hcju/isabelleps huggingface dataset:
compl_sup_top: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<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 x :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup (uminus x) x = top"
Vinv1I: fixes x :: "'a v_state_scheme" assumes "\<And>r. next_round x \<le> r \<Longrightarrow> votes x r = Map.empty" shows "x \<in> Vinv1"
prems_2: shows "w \<Turnstile>\<^sub>n \<phi>1 M\<^sub>n \<phi>2"
plus_sqmatrix_def: fixes A :: "('c, 'd) sqmatrix"   and B :: "('c, 'd) sqmatrix" shows "A + B \<equiv> sqmatrix_of_matrix (matrix_of_sqmatrix A + matrix_of_sqmatrix B)"
poly_deg_univariate: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<in> P[X]" shows "poly_deg p = deg_pm (lpp p)"
seqp_seq_eq: fixes ars :: "('a \<times> 'b \<times> 'a) set" shows "seqp (\<lambda>x xa xb. (x, xa, xb) \<in> ars) = (\<lambda>x xa. (x, xa) \<in> seq ars)"
ht_hash_in_dom_i_bounded_hashcode_nat_i: fixes l :: "('a \<times> 'b) list list"   and i :: "nat"   and k :: "'a" assumes "ht_hash l"   and "1 < length l"   and "i < length l"   and "k \<in> dom (map_of (l ! i))" shows "i = bounded_hashcode_nat (length l) k"
step0: shows "j < length M2 \<longrightarrow> (i < length v * row_length M2 \<and> j < length M2 * length (v # M)) \<and> mat (row_length M2) (length M2) M2 \<longrightarrow> (v # M \<otimes> M2) ! j ! i = (v # M) ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)"
scan_last: fixes a :: "'a"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b list" shows "last (a # scan f xs a) = fold f xs a"
coerce_eq_emb: shows "COERCE('c, udom) = EMB('c)"
skelInp_def: fixes inp :: "'e \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qTerm set option" shows "skelInp inp = qSkelInp (pickInp inp)"
Q'''Trans: shows "Q''' \<longmapsto> a<\<nu>x> \<prec> Q''"
bit_drop_bit_eq: fixes n :: "nat"   and a :: "'a" shows "bit (drop_bit n a) = bit a \<circ> (+) n"
insert_subsetI: fixes x :: "'b"   and A :: "'b set"   and X :: "'b set" assumes "(x::'b) \<in> (A::'b set)"   and "(X::'b set) \<subseteq> (A::'b set)" shows "(x::'b) \<triangleright> (X::'b set) \<subseteq> (A::'b set)"
find_None: fixes a :: "'c"   and xs :: "('c, 'd) bintree option list" assumes "Binomial_Queue.find a xs = None" shows "Some a \<notin> set (map (map_option priority) xs)"
stack_rel_internal_def: fixes lrel :: "('i \<times> 'j) set \<Rightarrow> ('f \<times> 'h list) set"   and srel :: "('i \<times> 'j) set \<Rightarrow> ('g \<times> 'h set) set"   and vrel :: "('i \<times> 'j) set" shows "stack_rel lrel srel vrel \<equiv> {((li, si), l). (li, l) \<in> \<langle>vrel\<rangle>lrel \<and> (si, set l) \<in> \<langle>vrel\<rangle>srel \<and> distinct l}"
part_UNpart01: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "properL cl"   and "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n" shows "part {..<brnL cl (length cl)} (UNpart01 cl dl P)"
set_intros_1: fixes x21 :: "'a"   and x22 :: "'a list" shows "x21 \<in> set (x21 \<Colon> x22)"
inner_diff_right: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "inner x (y - z) = inner x y - inner x z"
matrix_matrix_mult_inner_mult: fixes A :: "((real, 'd) vec, 'c) vec"   and B :: "((real, 'b) vec, 'd) vec"   and i :: "'c"   and j :: "'b" shows "(A ** B) $ i $ j = row i A \<bullet> column j B"
pcr_cr_eq: shows "pcr_commute (=) (=) = cr_commute"
le_inf_iff: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "(x \<subseteq># y \<inter># z) = (x \<subseteq># y \<and> x \<subseteq># z)"
simps_1: shows "del_min2 \<langle>\<rangle> = \<langle>\<rangle>"
lcrec: fixes n :: "nat" assumes "3 < n" shows "lcount n = lcount (n - 1) + 1 + (\<Sum>i = 3..<n. lcount (n - i - 1))"
