hcju/isabelleps huggingface dataset:
restrict_sublist: fixes ls :: "'c list"   and ks :: "'c list"   and ms :: "'c list"   and ns :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "is_sublist ls ks"   and "is_sublist ms ns" shows "ls \<langle>ks \<langle>f\<rangle> ns\<rangle> ms = ls \<langle>f\<rangle> ms"
decseq_harm_diff_ln: shows "decseq (\<lambda>n. harm (Suc n) - ln (real (Suc n)))"
bangRelSymetric: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q) \<in> bangRel Rel"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> (Q, P) \<in> Rel" shows "(Q, P) \<in> bangRel Rel"
rel_prod2_simps: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and c :: "'c"   and b :: "'b" shows "rel_prod2 R a (c, b) = R a b"
Exit_no_return_target: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Exit :: "'h"   and a :: "'g"   and Q :: "('i \<Rightarrow> 'j option) \<times> 'k \<Rightarrow> bool"   and p :: "'l"   and f :: "('i \<Rightarrow> 'j option) \<Rightarrow> ('i \<Rightarrow> 'j option) \<Rightarrow> 'i \<Rightarrow> 'j option" assumes "CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "targetnode a = Exit" shows "False"
single_LeadsTo_I: fixes A :: "'b set"   and F :: "'b program"   and B :: "'b set" assumes "\<And>s. s \<in> A \<Longrightarrow> F \<in> {s} \<longmapsto>w B" shows "F \<in> A \<longmapsto>w B"
rel_bset_bempty_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Y :: "'c set['e]" shows "rel_bset R Y bempty = (Y = bempty)"
assms_2: fixes k :: "nat" assumes "P k" shows "k \<le> l"
infsD_sset: fixes A :: "'b set"   and w :: "'b stream" assumes "infs A w" shows "sset w \<inter> A \<noteq> {}"
r_parallel_3: fixes i :: "nat"   and f :: "recf"   and j :: "nat"   and g :: "recf"   and x :: "nat" assumes "\<And>x. eval r_phi [i, x] = eval f [x]"   and "\<And>x. eval r_phi [j, x] = eval g [x]"   and "eval g [x] \<down> \<and> eval f [x] \<up>" shows "eval r_parallel [i, j, x] \<down>= prod_encode (1, the (eval g [x]))"
setsum_single: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c" shows "sum f {a} = f a"
in_homI: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f"   and "local.dom f = a"   and "cod f = b" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
holomorphic_fun_eq_0_on_connected: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and z :: "complex"   and w :: "complex" assumes "f holomorphic_on S"   and "open S"   and "connected S"   and "\<And>n. (deriv ^^ n) f z = 0"   and "z \<in> S"   and "w \<in> S" shows "f w = 0"
isomorphic_implies_same_tab_1: fixes r' :: "'a" assumes "r\<^sub>0\<^sup>* \<star> s\<^sub>1 \<cong> r'" shows "r\<^sub>0s\<^sub>1.p\<^sub>0 = tab\<^sub>0 r'"
is_processT9_tick: fixes P :: "'a process"   and s :: "'a event list" assumes "[tick] \<in> D P"   and "front_tickFree s" shows "s \<in> D P"
distinct_adj_Cons: fixes x :: "'a"   and xs :: "'a list" shows "distinct_adj (x # xs) = (xs = [] \<or> x \<noteq> hd xs \<and> distinct_adj xs)"
natpermute_max_card: fixes n :: "nat"   and k :: "nat" assumes "n \<noteq> 0" shows "card {xs \<in> natpermute n (k + 1). n \<in> set xs} = k + 1"
p: fixes x :: "'a" assumes "x \<in> set_pmf p" shows "integral\<^sup>N (count_space (set_pmf q)) (h x) = ennreal (pmf p x)"
shiftl_rev: fixes w :: "'b word"   and n :: "nat" shows "w << n = word_reverse (word_reverse w >> n)"
