "'m" assumes "countable_bipartite_web \<Gamma>" shows "(bi.Inner x3 = bi.Inner y3) = (x3 = y3)"
is_schedule_of_def: fixes A :: "('c, 'd) ioa"   and sch :: "'d list" shows "is_schedule_of A sch \<equiv> \<exists>e. is_exec_of A e \<and> sch = filter_act (snd e)"
lookup_env_cons: fixes b :: "'b"   and es :: "'c \<Rightarrow> ('a, 'b, 'c) env option"   and x :: "'c"   and xs :: "'c list" shows "lookup (Env b es) (x # xs) = (case es x of None \<Rightarrow> None | Some e \<Rightarrow> lookup e xs)"
rk_ax_couple: fixes P :: "Points"   and Q :: "Points" shows "P \<noteq> Q \<longrightarrow> 2 \<le> rk {P, Q}"
permute_char_def: fixes p :: "perm"   and c :: "char" shows "p \<bullet> c = c"
ideal_inc_set_mult: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<subseteq> carrier R"   and "ideal R B"   and "ideal R C"   and "set_mult R A B \<subseteq> C" shows "sum_mult R A B \<subseteq> C"
Arctan_of_real: fixes x :: "real" shows "Arctan (complex_of_real x) = complex_of_real (arctan x)"
simps_3: shows "eval_ternary_Not TernaryUnknown = TernaryUnknown"
unfold_optimize_ruleset_CHAIN_def: fixes optimize :: "'c match_expr \<Rightarrow> 'c match_expr"   and chain_name :: "char list"   and default_action :: "action"   and rs :: "char list \<Rightarrow> 'c rule list option" shows "unfold_optimize_ruleset_CHAIN optimize chain_name default_action rs = (let rs = repeat_stabilize 1000 (optimize_matches opt_MatchAny_match_expr) (optimize_matches optimize (rw_Reject (rm_LogEmpty (repeat_stabilize 10000 (process_call rs) [Rule MatchAny (Call chain_name), Rule MatchAny default_action])))) in if simple_ruleset rs then Some rs else None)"
inter_rbt_def: shows "inter_rbt \<equiv> Set2_Join.inter Set2_Join_RBT.join"
add_increasing2: fixes c :: "'a multiset"   and b :: "'a multiset"   and a :: "'a multiset" assumes "\<Empt> \<subseteq># c"   and "b \<subseteq># a" shows "b \<subseteq># a + c"
revis_inclusion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<^bold>* \<phi> \<subseteq> K \<oplus> \<phi>"
list_induct3: fixes xs :: "'b list"   and ys :: "'c list"   and zs :: "'d list"   and P :: "'b list \<Rightarrow> 'c list \<Rightarrow> 'd list \<Rightarrow> bool" assumes "length xs = length ys"   and "length ys = length zs"   and "P [] [] []"   and "\<And>x xs y ys z zs. \<lbrakk>length xs = length ys; length ys = length zs; P xs ys zs\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys) (z # zs)" shows "P xs ys zs"
genlist_length: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "length (genlist f n) = n"
set_subset_Cons: fixes xs :: "'e list"   and x :: "'e" shows "set xs \<subseteq> set (Cons x xs)"
red_preserves_defass: fixes 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"   and e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "\<D> e \<lfloor>dom l\<rfloor>" shows "\<D> e' \<lfloor>dom l'\<rfloor>"
elemrcos_carrier: fixes a :: "'a"   and a' :: "'a" assumes "Group.group G"   and "a \<in> carrier G"   and "a' \<in> derived G (carrier G) #> a" shows "a' \<in> carrier G"
ltl_prop_implies_def: fixes \<phi> :: "'c ltl"   and \<psi> :: "'c ltl" shows "\<phi> \<longrightarrow>\<^sub>P \<psi> \<equiv> \<forall>\<A>. \<A> \<Turnstile>\<^sub>P \<phi> \<longrightarrow> \<A> \<Turnstile>\<^sub>P \<psi>"
IH: fixes n :: "'node" assumes "n = hd ns \<and> m = last ns" shows "P n ns m"
not_col_permutation_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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C" shows "\<not> Tarski_neutral_dimensionless.Col Bet C B A"
extend_Join: fixes F :: "'a program"   and G :: "'a program" shows "extend h (F \<squnion> G) = extend h F \<squnion> extend h G"
ex_in_conv: fixes A :: "'c set" shows "(\<exists>x. x \<in> A) = (A \<noteq> {})"
a_assigned_slot_def: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog) \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog" shows "l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_assigned_slot get_parent get_shadow_root get_mode get_attribute get_tag_name first_in_tree_order = l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_find_slot get_parent get_shadow_root get_mode get_attribute get_tag_name first_in_tree_order True"
