hcju/isabelleps huggingface dataset:
induct: fixes P :: "('a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> ('a, 'c) rbt \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a1 :: "'a"   and a2 :: "'c"   and a3 :: "('a, 'c) rbt" assumes "\<And>f k v. P f k v rbt.Empty"   and "\<And>f k v l x y r. \<lbrakk>c k x = Lt \<Longrightarrow> P f k v l; c k x = Gt \<Longrightarrow> P f k v r\<rbrakk> \<Longrightarrow> P f k v (Branch color.B l x y r)"   and "\<And>f k v l x y r. \<lbrakk>c k x = Lt \<Longrightarrow> P f k v l; c k x = Gt \<Longrightarrow> P f k v r\<rbrakk> \<Longrightarrow> P f k v (Branch color.R l x y r)" shows "P a0 a1 a2 a3"
asIMOD_igWls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and s :: "'sort"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b" shows "igWls (asIMOD SEM) s X = (\<forall>val. (sWlsVal SEM val \<or> X val = undefined) \<and> (sWlsVal SEM val \<longrightarrow> sWls SEM s (X val)))"
mat_to_cols_list_is_not_Nil: fixes A :: "complex Matrix.mat" assumes "0 < dim_col A" shows "mat_to_cols_list A \<noteq> []"
snd_compose_pairself: fixes f :: "'a \<Rightarrow> 'b" shows "snd \<circ> pairself f = f \<circ> snd"
is_projection_on_reduces_norm: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set"   and h :: "'b" assumes "is_projection_on \<pi> M"   and "closed_csubspace M" shows "\<parallel>\<pi> h\<parallel> \<le> \<parallel>h\<parallel>"
vrestr_union: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "qmpt M T" shows "qmpt.vimage_restr M f (A \<union> B) = qmpt.vimage_restr M f A \<union> qmpt.vimage_restr M f B"
assms_3: shows "\<phi> \<in> set \<phi>s"
fls_X_power_times_comm: fixes n :: "nat"   and f :: "'a fls" shows "fls_X ^ n * f = f * fls_X ^ n"
r: fixes b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "red F a b" shows "thesis"
hd0: shows "\<forall>a\<in>set as. hd_coeff a \<noteq> 0"
ans3: fixes x :: "'a" shows "ad (ad x) + ad x = (1::'a)"
IH: assumes "x \<noteq> (0::'a)" shows "normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x"
delete_min_max_inv: fixes t :: "interval_tree" assumes "tree_max_inv t"   and "t \<noteq> Tip" shows "tree_max_inv (snd (del_min t))"
jump_eq: shows "jumpF_pathfinish (linepath a b) z = jumpF (\<lambda>t. f t / g t) (at_left 1)"
is_App_apps: fixes s :: "('a, 'b) tm"   and ss :: "('a, 'b) tm list" shows "is_App (apps s ss) = (args (apps s ss) \<noteq> [])"
density_embed_measure': fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "inj f"   and "\<And>x. f' (f x) = x"   and "g \<in> borel_measurable M" shows "density (embed_measure M f) (g \<circ> f') = embed_measure (density M g) f"
intros_3: fixes xps :: "('e \<times> 'f) list"   and Y :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and X :: "('e \<times> 'f) set" assumes "(xps, Y) \<in> c_failures step out s\<^sub>0"   and "X \<subseteq> Y" shows "(xps, X) \<in> c_failures step out s\<^sub>0"
assms_3: shows "\<alpha> \<subseteq>\<^sub>\<circ> \<beta>"
bet_neq21__neq: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C"   and "B \<noteq> A" shows "A \<noteq> C"
bag_finite: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set"   and t :: "'g" assumes "TreeDecomposition G T bag"   and "t \<in> V\<^bsub>T\<^esub>" shows "finite (bag t)"
False_imp_not_eventually: fixes P :: "'c \<Rightarrow> bool"   and net :: "'c filter" assumes "\<forall>x. \<not> P x"   and "net \<noteq> \<bottom>" shows "\<not> eventually P net"
length_greater_0_conv: fixes xs :: "'c list" shows "(0 < |xs|) = (xs \<noteq> [])"