ext_inject: fixes arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'b"   and arrays' :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs' :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim' :: "nat"   and more' :: "'b" shows "(\<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = \<lparr>arrays = arrays', refs = refs', lim = lim', \<dots> = more'\<rparr>) = (arrays = arrays' \<and> refs = refs' \<and> lim = lim' \<and> more = more')"
simps_1: fixes i :: "nat"   and divs :: "'b mod_ring poly list"   and v :: "'b mod_ring poly"   and vs :: "'b mod_ring poly list"   and n :: "nat" shows "berlekamp_factorization_main i divs (v # vs) n = (if v = 1 then berlekamp_factorization_main i divs vs n else if length divs = n then divs else let facts = concat (map (\<lambda>u. concat (map (\<lambda>s. concat (map (\<lambda>w. if w \<noteq> 1 then [w] else []) [gcd u (v - [:of_int s:])])) (map int [0..<CARD('b)]))) divs); (lin, nonlin) = List.partition (\<lambda>q. degree q = i) facts in lin @ berlekamp_factorization_main i nonlin vs (n - length lin))"
f: fixes x :: "'n"   and k :: "'n set" assumes "(x, k) \<in> p" shows "f integrable_on k"
bisimSubstScopeExt: fixes x :: "name"   and P :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi" assumes "x \<sharp> P" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<sim>\<^sub>s P \<parallel> \<lparr>\<nu>x\<rparr>Q"
induces_adjunction: shows "adjunction (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) replete_setcat.comp FG.\<phi>C FG.\<phi>D F G \<phi> \<psi> FG.\<eta> FG.\<epsilon> FG.\<Phi> FG.\<Psi>"
domains_comp: fixes g :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" assumes "seq g f" shows "domains (g \<cdot> f) = domains f"
L_nths_Lxy: fixes x :: "'b"   and init :: "'b list"   and y :: "'b" assumes "x \<in> set init"   and "y \<in> set init"   and "x \<noteq> y"   and "distinct init" shows "map_pmf (\<lambda>l. nths l {index init x, index init y}) (bv (length init)) = bv (length (Lxy init {x, y}))"
matchC_TMC_def: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and theta :: "(('b, 'c) com \<times> ('b, 'c) com) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.matchC_TMC tval aval indis theta c d \<equiv> \<forall>s t c' s'. PL.mustT tval aval c s \<and> PL.mustT tval aval d t \<and> (s, t) \<in> indis \<and> PL.transC tval aval (c, s) (c', s') \<longrightarrow> (\<exists>d' t'. PL.MtransC tval aval (d, t) (d', t') \<and> (s', t') \<in> indis \<and> (c', d') \<in> theta)"
mult2_alt_emptyL: fixes Y :: "'a multiset"   and b :: "bool"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "(({#}, Y) \<in> mult2_alt b ns s) = (b \<or> Y \<noteq> {#})"
eq_swap_extend: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and u :: "'item list"   and v :: "'item list" assumes "w\<^sub>1 =\<^sub>S w\<^sub>2" shows "u @ w\<^sub>1 @ v =\<^sub>S u @ w\<^sub>2 @ v"
trace_hide_append: fixes s :: "'a event list"   and t :: "'a event list"   and A :: "'a set"   and ta :: "'a event list" assumes "s @ t = trace_hide ta (ev ` A)" shows "\<exists>ss tt. ta = ss @ tt \<and> s = trace_hide ss (ev ` A) \<and> t = trace_hide tt (ev ` A)"
norm_nth_le: fixes x :: "('a, 'b) vec"   and i :: "'b" shows "norm (x $ i) \<le> norm x"
g: shows "recfn 1 g \<and> (\<forall>x. (eval g [x] \<down>) = (x \<in> - X))"
coeff_monom_mult: fixes c :: "'b"   and n :: "nat"   and p :: "'b poly"   and k :: "nat" shows "poly.coeff (Polynomial.monom c n * p) k = (if k < n then 0::'b else c * poly.coeff p (k - n))"
s0_defs_3: shows "s0_trans \<equiv> (\<Union> (range s0_add_secret) \<union> \<Union> (range s0_learn))\<^sup>="