ord_strict_p_transitive: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'t \<Rightarrow>\<^sub>0 'c" assumes "a \<prec>\<^sub>p b"   and "b \<prec>\<^sub>p c" shows "a \<prec>\<^sub>p c"
normal_cooper1: fixes \<phi> :: "atom fm" assumes "Z.normal \<phi>" shows "Z.normal (qe_cooper\<^sub>1 \<phi>)"
"4_2": shows "card {p. p partitions m - k \<and> sum p {..m - k} = Suc k} = Partition (m - k) (Suc k)"
infsetsum_Times: fixes A :: "'d set"   and B :: "'e set"   and f :: "'d \<times> 'e \<Rightarrow> 'f" assumes "countable A"   and "countable B"   and "f abs_summable_on A \<times> B" shows "infsetsum f (A \<times> B) = (\<Sum>\<^sub>ax\<in>A. \<Sum>\<^sub>ay\<in>B. f (x, y))"
stutter_equiv_refl: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "\<sigma> \<approx> \<sigma>"
Ain: shows "A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)) \<noteq> (0::'a)"
cfcomp1: fixes f :: "'c \<rightarrow> 'b"   and g :: "'a \<rightarrow> 'c" shows "f oo g = (\<Lambda> x. f\<cdot>(g\<cdot>x))"
induct: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_bochner_integrable M f" obtains "simple_function M f" and "emeasure M {y \<in> space M. f y \<noteq> (0::'d)} \<noteq> \<infinity>"
assms: shows "\<Gamma> \<Rightarrow>\<^sub>n"
in_Mne_lt_n: shows "\<forall>m\<in>#{#m \<in># M. m \<noteq> n#}. m < n"
3: fixes v :: "nat \<times> 'b"   and u :: "nat \<times> 'b" assumes "v \<in> gunodes A w"   and "u \<in> gusuccessors A w v" shows "f u \<le> f v"
card_of_countable_sets_range: fixes A :: "'b set" shows "|{X. X \<subseteq> A \<and> countable X \<and> X \<noteq> {}}| \<le>o |{f. range f \<subseteq> A}|"
skip_elim: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((Skip, mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "c' = Stop \<and> mds = mds' \<and> mem = mem'"
rm_LogEmpty_filter: fixes rs :: "'b rule list" shows "rm_LogEmpty rs = filter (\<lambda>r. get_action r \<noteq> Log \<and> get_action r \<noteq> Empty) rs"
Zp_defs_3: shows "carrier Zp = padic_set p"
distinct_11: shows "SizeOp \<noteq> ToRealOp"
bind_RES: fixes X :: "'d set"   and f :: "'d \<Rightarrow> 'c nres" shows "RES X \<bind> f = Sup (f ` X)"
has_bottomI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and P :: "'b set" assumes "ordering less_eq less"   and "z \<in> P"   and "\<And>x. x \<in> P \<Longrightarrow> less_eq z x" shows "ordering.has_bottom less_eq P"
l_1: shows "(u \<circ> a) \<longlonglongrightarrow> liminf u"
nuc_comp_distl: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and Y :: "'b set" assumes "nucleus f" shows "f (x \<cdot> f (\<Squnion> Y)) = f (\<Squnion>y\<in>Y. f (x \<cdot> y))"
index_mat_swaprows_2: fixes k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "dim_row (swaprows k l A) = dim_row A"
simps_2: fixes s1 :: "'b stream"   and s2 :: "'c stream" shows "stl (szip s1 s2) = szip (stl s1) (stl s2)"
inject_1: fixes x11 :: "'a"   and x12 :: "'b"   and y11 :: "'a"   and y12 :: "'b" shows "(x11 \<leftarrow> x12 = y11 \<leftarrow> y12) = (x11 = y11 \<and> x12 = y12)"
Rice_1: fixes A :: "nat set" assumes "index_set A"   and "A \<noteq> {}"   and "A \<noteq> UNIV" shows "one_reducible_to univ_ce A \<or> one_reducible_to univ_ce (- A)"
valid_add_new_ts_mmsaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and auxlist :: "(nat \<times> 'a option list set) list"   and nt :: "nat" assumes "valid_mmsaux args cur aux auxlist"   and "cur \<le> nt" shows "valid_mmsaux args nt (add_new_ts_mmsaux args nt aux) (filter (\<lambda>(t, rel). enat (nt - t) \<le> right (args_ivl args)) auxlist)"
