hcju/isabelleps huggingface dataset:
vimage_eq_UN: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'d set" shows "f -` B = (\<Union>y\<in>B. f -` {y})"
digraph_map_axioms_def: fixes G :: "('e, 'f) pre_digraph"   and M :: "'f pre_map" shows "digraph_map_axioms G M \<equiv> edge_succ M permutes arcs G \<and> (\<forall>v. v \<in> verts G \<longrightarrow> out_arcs G v \<noteq> {} \<longrightarrow> cyclic_on (edge_succ M) (out_arcs G v))"
per_distinct: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Per A B C"   and "A \<noteq> B" shows "A \<noteq> C"
Goal: fixes P'' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> \<tau>.P \<Longrightarrow>\<^sub>\<tau> P''"   and "\<Psi> \<rhd> P' \<sim> P''" shows "thesis"
conts': fixes \<epsilon> :: "real"   and x1 :: "'b"   and i :: "'a" assumes "0 < \<epsilon>" shows "\<exists>\<delta>>0. \<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon>"
Int_Union_Union: fixes B :: "'d set set"   and A :: "'d set" shows "\<Union> B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
elims_3: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('fsym, 'psym) atm" assumes "\<not> F.satA x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and T1 :: "'fsym trm" and T2 :: "'fsym trm" where "x = \<xi>'__" and "xa = Eq T1 T2" and "F.int \<xi>'__ T1 \<noteq> F.int \<xi>'__ T2"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and p :: "'psym" and Tl :: "'fsym trm list" where "x = \<xi>" and "xa = Pr p Tl" and "\<not> intPF p (map (F.int \<xi>) Tl)"
simps: fixes z :: "name"   and r :: "tm"   and z' :: "name"   and x :: "name"   and y :: "name"   and x' :: "name"   and y' :: "name" assumes "atom z \<sharp> (r, z', x, y, x', y')"   and "atom z' \<sharp> (r, x, y, x', y')"   and "atom x \<sharp> (r, y, x', y')"   and "atom y \<sharp> (r, x', y')"   and "atom x' \<sharp> (r, y')"   and "atom y' \<sharp> r" shows "HFun_Sigma r = All2 z r (All2 z' r (SyntaxN.Ex x (SyntaxN.Ex y (SyntaxN.Ex x' (SyntaxN.Ex y' (Var z EQ HPair (Var x) (Var y) AND Var z' EQ HPair (Var x') (Var y') AND OrdP (Var x) AND OrdP (Var x') AND (Var x EQ Var x' IMP Var y EQ Var y')))))))"
esqrt_eq_iff_square: fixes x :: "ereal"   and y :: "ereal" assumes "0 \<le> x"   and "0 \<le> y" shows "(esqrt x = y) = (x = y * y)"
monomial_inj: fixes s :: "'b"   and c :: "'c"   and t :: "'b"   and d :: "'c" assumes "monomial c s = monomial d t" shows "c = (0::'c) \<and> d = (0::'c) \<or> c = d \<and> s = t"
tt_mult_scalar: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "tt (p \<odot> q) = punit.tt p \<oplus> tt q"
fold_Option_bind_eq_Some_start_not_None': fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a option"   and list :: "('b \<times> 'c) list"   and start :: "'a option"   and res :: "'a" assumes "fold (\<lambda>(x, y) option. Option.bind option (f x y)) list start = Some res" shows "start \<noteq> None"
prism_diff_intro: fixes X :: "'c \<Longrightarrow>\<^sub>\<triangle> 'e"   and Y :: "'d \<Longrightarrow>\<^sub>\<triangle> 'e" assumes "\<And>s\<^sub>1 s\<^sub>2. build\<^bsub>X\<^esub> s\<^sub>1 = build\<^bsub>Y\<^esub> s\<^sub>2 \<Longrightarrow> False" shows "X \<nabla> Y"
rem_unPath_com: fixes ps :: "('a \<times> 'b \<times> 'a) list"   and v :: "'a"   and w :: "'b"   and v' :: "'a"   and g :: "('a, 'b) graph" shows "rem_unPath ps (del_unEdge v w v' g) = del_unEdge v w v' (rem_unPath ps g)"
assms: shows "f = M.sumlist (map (\<lambda>i. of_int (c i) \<cdot>\<^sub>v fs ! i) [0..<length fs])"