ok2_shift2: fixes r :: "'b mrexp2"   and m :: "bool"   and c :: "'b" assumes "ok2 r" shows "ok2 (shift2 m r c)"
preserves_reflects_arr: fixes f :: "'a" shows "R'.C.arr (H\<^sub>R a' f) = R'.C.arr f"
nsqn\<^sub>r_def: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat" shows "nsqn\<^sub>r r \<equiv> if \<pi>\<^sub>4 r = val \<or> \<pi>\<^sub>2 r = 0 then \<pi>\<^sub>2 r else \<pi>\<^sub>2 r - 1"
fb_graph_subset: fixes G' :: "('v, 'a) graph_rec_scheme" assumes "g_V G' = V"   and "g_E G' \<subseteq> E"   and "finite (g_V0 G')"   and "g_V0 G' \<subseteq> reachable" shows "fb_graph G'"
ASSERT_eq: shows "ASSERT \<equiv> iASSERT RETURN"
emb_step_is_emb: fixes t :: "('a, 'b) tm"   and s :: "('a, 'b) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
mojmir_to_rabin_axioms: shows "mojmir_to_rabin \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}"
set_mset_add_mset_insert: fixes a :: "'k"   and A :: "'k multiset" shows "set_mset (add_mset a A) = insert a (set_mset A)"
maxR_def: fixes d :: "D2"   and x :: "X4"   and y :: "X4" shows "StableNoDecomp.maxR d x y = (if (x, y) \<in> PX4d d then y else x)"
sup_assoc: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type"   and z :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (sup (x::'b::type) (y::'b::type)) (z::'b::type) = sup x (sup y z)"
subseqs_subformulas\<^sub>\<nu>_list: fixes Y :: "'b ltln set"   and \<phi> :: "'b ltln" shows "(Y \<subseteq> subformulas\<^sub>\<nu> \<phi>) = (\<exists>ys. Y = set ys \<and> ys \<in> set (subseqs (subformulas\<^sub>\<nu>_list \<phi>)))"
prod_zero_func: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_zero I A \<in> carr_prodag I A"
inv1: shows "callee_invariant_on callee2 I2 \<I>"
partition_def: fixes r :: "'b \<times> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "List_local.partition r xs \<equiv> snd (partition_aux r xs)"
proj_Eps: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set" assumes "equiv A r"   and "X \<in> A // r" shows "Equiv_Relations.proj r (SOME x. x \<in> X) = X"
cmult_in_iff': fixes c :: "'b"   and f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "((\<lambda>x. f x * c) \<in> L F g) = (f \<in> L F g)"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vint_mult" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> vint_mult). vint_mult\<lparr>a\<rparr> = b"
comp_single_set_bd: fixes fbd :: "('k \<times> 'k) set"   and fset :: "'l \<Rightarrow> 'm set"   and gset :: "'n \<Rightarrow> 'l set"   and gbd :: "('o \<times> 'o) set"   and x :: "'n" assumes "Card_order fbd"   and "\<And>x. |fset x| \<le>o fbd"   and "\<And>x. |gset x| \<le>o gbd" shows "|\<Union> (fset ` gset x)| \<le>o gbd *c fbd"
m_defs_1: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_C theta c d s t P F \<equiv> mC_C_part c d P F \<and> inj_on F P \<and> mC_C_wt c d s t P F \<and> mC_C_eff_cont theta c d s t P F"
strict_prefix_simps_1: fixes xs :: "'b list" shows "strict_prefix xs [] = False"
fds_abs_converges_moebius_mu_real: fixes s :: "real" assumes "1 < s" shows "fds_abs_converges (fds moebius_mu) s"
mirror_elem_mirror_elem_ident: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "n \<le> iMin I + Max I" shows "mirror_elem (mirror_elem n I) I = n"
triangle_in_hom_2: shows "\<guillemotleft>(g \<star>\<^sub>C \<epsilon>) \<cdot>\<^sub>C \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>C (\<eta> \<star>\<^sub>C g) : trg\<^sub>C g \<star>\<^sub>C g \<Rightarrow>\<^sub>C g \<star>\<^sub>C src\<^sub>C g\<guillemotright>"
bind_RES: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'b nres" shows "RES X \<bind> f = Sup (f ` X)"
