hcju/isabelleps huggingface dataset:
simps_2: fixes cs :: "'edge list"   and a :: "'edge"   and as :: "'edge list" shows "upd_cs cs (a # as) = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> upd_cs (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> upd_cs cs as | c' # cs' \<Rightarrow> upd_cs cs' as | _ \<Rightarrow> upd_cs cs as)"
diff_left_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq b a" shows "less_eq (minus c a) (minus c b)"
distinct_indices_def: fixes as :: "('c \<times> 'd) list" shows "distinct_indices as = distinct (map fst as)"
intro: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and A :: "('a \<times> 'a) set" assumes "Prim.prim_invar1 w g r A" shows "Prim_Invar1_loc w g r A"
intersecting_closure: fixes x :: "'a set" shows "intersecting (closure x) = intersecting x"
comp_cod_arr: fixes f :: "(bool, bool list) concrete_category.arr"   and b :: "(bool, bool list) concrete_category.arr" assumes "J.arr f"   and "J.cod f = b" shows "b \<cdot>\<^sub>J f = f"
Class_in_Partition: fixes a :: "'a" assumes "a \<in> G" shows "contained.Class a \<in> contained.Partition"
card: fixes X :: "'a set" shows "card (linear_cf X) = (if X \<inter> Field r = {} then 0 else 1)"
Inters_Some: fixes r :: "'c rexp"   and xs :: "'c rexp list" shows "(Inters r = Some xs) = (Zero \<notin> set xs \<and> xs = sorted_list_of_set (toplevel_inters r - {Full}))"
simps_1: fixes xs :: "'a list" shows "List.n_lists 0 xs = [[]]"
intros_22: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and n :: "nat"   and vs :: "v list"   and c :: "i32" assumes "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "mem_size m = n" shows "\<lparr>s;vs;[$C ConstInt32 c, $Grow_memory]\<rparr> \<leadsto>_ i \<lparr>s;vs;[$C ConstInt32 int32_minus_one]\<rparr>"
mult_strict_right_mono_neg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less b a"   and "less c zero" shows "less (times a c) (times b c)"
sound: fixes L :: "'a set"   and ars :: "('b \<times> 'b) set" assumes "LD L ars" shows "CR ars"
tab_simps_2: shows "src \<tau>\<mu>.tab = src (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0)"
has_fields_fun: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> ty) list"   and FDTs' :: "((char list \<times> char list) \<times> ty) list" assumes "P \<turnstile> C has_fields FDTs"   and "P \<turnstile> C has_fields FDTs'" shows "FDTs' = FDTs"
pr_algo_term_maxflow: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" assumes "Height_Bounded_Labeling c s t f l"   and "(f, l) \<notin> Domain (Network.pr_algo_lts c t)" shows "Network.isMaxFlow c s t f"
rotation_matrix_exists_basis: fixes a :: "(real, 'a) vec"   and k :: "'a" assumes "2 \<le> CARD('a)"   and "norm a = 1" obtains A :: "((real, 'a) vec, 'a) vec" where "rotation_matrix A" and "A *v axis k 1 = a"
3: shows "rawpsubstT (rawpsubstT (tms r1 r2) (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = tms (rawpsubstT (rawpsubstT r1 (zip (map Var us) (map snd txs))) (zip (map fst txs) us)) (rawpsubstT (rawpsubstT r2 (zip (map Var us) (map snd txs))) (zip (map fst txs) us))"
card_rcosets_triv: assumes "finite (carrier (G\<lparr>carrier := H\<rparr>))" shows "card (rcosets\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> {\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub>}) = order (G\<lparr>carrier := H\<rparr>)"
simps_4: fixes \<phi> :: "'b pltl" shows "atoms_pltl (X\<^sub>p \<phi>) = atoms_pltl \<phi>"