MGU_Var: fixes v :: "'b"   and t :: "'b trm" assumes "\<not> Var v \<prec> t" shows "MGU [(v, t)] (Var v) t"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs"   and "l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs" shows "l_set_tag_name_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs"
split_spec_params_def: fixes d :: "nat"   and n :: "'c"   and X :: "'b list set" shows "split_spec_params d n X = SPEC (\<lambda>(A, B). env_len A d \<and> env_len B d \<and> X \<subseteq> A \<union> B)"
rtranclp_into_equivclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "r\<^sup>*\<^sup>* x y" shows "equivclp r x y"
of_nat_simps_1: shows "of_nat 0 = (0::'b)"
triangle: fixes a :: "'d"   and b :: "'d" assumes "D.ide a"   and "D.ide b" shows "(a \<otimes>\<^sub>D \<l>\<^sub>1[b]) \<cdot>\<^sub>D \<a>\<^sub>D[a, \<I>\<^sub>1, b] = \<r>\<^sub>1[a] \<otimes>\<^sub>D b"
group_comm_groupI: assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" shows "comm_group G"
star2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.left_near_kleene_algebra plus times one less_eq less star" shows "star (plus one x) = star x"
simps_3: fixes \<A> :: "'b ltln set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<A> \<Turnstile>\<^sub>P \<phi> and\<^sub>n \<psi> = (\<A> \<Turnstile>\<^sub>P \<phi> \<and> \<A> \<Turnstile>\<^sub>P \<psi>)"
l1_step1_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent" shows "l1_step1 Ra A B \<equiv> {(s, s'). Ra \<notin> dom (progress s) \<and> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xpkE, xskE})\<rparr>}"
norm_mult: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.real_normed_div_algebra divide inverse one times plus zero minus uminus scaleR dist norm sgn uniformity open" shows "norm (times x y) = norm x * norm y"
prems_2: fixes x :: "variable \<times> typ" assumes "x \<in> \<Union> (fv ` snd ` set (xs @ [x__]))" shows "x \<notin> fst ` set (xs @ [x__])"
token_stays_in_final_states: fixes x :: "nat"   and n :: "nat"   and m :: "nat" assumes "token_run x n \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q}" shows "token_run x (n + m) \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q}"
f: shows "(\<lambda>n. sum f {..<n}) \<longlonglongrightarrow> (0::'a)"
ax_Init_Skip_lemma: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and l' :: "lname \<Rightarrow> val option" shows "\<forall>l. G,A\<turnstile>{P\<leftarrow>\<diamondsuit> \<and>. (\<lambda>s. l = locals (snd s)) ;. set_lvars l'} .Skip. {set_lvars l .; P}"
trivial_subgroup_is_normal: shows "{\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub>} \<lhd> G\<lparr>carrier := H\<rparr>"
nsum_zeroA: fixes A :: "('d, 'e) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "aGroup A"   and "\<forall>j\<le>n. f j = \<zero>\<^bsub>A\<^esub>" shows "\<Sigma>\<^sub>e A f n = \<zero>\<^bsub>A\<^esub>"
norm_commute: fixes x :: "'a"   and y :: "'b" shows "norm (x, y) = norm (y, x)"
while_reds_induct: fixes b :: "expr"   and cx :: "com"   and s :: "char list \<Rightarrow> val option"   and s' :: "char list \<Rightarrow> val option"   and P :: "(char list \<Rightarrow> val option) \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> bool" assumes "\<langle>while (b) cx,s\<rangle> \<rightarrow>* \<langle>Skip,s'\<rangle>"   and "\<And>s. \<lbrakk>b\<rbrakk>s = Some false \<Longrightarrow> P s s"   and "\<And>s s''. \<lbrakk>\<lbrakk>b\<rbrakk>s = Some true; \<langle>cx,s\<rangle> \<rightarrow>* \<langle>Skip,s''\<rangle>; \<langle>while (b) cx,s''\<rangle> \<rightarrow>* \<langle>Skip,s'\<rangle>; P s'' s'\<rbrakk> \<Longrightarrow> P s s'" shows "P s s'"