in_AboveS_underS: fixes a :: "'c"   and r :: "('c \<times> 'c) set" assumes "a \<in> Field r" shows "a \<in> AboveS r (underS r a)"
pickBinp_asBinp: fixes qbinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs option" assumes "qGoodBinp qbinp" shows "(pickBinp (asBinp qbinp), qbinp) \<in> alphaBinp"
proj3_addpreRT: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip' :: "nat"   and npre :: "nat set" assumes "ip \<in> kD rt"   and "ip' \<in> kD rt" shows "\<pi>\<^sub>3 (the (the (addpreRT rt ip' npre) ip)) = \<pi>\<^sub>3 (the (rt ip))"
abs_e2: shows "\<bar>e2\<bar> \<le> snd (trunc_bound_pdevs p (scaleR_pdevs (fst (fst Y)) (snd (fst X))))"
notin_Lin_iff: fixes L :: "(alt \<times> alt) set"   and x :: "alt"   and y :: "alt" assumes "L \<in> Lin"   and "x \<noteq> y" shows "((x, y) \<notin> L) = (y <\<^bsub>L\<^esub> x)"
iteratei_rule_insert_P: fixes iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "'d set"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and I :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma>0 :: "'c"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool" assumes "set_iterator_genord iti S0 R"   and "I {} \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S0 - S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S0 - S - {x}. R x y; \<forall>y\<in>S. R y x\<rbrakk> \<Longrightarrow> I (insert x S) (f x \<sigma>)"   and "\<And>\<sigma>. I S0 \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> S0; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S0 - S. \<forall>y\<in>S. R y x\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iti c f \<sigma>0)"
Par_Arr_norm: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "monoidal_language.Arr C (free_monoidal_category.norm C t) \<and> monoidal_language.Arr C t \<and> monoidal_language.Dom C (free_monoidal_category.norm C t) = monoidal_language.Dom C t \<and> monoidal_language.Cod C (free_monoidal_category.norm C t) = monoidal_language.Cod C t"
ring_record_simps_11: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
s_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "get_S (cpu_reg_val PSR s1) = get_S (cpu_reg_val PSR s2)"
zariski_is_topological_space: shows "topological_space Spec is_zariski_open"
is_multi_arith_prog_onD: fixes l :: "nat"   and m :: "nat"   and start :: "int"   and steps :: "nat \<Rightarrow> nat"   and a :: "int"   and b :: "int"   and c :: "nat \<Rightarrow> nat" assumes "is_multi_arith_prog_on l m start steps a b"   and "c \<in> {0..<m} \<rightarrow> {0..<l}" shows "multi_arith_prog m start steps c \<in> {a..b}"
left_commute: fixes b :: "'e"   and a :: "'e"   and c :: "'e" shows "inf b (inf a c) = inf a (inf b c)"
assms_2: fixes \<sigma> :: "nat \<Rightarrow> 'g" assumes "P \<sigma>" shows "Q \<sigma>"
Call: fixes A :: "triple set"   and P :: "assn"   and e1 :: "expr"   and Q :: "vassn"   and e2 :: "expr"   and R :: "val \<Rightarrow> vassn"   and C :: "cname"   and m :: "mname"   and S :: "vassn" assumes "A \<turnstile>\<^sub>e {P} e1 {Q}"   and "\<forall>a. A \<turnstile>\<^sub>e {Q a} e2 {R a}"   and "\<forall>a p ls. A \<turnstile> {\<lambda>s'. \<exists>s. R a p s \<and> ls = s \<and> s' = lupd(This\<mapsto>a) (lupd(Par\<mapsto>p) (del_locs s))} Meth (C, m) {\<lambda>s. S (s<Res>) (set_locs ls s)}" shows "A \<turnstile>\<^sub>e {P} {C}e1..m(e2) {S}"
power_div_same_base: fixes b :: "'a"   and y :: "nat"   and x :: "nat" assumes "b ^ y \<noteq> (0::'a)"   and "y \<le> x" shows "b ^ x div b ^ y = b ^ (x - y)"