eval_simps'_1: fixes f :: "'a E.term" assumes "E.Arr f" shows "B.arr (EVAL f)"
hyps: assumes "x \<in> set (remdups_wrt f xs)"   and "y \<in> set (remdups_wrt f xs)" shows "f x \<noteq> f y"
Seq_ir_valid: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and d :: "com"   and d' :: "com"   and R :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid P c c' Q"   and "ir_valid Q d d' R" shows "ir_valid P (c;; d) (c';; d') R"
sum_dist: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "\<Sum> A \<cdot> \<Sum> B = \<Sum> {x \<cdot> y |x y. x \<in> A \<and> y \<in> B}"
assms_3: shows "wf\<^sub>e\<^sub>s\<^sub>t {} A"
mem_carrier: fixes x :: "'a" assumes "x \<in> A" shows "x \<in> carrier G"
Method_i_i_i_o_o_o_oE: fixes x :: "'a prog"   and xa :: "String.literal"   and xb :: "String.literal"   and xc :: "ty list"   and xd :: "ty"   and xe :: "'a option"   and xf :: "String.literal" assumes "pred.eval (Method_i_i_i_o_o_o_o x xa xb) (xc, xd, xe, xf)" obtains "x \<turnstile> xa sees xb: xc\<rightarrow>xd = xe in xf"
homogeneous_set_cone_decomp: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "cone_decomp T ps"   and "hom_decomp ps" shows "homogeneous_set T"
assms_5: shows "snd (rep_nat_term u) \<le> snd (rep_nat_term v)"
Op_preserves_good: fixes inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and binp :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs set option"   and delta :: "'c" assumes "goodInp inp"   and "goodBinp binp" shows "good (Op delta inp binp)"
good0: fixes w :: "letter list"   and ws :: "letter list list" assumes "L w ws" shows "good (w # ws)"
finite_extension_subring_incl: fixes K' :: "'a set"   and K :: "'a set"   and xs :: "'a list" assumes "subring K' R"   and "K \<subseteq> K'"   and "set xs \<subseteq> K'" shows "finite_extension K xs \<subseteq> K'"
accp_downward: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b" assumes "Wellfounded.accp r b"   and "r a b" shows "Wellfounded.accp r a"
CCSNil_def: shows "\<zero> \<equiv> Abs_ccs CCSNil_Rep"
leg1_simps_3: shows "src f\<^sub>0g\<^sub>1.p\<^sub>1 = src f\<^sub>0g\<^sub>1.p\<^sub>0"
q: shows "prefix t' t'' \<or> t' = t'' @ [ev]"
a_def: fixes n :: "nat" shows "a n = (SOME y. r (fst (u n)) (fst (u (Suc n))) (x n) y \<and> r' (snd (u n)) (snd (u (Suc n))) y (z n))"
irreducibleD': fixes a :: "'a"   and b :: "'a" assumes "irreducible a"   and "b dvd a" shows "a dvd b \<or> b dvd (1::'a)"
assms_4: shows "a \<otimes> fst h [^] e = \<^bold>g [^] z"
mult_scalar_minus_mult_right: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'b \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_scalar pair_of_term term_of_pair p (- q) = - term_powerprod.mult_scalar pair_of_term term_of_pair p q"
pvalid_admissible: fixes p :: "(symbol \<times> character list) list"   and x :: "item" assumes "pvalid p x" shows "admissible p"
listsp_conj_eq: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" shows "listsp (\<lambda>x. A x \<and> B x) = (\<lambda>x. listsp A x \<and> listsp B x)"
lim: shows "((inverse \<circ> f) \<longlongrightarrow> l) at_infinity"
simps_143: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x6 :: "char list" shows "Call x101 x102 x103 x104 \<noteq> Var x6"
reversepath_def: fixes g :: "real \<Rightarrow> 'a" shows "reversepath g = (\<lambda>x. g (1 - x))"
diff_eq_conv_nat: fixes x :: "nat"   and y :: "nat"   and z :: "nat" shows "(x - y = z) = (if y < x then x = y + z else z = 0)"