distinct_map_entry: fixes xs :: "('b \<times> 'c) list"   and k :: "'b"   and f :: "'c \<Rightarrow> 'c" assumes "distinct (map fst xs)" shows "distinct (map fst (AList.map_entry k f xs))"
top_sub_from_counter: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and tp :: "top0"   and s' :: "'b state"   and r :: "(v sem_env, v) result"   and cnt :: "nat"   and extra :: "nat"   and cnt' :: "nat" assumes "evaluate_top ck env s tp (s', r)"   and "ck = True"   and "clock s = cnt + extra"   and "clock s' = cnt' + extra" shows "evaluate_top ck env (update_clock (\<lambda>_. cnt) s) tp (update_clock (\<lambda>_. cnt') s', r)"
assms: shows "0\<^sub>h \<in> ocircline_set H \<and> 1\<^sub>h \<in> ocircline_set H \<and> \<infinity>\<^sub>h \<in> ocircline_set H"
assms_3: shows "circline_type (of_ocircline H1) < 0"
int_to_ipv6preferred_ipv6preferred_to_int: fixes ip :: "ipv6addr_syntax" shows "int_to_ipv6preferred (ipv6preferred_to_int ip) = ip"
in_idem: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "max_torder r x (Max_torder r A) = Max_torder r A"
preserves_iso: fixes f :: "'d" assumes "D.iso f" shows "C.iso (F f)"
intro: fixes F :: "('c, 'd, 'a, 'b, 'e, 'f, 'g) Functor_scheme" assumes "(##) F \<in> extensional (mor\<^bsub>CatDom F\<^esub>)" shows "FunctorExt F"
h4: shows "tr \<cdot> y \<le> y \<cdot> tq"
induct: fixes P :: "int \<Rightarrow> nat \<Rightarrow> term \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "nat"   and a2 :: "term" assumes "\<And>inc lev i. P inc lev (Bound i)"   and "\<And>inc lev u. P inc (lev + 1) u \<Longrightarrow> P inc lev (\<Lambda> u)"   and "\<And>inc lev t\<^sub>1 t\<^sub>2. \<lbrakk>P inc lev t\<^sub>1; P inc lev t\<^sub>2\<rbrakk> \<Longrightarrow> P inc lev (t\<^sub>1 $ t\<^sub>2)"   and "\<And>uu_ uv_ v. P uu_ uv_ (Const v)"   and "\<And>uu_ uv_ v. P uu_ uv_ (Free v)" shows "P a0 a1 a2"
assms_2: shows "finite {b. g b \<noteq> (0::'b)}"
insertE: fixes a :: "'b"   and b :: "'b"   and A :: "'b set" assumes "a \<in> b \<triangleright> A" obtains "a = b"   | "a \<in> A"
hoare_r_def: fixes p :: "'d upred"   and Q :: "'d hrel"   and r :: "'d upred" shows "\<lbrace>p\<rbrace> Q \<lbrace>r\<rbrace>\<^sub>u = ((\<lceil>p\<rceil>\<^sub>< \<Rightarrow> \<lceil>r\<rceil>\<^sub>>) \<sqsubseteq> Q)"
assms_6: shows "F = (\<lambda>x. \<forall>rel\<in>S. \<forall>(a, b, c)\<in>X rel. rel (a * x\<^sup>2 + b * x + c) 0)"
eqvt_2: fixes x1 :: "binding list"   and x2 :: "trm"   and x3 :: "ty"   and pi :: "(tyvrs \<times> tyvrs) list" assumes "x1 \<turnstile> x2 : x3" shows "pi \<bullet> x1 \<turnstile> pi \<bullet> x2 : pi \<bullet> x3"
indRelR_cond_reflection_iff_indRelL_cond_preservation: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "(\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_reflects_pred Rel Pred) = (\<exists>Rel. (\<forall>S. (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel) \<and> rel_preserves_pred Rel Pred)"
NAK: fixes A :: "'b set" assumes "\<not> zeroring R"   and "M fgover R"   and "ideal R A"   and "A \<subseteq> J_rad R"   and "A \<odot>\<^bsub>R\<^esub> M = carrier M" shows "carrier M = {\<zero>}"
simps_5: fixes e\<^sub>1 :: "expr"   and bop :: "bop"   and e\<^sub>2 :: "expr" shows "fv (e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2) = fv e\<^sub>1 \<union> fv e\<^sub>2"
trancl_induct3: fixes r :: "'d \<times> 'e \<times> 'f \<Rightarrow> 'd \<times> 'e \<times> 'f \<Rightarrow> bool"   and ax :: "'d"   and ay :: "'e"   and az :: "'f"   and bx :: "'d"   and "by" :: "'e"   and bz :: "'f"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ (ax, ay, az) (bx, by, bz)"   and "\<And>a aa b. r (ax, ay, az) (a, aa, b) \<Longrightarrow> P a aa b"   and "\<And>a aa b ab ac ba. \<lbrakk>r\<^sup>+\<^sup>+ (ax, ay, az) (a, aa, b); r (a, aa, b) (ab, ac, ba); P a aa b\<rbrakk> \<Longrightarrow> P ab ac ba" shows "P bx by bz"
