hcju/isabelleps huggingface dataset:
pprt_singleton: fixes j :: "nat"   and i :: "nat"   and x :: "'a" shows "pprt (singleton_matrix j i x) = singleton_matrix j i (pprt x)"
bvars\<^sub>s\<^sub>s\<^sub>t_unlabel_Cons: fixes l :: "'c"   and b :: "('b, 'a) stateful_strand_step"   and A :: "('c \<times> ('b, 'a) stateful_strand_step) list" shows "bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, b) # A) = set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p b) \<union> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
"4_2": shows "a \<notin> set v\<^sub>1"
intro: fixes M :: "'c set set"   and \<Omega> :: "'c set" assumes "M \<subseteq> Pow \<Omega>" shows "subset_class \<Omega> M"
emb_step_at_chop: fixes p :: "dir list"   and t :: "('a, 'b) tm"   and s :: "('a, 'b) tm" assumes "emb_step_at p dir.Right t = s"   and "position_of t (p @ [dir.Right])"   and "list_all (\<lambda>x. x = dir.Left) p" shows "chop t = s \<or> chop t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s"
fresh_subst': fixes z :: "name"   and y :: "name"   and t\<^sub>1 :: "trm"   and t\<^sub>2 :: "trm" assumes "z \<sharp> [y].t\<^sub>1"   and "z \<sharp> t\<^sub>2" shows "z \<sharp> t\<^sub>1[y::=t\<^sub>2]"
sphere_1D_doubleton: fixes r :: "real"   and a :: "'a" assumes "DIM('a) = 1"   and "0 < r" obtains x :: "'a" and y :: "'a" where "sphere a r = {x, y} \<and> dist x y = 2 * r"
l_set_disconnected_nodes_get_disconnected_nodes_axioms: shows "l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
statEqSim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<leadsto>[Rel] Q"   and "eqvt Rel'"   and "\<Psi> \<simeq> \<Psi>'"   and "\<And>\<Psi>'' R S \<Psi>'''. \<lbrakk>(\<Psi>'', R, S) \<in> Rel; \<Psi>'' \<simeq> \<Psi>'''\<rbrakk> \<Longrightarrow> (\<Psi>''', R, S) \<in> Rel'" shows "\<Psi>' \<rhd> P \<leadsto>[Rel'] Q"
l': shows "l' = ((3::'a) * x\<^sub>2\<^sup>2 + a) / ((2::'a) * - y\<^sub>2)"
r_inv_ex: fixes x :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)" shows "\<exists>y\<in>carrier (DirProds G I). x \<otimes>\<^bsub>DirProds G I\<^esub> y = \<one>\<^bsub>DirProds G I\<^esub>"
Liminf_PInfty: fixes net :: "'a filter"   and f :: "'a \<Rightarrow> ereal" assumes "net \<noteq> bot" shows "(f \<longlongrightarrow> \<infinity>) net = (Liminf net f = \<infinity>)"
indRelR_preserves_binary_pred_iff_indRelL_reflects_binary_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> 'a \<Rightarrow> bool" shows "rel_preserves_binary_pred indRelR Pred = rel_reflects_binary_pred indRelL Pred"
InitObject: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and sh' :: "sheap"   and P :: "(char list list \<times> char list exp) prog"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "sh C = \<lfloor>(sfs, Prepared)\<rfloor>"   and "C = Object"   and "sh' = sh(C \<mapsto> (sfs, Processing))"   and "P \<turnstile> \<langle>INIT C' (C # Cs,True) \<leftarrow> e,(h, l, sh')\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
inv2_rbtreeify_g: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> Suc (length kvs)" shows "inv2 (fst (rbtreeify_g n kvs))"
mayDiverge_transC: fixes c :: "('test, 'atom) com"   and s :: "'state" assumes "mayDiverge c s" shows "\<exists>c' s'. (c, s) \<rightarrow>c (c', s') \<and> mayDiverge c' s'"