h5: shows "\<turnstile>\<^sub>p {adjoint (exexH_k (n - 1)) * (adjoint (tensor_P mat_Ph (1\<^sub>m K)) * (adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1)) * tensor_P mat_Ph (1\<^sub>m K)) * exexH_k (n - 1)} hadamard_n n;; Utrans_P vars1 mat_Ph;; hadamard_n n {Q2}"
rep_eq: fixes x :: "nat"   and xa :: "float" shows "real_of_float (float_round_down x xa) = truncate_down x (real_of_float xa)"
app3: fixes env :: "v sem_env"   and s1 :: "'a state"   and es :: "exp0 list"   and s2 :: "'a state"   and vs :: "v list" assumes "Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked_Single.evaluate env) s1 (rev es) (s2, Rval vs)"   and "do_opapp (rev vs) = None" shows "Big_Step_Unclocked_Single.evaluate env s1 (App Opapp es) (s2, Rerr (Rabort Rtype_error))"
that: fixes x :: "'d" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h' \<turnstile> g x \<rightarrow>\<^sub>h h''" shows "thesis"
image_mset_single: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "image_mset f {#x#} = {#f x#}"
f: shows "f \<in> borel_measurable (lebesgue_on S)"
simps_5: fixes c :: "int"   and e :: "MIR.num" shows "iszlfm (Lt (CN 0 c e)) = (\<lambda>bs. 0 < c \<and> numbound0 e \<and> isint e bs)"
fmdom'_alt_def: fixes m :: "'c \<rightharpoonup>\<^sub>f 'd" shows "fmdom' m = fset (fmdom m)"
BIT_competitive3: assumes "init \<noteq> []"   and "\<forall>i<length qs. qs ! i \<in> set init" shows "T_BIT (length qs) \<le> (7 / 4 - 3 / real (4 * length init)) * real (T init qs acts)"
residue_bernoulli: fixes n :: "int"   and m :: "nat" assumes "n \<noteq> 0" shows "residue (\<lambda>z. 1 / (z ^ m * (exp z - 1))) (complex_of_real (2 * pi * real_of_int n) * \<i>) = 1 / (complex_of_real (2 * pi * real_of_int n) * \<i>) ^ m"
zero_pf_neg_uvp: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and n :: "nat"   and w :: "bool stream" assumes "stock_portfolio Mkt pf"   and "pf stk (Suc n) w = 0"   and "pf risk_free_asset (Suc n) w \<noteq> 0"   and "val_process Mkt pf n w \<le> 0" shows "cls_val_process Mkt pf (Suc n) w < 0"
assms_1: shows "c = ((S, xs, ls) # fs, rv)"
assms_4: shows "S.independent (h ` K) (map h Us)"
LocalSecretsComposition_neg_k: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and m :: "Keys" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionLoc PQ"   and "correctCompositionKS PQ"   and "kKS m \<notin> specKeysSecrets P"   and "kKS m \<notin> specKeysSecrets Q"   and "\<not> ine P (kE m)"   and "\<not> ine Q (kE m)"   and "kKS m \<notin> LocalSecrets P \<union> LocalSecrets Q" shows "kKS m \<notin> LocalSecrets PQ"
simps_5: fixes r1 :: "'b rexp"   and r2 :: "'b rexp" shows "nullable (Times r1 r2) = (nullable r1 \<and> nullable r2)"
xsMin_def: shows "(ARG_MIN length xs. xs \<in> {xs. \<exists>vs'\<in>L\<^sub>i\<^sub>n M1 V. sequence_to_failure M1 M2 (vs' @ xs)}) \<in> {xs. \<exists>vs'\<in>L\<^sub>i\<^sub>n M1 V. sequence_to_failure M1 M2 (vs' @ xs)} \<and> (\<forall>xs\<in>{xs. \<exists>vs'\<in>L\<^sub>i\<^sub>n M1 V. sequence_to_failure M1 M2 (vs' @ xs)}. length (ARG_MIN length xs. xs \<in> {xs. \<exists>vs'\<in>L\<^sub>i\<^sub>n M1 V. sequence_to_failure M1 M2 (vs' @ xs)}) \<le> length xs)"
prems: shows "\<forall>r. \<not> evaluate_decs False mn env s (d # ds) r"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) (g \<star> f)"
assms_4: fixes a :: "real"   and b :: "real"   and as :: "real list"   and bs :: "real list" assumes "a \<noteq> b" shows "P (a # as) (b # bs)"
