hcju/isabelleps huggingface dataset:
ticket_authentic: fixes B :: "agent"   and Tk :: "nat"   and A :: "agent"   and K :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>\<rbrace>) \<in> set evs"
all_is_branch_nominals': fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set" assumes "all_is_branch p i branch xs" shows "nominals p \<union> branch_nominals (omit_branch xs branch) = nominals p \<union> branch_nominals branch"
is_le_relI_3: shows "ord_term_lin.is_le_rel (\<prec>\<^sub>t)"
simps_4: fixes uu :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set"   and uv :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set"   and uw :: "((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set"   and C :: "acom"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "preT ({uu/uv/uw} CONSEQ C) e = preT C e"
powr_mono': fixes x :: "real"   and a :: "real"   and b :: "real" assumes "0 < x"   and "x \<le> 1"   and "a \<le> b" shows "x powr b \<le> x powr a"
checker_SNF_def: fixes A :: "int mat" shows "checker_SNF A = (let A' = mat_to_list A; m = dim_row A; n = dim_col A in case external_SNF A' of (P_ext, S_ext, Q_ext, P'_ext, Q'_ext) \<Rightarrow> let P = mat_of_rows_list m P_ext; S = mat_of_rows_list m S_ext; Q = mat_of_rows_list m Q_ext; P' = mat_of_rows_list m P'_ext; Q' = mat_of_rows_list m Q'_ext in if dim_row P = m \<and> dim_col P = m \<and> dim_row S = m \<and> dim_col S = n \<and> dim_row Q = n \<and> dim_col Q = n \<and> dim_row P' = m \<and> dim_col P' = m \<and> dim_row Q' = n \<and> dim_col Q' = n \<and> P * P' = 1\<^sub>m m \<and> Q * Q' = 1\<^sub>m n \<and> Smith_normal_form_mat S \<and> S = P * A * Q then (P, S, Q) else Code.abort STR ''Certification failed'' (\<lambda>_. Smith_ED_mxn A))"
filter_join_False_empty: fixes m :: "('a, 'b) mapping" shows "filter_join False {} m = m"
subcocycle_avg_ereal_def: fixes u :: "nat \<Rightarrow> (int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> real" shows "I.subcocycle_avg_ereal u = \<Sqinter> {ereal (I.expectation (u n) / real n) |n. 0 < n}"
assms: shows "n \<noteq> m \<and> dominates g n m"
Sr_def: fixes R :: "('c, 'd) Ring_scheme"   and S :: "'c set" shows "Sr R S = R\<lparr>carrier := S, pop := \<lambda>x\<in>S. restrict ((\<plusminus>\<^bsub>R\<^esub>) x) S, mop := restrict (mop R) S, zero := \<zero>\<^bsub>R\<^esub>, tp := \<lambda>x\<in>S. restrict ((\<cdot>\<^sub>r\<^bsub>R\<^esub>) x) S, un := 1\<^sub>r\<^bsub>R\<^esub>\<rparr>"
distrib_left_ereal_nn: fixes c :: "real"   and x :: "ereal"   and y :: "ereal" assumes "0 \<le> c" shows "(x + y) * ereal c = x * ereal c + y * ereal c"
assms_1: shows "X\<^sub>1 \<bowtie> X\<^sub>2"
id_subs: fixes x :: "name"   and t :: "lam" shows "t[x::=Var x] = t"
ev: shows "\<forall>\<^sub>F b' in at_left b. b' \<in> {a<..<b}"
isGlb_cInf: fixes S :: "'a set" assumes "S \<noteq> {}"   and "\<exists>b. b <=* S" shows "isGlb UNIV S (Inf S)"
lsorted_coinduct': fixes X :: "'c llist \<Rightarrow> bool"   and xs :: "'c llist"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "X xs"   and "\<And>xs. \<lbrakk>X xs; \<not> lnull xs; \<not> lnull (ltl xs)\<rbrakk> \<Longrightarrow> less_eq (lhd xs) (lhd (ltl xs)) \<and> (X (ltl xs) \<or> ord.lsorted less_eq (ltl xs))" shows "ord.lsorted less_eq xs"
add_mset_remove_trivial: fixes x :: "'b"   and M :: "'b multiset" shows "add_mset x M \<ominus> x = M"