closed_cdi_Compl: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and s :: "'c set" assumes "closed_cdi \<Omega> M"   and "s \<in> M" shows "\<Omega> - s \<in> M"
P_pow: fixes i :: "nat" shows "P ((f ^^ i) (f top))"
Frel_coind: fixes phi1 :: "'d JF1 \<Rightarrow> 'd JF1 \<Rightarrow> bool"   and phi2 :: "'d JF2 \<Rightarrow> 'd JF2 \<Rightarrow> bool"   and a1 :: "'d JF1"   and b1 :: "'d JF1"   and a2 :: "'d JF2"   and b2 :: "'d JF2" assumes "\<forall>a b. phi1 a b \<longrightarrow> F1rel (=) phi1 phi2 (dtor1 a) (dtor1 b)"   and "\<forall>a b. phi2 a b \<longrightarrow> F2rel (=) phi1 phi2 (dtor2 a) (dtor2 b)" shows "(phi1 a1 b1 \<longrightarrow> a1 = b1) \<and> (phi2 a2 b2 \<longrightarrow> a2 = b2)"
ln_exp: fixes x :: "real" shows "real_ln (exp x) = x"
olv_round_def: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and r_decisions :: "process \<Rightarrow> val option"   and Ob :: "process set" shows "olv_round r S v r_decisions Ob \<equiv> {(s, s'). r = opt_obsv_state.next_round s \<and> (S \<noteq> {} \<longrightarrow> opt_obs_safe (last_obs s) v) \<and> (S \<in> Quorum \<longrightarrow> Ob = UNIV) \<and> d_guard r_decisions (const_map v S) \<and> (Ob \<noteq> {} \<longrightarrow> S \<noteq> {}) \<and> s' = s\<lparr>opt_obsv_state.next_round := Suc r, opt_obsv_state.decisions := opt_obsv_state.decisions s ++ r_decisions, last_obs := last_obs s ++ const_map v Ob\<rparr>}"
open_cbox_convex: fixes x :: "'b"   and a :: "'b"   and b :: "'b"   and y :: "'b"   and e :: "real" assumes "x \<in> box a b"   and "y \<in> cbox a b"   and "0 < e"   and "e \<le> 1" shows "e *\<^sub>R x + (1 - e) *\<^sub>R y \<in> box a b"
linear_power_nonzero: fixes a :: "'a"   and n :: "nat" shows "[:a, 1::'a:] ^ n \<noteq> 0"
hd_coeff_is1_hd_coeffs1: fixes as :: "atom list"   and a :: "atom" assumes "\<forall>a\<in>set as. hd_coeff a \<noteq> 0"   and "a \<in> set (hd_coeffs1 as)" shows "hd_coeff_is1 a"
sorted_list_set: fixes t :: "('b, unit) RBT.rbt" shows "sorted_list_of_set (Set t) = RBT.keys t"
assms_2: shows "fst ` set txs \<subseteq> trm"
ide_cod: fixes f :: "'a" assumes "arr f" shows "ide (cod f)"
B_def: shows "B = (\<lambda>N K e. {x \<in> A. \<exists>n\<in>{1..N}. u n x - real n * F K e x < 0})"
filtersD1: fixes F :: "'b set set"   and T :: "'b set set" assumes "F \<in> Filters\<^bsub>T\<^esub>" shows "{} \<notin> F"
measurable_comp: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and g :: "'c \<Rightarrow> 'd"   and L :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> N \<rightarrow>\<^sub>M L" shows "g \<circ> f \<in> M \<rightarrow>\<^sub>M L"
real_of_ereal_ord_simps_3: fixes x :: "real"   and y :: "ereal" shows "(x < real_of_ereal y) = ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))"
cnf_disj_ex: fixes F :: "'b formula" assumes "is_disj F" shows "\<exists>R. cnf F = {R} \<or> cnf F = {}"
eventually_subseq: fixes r :: "nat \<Rightarrow> nat"   and P :: "nat \<Rightarrow> bool" assumes "strict_mono r"   and "eventually P sequentially" shows "\<forall>\<^sub>F n in sequentially. P (r n)"
hyps_2: shows "P \<parallel> !P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'"
height_balL: fixes l :: "('c \<times> nat) tree"   and r :: "('c \<times> nat) tree"   and a :: "'c" assumes "avl l"   and "avl r"   and "height l = height r + 2" shows "height (balL l a r) \<in> {height r + 2, height r + 3}"