simps_3: 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 prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c" assumes "ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw" shows "ProblemIkTpart.TE_wtFsym wtFsym resOf (Wit \<sigma>) = (\<not> ProblemIkTpart.isRes wtFsym resOf \<sigma>)"
healthy: fixes a :: "'a"   and s :: "'s" assumes "a \<in> supp (p s)" shows "nearly_healthy (wlp (f a))"
cross7_def: fixes a :: "(real, 7) vec"   and b :: "(real, 7) vec" shows "a \<times>\<^sub>7 b \<equiv> vector [a $ 2 * b $ 4 - a $ 4 * b $ 2 + a $ 3 * b $ 7 - a $ 7 * b $ 3 + a $ 5 * b $ 6 - a $ 6 * b $ 5, a $ 3 * b $ 5 - a $ 5 * b $ 3 + a $ 4 * b $ 1 - a $ 1 * b $ 4 + a $ 6 * b $ 7 - a $ 7 * b $ 6, a $ 4 * b $ 6 - a $ 6 * b $ 4 + a $ 5 * b $ 2 - a $ 2 * b $ 5 + a $ 7 * b $ 1 - a $ 1 * b $ 7, a $ 5 * b $ 7 - a $ 7 * b $ 5 + a $ 6 * b $ 3 - a $ 3 * b $ 6 + a $ 1 * b $ 2 - a $ 2 * b $ 1, a $ 6 * b $ 1 - a $ 1 * b $ 6 + a $ 7 * b $ 4 - a $ 4 * b $ 7 + a $ 2 * b $ 3 - a $ 3 * b $ 2, a $ 7 * b $ 2 - a $ 2 * b $ 7 + a $ 1 * b $ 5 - a $ 5 * b $ 1 + a $ 3 * b $ 4 - a $ 4 * b $ 3, a $ 1 * b $ 3 - a $ 3 * b $ 1 + a $ 2 * b $ 6 - a $ 6 * b $ 2 + a $ 4 * b $ 5 - a $ 5 * b $ 4]"
card_ip1_to_j_is_0_in_swapped_events: fixes t :: "('a, 'b, 'c) event list"   and j :: "nat"   and i :: "nat"   and n :: "nat" assumes "prerecording_event t j"   and "postrecording_event t i"   and "i < j"   and "j < length t"   and "count_violations t = Suc n"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "count_violations t = Suc n"   and "trace init t final" shows "card {k \<in> {i + 1..<j + 1}. prerecording_event (swap_events i j t) k} = 0"
nth_prime_le_iff: fixes m :: "nat"   and n :: "nat" shows "(nth_prime m \<le> nth_prime n) = (m \<le> n)"
next_sendupd'_def: fixes c0 :: "('c, 'd, 'e) configuration_scheme"   and c1 :: "('c, 'd, 'e) configuration_scheme"   and p :: "'c"   and tt :: "'d set" shows "next_sendupd' c0 c1 p tt = (let \<gamma> = {#t \<in>#\<^sub>z c_temp c0 p. t \<in> tt#} in \<gamma> \<noteq> {#}\<^sub>z \<and> upright (c_temp c0 p - \<gamma>) \<and> c1 = c0\<lparr>c_msg := (c_msg c0)(p := \<lambda>q. c_msg c0 p q @ [\<gamma>]), c_temp := (c_temp c0)(p := c_temp c0 p - \<gamma>)\<rparr>)"
bidel_arc_diff_face_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" shows "bidel_arc_diff_face G M a \<equiv> bidel_arc G M a \<and> bidel_arc_diff_face_axioms M a"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and xa :: "'a \<Rightarrow>\<^sub>0 'b" shows "(x < xa) = less_fun (lookup x) (lookup xa)"
compose_def: fixes ss :: "('c \<Rightarrow> ('d, 'c) Term.term) list" shows "compose ss = foldr (\<circ>\<^sub>s) ss Var"
simps: fixes bs :: "atom set"   and x :: "'a"   and cs :: "atom set"   and y :: "'a" shows "((bs, x) \<approx>abs_set (cs, y)) = (\<exists>p. (bs, x) \<approx>set (=) supp p (cs, y))"
int_ge_induct: fixes k :: "int"   and i :: "int"   and P :: "int \<Rightarrow> bool" assumes "k \<le> i"   and "P k"   and "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P (i + 1)" shows "P i"
of_weak_ranking_imp_in_set_1: fixes a :: "'alt"   and b :: "'alt" assumes "of_weak_ranking xs a b" shows "a \<in> \<Union> (set xs)"
independent_empty: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e" assumes "module scale" shows "\<not> module.dependent scale {}"
IH: shows "\<forall>ys. length ys < length xs \<longrightarrow> (\<forall>x xa. (x, xa) = closest_pair_rec ys \<longrightarrow> set x = set ys \<and> length x = length ys \<and> sorted_snd x)"
bdd_below_def: fixes A :: "'c set" shows "bdd_below A = (\<exists>m. \<forall>x\<in>A. m \<le> x)"