simple_action_ne_2: fixes b :: "simple_action" shows "(b \<noteq> Drop) = (b = Accept)"
small_tiny_digraphs: fixes \<alpha> :: "V" shows "small {\<CC>. tiny_digraph \<alpha> \<CC>}"
minimal_redex_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and p :: "indices list"   and t :: "'b trm"   and C :: "'b eclause"   and S :: "'b eclause set"   and t' :: "'b trm" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.minimal_redex trm_ord sel pos_ord p t C S t' = (\<forall>q s. (q, p) \<in> pos_ord C t' \<longrightarrow> subterm t q s \<longrightarrow> basic_superposition.trm_rep trm_ord sel s S = s)"
gtr_preserve: fixes c :: "'c multiset"   and w :: "'d list"   and c' :: "'c multiset"   and T :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'c \<times> 'c multiset) set"   and P :: "'c multiset \<Rightarrow> bool" assumes "(c, w, c') \<in> trcl (gtr T)"   and "P c"   and "\<And>s c s' c' e. \<lbrakk>P (add_mset s c); ((s, c), e, s', c') \<in> T\<rbrakk> \<Longrightarrow> P (add_mset s' c')" shows "P c'"
PivotUpdate_axioms: shows "PivotUpdate eq_idx_for_lvar pivot update"
prems_2: fixes xs :: "'a list"   and x :: "'a" shows "P (xs \<cdot> [x]) \<epsilon>"
complemented_regular: fixes x :: "'a" assumes "complemented x" shows "regular x"
pochhammer_eq_0_mono: fixes a :: "'b"   and n :: "nat"   and m :: "nat" assumes "pochhammer a n = (0::'b)"   and "n \<le> m" shows "pochhammer a m = (0::'b)"
simps_2: fixes x2 :: "'d"   and y2 :: "'d" shows "(Inr x2 = Inr y2) = (x2 = y2)"
Err_lift2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b err"   and x :: "'d err" shows "Err \<squnion>\<^bsub>lift2 f\<^esub> x = Err"
constraint_expand: fixes x :: "'b word"   and lower :: "'b word"   and upper :: "'b word" shows "(x \<in> {y. lower \<le> y \<and> y \<le> upper}) = (lower \<le> x \<and> x \<le> upper)"
arr_dom_iff_arr: fixes f :: "'s" shows "S.arr (S.dom f) = S.arr f"
chain_Union_is_ideal: fixes C :: "'a set set" assumes "subset.chain {I. ideal I R} C" shows "ideal (if C = {} then {\<zero>} else \<Union> C) R"
sappend_bottom_iff: fixes xs :: "[:'a:]"   and ys :: "[:'a:]" shows "(xs :@ ys = \<bottom>) = (xs = \<bottom> \<or> ys = \<bottom>)"
insort_key_left_comm: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'b"   and xs :: "'b list" assumes "f x \<noteq> f y" shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
reduction_inv_8: fixes t' :: "db"   and P :: "bool" assumes "DBtt \<rightarrow>\<^sub>v t'" shows "P"
D_def: shows "D = max (real_of_ereal C) (Max {u n |n. n \<le> N})"
list_all_scoc: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" shows "list_all P (xs @ [x]) = (P x \<and> list_all P xs)"
list_augment_twice: fixes xs :: "'b list"   and i :: "nat"   and u :: "'b"   and j :: "nat"   and v :: "'b" shows "list_augment (list_augment xs i u) j v = (list_pad_out xs (max i j))[i := u, j := v]"
distinct_125: fixes x5 :: "Syntax.term"   and x6 :: "Syntax.term" shows "Inj1 x5 \<noteq> Inj2 x6"
ub: shows "\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> x < ub"