order_le_opt: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "order r" shows "order (Opt.le r)"
cross3_simps_7: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "a + b + c = a + (b + c)"
language_state_for_inputs_alt_def: fixes M :: "('d, 'e, 'f) FSM"   and q :: "'f"   and ISeqs :: "'d list set" shows "LS\<^sub>i\<^sub>n M q ISeqs = \<Union> (language_state_for_input M q ` ISeqs)"
alpha_qSwap_preserves_qGood1: fixes zs :: "'varSort"   and y :: "'var"   and x :: "'var"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and zs' :: "'varSort"   and y' :: "'var"   and x' :: "'var"   and X' :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "X #[[y \<and> x]]_zs #= X' #[[y' \<and> x']]_zs'"   and "qGood X" shows "qGood X'"
threshold_def: fixes x :: "('a, 'b) IEEE.float itself" shows "threshold x = 2 ^ (emax x - 1) / 2 ^ bias x * (2 - 1 / 2 ^ Suc LENGTH('b))"
simps: fixes a :: "'a list" shows "local.sorted a = (a = [] \<or> (\<exists>xs x. a = x # xs \<and> (\<forall>y\<in>set xs. x \<prec> y) \<and> local.sorted xs))"
hyps: fixes f :: "'a \<Rightarrow> 'a" assumes "distincts xss"   and "\<And>xs. xs \<in> set xss \<Longrightarrow> list_succ xs = perm_restrict f (set xs)"   and "f permutes \<Union> (sset xss)" shows "f = lists_succ xss"
assms_1: shows "g \<in> \<Theta>(\<lambda>x. real x powr p')"
order_trans3: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.transitive less_eq less"   and "less x y"   and "less_eq y z" shows "less x z"
zip_rev: fixes xs :: "'d list"   and ys :: "'e list" assumes "length xs = length ys" shows "zip (rev xs) (rev ys) = rev (zip xs ys)"
expand_set: fixes g :: "'c \<Rightarrow> 'b" shows "Prod_any g = prod g {a. g a \<noteq> (1::'b)}"
path_components_of_def: fixes X :: "'a topology" shows "path_components_of X \<equiv> path_component_of_set X ` topspace X"
eq_norm: fixes x :: "int" shows "sint (word_of_int x) = signed_take_bit (LENGTH('b) - 1) x"
has_default_policy_fst: fixes rs :: "'a simple_rule list"   and r :: "'a simple_rule" assumes "has_default_policy rs" shows "has_default_policy (r # rs)"
E_ss: shows "G'.E \<subseteq> G'.V \<times> G'.V"
complex_unit_circle: fixes z :: "complex" assumes "z \<noteq> 0" shows "(Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1"
rec: fixes f :: "('b \<Rightarrow> bool) \<Rightarrow> 'a"   and x :: "'b \<Rightarrow> bool" shows "rec_pred f (Pred x) = f x"
Un_Int_distrib2: fixes B :: "'c set"   and C :: "'c set"   and A :: "'c set" shows "B \<inter> C \<union> A = (B \<union> A) \<inter> (C \<union> A)"
bisimE_4: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "(\<Psi>, Q, P) \<in> env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"
cd_split: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<not> i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k" obtains m :: "nat" where "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> m" and "m cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
preserves_ide: fixes a :: "'s" assumes "S.ide a" shows "S'.ide (\<Phi> a)"
reindex_cong: fixes l :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'c" assumes "bij l"   and "g \<circ> l = h" shows "Sum_any g = Sum_any h"
r_rev_aux_full: fixes e :: "nat"   and xs :: "nat list" assumes "Partial_Recursive.list_decode e = xs"   and "0 < length xs" shows "eval r_rev_aux [length xs - 1, e] \<down>= tuple_encode (length xs - 1) (rev xs)"