wellformed_item_y: shows "wellformed_item (init_item (item_nonterminal x, item_rhs x) 0)"
Init_simps2_6: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "Init P = Q \<equiv> Init P = (Init Q)"
monoI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f x \<sqsubseteq> f y" shows "order.mono less_eq f"
cauchy_isometric: fixes e :: "real"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "nat \<Rightarrow> 'a" assumes "0 < e"   and "subspace s"   and "bounded_linear f"   and "\<forall>x\<in>s. e * norm x \<le> norm (f x)"   and "\<forall>n. x n \<in> s"   and "Cauchy (f \<circ> x)" shows "Cauchy x"
eta_confluent: shows "confluent (\<rightarrow>\<^sub>\<eta>)"
r_into_trancl: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "r a b" shows "r\<^sup>+\<^sup>+ a b"
le_less_trans: fixes x :: "'t"   and y :: "'t"   and z :: "'t" assumes "x \<preceq>\<^sub>t y"   and "y \<prec>\<^sub>t z" shows "x \<prec>\<^sub>t z"
type_less_eq_x_OrderedSet: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<le> OrderedSet \<sigma>" obtains \<rho> :: "'a OCL_Types.type" where "\<tau> = OrderedSet \<rho>" and "\<rho> \<le> \<sigma>"
ap_state_rev_B: fixes f :: "('c, 'e \<Rightarrow> 'd) state"   and x :: "('c, 'e) state" shows "B f \<diamondop> B x = B (pure (\<lambda>x f. f x) \<diamondop> x \<diamondop> f)"
lappend_llist_of_LCons: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a llist" shows "lappend (llist_of xs) (LCons y ys) = lappend (llist_of (xs @ [y])) ys"
triangle_left_implies_right: assumes "(\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]" shows "(g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
a: shows "\<forall>x. M \<noteq> Ax x c"
rel_inject_2: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x2 :: "'a"   and y2 :: "'b" shows "rel_hd R1 R2 (Sym x2) (Sym y2) = R1 x2 y2"
rule_sub_lem_1_b: fixes v :: "i"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>"   and \<Theta> :: "\<o>" assumes "[\<^bold>\<box>(\<psi> \<^bold>\<equiv> \<chi>) in v]" shows "[\<psi> \<^bold>\<rightarrow> \<Theta> \<^bold>\<equiv> (\<chi> \<^bold>\<rightarrow> \<Theta>) in v]"
r_def: shows "r = (\<lambda>n. r0 (n + 1))"
converse_Image: fixes R :: "('a \<times> 'b) set"   and X :: "'a set" assumes "runiq R"   and "runiq (R\<inverse>)" shows "R\<inverse> `` R `` X \<subseteq> X"
dim_col_take_cols: fixes inds :: "nat list"   and A :: "'a mat" assumes "\<And>j. j \<in> set inds \<Longrightarrow> j < dim_col A" shows "dim_col (take_cols A inds) = length inds"
Fract_cases: fixes q :: "'a fract" obtains a :: "'a" and b :: "'a" where "q = Fraction_Field.Fract a b" and "b \<noteq> (0::'a)"
gm_int: shows "integrable M (\<lambda>x. f x * enn2real (nn_cond_exp M F gm x))"
Switch2: fixes p :: "'b"   and c :: "'c"   and av :: "'a" assumes "s = s' \<and> Abort p c av t t'"   and "a = Switch id2 p c av" shows "thesis"
genset_sub: fixes G :: "'b list set" shows "G \<subseteq> \<langle>G\<rangle>"
univ_commute: fixes P :: "'a \<Rightarrow> bool"   and \<phi> :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "equivP P \<phi>"   and "f respectsP \<phi>"   and "P x" shows "Equiv_Relation2.univ f (proj \<phi> x) = f x"
left_neutral: fixes a :: "('a \<times> 'a) set" shows "Id O a = a"
adjoint_dim_row: fixes A :: "'a mat" shows "dim_row (adjoint A) = dim_col A"
canonical_tabulation: fixes g :: "'a" assumes "ide g"   and "src (tab\<^sub>0 r) = src g"   and "\<exists>\<rho>. tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (g \<star> (tab\<^sub>0 r)\<^sup>*) \<rho> (tab\<^sub>0 r) g" shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (g \<star> (tab\<^sub>0 r)\<^sup>*) (tab\<^sub>0.trnr\<^sub>\<eta> g (g \<star> (tab\<^sub>0 r)\<^sup>*)) (tab\<^sub>0 r) g"