pqs_def: shows "pqs \<equiv> {(p, q). 0 < q \<and> coprime p q \<and> \<bar>\<xi> - real_of_int p / real_of_int q\<bar> < 1 / real_of_int q powr (2 + \<delta>)}"
inj_on_adjoin: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H G"   and "a \<in> carrier G"   and "a \<notin> H" shows "inj_on (\<lambda>(x, k). x \<otimes> a [^] k) (H \<times> {..<subgroup_indicator G H a})"
nat_minus_mod: fixes n :: "nat"   and m :: "nat" shows "(n - n mod m) mod m = 0"
between_trans: fixes b :: "'b"   and c :: "'b"   and a :: "'b"   and d :: "'b" assumes "between (b, c) a"   and "between (a, c) d" shows "between (b, c) d"
mprev: fixes I :: "\<I>"   and xs :: "event_data option list set list"   and ts :: "nat list"   and ys :: "event_data option list set list"   and xs' :: "event_data option list set list"   and ts' :: "nat list"   and P :: "nat \<Rightarrow> event_data option list set \<Rightarrow> bool"   and i :: "nat"   and j' :: "nat"   and \<sigma> :: "'a trace"   and j :: "nat" assumes "mprev_next I xs ts = (ys, xs', ts')"   and "list_all2 P [i..<j'] xs"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [i..<j] ts"   and "i \<le> j'"   and "i < j" shows "list_all2 (\<lambda>i X. if mem (\<tau> \<sigma> (Suc i) - \<tau> \<sigma> i) I then P i X else X = empty_table) [i..<min j' (j - 1)] ys \<and> list_all2 P [min j' (j - 1)..<j'] xs' \<and> list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min j' (j - 1)..<j] ts'"
rep_number_alt_def_all: shows "\<forall>x\<in>\<V>. \<B> rep x = \<Lambda> * (\<v> - 1) div (\<k> - 1)"
distinct_11: fixes x1 :: "agent"   and x71 :: "key"   and x72 :: "msg" shows "msg.Agent x1 \<noteq> Crypt x71 x72"
map_iterator_product_def: fixes it_a :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<times> 'h \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and it_b :: "'h \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> ('g \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" shows "map_iterator_product it_a it_b = set_iterator_image (\<lambda>kvv'. (fst (fst kvv'), snd kvv')) (set_iterator_product it_a (\<lambda>kv. it_b (snd kv)))"
hd_append2: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<noteq> []" shows "hd (xs @ ys) = hd xs"
nth_tl: fixes n :: "nat"   and xs :: "'b list" assumes "n < length (tl xs)" shows "tl xs ! n = xs ! Suc n"
holomorphic_deriv: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "f holomorphic_on S"   and "open S" shows "deriv f holomorphic_on S"
fps_ext: fixes p :: "'a fps"   and q :: "'a fps" assumes "\<And>n. p $ n = q $ n" shows "p = q"
lexordp_irreflexive': fixes xs :: "'a list" shows "\<not> xs <lex xs"
bet_le_eq: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C"   and "A C Le B C" shows "A = B"
fps_div_zero: fixes g :: "'a fps" shows "0 / g = 0"
d1: assumes "(\<forall>p\<in>set (retrieve_polys qs (fst I)). poly p x = 0) \<and> poly (prod_list (retrieve_polys qs (snd I))) x < 0" shows "z_R I sign = - 1"
IH_1: fixes n :: "nat" shows "cut_bound' n F1 \<le> cbnd n (depth F1)"
is_acc_run_gbg: fixes m :: "'b" shows "gb_graph.is_acc_run (to_gbg_ext m) = is_acc_run"
cexp_def: fixes r1 :: "('e \<times> 'e) set"   and r2 :: "('d \<times> 'd) set" shows "r1 ^c r2 \<equiv> |Func (Field r2) (Field r1)|"
exhaust: fixes y :: "status" obtains "y = FAILED"   | "y = SUCCESS"   | "y = FOUND"
preserves_dom: fixes f :: "'a" assumes "\<tau>.A.arr f" shows "\<tau>.B.dom (\<tau>''.map f) = F (\<tau>.A.dom f)"