CallThrowObj: fixes E :: "char list \<Rightarrow> ty option"   and r :: "nat \<times> char list list"   and Copt :: "char list option"   and M :: "char list"   and es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>Call (Throw r) Copt M es,s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
simps_7: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Row of Row \<Rightarrow> f1 | Col \<Rightarrow> f2 | Num \<Rightarrow> f3) = f1"
case_1: fixes f1 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and f2 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> bool) \<Rightarrow> 'a"   and f3 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list \<Rightarrow> 'a"   and f4 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and x1 :: "('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option" shows "(case \<Up>x1 of \<Up>x \<Rightarrow> f1 x | (x)\<^sub>\<surd> \<Rightarrow> f2 x | x:xa\<hookrightarrow>\<^bsub>xb\<^esub>xc \<Rightarrow> f3 x xa xb xc | x\<hookleftarrow>\<^bsub>xa\<^esub>xb \<Rightarrow> f4 x xa xb) = f1 x1"
t_heap_of_A_bound: fixes xs :: "'b list" shows "t_heap_of_A xs \<le> length xs * (height (heap_of_A xs) + 1)"
comp_converter_parallel2'_1: fixes conv1l :: "('i, 'k, 'q, 's) converter"   and conv1r :: "('j, 'l, 'r, 't) converter"   and conv2l :: "('q, 's, 'm, 'o) converter"   and conv2r :: "('r, 't, 'n, 'p) converter" shows "(conv1l |\<^sub>= conv1r) \<odot> (conv2l |\<^sub>= conv2r) = conv1l \<odot> conv2l |\<^sub>= conv1r \<odot> conv2r"
Card_order_ordIso2: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Card_order r"   and "r =o r'" shows "Card_order r'"
neighboring_events_shifted: shows "\<forall>k. i + 1 < k \<and> k < j \<longrightarrow> \<not> regular_event (swap_events i (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)]) ! k)"
map_fixed_idle: fixes t :: "Syntax.term"   and l :: "Syntax.var list"   and v :: "Syntax.var" assumes "{x. \<not> atom x \<sharp> t} \<subseteq> set l" shows "map_fixed v l t = t"
closure_complement: fixes A :: "'a list set" assumes "regular A" shows "regular (- A)"
divide_poly_def: fixes f :: "'a poly"   and g :: "'a poly" shows "f div g = (if g = 0 then 0 else local.divide_poly_main (lead_coeff g) 0 f g (degree f) (1 + length (coeffs f) - length (coeffs g)))"
select_convs_4: fixes orderA :: "('e, 'f) gorder_scheme"   and orderB :: "('d, 'g) gorder_scheme"   and lower :: "'e \<Rightarrow> 'd"   and upper :: "'d \<Rightarrow> 'e"   and more :: "'h" shows "\<pi>\<^sub>*\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = upper"
simps_3: fixes n :: "nat"   and y :: "'a"   and x :: "'a" assumes "n \<noteq> 0"   and "even n" shows "local.efficient_funpow y x n = local.efficient_funpow y (f x x) (n div 2)"
borel_measurable_if_lebesgue_on: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> sets lebesgue"   and "T \<in> sets lebesgue"   and "S \<subseteq> T" shows "((\<lambda>x. if x \<in> S then f x else (0::'b)) \<in> borel_measurable (lebesgue_on T)) = (f \<in> borel_measurable (lebesgue_on S))"
hom_component_lpp_1: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0" shows "hom_component p (deg_pm (lpp p)) \<noteq> 0"