in_all_resolvents_upon_def: fixes S :: "'at Literal set set"   and A :: "'at"   and C :: "'at Literal set" shows "in_all_resolvents_upon S A C = (\<exists>P1 P2. P1 \<in> S \<and> P2 \<in> S \<and> C = resolvent_upon P1 P2 A)"
asympartpI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "r x y"   and "\<not> r y x" shows "asympartp r x y"
proj_proj_set_subset_3: fixes n :: "'e"   and m :: "'e"   and A :: "('e strand_label \<times> 'd) list" shows "set (proj_unl n (proj m A)) \<subseteq> set (proj_unl n A)"
delete_edges_list_set: fixes G :: "'c graph"   and E :: "('c \<times> 'c) list" shows "delete_edges_list G E = FiniteGraph.delete_edges G (set E)"
inv_mult: fixes x :: "'a \<Rightarrow> complex"   and y :: "'a \<Rightarrow> complex" assumes "x \<in> carrier C"   and "y \<in> carrier C" shows "inv\<^bsub>C\<^esub> (x \<otimes>\<^bsub>C\<^esub> y) = inv\<^bsub>C\<^esub> x \<otimes>\<^bsub>C\<^esub> inv\<^bsub>C\<^esub> y"
diff_is_0_eq: fixes m :: "nat"   and n :: "nat" shows "(m - n = 0) = (m \<le> n)"
intros_1: fixes n :: "'a"   and \<alpha>n :: "'b \<Rightarrow> 'a list"   and g :: "'b"   and invar :: "'b \<Rightarrow> bool"   and inEdges' :: "'b \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'c) list" assumes "n \<in> set (\<alpha>n g)"   and "invar g" shows "graph_path_base.path \<alpha>n invar inEdges' g [n]"
validSubst: fixes P :: "pi"   and a :: "name"   and b :: "name" shows "valid (P[a::=b]) = valid P"
p_subid_par_eq_meet: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (0::'a) \<parallel> (y \<cdot> (0::'a)) = x \<cdot> (0::'a) \<sqinter> y \<cdot> (0::'a)"
scc_root_is_discovered: fixes v :: "'a"   and scc :: "'a set" assumes "scc_root s v scc" shows "v \<in> dom (discovered s)"
m_lcomm: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "z \<in> carrier Zp" shows "x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
ctr_transfer_1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_option R None None"
Kl: fixes x :: "'a" shows "(1::'a) + x \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
simps_3: fixes V :: "'a set"   and t :: "('b, 'a) Term.term"   and S :: "('b, 'a) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (send\<langle>t\<rangle> # S) = wf'\<^sub>s\<^sub>s\<^sub>t (V \<union> fv t) S"
prems: shows "eval\<^sub>c\<^sub>s F G Cs"
orbit_FOO: fixes a :: "'b"   and g :: "'b \<Rightarrow> 'b"   and g' :: "'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "a \<in> orbit g a"   and "\<And>x. x \<in> orbit g a \<Longrightarrow> g' (f x) = f (g x)" shows "f ` orbit g a = orbit g' (f a)"
img_dce_def: fixes M :: "'d measure"   and Y :: "'d \<Rightarrow> 'c"   and X :: "'d \<Rightarrow> real" shows "img_dce M Y X = (\<lambda>y. if Sigma_Algebra.measure M (Y -` {y} \<inter> space M) = 0 then 0 else (LINT w|M. X w * indicat_real (Y -` {y} \<inter> space M) w) / Sigma_Algebra.measure M (Y -` {y} \<inter> space M))"
continuous_on_add_ennreal: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "continuous_on A f"   and "continuous_on A g" shows "continuous_on A (\<lambda>x. f x + g x)"
eq0_iff: fixes x :: "V" shows "(x = []\<^sub>\<circ>) = (\<forall>y. y \<notin>\<^sub>\<circ> x)"
foldli_not_cond: fixes c :: "'g \<Rightarrow> bool"   and \<sigma> :: "'g"   and xs :: "'h list"   and f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g" assumes "\<not> c \<sigma>" shows "foldli xs c f \<sigma> = \<sigma>"