simps_2: fixes b :: "'bool" shows "generalize_instr (IPushUbx2 b) = IPush (box_ubx2 b)"
elims_3: fixes x :: "fterm" assumes "\<not> ground\<^sub>t x" obtains xa :: "char list" where "x = Var xa"   | f :: "char list" and ts :: "fterm list" where "x = Fun f ts" and "\<not> ground\<^sub>t\<^sub>s ts"
a: shows "x \<sharp> M[a\<turnstile>c>b]"
supp_conv_fresh: fixes x :: "'b" shows "supp x = {a. \<not> a \<sharp> x}"
FW_fixed_preservation: fixes i :: "nat"   and n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c" assumes "i \<le> n"   and "M 0 i + M i 0 = \<one>"   and "canonical (FW M n) n"   and "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len (FW M n) i i xs" shows "FW M n 0 i + FW M n i 0 = \<one>"
conjunctive_def: shows "conjunctive = {x. \<forall>y z. times_abc x (inf_b y z) = inf_c (times_abc x y) (times_abc x z)}"
xor_assoc: fixes x :: "'e"   and y :: "'e"   and z :: "'e" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "(x \<oplus> y) \<oplus> z = x \<oplus> y \<oplus> z"
wellorder_Least_lemma_1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and k :: "'c" assumes "class.wellorder less_eq less"   and "P k" shows "P (ord.Least less_eq P)"
not_\<omega>\<^sub>z_le_1: shows "\<not> \<omega>\<^sub>z \<le> 1"
reds_preserves_hconf: fixes es :: "char list exp list"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and es' :: "char list exp list"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and Ts :: "ty list" assumes "P \<turnstile> \<langle>es,(h, l, sh),b\<rangle> [\<rightarrow>] \<langle>es',(h', l', sh'),b'\<rangle>"   and "P,E,h,sh \<turnstile> es [:] Ts"   and "P \<turnstile> h \<surd>" shows "P \<turnstile> h' \<surd>"
ASSERT_le_iff: fixes \<Phi> :: "bool"   and S :: "'b nres"   and S' :: "'b nres" shows "(ASSERT \<Phi> \<bind> (\<lambda>_. S) \<le> S') = ((S' \<noteq> FAIL \<longrightarrow> \<Phi>) \<and> S \<le> S')"
left_commute: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "b \<sqinter> (a \<sqinter> c) = a \<sqinter> (b \<sqinter> c)"
N_1: fixes k :: "nat"   and j :: "nat" assumes "N k \<le> j" shows "real (card (A k \<inter> {..<j})) \<le> (l + (1 / 2) ^ k) * real j"
card_gr0_imp_not_empty: fixes A :: "'a set" assumes "0 < card A" shows "A \<noteq> {}"
eq: shows "outs_\<I> \<I> \<turnstile>\<^sub>R res1 \<approx> res2"
rcp_spec_def: fixes rcp :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" shows "rcp_spec rcp = (\<forall>gs bs ps sps data. 0 \<notin> fst ` set (fst (rcp gs bs ps sps data)) \<and> (\<forall>h b. h \<in> set (fst (rcp gs bs ps sps data)) \<longrightarrow> b \<in> set gs \<union> set bs \<longrightarrow> fst b \<noteq> 0 \<longrightarrow> \<not> lt (fst b) adds\<^sub>t lt (fst h)) \<and> (\<forall>d. dickson_grading d \<longrightarrow> dgrad_p_set_le d (fst ` set (fst (rcp gs bs ps sps data))) (args_to_set (gs, bs, sps))) \<and> component_of_term ` Keys (fst ` set (fst (rcp gs bs ps sps data))) \<subseteq> component_of_term ` Keys (args_to_set (gs, bs, sps)) \<and> (is_Groebner_basis (fst ` set gs) \<longrightarrow> unique_idx (gs @ bs) data \<longrightarrow> fst ` set (fst (rcp gs bs ps sps data)) \<subseteq> pmdl (args_to_set (gs, bs, sps)) \<and> (\<forall>(p, q)\<in>set sps. set sps \<subseteq> set bs \<times> (set gs \<union> set bs) \<longrightarrow> (red (fst ` (set gs \<union> set bs) \<union> fst ` set (fst (rcp gs bs ps sps data))))\<^sup>*\<^sup>* (spoly (fst p) (fst q)) 0)))"