set_disconnected_nodes_get_tag_name: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" assumes "l_set_disconnected_nodes_get_tag_name type_wf set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
third_gt0: shows "0 < 1 / 3 + 1"
f2r2f_inv2: shows "\<F> \<circ> \<R> = id"
Nml_Nmlize_1: fixes t :: "'c local.term" assumes "Arr t" shows "Nml \<^bold>\<lfloor>t\<^bold>\<rfloor>"
simps_1: fixes x :: "Preliminaries.var" shows "wt (Var x) = True"
simps_3: fixes x :: "'b"   and hs :: "'b heap list" shows "cost Del_min [Hp x hs] = T\<^sub>p\<^sub>a\<^sub>s\<^sub>s\<^sub>2 (pass\<^sub>1 hs) + T\<^sub>p\<^sub>a\<^sub>s\<^sub>s\<^sub>1 hs"
valid_shift_end_mmsaux_unfolded: fixes args :: "Monitor.args"   and cur :: "nat"   and ot :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'b option list set) queue"   and data_in :: "(nat \<times> 'b option list set) queue"   and tuple_in :: "('b option list, nat) mapping"   and tuple_since :: "('b option list, nat) mapping"   and auxlist :: "(nat \<times> 'b option list set) list"   and nt :: "nat" assumes "valid_mmsaux args cur (ot, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) auxlist"   and "cur \<le> nt" shows "valid_mmsaux args cur (shift_end args nt (ot, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)) (filter (\<lambda>(t, rel). enat (nt - t) \<le> right (args_ivl args)) auxlist)"
Fs_dim_1: shows "dim_row (mat_of_rows n (map gs1.gso [0..<length (map of_int_hom.vec_hom fs)])) = length (map of_int_hom.vec_hom fs)"
lex_ord_pair_valI: fixes xs :: "('a \<times> 'd) list"   and ys :: "('a \<times> 'e) list"   and aux :: "order option"   and k :: "'a"   and f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> order option" assumes "oalist_inv_raw xs"   and "oalist_inv_raw ys"   and "aux \<noteq> Some Eq"   and "k \<in> fst ` set xs \<union> fst ` set ys"   and "aux = f k (lookup_pair xs k) (lookup_pair ys k)"   and "\<And>k'. \<lbrakk>k' \<in> fst ` set xs \<union> fst ` set ys; lt k' k\<rbrakk> \<Longrightarrow> f k' (lookup_pair xs k') (lookup_pair ys k') = Some Eq" shows "lex_ord_pair f xs ys = aux"
same_path_reachable2: fixes Q :: "'a set"   and R :: "'a set"   and a :: "'a"   and b :: "'a" assumes "Q \<in> \<P>"   and "R \<in> \<P>"   and "a \<in> Q"   and "a \<in> R"   and "b \<in> Q" shows "a \<in> R - \<emptyset> R b"
singular_relboundary_ss: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and x :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_relboundary p X S x" shows "Poly_Mapping.keys x \<subseteq> singular_simplex_set p X"
DSTEP_trancl_term: fixes S :: "'c"   and t :: "('c + 'd) list"   and G :: "('c \<times> ('c, 'd) RHS) list"   and x :: "'d" assumes "[Inl S] -G\<rightarrow>\<^sup>+ t"   and "Inr x \<in> set t" shows "\<exists>N. (N, Leaf x) \<in> set G"
segment_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "MinkowskiBetweenness.segment \<P> betw a b \<equiv> {x. \<exists>ab. betw a x b \<and> x \<in> ab \<and> ab \<in> \<P> \<and> a \<in> ab \<and> b \<in> ab \<and> a \<noteq> b}"
simps_2: fixes r :: "'form" shows "inPorts' (Assumption r) = []"
properfactor_prod_r: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "properfactor G a b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G a (b \<otimes> c)"