linear_scale: fixes f :: "'a \<Rightarrow> 'b"   and r :: "real"   and b :: "'a" assumes "linear f" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
cast_below_imp_below: fixes A :: "'e defl"   and B :: "'e defl" assumes "cast\<cdot>A \<sqsubseteq> cast\<cdot>B" shows "A \<sqsubseteq> B"
univ_poly_is_domain: fixes K :: "'a list set" assumes "subring K (K [X])" shows "domain (K [X]\<^bsub>K [X]\<^esub>)"
distinct_35: fixes x3 :: "nat"   and x4 :: "key" shows "Pwd x3 \<noteq> Key x4"
norm1_nonzero: fixes v :: "('c, 'd) Finite_Cartesian_Product.vec" assumes "v \<noteq> 0" shows "0 < norm1 v"
runit_simps_4: fixes f :: "'a" assumes "C.ide f" shows "C.dom \<r>\<^sub>C[f] = f \<star>\<^sub>C src\<^sub>C f"
zero_not_one: shows "(0::'c) \<noteq> (1::'c)"
IH_2: fixes Ts :: "typ list"   and T :: "typ" assumes "typ_of1 Ts t2 = Some T" shows "typ_of1 (map (subst_typ insts) Ts) (subst_typ' insts t2) = Some (subst_typ insts T)"
indspace_def: fixes G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set" shows "indspace G FH V = addfunset FH V \<inter> smultfunset G FH"
rep_eq: fixes x :: "'c key_order" shows "lt_of_key_order x = lt_of_comp (key_compare x)"
assms_4: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "lt (seq i) \<prec>\<^sub>t lt (seq j)"
if_split: fixes P :: "'b \<Rightarrow> bool"   and Q :: "bool"   and x :: "'b"   and y :: "'b" shows "P (if Q then x else y) = ((Q \<longrightarrow> P x) \<and> (\<not> Q \<longrightarrow> P y))"
elims: fixes x :: "'a btnode"   and y :: "nat" assumes "btnode_encode x = y" obtains ts :: "('a btnode ref option \<times> 'a) array \<times> nat" and t :: "'a btnode ref option" where "x = Btnode ts t" and "y = to_nat (ts, t)"
interaction_bounded_byI: fixes "consider" :: "'h \<Rightarrow> bool"   and gpv :: "('i, 'h, 'j) gpv"   and n :: "enat" assumes "interaction_bound consider gpv \<le> n" shows "interaction_bounded_by consider gpv n"
ss_exps_Vals_NoneI: fixes es :: "'a exp list" assumes "ss_exps es = None" shows "\<exists>vs. es = map Val vs"
assoc'_eq_inv_assoc: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "\<a>\<^sub>C\<^sup>-\<^sup>1[f, g, h] = C.inv \<a>\<^sub>C[f, g, h]"
perm_vars_inj_on_nth: fixes n :: "nat"   and vs :: "nat list" assumes "perm_vars n vs" shows "inj_on ((!) vs) {0..<n}"
group_add_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "group_add_hom hom \<equiv> monoid_add_hom hom"
simps_4: fixes x2 :: "'d"   and x1 :: "'c" shows "Inr x2 \<noteq> Inl x1"
bind_return': fixes M :: "'c measure" shows "M \<bind> return M = M"
assms_5: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) B = Bv"
factorization_property: fixes a :: "'a" assumes "a \<in> carrier R - {\<zero>}"   and "a \<notin> Units R" shows "\<exists>fs. set fs \<subseteq> carrier (Ring_Divisibility.mult_of R) \<and> wfactors (Ring_Divisibility.mult_of R) fs a"
wf_matchF_aux_Cons1: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and t :: "nat"   and rels :: "event_data option list set list"   and rel :: "event_data option list set"   and aux :: "(nat \<times> event_data option list set list \<times> event_data option list set) list"   and ne :: "nat"   and i :: "nat" assumes "wf_matchF_aux \<sigma> V n R I r ((t, rels, rel) # aux) ne i" shows "t = \<tau> \<sigma> ne"
incidence_system_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" shows "incidence_system point_set block_collection \<equiv> \<forall>b. b \<in># block_collection \<longrightarrow> b \<subseteq> point_set"