intro: fixes step1 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and final1 :: "'d \<Rightarrow> bool"   and step2 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and final2 :: "'e \<Rightarrow> bool"   and order :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and match :: "'f \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and load1 :: "'g \<Rightarrow> 'd \<Rightarrow> bool"   and load2 :: "'h \<Rightarrow> 'e \<Rightarrow> bool"   and compile :: "'g \<Rightarrow> 'h option" assumes "language step1 final1"   and "language step2 final2"   and "backward_simulation step1 step2 final1 final2 order match"   and "compiler_axioms load1 load2 match compile" shows "compiler step1 step2 final1 final2 load1 load2 order match compile"
i_Exec_Comp_Stream_Acc_Output_def: fixes k :: "nat"   and output_fun :: "'b \<Rightarrow> 'a message_af"   and trans_fun :: "'c message_af \<Rightarrow> 'b \<Rightarrow> 'b"   and input :: "nat \<Rightarrow> 'c message_af"   and c :: "'b" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c \<equiv> (output_fun \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) \<div> k"
prefixes_Suc_nth: fixes x :: "nat"   and j :: "nat"   and t :: "nat" assumes "x < e_length (local.prefixes j t)" shows "e_nth (local.prefixes j t) x = e_nth (local.prefixes j (Suc t)) x"
ordinal_exp_0: fixes x :: "ordinal" shows "x ** 0 = 1"
axioms_2: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd" assumes "identity_pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F" shows "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"
indis_trans: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and s :: "'b"   and s' :: "'b"   and s'' :: "'b" assumes "PL_Indis cval indis"   and "(s, s') \<in> indis"   and "(s', s'') \<in> indis" shows "(s, s'') \<in> indis"
case_ifexi_def: fixes fti :: "'s \<Rightarrow> 'e option"   and ffi :: "'s \<Rightarrow> 'e option"   and fii :: "'a \<Rightarrow> 'ni \<Rightarrow> 'ni \<Rightarrow> 's \<Rightarrow> 'e option"   and ni :: "'ni"   and s :: "'s" shows "case_ifexi fti ffi fii ni s \<equiv> DESTRimpl ni s \<bind> (\<lambda>dest. case dest of TD \<Rightarrow> fti s | FD \<Rightarrow> ffi s | IFD v ti ei \<Rightarrow> fii v ti ei s)"
empty: fixes A :: "'b set" assumes "A = {}" shows "topo (\<Union>s\<in>A\<^sup>\<star>. {infsuff A s}) = {{}}"
fmadd_singletons_comm: fixes k\<^sub>1 :: "'c"   and k\<^sub>2 :: "'c"   and v\<^sub>1 :: "'d"   and v\<^sub>2 :: "'d" assumes "k\<^sub>1 \<noteq> k\<^sub>2" shows "{k\<^sub>1 $$:= v\<^sub>1} ++\<^sub>f {k\<^sub>2 $$:= v\<^sub>2} = {k\<^sub>2 $$:= v\<^sub>2} ++\<^sub>f {k\<^sub>1 $$:= v\<^sub>1}"
simps: fixes xs :: "('a \<times> real) msllist"   and uu :: "real \<Rightarrow> real" shows "trimmed (MS xs uu) = trimmed_ms_aux xs"
diff_diff_eq2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.group_add minus plus zero uminus" shows "minus a (minus b c) = minus (plus a c) b"
per_init_self: fixes a :: "'a"   and b :: "'a"   and D :: "'a set" assumes "(a, b) \<in> per_init D" shows "a = b"
ord_strict_p_recE1: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<prec>\<^sub>p q" shows "q \<noteq> 0"
head_in_verts: fixes G :: "('c, 'd) pre_digraph"   and e :: "'d" assumes "wf_digraph G"   and "e \<in> arcs G" shows "head G e \<in> verts G"
id_eq_prop_prop_8_b: fixes v :: "i"   and p :: "\<o>"   and q :: "\<o>" shows "[p \<^bold>= q \<^bold>\<rightarrow> q \<^bold>= p in v]"
LIN: fixes r :: "('b \<times> 'b) set" assumes "wo_rel r" shows "Linear_order r"
nprovRuleCharacterise: fixes Ps :: "'a list"   and C :: "sequent" assumes "(Ps, C) \<in> nprovRules" shows "\<exists>F x A. C = ( \<Empt> \<Rightarrow>* \<LM> F \<nabla> [x].A  \<RM>) \<or> C = ( \<LM> F \<nabla> [x].A  \<RM> \<Rightarrow>* \<Empt>)"
