hcju/isabelleps huggingface dataset:
sorted_wrt_imp_le_last: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list"   and x :: "'b" assumes "Linorder_Relations.sorted_wrt R xs"   and "xs \<noteq> []"   and "x \<in> set xs"   and "x \<noteq> last xs" shows "(x, last xs) \<in> R"
assms: shows "\<forall>x. proper_it (it x) (it' x)"
lossless_sample_uniform_units: fixes q :: "nat" assumes "1 < q" shows "lossless_spmf (sample_uniform_units q)"
assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "sup (sup a b) c = sup a (sup b c)"
elims_3: fixes x :: "'a formula" assumes "\<not> is_mini_formula x" obtains F :: "'a formula" and G :: "'a formula" where "x = F \<^bold>\<rightarrow> G" and "\<not> (is_mini_formula F \<and> is_mini_formula G)"   | v :: "'a formula" where "x = \<^bold>\<not> v__"   | v :: "'a formula" and va :: "'a formula" where "x = va__ \<^bold>\<and> vaa__"   | v :: "'a formula" and va :: "'a formula" where "x = v \<^bold>\<or> va"
map_of_rev_distinct: fixes m :: "('a \<times> 'b) list" assumes "distinct (map fst m)" shows "map_of (rev m) = map_of m"
simps: fixes a1 :: "cld"   and a2 :: "cl" shows "superclass_name a1 a2 = (\<exists>dcl cl fds meth_defs. a1 = cld_def dcl cl fds meth_defs \<and> a2 = cl)"
distinct_7: fixes x11 :: "String.literal"   and x12 :: "typ"   and x51 :: "Core.term"   and x52 :: "Core.term" shows "Ct x11 x12 \<noteq> x51 $ x52"
set_4_disj: fixes i :: "nat" assumes "i < 4" shows "i = 0 \<or> i = 1 \<or> i = 2 \<or> i = 3"
implConfid_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> asym.valid \<union> range LtK \<union> Tags"   and "asym.implConfid A B M \<in> synth (analz H)" shows "asym.implConfid A B M \<in> H \<or> M \<in> synth (analz H)"
coeff_subresultant_0_0_resultant: fixes f :: "'b poly"   and g :: "'b poly" shows "coeff (subresultant 0 f g) 0 = resultant f g"
inj_on_fst_map_to_set: fixes m :: "'d \<Rightarrow> 'e option" shows "inj_on fst (map_to_set m)"
that: assumes "a = \<aa>"   and "b = \<aa>"   and "f = \<ff>" shows "thesis"
frequently_mono: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'b filter" assumes "\<forall>x. P x \<longrightarrow> Q x"   and "frequently P F" shows "frequently Q F"
not_final_thread_final_thread_conv: fixes s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c"   and final :: "'a \<Rightarrow> bool" assumes "thr s t \<noteq> None" shows "(\<not> final_thread.final_thread final s t) = final_thread.not_final_thread final s t"
eval_add: fixes n :: "nat"   and P :: "'v" shows "eval (add n P) n"
subcocycle_lim_meas_Inv_1: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" shows "subcocycle_lim_ereal u \<in> borel_measurable Invariants"
permutation_lists_fc_succ: shows "permutation (lists_fc_succ M)"
arith_prog_def: fixes a :: "int"   and b :: "nat" shows "arith_prog a b = {x. [x = a] (mod int b)}"
eq_eqI: fixes x :: "tr"   and y :: "tr" assumes "x = \<bottom> \<Longrightarrow> y = \<bottom>"   and "y = \<bottom> \<Longrightarrow> x = \<bottom>"   and "x = TT \<Longrightarrow> y = TT"   and "y = TT \<Longrightarrow> x = TT" shows "x = y"
simulation1_\<tau>rtrancl3p: fixes s1 :: "'s1"   and tls1 :: "'tl1 list"   and s1' :: "'s1"   and s2 :: "'s2" assumes "s1 -\<tau>1-tls1\<rightarrow>* s1'"   and "s1 \<approx> s2" shows "\<exists>tls2 s2'. s2 -\<tau>2-tls2\<rightarrow>* s2' \<and> s1' \<approx> s2' \<and> tls1 [\<sim>] tls2"
simps_2: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and y31 :: "('a, 'b, 'c) Stmt"   and y32 :: "'a ModeUpd" shows "(x31@[x32] = y31@[y32]) = (x31 = y31 \<and> x32 = y32)"
inv_DINING: fixes s :: "nat list \<times> nat list" assumes "s \<in> \<RR> (Tr\<^sub>D N) Up\<^sub>D (replicate N 0, replicate N 0)" shows "inv_dining (fst s) (snd s)"