bounded_norm_comp: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" shows "bounded ((\<lambda>x. norm (f x)) ` S) = bounded (f ` S)"
map_option_param: fixes R :: "('a \<times> 'd) set"   and S :: "('c \<times> 'e) set" shows "(map_option, map_option) \<in> (R \<rightarrow> S) \<rightarrow> \<langle>R\<rangle>option_rel \<rightarrow> \<langle>S\<rangle>option_rel"
dtor2_diff: fixes x :: "'d JF2"   and y :: "'d JF2" shows "(dtor2 x = dtor2 y) = (x = y)"
SubstTermP_graph_aux_def: shows "SubstTermP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SubstTermP_aux s v i t u k. x1 = (v, i, t, u) \<and> x2 = TermP i AND SyntaxN.Ex s (SyntaxN.Ex k (SeqStTermP v i t u (Var s) (Var k))) \<and> atom s \<sharp> (v, i, t, u, k) \<and> atom k \<sharp> (v, i, t, u))"
vcons_eqD_1: fixes xs\<^sub>1 :: "V"   and xs\<^sub>2 :: "V"   and x\<^sub>1 :: "V"   and x\<^sub>2 :: "V" assumes "vfsequence_pair xs\<^sub>1 xs\<^sub>2"   and "xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2" shows "xs\<^sub>1 = xs\<^sub>2"
WT_calleeI: fixes \<I> :: "('h, 'i) \<I>"   and callee :: "'h \<Rightarrow> ('i \<times> 'j) spmf" assumes "\<And>call ret s. \<lbrakk>call \<in> outs_\<I> \<I>; (ret, s) \<in> set_spmf (callee call)\<rbrakk> \<Longrightarrow> ret \<in> responses_\<I> \<I> call" shows "\<I> \<turnstile>c callee \<surd>"
ring_hom_nat_pow: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and a :: "'e"   and n :: "nat" assumes "ring R"   and "ring S"   and "h \<in> ring_hom R S"   and "a \<in> carrier R" shows "h (a [^]\<^bsub>R\<^esub> n) = h a [^]\<^bsub>S\<^esub> n"
ide_char: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.ide \<mu> = (arrow_of_spans (\<odot>) \<mu> \<and> Maps.ide (Chn \<mu>))"
hyps_4: fixes x :: "'a" assumes "x \<in> M" shows "\<exists>j. x = (f ^^ j) a"
lt: fixes p :: "tm"   and vs :: "'b list"   and bs :: "'b list" assumes "allpolys isnpoly p" shows "Ifm vs bs (lt p) = Ifm vs bs (Lt p)"
If_True_ir_valid: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c\<^sub>1 :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com" assumes "ir_valid (\<lambda>a c. P a c \<and> bval b a) c\<^sub>1 c' Q" shows "ir_valid P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) c' Q"
faxCD_def: fixes f :: "'fsym" shows "faxCD f \<equiv> map Neg (hOfFax f) @ fax f"
A: shows "A \<in> carrier_mat CARD('nr mod_ring) CARD('nc mod_ring)"
Exp_def: shows "Exp \<equiv> map_fun Re (map_fun Re Ab) cExp"
parabolic_isometryD_3: fixes f :: "'b \<Rightarrow> 'b" assumes "parabolic_isometry f" shows "stable_translation_length f = 0"
s_prod_idr: fixes x :: "'a" shows "x \<cdot> 1\<^sub>\<sigma> = x"
inf_top_left: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "inf top x = x"
chine_simps_3: shows "C.cod \<epsilon>.chine = \<epsilon>.cod.apex"
rem_comps_spec_count_rem_components: fixes bs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "'g" shows "punit.rem_comps_spec bs (punit.count_rem_components bs, data)"
TS_b1'': fixes x :: "nat"   and y :: "nat"   and x0 :: "nat"   and y0 :: "nat"   and s :: "(nat list \<times> nat list) pmf"   and qs :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "TS_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "TS_inv (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h0)) s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' (Partial_Cost_Model.embed (rTS h0)) s qs = real (length qs - 2)"
map_add_upd_left: fixes m :: "'d"   and e2 :: "'d \<Rightarrow> 'e option"   and e1 :: "'d \<Rightarrow> 'e option"   and u1 :: "'e" assumes "m \<notin> dom e2" shows "e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
LPDs_Times: fixes r :: "mregex"   and s :: "mregex" shows "LPDs (MTimes r s) \<subseteq> {MTimes r s} \<union> MTimesR (LPDs r) s \<union> LPDs s"