arev_in_arcs: fixes G :: "('e, 'f) pre_digraph"   and arev :: "'f \<Rightarrow> 'f"   and a :: "'f" assumes "bidirected_digraph G arev"   and "a \<in> arcs G" shows "arev a \<in> arcs G"
trg_inv: fixes \<mu> :: "'b" assumes "D.iso \<mu>" shows "trg\<^sub>D (D.inv \<mu>) = trg\<^sub>D \<mu>"
sup_synlr_def: fixes f :: "('a, 'b) synlr"   and g :: "('a, 'b) synlr" shows "sup f g = mksynlr (unsynlr f \<union> unsynlr g)"
tautci_def: fixes v :: "nat"   and bdd :: "bddi" shows "tautci v bdd \<equiv> destrci v bdd \<bind> (\<lambda>d. return (d = TD))"
holds_mono: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a stream"   and Q :: "'a \<Rightarrow> bool" assumes "holds P xs"   and "\<And>x. P x \<Longrightarrow> Q x" shows "holds Q xs"
assms_1: shows "\<turnstile>\<^sub>3\<^sub>a {P} c { Q}"
Nat_case_suc: fixes z :: "'a"   and s :: "nat discr Box \<rightarrow> 'a"   and n :: "nat" shows "Nat_case\<cdot>z\<cdot>s\<cdot>(box\<cdot>(Discr (Suc n))) = s\<cdot>(box\<cdot>(Discr n))"
incoming_edges: fixes u :: "nat" shows "cf.incoming u \<subseteq> cf.E"
ipurge_ref_subset: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'b list"   and X :: "'b set" shows "ipurge_ref I D u xs X \<subseteq> X"
inv_inj_on_unit: shows "inj_on local.inv (Collect unit)"
range_eqI: fixes b :: "'b"   and f :: "'d \<Rightarrow> 'b"   and x :: "'d" assumes "b = f x" shows "b \<in> range f"
divTr5: fixes r :: "nat"   and s :: "nat"   and l :: "nat" assumes "0 < r"   and "0 < s"   and "l < r * s" shows "l div s * s \<le> l \<and> l \<le> Suc (l div s) * s"
mult_smult_distrib: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat"   and k :: "'b" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "A * (k \<cdot>\<^sub>m B) = k \<cdot>\<^sub>m (A * B)"
cobounded1: fixes a :: "'a word"   and b :: "'a word" shows "signed.min a b \<le>s a"
ListRed1: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and es :: "expr list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e # es,s\<rangle> [\<rightarrow>] \<langle>e' # es,s'\<rangle>"
assms_1: shows "a \<longlonglongrightarrow> 0"
nAct_exists: fixes x :: "nat"   and c :: "'id"   and t :: "cnf llist" assumes "enat x < llength (\<pi>\<^bsub>c\<^esub>t)" shows "\<exists>n'. enat x = \<langle>c #\<^bsub>enat n'\<^esub>t\<rangle>"
prj_chine_assoc_2: shows "\<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1\<^sub>0 \<cdot> \<mu>_\<nu>_\<pi>\<rho>.chine_assoc = \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0\<^sub>1"
Collect_mem_eq: fixes A :: "'l set" shows "{x. x \<in> A} = A"
\<Phi>_p_def: fixes x :: "real" shows "\<Phi>_p x = x\<^sup>2 / 2 * ((real_of_int i' + 1) * 2 ^ (l + 1) + (real_of_int i - 1) * 2 ^ (l' + 1)) - x ^ 3 / 3 * 2 ^ (l' + 1) * 2 ^ (l + 1) - x * (real_of_int i - 1) * (real_of_int i' + 1)"
run_reachable': fixes x :: "'a"   and xs :: "'a stream"   and x\<^sub>0 :: "'a" assumes "run (x ## xs)"   and "x\<^sub>0 \<rightarrow>* x" shows "pred_stream (reaches x\<^sub>0) xs"
pnderiv_alt: fixes a :: "'b"   and r :: "'b rexp" shows "pnderiv a r = pnorm (deriv a r)"
Diff_Diff_Int: fixes A :: "'c set"   and B :: "'c set" shows "A - (A - B) = A \<inter> B"
subspace_Int: fixes I :: "'d set"   and s :: "'d \<Rightarrow> 'b set" assumes "\<And>i. i \<in> I \<Longrightarrow> subspace (s i)" shows "subspace (\<Inter> (s ` I))"