fBex_triv_one_point2: fixes A :: "'a fset"   and a :: "'a" shows "fBex A ((=) a) = (a |\<in>| A)"
assms: shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<bullet> r"
zero_less_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "Num.num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (numeral.numeral one plus n)"
semilattice_set_apply: fixes x :: "'b semilattice_set" shows "semilattice_set_apply x \<in> {f. semilattice_set f}"
wt_instr_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and T :: "ty"   and mxs :: "nat"   and mpc :: "nat"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and i :: "instr"   and pc :: "nat"   and \<tau>s :: "(ty list \<times> ty err list) option list" shows "P,T,mxs,mpc,xt \<turnstile> i,pc :: \<tau>s \<equiv> app i P mxs T pc mpc xt (\<tau>s ! pc) \<and> (\<forall>(pc', \<tau>')\<in>set (eff i P pc xt (\<tau>s ! pc)). P \<turnstile> \<tau>' \<le>' \<tau>s ! pc')"
direct_sum_is_vs: fixes K :: "('g, 'h) ring_scheme"   and V1 :: "('g, 'i, 'j) module_scheme"   and V2 :: "('g, 'k, 'l) module_scheme" assumes "vectorspace K V1"   and "vectorspace K V2" shows "vectorspace K (direct_sum V1 V2)"
lucas_lehmer_test_code_arithmetic: fixes p :: "nat" shows "lucas_lehmer_test p = (2 < p \<and> gen_lucas_lehmer_sequence' (2 ^ p - 1) 4 (p - 2) = 0)"
i_Exec_Stream_expand_aggregate_take: fixes k :: "nat"   and n :: "nat"   and trans_fun :: "'e message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e message_af"   and c :: "'d"   and ag :: "'d list \<Rightarrow> 'd" assumes "0 < k" shows "i_aggregate (i_Exec_Comp_Stream trans_fun (input \<odot> k) c) k ag \<Down> n = f_aggregate (f_Exec_Comp_Stream trans_fun (input \<Down> n \<odot> k) c) k ag"
elims: fixes x :: "nat"   and xa :: "real mpoly"   and xb :: "real mpoly"   and xc :: "real mpoly"   and xd :: "real mpoly"   and xe :: "atom fm"   and xf :: "nat"   and y :: "atom fm" assumes "quadratic_sub_fm_helper x xa xb xc xd xe xf = y" obtains var :: "nat" and a :: "real mpoly" and b :: "real mpoly" and c :: "real mpoly" and d :: "real mpoly" and F :: "atom fm" and z :: "nat" where "x = var" and "xa = a" and "xb = b" and "xc = c" and "xd = d" and "xe = F" and "xf = z" and "y = liftmap (\<lambda>x. quadratic_sub (var + x) (liftPoly 0 x a) (liftPoly 0 x b) (liftPoly 0 x c) (liftPoly 0 x d)) F z"
le_less_linear: fixes x :: "'a"   and y :: "'a" shows "x \<le> y \<or> y < x"
units_of_mult: fixes G :: "('c, 'd) monoid_scheme" shows "(\<otimes>\<^bsub>units_of G\<^esub>) = (\<otimes>\<^bsub>G\<^esub>)"
integrable_suminf: fixes M :: "'b measure"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>i. integrable M (f i)"   and "AE x in M. summable (\<lambda>i. norm (f i x))"   and "summable (\<lambda>i. LINT x|M. norm (f i x))" shows "integrable M (\<lambda>x. \<Sum>i. f i x)"
preserves_cod_def: fixes G :: "('a, 'b, 'd, 'e) functor" shows "preserves_cod G \<equiv> \<forall>f\<in>Ar. G \<^bsub>\<o>\<^esub> (Cod f) = Cod\<^bsub>BB\<^esub> G \<^bsub>\<a>\<^esub> f"
tree_chop_pure_tree: fixes x :: "'a" shows "tree_chop (pure x) = pure x"
set_Collect_arr_Set: fixes T :: "V"   and \<alpha> :: "V" shows "(T \<in>\<^sub>\<circ> ZFC_in_HOL.set (Collect (arr_Set \<alpha>))) = arr_Set \<alpha> T"
eq_refl: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" assumes "x = y" shows "less_eq_prod x y"