b: shows "abc_steps_l (0, lm) (A @ B) n = abc_steps_l (0, lm) A n"
abs_eq: fixes x :: "'a" shows "blinfun_mult_left x = Blinfun (\<lambda>y. y * x)"
product_integral_insert: fixes I :: "'i set"   and i :: "'i"   and f :: "('i \<Rightarrow> 'a) \<Rightarrow> 'b" assumes "finite I"   and "i \<notin> I"   and "integrable (Pi\<^sub>M (insert i I) M) f" shows "integral\<^sup>L (Pi\<^sub>M (insert i I) M) f = LINT x|Pi\<^sub>M I M. LINT y|M i. f (x(i := y))"
insert: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a" assumes "ListMem x xs" shows "ListMem x (y # xs)"
homeomorphic_empty_space_eq: fixes X :: "'a topology"   and Y :: "'b topology" assumes "topspace X = {}" shows "(X homeomorphic_space Y) = (topspace Y = {})"
intros_16: fixes tf :: "tf"   and tn :: "t list"   and tm :: "t list"   and \<C> :: "t_context"   and es :: "b_e list" assumes "tf = tn _> tm"   and "\<C>\<lparr>label := [tn] @ label \<C>\<rparr> \<turnstile> es : tn _> tm" shows "\<C> \<turnstile> [Loop tf es] : tn _> tm"
abs_convergent_prod_def: fixes f :: "nat \<Rightarrow> 'a" shows "abs_convergent_prod f = convergent_prod (\<lambda>i. 1 + norm (f i - (1::'a)))"
inf2D1: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" assumes "(A \<sqinter> B) x y" shows "A x y"
map_resource_parametric: fixes A' :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and B' :: "'h \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_fun (rel_fun A' A) (rel_fun (rel_fun B B') (rel_fun (rel_resource A B) (rel_resource A' B'))) map_resource map_resource"
sub_num_simps_3: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and l :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one num.One (num.Bit1 l) = uminus (numeral.numeral one plus (num.Bit0 l))"
tranclp_unfold: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "r\<^sup>+\<^sup>+ a b = ((a, b) \<in> {(x, y). r x y}\<^sup>+)"
H_fcs_eq: fixes a :: "'b" assumes "a \<in> arcs H" shows "proj_arcs_H ` H.face_cycle_set a = GM.face_cycle_set (proj_arcs_H a)"
1: shows "infinite (range ((*) 2)) \<and> infinite (range (\<lambda>k. Suc (2 * k)))"
intersect'_nodes_finite: fixes A :: "('e, 'd) nbta"   and B :: "('e, 'f) nbta" assumes "finite (NBTA.nodes A)"   and "finite (NBTA.nodes B)" shows "finite (NGBTA.nodes (intersect' A B))"
simps_2: fixes b :: "bit"   and x :: "bit"   and xs :: "bit list" shows "rem_initial b (x # xs) = (if b = x then rem_initial b xs else x # xs)"
superset_preserves_redundancy: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'b Literal set"   and S :: "'b Literal set set"   and S' :: "'b Literal set set" assumes "propositional_atoms atom_ordering"   and "redundant C S"   and "S \<subseteq> S'" shows "redundant C S'"
wf_Int_Times: fixes A :: "'b set"   and B :: "'b set" assumes "A \<inter> B = {}" shows "wf (A \<times> B)"
C2_b: fixes x :: "'a" shows "(0::'a) r\<rightarrow> x = (1::'a)"
s_TS_append: fixes i :: "nat"   and as :: "nat list"   and init :: "nat list"   and h :: "nat list"   and bs :: "nat list" assumes "i \<le> length as" shows "s_TS init h (as @ bs) i = s_TS init h as i"
size_11: fixes x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm" shows "size (PFn x41 x42 x43) = size x43 + Suc 0"
Block_closed: fixes a :: "'a set \<times> 'b" assumes "a \<in> Sigma I \<FF>" shows "rel.Class a \<in> rel.Partition"
assert_fun_le_id: fixes p :: "'b" shows "assert_fun p \<le> id"
continuous_map_of_fst: fixes X :: "'a topology"   and Y :: "'b topology"   and Z :: "'c topology"   and f :: "'a \<Rightarrow> 'c" shows "continuous_map (prod_topology X Y) Z (f \<circ> fst) = (topspace Y = {} \<or> continuous_map X Z f)"