comp_char: fixes g :: "'d"   and f :: "'d" shows "D'.comp g f = (if D'.arr f \<and> D'.arr g \<and> D.seq g f then g \<cdot>\<^sub>D f else D.null)"
poincare_distance_sym: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "poincare_distance u v = poincare_distance v u"
deg_add_eq: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "deg R q \<noteq> deg R p" shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) = max (deg R p) (deg R q)"
cases: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b tree" assumes "class.linorder less_eq less" obtains "x = \<langle>\<rangle>"   | l :: "'b tree" and m :: "'b" and r :: "'b tree" where "x = \<langle>l, m, r\<rangle>"
return_bind: fixes x :: "'d"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog" shows "Heap_Error_Monad.bind (return x) g = g x"
SUP_empty: fixes f :: "'d \<Rightarrow> 'b \<times> 'c" shows "Sup_prod (f ` {}) = bot_prod"
m_def: shows "m = root (traverse_tree path (stern_brocot_iterate_aux I))"
simps_1: fixes w :: "nat"   and a :: "'a" shows "consistent (Leaf w a) = True"
conj_disj_abs: fixes P :: "'a upred"   and Q :: "'a upred" shows "(P \<and> (P \<or> Q)) = P"
1: shows "\<forall>x. x |\<in>| |dom|f \<longrightarrow> f |@| x |\<in>| |cod|f"
eventually_at_right: fixes x :: "'b"   and y :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "x < y" shows "eventually P (at_right x) = (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
that: fixes d :: "'b" assumes "D = (\<lambda>x. x *s d)" shows "thesis"
isOmax_def: fixes R :: "('c \<times> 'c) set set"   and r :: "('c \<times> 'c) set" shows "isOmax R r \<equiv> r \<in> R \<and> (\<forall>r'\<in>R. r' \<le>o r)"
assms: shows "AR S \<and> closedin (top_of_set U) S"
comm: shows "u \<^sup>@ Suc k = v \<^sup>@ Suc l"
z_def: shows "z \<equiv> getFr [x] [t] [\<phi>]"
nodeToList_def: shows "nodeToList \<equiv> rec_Node (\<lambda>e a. [(e, a)]) (\<lambda>_ a b. (@)) (\<lambda>_ a b c aa ba ca. aa @ ba @ ca)"
subseq_conv_nths: fixes xs :: "'b list"   and ys :: "'b list" shows "subseq xs ys = (\<exists>N. xs = nths ys N)"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> 'j" assumes "l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
case_eq_if: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> ('e, 'd) Term.term list \<Rightarrow> 'c"   and "term" :: "('e, 'd) Term.term" shows "(case term of Var x \<Rightarrow> f1 x | Fun x xa \<Rightarrow> f2 x xa) = (if is_Var term then f1 (the_Var term) else f2 (un_Fun1 term) (args term))"
cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_none: fixes obj :: "(_) Object" shows "(cast obj = None) = (\<nexists>document. cast document = obj)"
row_four_block_mat_1: fixes A :: "'b mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'b mat"   and nc2 :: "nat"   and C :: "'b mat"   and nr2 :: "nat"   and D :: "'b mat"   and i :: "nat" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2"   and "i < nr1" shows "row (four_block_mat A B C D) i = row A i @\<^sub>v row B i"
integrate_insert_result_helper: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and m :: "('a, 'b) insert_message"   and pm :: "'a extended"   and sm :: "'a extended"   and m' :: "('a, 'b) insert_message"   and s' :: "('a, 'b option) insert_message list" assumes "integrate_insert_commute_insert M a s m"   and "integrate_insert_commute_insert.invariant a s m pm sm"   and "m' = m"   and "s' = s" shows "integrate_insert_commute_insert.is_certified_associated_string' M a m (integrate_insert m' s' pm sm)"
index_mat_multrow_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and mul :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "nat"   and a :: "'a" assumes "i < dim_row A"   and "j < dim_col A" shows "mat_multrow_gen mul k a A $$ (i, j) = (if k = i then mul a (A $$ (i, j)) else A $$ (i, j))"