comp: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "ChamberComplexAutomorphism X f"   and "ChamberComplexAutomorphism X g" shows "ChamberComplexAutomorphism X (g \<circ> f)"
subtensor_combine_vec: fixes As :: "'b tensor list"   and ds :: "nat list" assumes "\<And>A. A \<in> set As \<Longrightarrow> dims A = ds" shows "vec (subtensor_combine ds As) = concat (map vec As)"
sup_state_opt_def: fixes P :: "'b prog" shows "sup_state_opt P \<equiv> Opt.le (sup_state P)"
P1: shows "Per A' B' C' \<and> A' B' C' LtA D E F"
eq_ffilter: fixes P :: "'a \<Rightarrow> bool"   and A :: "'a fset"   and Q :: "'a \<Rightarrow> bool" shows "(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
nth_least_card: fixes k :: "nat"   and A :: "nat set" assumes "k \<in> A" shows "nth_least A (card {i \<in> A. i < k}) = k"
neqE: fixes x :: "'a word"   and y :: "'a word" assumes "x \<noteq> y" obtains "x <s y"   | "y <s x"
basis_indep: fixes X :: "'a set" assumes "basis X" shows "indep X"
Quotient_fin_string: shows "Quotient (eq_onp (\<lambda>s. length s \<le> MAX_STR)) Abs_fin_string Rep_fin_string cr_fin_string"
cobounded1: fixes a :: "'d upred"   and b :: "'d upred" shows "a \<sqsubseteq> (a \<and> b)"
lm036: fixes P :: "('a \<times> 'b) set"   and X :: "'a set" shows "P || X = Outside' (Domain P - X) P"
in_set_cyc_permutationss: fixes xss :: "'b list list"   and yss :: "'b list list" assumes "distincts xss" shows "(yss \<in> set (cyc_permutationss xss)) = (distincts yss \<and> map set xss = map set yss \<and> map hd xss = map hd yss)"
stable_def: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set"   and \<mu> :: "('a \<times> 'b) set" assumes "StableMarriage Pm Pw" shows "StableMarriage.stable Pm Pw \<mu> = (StableMarriage.match Pm Pw \<mu> \<and> (\<forall>m w. \<not> StableMarriage.blocks Pm Pw m w \<mu>))"
assms_4: shows "\<forall>x\<in>U\<^sub>2. f x \<in> U"
match_list_matches: fixes d :: "'d \<Rightarrow> ('e, 'f) Term.term"   and P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) list"   and \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term"   and p :: "('e, 'd) Term.term"   and t :: "('e, 'f) Term.term" assumes "match_list d P = Some \<sigma>"   and "(p, t) \<in> set P" shows "p \<cdot> \<sigma> = t"
relabel': fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "Height_Bounded_Labeling c s t f l"   and "relabel_precond f l u" shows "((f, l), RELABEL', f, relabel_effect f l u) \<in> pr_algo_lts'"
HLD_iff: fixes s :: "'c set"   and \<omega> :: "'c stream" shows "HLD s \<omega> = (shd \<omega> \<in> s)"
s_ce_inj: fixes e1 :: "nat"   and x1 :: "nat"   and e2 :: "nat"   and x2 :: "nat" assumes "s_ce e1 x1 = s_ce e2 x2" shows "e1 = e2 \<and> x1 = x2"
flat_lub_def: fixes b :: "'f"   and A :: "'f set" shows "flat_lub b A = (if A \<subseteq> {b} then b else THE x. x \<in> A - {b})"
splitqs: fixes a :: "'a"   and x :: "'a"   and y :: "'a"   and rest1 :: "'a list" assumes "a \<in> {x, y}" shows "T\<^sub>p [x, y] (a # rest1) (OPT2 (a # rest1) [x, y]) = t\<^sub>p [x, y] a (hd (OPT2 (a # rest1) [x, y])) + (let L = step [x, y] a (hd (OPT2 (a # rest1) [x, y])) in T\<^sub>p L rest1 (OPT2 rest1 L))"
sorted_rev_iff_nth_mono: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (rev xs) = (\<forall>i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> less_eq (xs ! j) (xs ! i))"
init_C_def: fixes am :: "nat \<Rightarrow> nat list" shows "init_C am \<equiv> Let (0::'a) (nfoldli [0..<N] (\<lambda>_. True) (\<lambda>v cardV. assert (v < N) \<bind> (\<lambda>_. am_is_in_V am v \<bind> (\<lambda>inV. if inV then return (cardV + (1::'a)) else return cardV))))"