inv_is_inverse: fixes f :: "'b" assumes "B.iso f" shows "B.inverse_arrows f (B.inv f)"
rivest: fixes \<eta> :: "nat" shows "rivest (q \<eta>)"
1: shows "prv (imp (dsj \<phi>1 (ldsj (\<phi>1s @ \<phi> # \<phi>2s))) (dsj \<phi>1 (dsj \<phi> (ldsj (\<phi>1s @ \<phi>2s)))))"
Standard_scaleR: fixes x :: "'b star"   and r :: "real" assumes "x \<in> Standard" shows "r *\<^sub>R x \<in> Standard"
set_iterator_map_linord_def: fixes it :: "('f \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'g \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f"   and S0 :: "('a \<times> 'g) set" shows "set_iterator_map_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>(k, uu_) (k', uu_). k \<le> k')"
sorted_ss_hdw: shows "Sorted_Less.sorted (rev (ss ! hd w))"
subset_image_iff: fixes B :: "'b set"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set" shows "(B \<subseteq> f ` A) = (\<exists>AA\<subseteq>A. B = f ` AA)"
eq_Min_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and m :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(m = linorder.Min less_eq A) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq m a))"
alpha_refl: fixes X :: "('e, 'f, 'varSort, 'var, 'g) qTerm" assumes "qGood X" shows "X #= X"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Dom t) (Cod t') t'ot.map"
subsetCE: fixes A :: "'e set"   and B :: "'e set"   and c :: "'e" assumes "A \<subseteq> B" obtains "c \<notin> A"   | "c \<in> B"
filtermap_sequentually_ne_bot: fixes f :: "nat \<Rightarrow> 'a" shows "filtermap f sequentially \<noteq> bot"
pseudominimal_inD1: fixes P :: "'a set"   and x :: "'a" assumes "pseudominimal_in P x" shows "x \<in> P"
add: fixes x :: "real"   and y :: "real"   and xa :: "'d" shows "(x + y) *\<^sub>R xa = x *\<^sub>R xa + y *\<^sub>R xa"
mult_col_div_row_similar: fixes A :: "'b mat"   and n :: "nat"   and k :: "nat"   and a :: "'b" assumes "A \<in> carrier_mat n n"   and "k < n"   and "a \<noteq> (0::'b)" shows "similar_mat (mult_col_div_row a k A) A"
delta_sol: shows "ca.delta = ca_lr (dual ca.delta, ca.delta)"
that_2: fixes t :: "real" assumes "k \<le> t" shows "f t = g t"
tensor_preserves_can: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.can C f"   and "free_monoidal_category.can C g" shows "free_monoidal_category.can C (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C f g)"
assms: fixes i :: "'b" assumes "i \<in> I" shows "Int_stable (F i)"
that: fixes e' :: "nat \<Rightarrow> real" assumes "e' \<in> UNIV \<rightarrow> I"   and "pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list (ccw.selsort 0 xs))" shows "thesis"
simps_3: fixes x2 :: "'a" shows "OK x2 \<noteq> Err"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) A.F.map A.F.map A.F.map"
C1: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', \<tau>.P, P \<oplus>\<^sub>\<top> \<tau>.P) \<in> Rel"
assms_2: shows "f maps\<^bsub>NTCatDom \<eta>\<^esub> X to Y"
intros: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "\<And>y. r y x \<Longrightarrow> Wellfounded.accp r y" shows "Wellfounded.accp r x"
sets_return: fixes M :: "'c measure"   and x :: "'c" shows "sets (return M x) = sets M"
iia_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "iia_on A f = (\<forall>B\<subseteq>A. \<forall>C\<subseteq>B. \<forall>a\<in>C. a \<in> f B \<longrightarrow> a \<in> f C)"