string_of_word_single_def: fixes lc :: "bool"   and w :: "'a word" shows "string_of_word_single lc w \<equiv> if w < 10 then [char_of (48 + unat w)] else if w < 36 then [char_of ((if lc then 87 else 55) + unat w)] else undefined"
not_outside_1: fixes x :: "'b"   and y :: "'b" assumes "Pareto R' x y" shows "x \<in> alts'"
is_processT4_S1: fixes x :: "'a event list \<times> 'a event set"   and P :: "'a process"   and X :: "'a event set" assumes "x \<in> F P"   and "X \<subseteq> snd x" shows "(fst x, X) \<in> F P"
WCR_SN_on_imp_CR_on: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "WCR r"   and "SN_on r A" shows "CR_on r A"
prems_10: shows "k' = k + length (chars_of_token X)"
has_default_policy_simple_conn_matches: fixes rs :: "'a simple_rule list"   and c :: "parts_connection" assumes "has_default_policy rs" shows "has_default_policy (filter (\<lambda>r. simple_conn_matches (match_sel r) c) rs)"
sumpos: fixes k :: "nat" shows "0 < Re (\<Sum>i\<in>I. b i * x i ^ k)"
IH_2: fixes p :: "'a"   and n :: "'a" assumes "p \<notin> snd (pn_atoms G)"   and "n \<notin> fst (pn_atoms G)" shows "(M \<Turnstile> G \<longrightarrow> M(p := True) \<Turnstile> G \<and> M(n := False) \<Turnstile> G) \<and> (\<not> M \<Turnstile> G \<longrightarrow> \<not> M(n := True) \<Turnstile> G \<and> \<not> M(p := False) \<Turnstile> G)"
assms_4: shows "(i2, Insert e2 n2) \<in> set (node_deliver_messages xs)"
sds_lowering_neutral_axioms: shows "sds_lowering_neutral agents alts sds agents' alts'"
lexordp_def: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "ord.lexordp less \<equiv> lfp (\<lambda>p x1 x2. (\<exists>y ys. x1 = [] \<and> x2 = y # ys) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> less x y) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> \<not> less x y \<and> \<not> less y x \<and> p xs ys))"
one_le_HNatInfinite: fixes x :: "nat star" assumes "x \<in> HNatInfinite" shows "1 \<le> x"
add_commute: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<oplus> b = b \<oplus> a"
var_not_in_linear: fixes var :: "nat"   and b :: "real mpoly"   and c :: "real mpoly"   and A :: "atom" assumes "var \<notin> vars b"   and "var \<notin> vars c" shows "freeIn var (fm.Atom (linear_substitution var b c A))"
whileDecrIf: shows "\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. enat (nat (s ''x'')) + 1} WHILE Less (N 0) (V ''x'') DO (IF Less (N 0) (V ''z'') THEN SKIP;; SKIP ELSE SKIP;; ''x'' ::= Plus (V ''x'') (N (- 1))) {\<lambda>s. enat 0}"
adj_in_verts_2: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^bsub>G\<^esub> v" shows "v \<in> verts G"
Classes_def: shows "Classes = [(Base, BaseCl), (Ext, ExtCl), (Main, MainCl)] @ standard_classes"
square_free_factorizationD_1: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list" assumes "square_free_factorization p (c, bs)" shows "p = smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
hbt_delete_2: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "hbt t" shows "height t \<in> {height (delete x t), height (delete x t) + 1}"
D1: fixes a :: "'a" assumes "Binomial_Queue.min xs = Some a" shows "\<exists>t. find_min xs = Some t \<and> priority t = a"
basic_monos_7: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
deg_aboveD: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a"   and m :: "nat" assumes "UP_ring R"   and "deg R p < m"   and "p \<in> carrier (UP R)" shows "coeff (UP R) p m = \<zero>\<^bsub>R\<^esub>"
fw_upd_canonical_id: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "canonical M n"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "fw_upd M k i j = M"