fps_of_ratfps_unit_factor: fixes x :: "'b ratfps" shows "fps_of_ratfps (unit_factor x) = unit_factor (fps_of_ratfps x)"
hyp_ind: shows "\<forall>y. (y, x) \<in> trm_ord \<longrightarrow> y \<noteq> trm_rep y S \<longrightarrow> (trm_rep y S, y) \<in> trm_ord"
nonabort_rg: shows "chaos \<sqsubseteq> (rg::'a::refinement_lattice)"
copr_neg_inf: shows "algebraic_semidom_class.coprime f [:real_of_int (crb (prod_list fss)), 1:]"
smc_SemiCAT_is_arrD: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<^bsub>smc_SemiCAT \<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>"
assms_2: fixes I :: "'a set"   and a :: "'a"   and b :: "'a"   and J :: "'a set"   and c :: "'a"   and d :: "'a" assumes "I = interval a b"   and "J = interval c d" shows "([[a b c d]] \<longrightarrow> P I J) \<and> ([[a c b d]] \<longrightarrow> P I J) \<and> ([[a c d b]] \<longrightarrow> P I J)"
dom_fun_upd: fixes f :: "'c \<Rightarrow> 'd option"   and x :: "'c"   and y :: "'d option" shows "dom (f(x := y)) = (if y = None then dom f - {x} else {x} \<union> dom f)"
less_eq_poly_def: fixes x :: "'b poly"   and y :: "'b poly" shows "(x \<le> y) = (x = y \<or> pos_poly (y - x))"
af_imp_almost_full_on: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "af A P" shows "almost_full_on P A"
invar_meldUniq: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and q' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.queue_invar q"   and "SkewBinomialHeapStruc.queue_invar q'" shows "SkewBinomialHeapStruc.queue_invar (SkewBinomialHeapStruc.meldUniq q q')"
FORK_rec: fixes i :: "nat" shows "FORK i = (picks i i \<rightarrow> putsdown i i \<rightarrow> FORK i [+] picks ((i + 1) mod N) i \<rightarrow> putsdown ((i + 1) mod N) i \<rightarrow> FORK i)"
gen_tlength_code_1: fixes n :: "nat"   and b :: "'b" shows "gen_tlength n (TNil b) = enat n"
simps_4: fixes uu :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and uv :: "'b"   and uw :: "'b list" shows "uv # uw \<cong> {[], [], uu} = False"
nhds_discrete_open: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.topological_space open"   and "open {x}" shows "topological_space.nhds open x = principal {x}"
powr_of_real: fixes x :: "real"   and y :: "real" assumes "0 \<le> x" shows "complex_of_real x powr complex_of_real y = complex_of_real (x powr y)"
rel_dom_dgrad_set: shows "pp_of_term ` mlt ` rel_dom \<subseteq> dgrad_set d m'"
pos_recurrent_iffI_communicating: fixes y :: "'a"   and x :: "'a"   and K :: "'a \<Rightarrow> 'a pmf" assumes "(y, x) \<in> MC_syntax.communicating K" shows "MC_syntax.pos_recurrent K y = MC_syntax.pos_recurrent K x"
that: fixes g :: "'a" assumes "Finite_And_Cyclic_Groups.cyclic_group G g" shows "thesis"
assms: shows "as1 @ as2 = filter f1 (map f2 p)"
CondRed: 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 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 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"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow> \<langle>if (e') e\<^sub>1 else e\<^sub>2,s'\<rangle>"
orthogonal_comp_null: shows "{0::'a}\<^sup>\<bottom> = UNIV"
index_uminus_mat_3: fixes A :: "'c mat" shows "dim_col (- A) = dim_col A"
liftAll_cong: fixes inp :: "'d \<Rightarrow> 'e option"   and phi :: "'e \<Rightarrow> bool"   and phi' :: "'e \<Rightarrow> bool" assumes "\<And>i v. inp i = Some v \<Longrightarrow> phi v = phi' v" shows "liftAll phi inp = liftAll phi' inp"
inf_prefix_length_commute: fixes xs :: "'b list"   and ys :: "'b list" shows "inf_prefix_length xs ys = inf_prefix_length ys xs"
in_tl_gen2: fixes x :: "nat list \<times> nat list"   and A :: "nat"   and m :: "nat"   and B :: "nat"   and n :: "nat" assumes "x \<in> set (tl (gen2 A B m n))" shows "x \<in> set (gen2 A B m n)"
linear_span_def: fixes R :: "('f, 'g) Ring_scheme"   and M :: "('e, 'f, 'h) Module_scheme"   and A :: "'f set"   and H :: "'e set" shows "linear_span R M A H = (if H = {} then {\<zero>\<^bsub>M\<^esub>} else {x. \<exists>n. \<exists>f\<in>{j. j \<le> n} \<rightarrow> H. \<exists>s\<in>{j. j \<le> n} \<rightarrow> A. x = l_comb R M n s f})"