get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('z, 'x, 'y, 'ag option) RShadowRoot_scheme \<Rightarrow> 'aj" assumes "ShadowRootClass.type_wf h"   and "ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter"
axioms_2: fixes R :: "('a, 'b) ring_scheme" assumes "ring R" shows "Group.monoid R"
eexp_add: fixes x :: "ereal"   and y :: "ereal" assumes "\<not> (x = \<infinity> \<and> y = - \<infinity> \<or> x = - \<infinity> \<and> y = \<infinity>)" shows "eexp (x + y) = eexp x * eexp y"
Quotient_abs_rep: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and T :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'f" assumes "Quotient R Abs Rep T" shows "Abs (Rep a) = a"
commute: fixes a :: "'d"   and b :: "'d" shows "gcd a b = gcd b a"
minimal_APR: fixes M :: "'b dfa" assumes "dfa M"   and "dfa.states M = dfa.accessible M" shows "dfa.minimal (dfa.Accessible_dfa (nfa.Power_dfa (Reverse_nfa M)))"
zor_poly_pole_eqI: fixes f :: "complex \<Rightarrow> complex"   and z0 :: "complex"   and g :: "'b \<Rightarrow> complex"   and c :: "complex"   and F :: "'b filter" assumes "isolated_singularity_at f z0"   and "is_pole f z0"   and "((\<lambda>x. f (g x) * (g x - z0) ^ nat (- zorder f z0)) \<longlongrightarrow> c) F"   and "filterlim g (at z0) F"   and "F \<noteq> bot" shows "zor_poly f z0 z0 = c"
Un_C_submodule: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and C :: "('a set \<times> ('a \<Rightarrow> 'g)) set" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "C \<subseteq> fsps R M N f H"   and "C \<noteq> {}"   and "\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a" shows "submodule R M (fgs R M (\<Union> {a. \<exists>b. (a, b) \<in> C}))"
length_takeWhile_less_P_nth: fixes j :: "nat"   and P :: "'c \<Rightarrow> bool"   and xs :: "'c list" assumes "\<And>i. i < j \<Longrightarrow> P (xs ! i)"   and "j \<le> length xs" shows "j \<le> length (takeWhile P xs)"
simps_18: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "small_step \<pi> (WHILE b DO c, s) = Some (IF b THEN c;; WHILE b DO c ELSE SKIP, s)"
induct: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a0 :: "'b"   and a1 :: "'b"   and a2 :: "nat" assumes "\<And>y x. P y x 0"   and "\<And>y x. P y x (Suc 0)"   and "\<And>n y x. \<lbrakk>n \<noteq> 0; even n; P y (f x x) (n div 2)\<rbrakk> \<Longrightarrow> P y x n"   and "\<And>n y x. \<lbrakk>n \<noteq> 1; odd n; P (f x y) (f x x) (n div 2)\<rbrakk> \<Longrightarrow> P y x n" shows "P a0 a1 a2"
g_def: fixes s :: "real" shows "g s = (b - s) ^ (p - 1) / fact (p - 1)"
eval_tpoly_eq_eval_ztpoly: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and A :: "'c \<Rightarrow> hmultiset"   and p :: "('c, hmultiset) tpoly" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "zhmset_of (eval_tpoly A p) = eval_tpoly (\<lambda>v. zhmset_of (A v)) (map_tpoly (\<lambda>x. x) zhmset_of p)"
csols_def: fixes t0 :: "real"   and x0 :: "'a" shows "csols t0 x0 = {(x, t1). {t0--t1} \<subseteq> T \<and> x t0 = x0 \<and> (x solves_ode f) {t0--t1} S}"
tendsto_norm: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
tuple_pr: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "tuple (pr1 a b) (pr0 a b) = local.prod a b"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "\<And>n. \<lbrakk>\<not> 100 < n \<Longrightarrow> P (n + 11); \<not> 100 < n \<Longrightarrow> P (f91 (n + 11))\<rbrakk> \<Longrightarrow> P n" shows "P a0"