appNone: fixes i :: "'c instr"   and P :: "'d prog"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and et :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list" shows "app i P mxs T\<^sub>r pc mpc et None = True"
gen_divisions_1: shows "gen_division s (cubeImage ` two_chain_typeI)"
surj_imp_ordLeq: fixes B :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "B \<subseteq> f ` A" shows "|B| \<le>o |A|"
FF_dim_1: shows "dim_row (mat_of_rows n (map of_int_hom.vec_hom fs')) = length (map of_int_hom.vec_hom fs')"
simps_7: fixes t :: "MIR.num" shows "not (Ge t) = Lt t"
card_of_lists_mono: fixes A :: "'c set"   and B :: "'d set" assumes "|A| \<le>o |B|" shows "|lists A| \<le>o |lists B|"
bound: fixes a :: "'a" shows "deriv_bound {(a, b). (0::'a) \<le> b \<and> a \<succ> b} a (bound a)"
case_prod_Pair_iden: fixes p :: "'c \<times> 'd" shows "(case p of (x, y) \<Rightarrow> (x, y)) = p"
fmdom_notI: fixes m :: "('e, 'd) fmap"   and x :: "'e" assumes "fmlookup m x = None" shows "x |\<notin>| fmdom m"
fin: shows "emeasure M (\<Union> (range A)) \<noteq> \<infinity>"
conv_abscissa_const: fixes c :: "'a" shows "conv_abscissa (fds_const c) = - \<infinity>"
dX_range': fixes x :: "'a"   and X :: "'a set"   and d :: "'b" assumes "x \<in> dX X d" shows "x \<in> X \<and> Xd x = d"
mono_root_real: fixes n :: "nat" assumes "n \<equiv> n" shows "mono (root n)"
coprime_diff_one_left_nat: fixes n :: "nat" assumes "0 < n" shows "coprime (n - 1) n"
hyps_3: assumes "S \<noteq> {}" shows "\<exists>m\<in>S. \<forall>s\<in>S. f s \<le> f m"
map_semantics_1: fixes \<xi> :: "nat \<Rightarrow> expr set" shows "\<xi> \<Turnstile>\<^sub>c \<phi> = (map_props (idx (IArray.list_of APs)) \<circ> \<xi>) \<Turnstile>\<^sub>c map_ltlc ((the \<circ>\<circ> idx) (IArray.list_of APs)) \<phi>"
partition_factors: fixes xs :: "('a \<times> nat) list"   and a :: "('a \<times> nat) list"   and b :: "('a \<times> nat) list" assumes "partition_factors xs = (a, b)" shows "mset xs = mset a + mset b"
alignUp_is_aligned_nz: fixes x :: "'b word"   and n :: "nat"   and a :: "'b word" assumes "is_aligned x n"   and "n < LENGTH('b)"   and "a \<le> x"   and "a \<noteq> 0" shows "alignUp a n \<noteq> 0"
simps_27: fixes f1 :: "value \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'b aexp \<Rightarrow> 'b aexp \<Rightarrow> 'a"   and f4 :: "'b aexp \<Rightarrow> 'b aexp \<Rightarrow> 'a"   and f5 :: "'b aexp \<Rightarrow> 'b aexp \<Rightarrow> 'a"   and x2 :: "'b" shows "(case V x2 of L x \<Rightarrow> f1 x | V x \<Rightarrow> f2 x | Plus x xa \<Rightarrow> f3 x xa | Minus x xa \<Rightarrow> f4 x xa | Times x xa \<Rightarrow> f5 x xa) = f2 x2"
fmdrop_lookup: fixes a :: "'d"   and m :: "('d, 'c) fmap" shows "fmlookup (fmdrop a m) a = None"
rtranclD: fixes a :: "'d"   and b :: "'d"   and R :: "('d \<times> 'd) set" assumes "(a, b) \<in> R\<^sup>*" shows "a = b \<or> a \<noteq> b \<and> (a, b) \<in> R\<^sup>+"
planning_dimacs_complete: fixes \<Pi> :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and \<pi>s :: "char list list"   and h :: "nat" assumes "sat_solve_sasp \<Pi>"   and "ast_problem.valid_plan \<Pi> \<pi>s"   and "length \<pi>s \<le> h" shows "let cnf_formula = map_formula (cnf_to_dimacs.var_to_dimacs (Suc h) (Suc (length (ast_problem.ast\<delta> \<Pi>)))) (\<Phi>\<^sub>\<forall> \<phi> prob_with_noop (ast_problem.abs_prob \<Pi>)  h) in \<exists>dimacs_M. dimacs_model dimacs_M (cnf_to_dimacs cnf_formula)"