assms_1: shows "T \<in> GRepHomSet (\<star>) W"
fps_to_fls_eq_imp_fps_eq: fixes f :: "'b fps"   and g :: "'b fps" assumes "fps_to_fls f = fps_to_fls g" shows "f = g"
iso_iff3: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" assumes "Well_order r"   and "Well_order r'" shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and> compat r r' f)"
real_vector_space_on_axioms: shows "real_vector_space_on S2"
convexD: fixes s :: "'b set"   and x :: "'b"   and y :: "'b"   and u :: "real"   and v :: "real" assumes "convex s"   and "x \<in> s"   and "y \<in> s"   and "0 \<le> u"   and "0 \<le> v"   and "u + v = 1" shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
vec_of_ell2_def: shows "vec_of_ell2 = vec_of_basis_enum"
monotone_def: fixes orda :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and ordb :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i" shows "monotone orda ordb f = (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
offs_update: fixes j :: "nat"   and ns :: "nat list"   and i :: "nat"   and k :: "nat" assumes "j < length ns" shows "offs (ns[i := Suc (ns ! i)]) k ! j = (if j \<le> i then id else Suc) (offs ns k ! j)"
prems_1: shows "executionally_equivalent e1 s1 r1 e2 s2 r2 (a # x)"
map_upd_Some_unfold: fixes m :: "'e \<Rightarrow> 'd option"   and a :: "'e"   and b :: "'d"   and x :: "'e"   and y :: "'d" shows "((m(a \<mapsto> b)) x = \<lfloor>y\<rfloor>) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = \<lfloor>y\<rfloor>)"
fg: shows "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
splits_at_\<beta>: fixes \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and s :: "symbol list"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" assumes "Derives1 \<delta> i r s"   and "splits_at \<delta> i \<alpha> N \<beta>" shows "\<beta> = drop (Suc i) \<delta> \<and> \<beta> = drop (i + length (snd r)) s \<and> length \<beta> = length \<delta> - i - 1"
BIT_b_2: fixes x :: "'a"   and y :: "'a"   and init :: "'a list"   and v :: "'a list" assumes "x \<noteq> y"   and "init \<in> {[x, y], [y, x]}"   and "v \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "Partial_Cost_Model.config'_rand BIT (type0 init x y) v = type0 init y x"
G_parts_5: shows "arcs_ends G = GU \<times> GV \<union> GV \<times> GU"
lemma3_5_DD: fixes r :: "('a \<times> 'a) set"   and ars :: "('b \<times> 'a \<times> 'b) set"   and \<tau> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma>' :: "'b \<times> ('a \<times> 'b) list"   and \<tau>' :: "'b \<times> ('a \<times> 'b) list"   and \<upsilon> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma>'' :: "'b \<times> ('a \<times> 'b) list"   and \<upsilon>' :: "'b \<times> ('a \<times> 'b) list" assumes "trans r"   and "irrefl r"   and "DD ars r (\<tau>, \<sigma>, \<sigma>', \<tau>')"   and "DD ars r (\<upsilon>, \<sigma>', \<sigma>'', \<upsilon>')" shows "DD ars r ((fst \<tau>, snd \<tau> @ snd \<upsilon>), \<sigma>, \<sigma>'', fst \<tau>', snd \<tau>' @ snd \<upsilon>')"
istopology_openin: fixes U :: "'c topology" shows "istopology (openin U)"
inject: fixes x :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list, 'b \<times> 'b rose_tree\<^sub>h list) blindable\<^sub>m"   and ya :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list, 'b \<times> 'b rose_tree\<^sub>h list) blindable\<^sub>m" shows "(Tree\<^sub>m x = Tree\<^sub>m ya) = (x = ya)"
CALL: fixes \<phi> :: "(((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool) \<Rightarrow> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and X :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" assumes "({Sec (FIX \<phi>)} \<union> X) \<rhd> body : Sec (\<phi> (FIX \<phi>))"   and "Monotone \<phi>" shows "X \<rhd> Call : Sec (FIX \<phi>)"
AuthReq_msg_in_parts_spies: fixes M :: "agent"   and P_I :: "msg"   and OIData :: "msg"   and HPIData :: "msg"   and evs :: "event list" assumes "Gets M \<lbrace>P_I, OIData, HPIData\<rbrace> \<in> set evs"   and "evs \<in> set_pur" shows "P_I \<in> parts (knows Spy evs)"
column_def: fixes j :: "'g"   and A :: "(('e, 'g) vec, 'f) vec" shows "column j A = (\<chi>i. A $ i $ j)"