rcos_equation: fixes H :: "'a set"   and ha :: "'a"   and a :: "'a"   and h :: "'a"   and b :: "'a"   and hb :: "'a" assumes "subgroup H G"   and "ha \<otimes> a = h \<otimes> b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "h \<in> H"   and "ha \<in> H"   and "hb \<in> H" shows "hb \<otimes> a \<in> (\<Union>h\<in>H. {h \<otimes> b})"
localRelSS'_def: shows "localRelSS' = (\<lambda>(u, v) (u', v') x z. u = 0 \<and> u' = 1)"
RDirProd_iso1: fixes R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme" shows "(\<lambda>(x, y). (y, x)) \<in> ring_iso (RDirProd R S) (RDirProd S R)"
transitionsPreserveInvariants: fixes stateA :: "State"   and stateB :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" assumes "transition stateA stateB F0 decisionVars"   and "invariantsHoldInState stateA F0 decisionVars" shows "invariantsHoldInState stateB F0 decisionVars"
extended_Gromov_product_inside_boundary_aux: fixes y :: "'a Gromov_completion"   and e :: "'a"   and x :: "'a" assumes "y \<in> Gromov_boundary" shows "extended_Gromov_product_at e (to_Gromov_completion x) y = Inf {liminf (\<lambda>n. ereal (Gromov_product_at e x (v n))) |v. (\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> y}"
iso_in_hom_def: fixes f :: "'c"   and a :: "'c"   and b :: "'c" shows "\<guillemotleft>f : a \<cong>\<^sub>C b\<guillemotright> \<equiv> \<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright> \<and> C.iso f"
eval_convert: fixes var :: "nat"   and L :: "atom list"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and L' :: "atomUni list" assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"   and "length xs' = var" shows "(\<forall>f\<in>set L. aEval f (xs' @ x # xs)) = (\<forall>f\<in>set L'. aEvalUni f x)"
MM4b: fixes F :: "bool" shows "\<turnstile> \<box>\<not> #F = \<not> #F"
size_eq_Suc_imp_eq_union: fixes M :: "'c multiset"   and n :: "nat" assumes "size M = Suc n" shows "\<exists>a N. M = add_mset a N"
jkbpCn_length: fixes t :: "('a, 'es, 'as) BEState Trace"   and n :: "nat" assumes "t \<in> SPR.jkbpCn n" shows "tLength t = n"
powr_mono_both: fixes a :: "real"   and b :: "real"   and x :: "real"   and y :: "real" assumes "0 \<le> a"   and "a \<le> b"   and "1 \<le> x"   and "x \<le> y" shows "x powr a \<le> y powr b"
cast_shadow_root_ptr_not_document_ptr_1: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and document_ptr :: "(_) document_ptr" shows "cast shadow_root_ptr \<noteq> cast document_ptr"
web_flow_iff: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "web_flow \<Gamma> f = current \<Gamma> f"
Suc: shows "bound (a * of_nat n) \<le> bound a * bound (of_nat n)"
IH_2: fixes l :: "float poly"   and r :: "float poly" assumes "(l, r) = split_by_prec prec p2" shows "Ipoly args (map_poly real_of_float p2) = Ipoly args (map_poly real_of_float l) + Ipoly args (map_poly real_of_float r) \<and> num_params l \<le> num_params p2 \<and> num_params r \<le> num_params p2"
exeterm_ok_def: fixes \<Theta> :: "exetheory"   and t :: "Core.term" shows "exeterm_ok \<Theta> t \<equiv> exeterm_ok' (exesig \<Theta>) t \<and> typ_of t \<noteq> None"
S_transform_Pred: fixes \<phi> :: "'c" shows "S_transform (Formula.Pred \<phi>) = Formula.Act (S_action.Pred \<phi>) (Conj bempty)"
intros: fixes s :: "'a + 'b"   and x :: "'b" assumes "s = Inr x" shows "x \<in> Basic_BNFs.setr s"
map_of_map_Iface: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "(char list \<times> 'd) list"   and ifce :: "char list" shows "map_of (map (\<lambda>x. (Iface (fst x), f (snd x))) xs) (Iface ifce) = map_option f (map_of xs ifce)"