neg_Gmult: fixes g :: "'g"   and v :: "'v" assumes "g \<in> G"   and "v \<in> V" shows "g *\<cdot> - v = - (g *\<cdot> v)"
base: fixes C :: "'b sequent"   and R :: "('b sequent list \<times> 'b sequent) set" assumes "([], C) \<in> R" shows "(C, 0) \<in> derivable R"
simps_11: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "flatten (Catch c\<^sub>1 c\<^sub>2) = [Catch c\<^sub>1 c\<^sub>2]"
sub3_divisor: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and d :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r'' :: "'a"   and A'' :: "'a mat" assumes "mute_fun mf"   and "det_selection_fun sel_fun"   and "sub3 sel_fun mf d l (r, A) = (r'', A'')"   and "l \<le> d"   and "d \<le> dim_row A"   and "l \<le> dim_col A"   and "r \<noteq> (0::'a)" shows "r'' \<noteq> (0::'a)"
InitialIsInitReachable: fixes c :: "('p, 'v, 's) configuration" assumes "initial c" shows "initReachable c"
weight_w0_unary_2: fixes w :: "'b \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and pr_weak :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and least :: "'b \<Rightarrow> bool"   and scf :: "'b \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and t :: "('b, 'c) Term.term"   and f :: "'b"   and ts :: "('b, 'c) Term.term list"   and t1 :: "('b, 'c) Term.term"   and ts' :: "('b, 'c) Term.term list" assumes "admissible_kbo w w0 pr_strict pr_weak least scf"   and "weight_fun.weight w w0 scf t = w0"   and "t = Fun f ts"   and "ts = t1 # ts'" shows "w (f, 1) = 0"
aligned: shows "is_aligned (of_bl (to_bl b) << n) m"
preallocated_upd_obj: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a :: "nat"   and C :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and S' :: "(char list list \<times> (char list \<Rightarrow> val option)) set" assumes "preallocated h"   and "h a = \<lfloor>(C, S)\<rfloor>" shows "preallocated (h(a \<mapsto> (C, S')))"
sel_2: fixes x1 :: "'b"   and x2 :: "'b stream" shows "stl (x1 ## x2) = x2"
WhileTrue: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com" assumes "s \<in> b"   and "\<Gamma>\<turnstile>c \<down> Normal s"   and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>While b c \<down> s'" shows "\<Gamma>\<turnstile>While b c \<down> Normal s"
ceiling_less_zero: fixes x :: "'a" shows "(\<lceil>x\<rceil> < 0) = (x \<le> - (1::'a))"
enat_le_coinduct: fixes P :: "enat \<Rightarrow> enat \<Rightarrow> bool"   and m :: "enat"   and n :: "enat" assumes "P m n"   and "\<And>m n. P m n \<Longrightarrow> (n = 0 \<longrightarrow> m = 0) \<and> (m \<noteq> 0 \<longrightarrow> n \<noteq> 0 \<longrightarrow> (\<exists>k n'. P (epred m) n' \<and> epred n = n' + k) \<or> epred m \<le> epred n)" shows "m \<le> n"
tendsto_divide_zero: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and c :: "'c" assumes "(f \<longlongrightarrow> (0::'c)) F" shows "((\<lambda>x. f x / c) \<longlongrightarrow> (0::'c)) F"
that: fixes j :: "nat" assumes "is_kth_obs \<pi> l j" shows "thesis"
simps_2: fixes A :: "'b wordinterval"   and B :: "'b wordinterval" shows "wordinterval_lowest_element (RangeUnion A B) = (case (wordinterval_lowest_element A, wordinterval_lowest_element B) of (None, None) \<Rightarrow> None | (None, Some x) \<Rightarrow> Some x | (Some a, None) \<Rightarrow> Some a | (Some a, Some b) \<Rightarrow> Some (if a < b then a else b))"