rep_eq: fixes x :: "linear_poly"   and xa :: "linear_poly" shows "Rep_linear_poly (x + xa) = fun_plus (Rep_linear_poly x) (Rep_linear_poly xa)"
right_unique_rel_lcI: assumes "right_unique rel_c" shows "right_unique rel_lc"
at_neq_bot: fixes "open" :: "'a set \<Rightarrow> bool"   and a :: "'a" assumes "class.perfect_space open" shows "topological_space.at_within open a UNIV \<noteq> bot"
sum_strict_mono_ex1: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "\<forall>x\<in>A. f x \<le> g x"   and "\<exists>a\<in>A. f a < g a" shows "sum f A < sum g A"
drop_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'g list" shows "drop n (take m xs) = take (m - n) (drop n xs)"
sup_aci_2: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x + y + z = x + (y + z)"
k_def: fixes Q :: "nat \<Rightarrow> bool" shows "k Q = (LEAST k. j1 < k \<and> Q k)"
lookup_def: fixes return :: "int \<Rightarrow> 'b"   and fail :: "'b"   and ask :: "(('c \<Rightarrow> int option) \<Rightarrow> 'b) \<Rightarrow> 'b"   and x :: "'c" shows "reader_exp_base.lookup return fail ask x = ask (\<lambda>s. case s x of None \<Rightarrow> fail | Some y \<Rightarrow> return y)"
subclsRI: fixes 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 C :: "char list"   and Bs :: "base list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "Repeats D \<in> set Bs" shows "P \<turnstile> C \<prec>\<^sub>R D"
openin_open_Int: fixes S :: "'a set"   and U :: "'a set" assumes "open S" shows "openin (top_of_set U) (U \<inter> S)"
ennreal_minus_cancel: fixes c :: "ennreal"   and a :: "ennreal"   and b :: "ennreal" assumes "c \<noteq> \<top>"   and "a \<le> c"   and "b \<le> c"   and "c - a = c - b" shows "a = b"
elims_2: fixes x :: "real_min"   and xa :: "real_min" assumes "x < xa" obtains x :: "real" and y :: "real" where "xa__ = Rmin x" and "xa = Rmin y" and "x < y"
onode_sos_dest_is_net_state'': fixes \<sigma> :: "nat \<Rightarrow> 'a"   and i :: "nat"   and p :: "'b"   and R :: "nat set"   and a :: "'c node_action"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and s' :: "'b net_state"   and S :: "(((nat \<Rightarrow> 'a) \<times> 'b) \<times> 'c seq_action \<times> (nat \<Rightarrow> 'a) \<times> 'b) set" assumes "((\<sigma>, NodeS i p R), a, \<sigma>', s') \<in> onode_sos S" shows "\<exists>\<zeta>' R'. s' = NodeS i \<zeta>' R'"
not_le: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "(\<not> ko.le ko x y) = ko.lt ko y x"
fin: fixes A :: "'a set" assumes "A \<in> M" shows "f A \<noteq> \<infinity>"
assms_1: shows "f' = of_int_poly_i ff_ops (Mp f)"
phiUses_exI': fixes g :: "'g"   and p :: "'val"   and q :: "'val" assumes "phiArg g p q"   and "p \<in> allVars g" obtains m :: "'node" where "q \<in> phiUses g m" and "m \<in> set (old.predecessors g (defNode g p))"
LstSeqP_imp_Mem: fixes H :: "fm set"   and s :: "tm"   and k :: "tm"   and y :: "tm" assumes "H \<turnstile> LstSeqP s k y" shows "H \<turnstile> HPair k y IN s"
filter_compliant_stateful_ACS_def: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and Es :: "('b \<times> 'b) list" shows "filter_compliant_stateful_ACS G M Es = filter_compliant_stateful_ACS_accu G M [] Es"
shiftl_x_0: fixes x :: "'b word" shows "x << 0 = x"
inorder_baliR: fixes l :: "('b \<times> color) tree"   and a :: "'b"   and r :: "('b \<times> color) tree" shows "Tree2.inorder (baliR l a r) = Tree2.inorder l @ a # Tree2.inorder r"