eval_work_final_def: shows "eval_work_final \<equiv> fix\<cdot>eval_body_final"
fa: shows "\<forall>(r, a)\<in>set C. v \<Turnstile>\<^sub>a a"
set_ab: fixes x :: "'Fab"   and y :: "'Fab" assumes "Eab x y" shows "set_Fab x = set_Fab y"
less: shows "(nat \<lceil>real (length xs) / 5\<rceil> - Suc 0) div 2 < nat \<lceil>real (length xs) / 5\<rceil>"
to_fun_Fun_mult: fixes x :: "nat \<Rightarrow> nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "x \<in> carrier (UP Zp)"   and "y \<in> carrier (UP Zp)" shows "(\<bullet>) (x \<otimes>\<^bsub>UP Zp\<^esub> y) = (\<bullet>) x \<otimes>\<^bsub>function_ring (carrier Zp) Zp\<^esub> (\<bullet>) y"
fps_shift_nth: fixes n :: "nat"   and f :: "'b fps"   and i :: "nat" shows "fps_shift n f $ i = f $ (i + n)"
hd: shows "{x EQ x'} \<turnstile> P x y z IFF P x' y z"
simps_3: fixes f :: "dB"   and n :: "'a itrm"   and x :: "'a itrm" shows "norm_pn f (n \<diamondop> x) = norm_pn (\<B> \<degree> f) n \<diamondop> x"
wt_elim_cases_6: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and T' :: "ty"   and e :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (Cast T' e)\<Colon>T" obtains T :: "ty" where "Ta__ = Inl T'" and "E,dt\<Turnstile>e\<Colon>-T" and "is_acc_type (prg E) (pkg E) T'" and "prg E\<turnstile>T\<preceq>? T'"
ah_il_commute: fixes h1 :: "'b \<Rightarrow> 'b set"   and h2 :: "'b \<Rightarrow> 'b set" assumes "h1 [*] h2" shows "h2 [*] h1"
shows_list_gen_cong: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> char list \<Rightarrow> char list"   and g :: "'a \<Rightarrow> char list \<Rightarrow> char list"   and e :: "char list"   and l :: "char list"   and sep :: "char list"   and r :: "char list" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "shows_list_gen f e l sep r xs = shows_list_gen g e l sep r ys"
rev_bexI: fixes x :: "'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> A"   and "P x" shows "\<exists>x\<in>A. P x"
backlinks_alt: fixes E :: "('a \<times> 'a) list" shows "backlinks E = map (\<lambda>e. (snd e, fst e)) E"
is_image_spec_col_cong: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and P' :: "'a"   and A :: "'a"   and B :: "'a"   and X :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.ReflectL Bet Cong P P' A B"   and "Tarski_neutral_dimensionless.Col Bet A B X" shows "Cong P X P' X"
hyps: assumes "\<And>p. p \<in> set ps \<Longrightarrow> p \<noteq> 0" shows "degree (prod_list ps) = sum_list (map degree ps)"
truth_provable: fixes H :: "fmla set" shows "H \<turnstile> neg fls"
tt': shows "Inum (a # bs) t = Inum (a # bs) t'"
tabulation_axioms: shows "tabulation (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F r) (D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho>) (F f) (F g)"
find_index_leq_length: fixes xs :: "'a list"   and x :: "'a" shows "(find_index xs x < length xs) = (x \<in> set xs)"
DEP_REFL: fixes PROB :: "(('b, 'c) fmap \<times> ('b, 'd) fmap) set" shows "reflexive (dep PROB)"
list_same: fixes f :: "'e \<Rightarrow> 'd"   and s_set :: "'e list"   and z' :: "'d"   and x :: "'e" assumes "f ` set s_set = {z'}"   and "List.member s_set x" shows "f x = z'"
Sup_upper2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and u :: "'b"   and A :: "'b set"   and v :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "u \<in> A"   and "less_eq v u" shows "less_eq v (Sup A)"