induct: fixes P :: "('a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'g) \<Rightarrow> (('a \<times> 'e) list \<Rightarrow> ('a \<times> 'g) list) \<Rightarrow> (('a \<times> 'f) list \<Rightarrow> ('a \<times> 'g) list) \<Rightarrow> ('a \<times> 'e) list \<Rightarrow> ('a \<times> 'f) list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'g"   and a1 :: "('a \<times> 'e) list \<Rightarrow> ('a \<times> 'g) list"   and a2 :: "('a \<times> 'f) list \<Rightarrow> ('a \<times> 'g) list"   and a3 :: "('a \<times> 'e) list"   and a4 :: "('a \<times> 'f) list" assumes "\<And>f g h xs. P f g h xs []"   and "\<And>f g h v va. P f g h [] (v # va)"   and "\<And>f g h kx vx xs ky vy ys. \<lbrakk>\<And>x. \<lbrakk>comp kx ky = Eq; x = f kx vx vy\<rbrakk> \<Longrightarrow> P f g h xs ys; \<And>x. \<lbrakk>comp kx ky = Lt; x = f kx vx (0::'f)\<rbrakk> \<Longrightarrow> P f g h xs ((ky, vy) # ys); \<And>x. \<lbrakk>comp kx ky = Gt; x = f ky (0::'e) vy\<rbrakk> \<Longrightarrow> P f g h ((kx, vx) # xs) ys\<rbrakk> \<Longrightarrow> P f g h ((kx, vx) # xs) ((ky, vy) # ys)" shows "P a0 a1 a2 a3 a4"
prefix_map_inj: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and ys :: "'c list" assumes "inj_on f (set xs \<union> set ys)"   and "prefix (map f xs) (map f ys)" shows "prefix xs ys"
list_mean_eq_iff: fixes one :: "real list"   and two :: "real list" assumes "sum_list one = sum_list two"   and "length one = length two" shows "mean one = mean two"
add_less_le_mono: fixes a :: "'b zmultiset"   and b :: "'b zmultiset"   and c :: "'b zmultiset"   and d :: "'b zmultiset" assumes "a \<subset>#\<^sub>z b"   and "c \<subseteq>#\<^sub>z d" shows "a + c \<subset>#\<^sub>z b + d"
nth_Suc_zero: shows "nth_derivative (Suc k) f 0 1 = 0"
assms_6: shows "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c'"
Cons_in_lists_iff: fixes x :: "'g"   and xs :: "'g list"   and A :: "'g set" shows "(x # xs \<in> lists A) = (x \<in> A \<and> xs \<in> lists A)"
longest_common_prefix_prefix1: fixes xs :: "'c list"   and ys :: "'c list" shows "prefix (longest_common_prefix xs ys) xs"
right_adjoint_determines_left_up_to_iso: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f' :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.adjoint_pair f g"   and "Span.adjoint_pair f' g" shows "Span.isomorphic f f'"
eliminate_swap_global_rid: fixes r :: "'a"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r' :: "'a" assumes "r \<notin> RID\<^sub>G s"   and "r' \<notin> RID\<^sub>G s" shows "\<R>\<^sub>G (id(r := r', r' := r)) id s = s"
h'_def: shows "h' \<equiv> interpret_floatarith h vs"
freshInp_def2: fixes xs :: "'varSort"   and x :: "'var"   and inp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" shows "freshInp xs x inp = qFreshInp xs x (pickInp inp)"
pderivs_lang_UNIV: fixes r :: "'a rexp" shows "pderivs_lang UNIV r = pderivs [] r \<union> pderivs_lang UNIV1 r"
powser_times_n_limit_0: fixes x :: "'a" assumes "norm x < 1" shows "(\<lambda>n. of_nat n * x ^ n) \<longlonglongrightarrow> (0::'a)"
TOTAL: fixes r :: "('b \<times> 'b) set" assumes "wo_rel r" shows "Total r"
sb': shows "sb' = dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb"
content_cbox_if_cart: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "content (cbox a b) = (if cbox a b = {} then 0 else \<Prod>i\<in>UNIV. b $ i - a $ i)"
evaluate_iff_3: fixes eval :: "v sem_env \<Rightarrow> exp0 \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v, v) result"   and eval_list :: "v sem_env \<Rightarrow> exp0 list \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v list, v) result"   and eval_match :: "v sem_env \<Rightarrow> v \<Rightarrow> (pat \<times> exp0) list \<Rightarrow> v \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v, v) result"   and env :: "v sem_env"   and st :: "'b state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and v' :: "v"   and r :: "'b state \<times> (v, v) result" assumes "eval eval eval_list eval_match" shows "evaluate_match True env st v pes v' r = (r = eval_match env v pes v' st)"