lexordp_eq_antisym: fixes xs :: "'d word list"   and ys :: "'d word list" assumes "signed.lexordp_eq xs ys"   and "signed.lexordp_eq ys xs" shows "xs = ys"
that: fixes Q' :: "('a, 'b, 'c) psi"   and Q'' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'"   and "insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q') \<Psi>"   and "\<Psi> \<otimes> \<Psi>' \<rhd> Q' \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q''"   and "(\<Psi> \<otimes> \<Psi>', P, Q'') \<in> Rel" shows "thesis"
size_neq_size_imp_neq: fixes x :: "'k"   and y :: "'k" assumes "size x \<noteq> size y" shows "x \<noteq> y"
push_bit_eq_mult: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n a = times a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
assms: shows "n -as @ a # as'\<rightarrow>\<^sub>\<surd>* n'"
assms_2: shows "x \<in> rvars (\<T> s')"
kruskal_correct_forest: shows "<emp> kruskal getEdges_impl (\<lambda>(u, w, v). return (u, v)) () <\<lambda>r. \<up> (distinct r \<and> set r \<subseteq> csuper_E \<and> s.MSF (set (map \<alpha> r)))>\<^sub>t"
r_comp_rtrancl_eq: fixes r :: "('a \<times> 'a) set" shows "r ; r\<^sup>* = r\<^sup>* ; r"
assms_2: shows "b \<in>\<^sub>\<circ> A"
ab_group_add_ow_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" shows "ab_group_add_ow U plus zero minus uminus \<equiv> (minus_ow U minus \<and> uminus_ow U uminus) \<and> comm_monoid_add_ow U plus zero \<and> ab_group_add_ow_axioms U plus zero minus uminus"
linordered_field_no_lb: shows "\<forall>x. \<exists>y. y < x"
add_ac_3: fixes b :: "'k"   and a :: "'k"   and c :: "'k" shows "b + (a + c) = a + (b + c)"
unit_vecs_basis: fixes n :: "nat" shows "vectorspace.basis class_ring (module_vec TYPE('b) n) (set (unit_vecs n))"
convex_Int: fixes s :: "'b set"   and t :: "'b set" assumes "convex s"   and "convex t" shows "convex (s \<inter> t)"
pr1_def: fixes a :: "'a"   and b :: "'a" shows "\<p>\<^sub>1[a, b] \<equiv> if ide a \<and> ide b then \<l>[b] \<cdot> (\<t>[a] \<otimes> b) else null"
is_Prim\<^sub>0_def: fixes "term" :: "'b local.term" shows "is_Prim\<^sub>0 term = (\<exists>x1. term = \<^bold>\<langle>x1\<^bold>\<rangle>\<^sub>0)"
fps_div_by_zero: fixes g :: "'a fps" shows "g / 0 = 0"
chain_rule: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "g \<in> carrier P" shows "pderiv (Cring_Poly.compose R f g) = Cring_Poly.compose R (pderiv f) g \<otimes>\<^bsub>UP R\<^esub> pderiv g"
VAR_def: fixes v :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "VAR v f \<equiv> f v"
sum_eq: fixes I :: "nat set" assumes "I \<in> P" shows "(\<Sum>b\<in>I. wt (fst cf1) (snd cf1) b * T.prob (cont_eff cf1 b) {cfT \<in> space (T (cont_eff cf1 b)). snd ((cont_eff cf1 b ## cfT) !! n) \<approx> s'}) = (\<Sum>b\<in>F I. wt (fst cf2) (snd cf2) b * T.prob (cont_eff cf2 b) {cfT \<in> space (T (cont_eff cf2 b)). snd ((cont_eff cf2 b ## cfT) !! n) \<approx> s'})"
mkarr_Unity: shows "\<F>C.mkarr \<^bold>\<I> = \<I>"
integral_differentiable: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and x :: "real" assumes "continuous_on {a..b} f"   and "x \<in> {a..b}" shows "(\<lambda>u. integral {a..u} f) differentiable at x within {a..b}"
tan_inverse: fixes y :: "real" shows "1 / tan y = tan (pi / 2 - y)"
interval_size_3_tendsto_0: fixes x :: "real_alg_3" shows "(\<lambda>i. upper_3 x i - lower_3 x i) \<longlonglongrightarrow> 0"