int_of_real_sub: fixes a :: "real"   and b :: "real" assumes "real_is_int a"   and "real_is_int b" shows "int_of_real (a - b) = int_of_real a - int_of_real b"
real_hyp2_B_hyp2_cltn2: fixes J :: "cltn2"   and a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2" assumes "is_K2_isometry J"   and "B\<^sub>K a b c" shows "B\<^sub>K hyp2_cltn2 a J hyp2_cltn2 b J hyp2_cltn2 c J"
insert_Diff: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "insert a (A\<setminus>{a}) = A"
redT_updTs_None: fixes ts :: "'f \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and tas :: "('f, 'd, 'g) new_thread_action list"   and t :: "'f" assumes "redT_updTs ts tas t = None" shows "ts t = None"
invertible_times_eq_zero: fixes A :: "((real, 'a) vec, 'a) vec"   and x :: "(real, 'a) vec" assumes "invertible A"   and "A *v x = 0" shows "x = 0"
wt_db_db: fixes j :: "nat"   and i :: "nat" shows "wt_sym (db j) = wt_sym (db i)"
coupling_def: fixes Q :: "'d \<Rightarrow> nat option"   and V :: "'d \<Rightarrow> nat option"   and D :: "'d \<Rightarrow> enat"   and S :: "'d set" shows "coupling Q V D S \<equiv> D = Dijkstra_Abstract.enat_of_option \<circ> V ++ Q \<and> S = dom V \<and> dom V \<inter> dom Q = {}"
sep_disj_commuteI: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj x y" shows "sep_disj y x"
restrict_relation_carrier: shows "restrict_relation alts' (Pareto R') = Pareto R'"
assms_2: shows "is_heap_list (priority t2) (children t2)"
simps_2: shows "\<lbrace>\<^bold>\<I>\<rbrace> = D.unity"
field_sum_of_halves: fixes x :: "'c" shows "x / (2::'c) + x / (2::'c) = x"
inverse_arrow_unique: fixes f :: "'d"   and g :: "'d"   and g' :: "'d" assumes "D.inverse_arrows f g"   and "D.inverse_arrows f g'" shows "g = g'"
10: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z \<le> x" shows "y \<squnion> z \<le> x"
assms_1: shows "[:- a, 1::'a:] ^ k dvd p"
Class_widen: fixes P :: "'b prog"   and C :: "String.literal"   and T :: "ty" assumes "P \<turnstile> Class C \<le> T" shows "\<exists>D. T = Class D"
\<Delta>\<Phi>_pass2: fixes hs :: "'b tree" assumes "hs \<noteq> \<langle>\<rangle>" shows "\<Phi> (pass\<^sub>2 hs) - \<Phi> hs \<le> log 2 (real (size hs))"
map_upds_None_cut: fixes tab :: "'b \<Rightarrow> 'a option"   and xs :: "'b list"   and ys :: "'a list"   and vn :: "'b" assumes "(tab(xs [\<mapsto>] ys)) vn = None" shows "tab vn = None"
induct: fixes P :: "'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> (nat \<times> nat) list \<Rightarrow> 'b mat \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "nat"   and a2 :: "nat"   and a3 :: "(nat \<times> nat) list"   and a4 :: "'b mat" assumes "\<And>x l k A. P x l k [] A"   and "\<And>x l k i_begin i_end blocks A. (\<And>xa. xa = (if i_end = l then A else ??.Jordan_Normal_Form_Existence.step_3_c_inner_loop (A $$ (i_end, k) / x) l i_end (Suc i_end - i_begin) A) \<Longrightarrow> P x l k blocks xa) \<Longrightarrow> P x l k ((i_begin, i_end) # blocks) A" shows "P a0 a1 a2 a3 a4"
weak_reduction_coupled_simulation_and_closures_3: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_coupled_simulation Rel Cal" shows "weak_reduction_coupled_simulation (Rel\<^sup>*) Cal"
mk_less_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "mk_less R = (\<lambda>x y. R x y \<and> \<not> R y x)"
1: shows "D = P1 * A * Q1 \<and> invertible_mat P1 \<and> invertible_mat Q1 \<and> isDiagonal_mat D \<and> P1 \<in> carrier_mat 2 2 \<and> Q1 \<in> carrier_mat 2 2 \<and> D \<in> carrier_mat 2 2"