nlcons_invlevel: fixes n :: "nat"   and t :: "('c, 'd) FingerTreeStruc"   and nd :: "('c, 'd) Node" assumes "is_leveln_ftree n t"   and "is_leveln_node n nd" shows "is_leveln_ftree n (nlcons nd t)"
exec_1p_exec_1_eq: fixes P :: "jvm_prog" shows "exec_1p P = exec_1' P"
assms_2: shows "\<forall>y\<in>A \<inter> Field r. (y, x) \<in> r"
list_all2_Cons: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list"   and y :: "'d"   and ys :: "'d list" shows "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
no_solution_0_fst_int: fixes b :: "int" shows "\<not> solution (0, b)"
that: fixes x :: "hf" assumes "x \<^bold>\<in> A"   and "b \<^bold>\<in> B x" shows "thesis"
elims_2: fixes x :: "'b mat" assumes "square_mat x" obtains A :: "'b mat" where "x = A" and "dim_col A = dim_row A"
Re_divide_numeral: fixes z :: "complex"   and w :: "num" shows "Re (z / numeral w) = Re z / numeral w"
Span_m_inv_simprule: fixes Us :: "'a list"   and k :: "'a"   and a :: "'a" assumes "set Us \<subseteq> carrier R"   and "k \<in> K - {\<zero>}"   and "a \<in> carrier R"   and "k \<otimes> a \<in> Span K Us" shows "a \<in> Span K Us"
card_eq_implies_surjective_on: fixes B :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "finite B"   and "f \<in> A \<rightarrow>\<^sub>E B"   and "card ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}) = card B" shows "f ` A = B"
starfun_ln_gt_zero: fixes x :: "real star" assumes "1 < x" shows "0 < (*f* real_ln) x"
take_bit_length_eq: fixes w :: "'b word" shows "take_bit LENGTH('b) w = w"
restricted_once_def: fixes v :: "'a"   and G :: "'a gexp list" shows "restricted_once v G = (length (filter (\<lambda>g. gexp_constrains g (V v)) G) = 1)"
stable_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "stable P = ((\<exists>\<sigma>. P \<sigma>) \<and> (\<forall>\<sigma> i. P \<sigma> \<longrightarrow> P (\<sigma> |\<^sub>s i)))"
menger: fixes n :: "nat" assumes "\<And>S. Separation G v0 v1 S \<Longrightarrow> n \<le> card S" shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = n"
higher_differentiable_on_real_Suc': fixes S :: "real set"   and f :: "real \<Rightarrow> 'a"   and n :: "nat" assumes "open S" shows "higher_differentiable_on S f (Suc n) = ((\<forall>v. continuous_on S (\<lambda>x. nth_derivative (Suc n) f x 1)) \<and> (\<forall>x\<in>S. \<forall>v. (\<lambda>x. nth_derivative n f x 1) differentiable at x) \<and> higher_differentiable_on S f n)"
simps_1: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool" shows "eval e f g FF = False"
dom_char: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" shows "Cop_S.dom f = (if Cop_S.arr f then Cop_S.MkIde (Cop_S.Dom f) else Cop_S.null)"
HaddP_SUCC_Ex1: fixes i :: "name"   and x :: "tm"   and y :: "tm"   and z :: "tm"   and H :: "fm set" assumes "atom i \<sharp> (x, y, z)" shows "insert (HaddP (SUCC x) y z) (insert (OrdP x) H) \<turnstile> SyntaxN.Ex i (HaddP x y (Var i) AND z EQ SUCC (Var i))"
cons1: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s1 :: "'b state"   and d :: "dec"   and s2 :: "'b state"   and e :: "v error_result"   and ds :: "dec list" assumes "evaluate_dec ck mn env s1 d (s2, Rerr e)" shows "evaluate_decs ck mn env s1 (d # ds) (s2, Rerr e)"
bisims: shows "list_all2 (bisim1_fr P h) exs FRS"
map_ds_prop: fixes f :: "'a \<Rightarrow> 'b" shows "map_ds f \<circ> ds = ds \<circ> f"
intro: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and N :: "'c set" assumes "RModuleHom R smult M smult' T"   and "RModuleIso_axioms M T N" shows "RModuleIso R smult M smult' T N"
inside_not_bound_1: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set"   and j :: "nat" assumes "[f[a .. b ..  c]X]"   and "j < card X"   and "0 < j" shows "f j \<noteq> a"