exhaust: fixes y :: "'g \<times> 'h" obtains x1 :: "'g" and x2 :: "'h" where "y = (x1, x2)"
ij_2: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a < b" shows "{i a b c..j__ a b c} \<subseteq> {a..b}"
valid: fixes g :: "'G" assumes "g.invar g" shows "valid_graph (g.\<alpha> g)"
bounded_linear_Blinfun_apply: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "blinfun_apply (Blinfun f) = f"
map_upt_zip: fixes xs :: "'d list"   and ys :: "'e list"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "length xs = length ys" shows "map (\<lambda>i. f (xs ! i) (ys ! i)) [0..<length ys] = map2 f xs ys"
less_multiset\<^sub>H\<^sub>O_imp_less_multiset\<^sub>D\<^sub>M: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "less_multiset\<^sub>H\<^sub>O M N" shows "less_multiset\<^sub>D\<^sub>M M N"
simps_98: fixes x2 :: "'b" shows "set2_fm (Nom x2) = {x2}"
cbvs2: fixes e1 :: "lam"   and e2 :: "lam"   and e3 :: "lam" assumes "e1 \<longrightarrow>cbv e2"   and " e2 \<longrightarrow>cbv* e3" shows " e1 \<longrightarrow>cbv* e3"
all_is_branch: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set"   and v :: "nat"   and ps :: "('b, 'c) fm list"   and a :: "'c" assumes "all_is_branch p i branch xs"   and "branch !. v = Some (ps, a)" shows "all_is p ps (proj xs v)"
prems_2: shows "k \<notin> set (map fst (p # ps))"
composite_simps_2: shows "trg (\<nu> \<star> \<pi>) = trg \<nu>"
proj_id: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c"   and a :: "univ" assumes "ModelIkMcalc wtFsym wtPsym arOf resOf parOf \<Phi> infTp intT intF intP"   and "intT \<sigma> a" shows "ModelIkMcalc.proj intT \<sigma> a = a"
foldl_conc_empty_eq: fixes i :: "'a list"   and ww :: "'a list list" shows "foldl (@) i ww = i @ foldl (@) [] ww"
rewrite_negated_primitives: fixes m :: "'b match_expr"   and disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and C :: "'c \<Rightarrow> 'b"   and \<gamma> :: "('b \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and negate_f :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b match_expr"   and a :: "action"   and p :: "'d" assumes "normalized_nnf_match m"   and "wf_disc_sel disc_sel C"   and "\<forall>pts. matches \<gamma> (negate_f C pts) a p = matches \<gamma> (MatchNot (Match (C pts))) a p" shows "matches \<gamma> (rewrite_negated_primitives disc_sel C negate_f m) a p = matches \<gamma> m a p"
valid_add_new_muaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "'muaux"   and auxlist :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and rel1 :: "event_data option list set"   and rel2 :: "event_data option list set"   and nt :: "nat" assumes "valid_muaux args cur aux auxlist"   and "table (args_n args) (args_L args) rel1"   and "table (args_n args) (args_R args) rel2"   and "cur \<le> nt" shows "valid_muaux args nt (add_new_muaux args rel1 rel2 nt aux) (update_until args rel1 rel2 nt auxlist)"
min_dist_is_dist: fixes v :: "nat"   and v' :: "nat" assumes "local.connected v v'" shows "local.dist v (min_dist v v') v'"
ncop__ncol: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Coplanar A B C D" shows "\<not> Col A B C"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs" shows "l_new_character_data_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
dim_basis: fixes A :: "'e set" assumes "finite A"   and "im.basis A" shows "im.dim = card A"