det_Gauss_Jordan_in_ij_det_P: fixes A :: "(('b, 'c) vec, 'c) vec"   and i :: "'c"   and j :: "'c" shows "fst (Gauss_Jordan_in_ij_det_P A i j) * det A = det (snd (Gauss_Jordan_in_ij_det_P A i j))"
le_add_diff_inverse2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and b :: "'d"   and a :: "'d" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus (minus a b) b = a"
is_SNF_intro: fixes P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat"   and S :: "'b mat" assumes "P \<in> carrier_mat (dim_row A) (dim_row A)"   and "Q \<in> carrier_mat (dim_col A) (dim_col A)"   and "invertible_mat P"   and "invertible_mat Q"   and "Smith_normal_form_mat S"   and "S = P * A * Q" shows "is_SNF A (P, S, Q)"
rel_nondetT_def: shows "rel_nondetT = rel_nondetT' \<top>"
neg_add_closed: fixes G :: "'a set"   and g :: "'a"   and h :: "'a" assumes "Group G"   and "g \<in> G"   and "h \<in> G" shows "- g + h \<in> G"
verts_app_iso: fixes hom :: "('g, 'h, 'e, 'f) digraph_isomorphism"   and G :: "('g, 'h) pre_digraph" shows "verts (app_iso hom G) = iso_verts hom ` verts G"
nt_match_list_simp: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and a :: "action"   and p :: "'b"   and ms :: "'a negation_type list" shows "nt_match_list \<gamma> a p ms = ((\<forall>m\<in>set (getPos ms). matches \<gamma> (Match m) a p) \<and> (\<forall>m\<in>set (getNeg ms). matches \<gamma> (MatchNot (Match m)) a p))"
exec_drest_5: fixes x :: "('d, 'e) fmap"   and vs :: "'d set" assumes "fmdom' x \<subseteq> vs" shows "fmrestrict_set vs x = x"
Rosser_Form_Proofs_Minimal_Truth_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and Lq :: "'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and N :: "'c"   and S :: "'c"   and "proof" :: "'d set"   and prfOf :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and encPf :: "'d \<Rightarrow> 'b"   and Pf :: "'c"   and isTrue :: "'c \<Rightarrow> bool" shows "Rosser_Form_Proofs_Minimal_Truth var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num Lq prv bprv enc N S proof prfOf encPf Pf isTrue \<equiv> Rosser_Form_Proofs var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi fls prv bprv Lq dsj proof prfOf enc N S encPf Pf \<and> Minimal_Truth_Soundness var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num bprv isTrue"
mat_row_def: fixes k :: "'c"   and i :: "'d" shows "mat_row k i = (\<chi>j. if i = j then k else (0::'c))"
arith_geo_mean_sqrt: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y" shows "sqrt (x * y) \<le> (x + y) / 2"
termFSwImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) model_scheme" shows "termFSwImorph h hA MOD \<equiv> ipresWlsAll h hA MOD \<and> ipresCons h hA MOD \<and> ipresFreshAll h hA MOD \<and> ipresSwapAll h hA MOD"
line_integral_sum_gen_1: fixes basis :: "'a set"   and F :: "'a \<Rightarrow> 'a"   and basis1 :: "'a set"   and \<gamma> :: "real \<Rightarrow> 'a"   and basis2 :: "'a set" assumes "finite basis"   and "line_integral_exists F basis1 \<gamma>"   and "line_integral_exists F basis2 \<gamma>"   and "basis1 \<union> basis2 = basis"   and "basis1 \<inter> basis2 = {}" shows "line_integral F basis \<gamma> = line_integral F basis1 \<gamma> + line_integral F basis2 \<gamma>"
WPO_S_subset_WPO_NS: shows "WPO_S \<subseteq> WPO_NS"
trancl_on_def: shows "trancl_on \<equiv> \<lambda>U r. {(xa, x). on (\<lambda>x. x \<in> U) (\<lambda>x xa. (x, xa) \<in> r)\<^sup>+\<^sup>+ xa x}"