hcju/isabelleps huggingface dataset:
simps: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and s :: "'e" shows "terminates_within g s = (case g s of Done \<Rightarrow> Some 0 | Skip s' \<Rightarrow> map_option (\<lambda>n. n + 1) (terminates_within g s') | Yield a s' \<Rightarrow> map_option (\<lambda>n. n + 1) (terminates_within g s'))"
mat_det_left_def: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "Determinant.det A = (\<Sum>p | p permutes {0..<dim_row A}. signof p * (\<Prod>i = 0..<dim_row A. A $$ (p i, i)))"
dom_eff_subset_imp_dom_succ_eq_proj: fixes h :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and s :: "('d, 'e) fmap"   and vs :: "'d set" assumes "fmdom' (snd h) \<subseteq> fmdom' s" shows "fmdom' (state_succ s (action_proj h vs)) = fmdom' (state_succ s h)"
assms_1: shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P"
bounded: shows "\<exists>K. \<forall>a b. norm (a *\<^sub>v b) \<le> norm a * norm b * K"
lformula_norm: fixes \<phi> :: "('a, 'k) aformula" assumes "lformula \<phi>" shows "lformula (norm \<phi>)"
prems: fixes i :: "nat" assumes "i < length (ts @ [t])" shows "0 < f i"
seq_app_first: fixes s :: "'a"   and \<sigma> :: "nat \<Rightarrow> 'a" shows "(s ## \<sigma>) 0 = s"
assms: shows "mopup_aft_erase_a (Suc (Suc (2 * n)), l, Oc # xs) lm n ires"
table_of_mapconst_SomeI: fixes t :: "('b \<times> 'a) list"   and k :: "'b"   and y' :: "'a"   and y :: "'c \<times> 'a"   and c :: "'c" assumes "table_of t k = Some y'"   and "snd y = y'"   and "fst y = c" shows "table_of (map (\<lambda>(k, x). (k, c, x)) t) k = Some y"
streamValue10: fixes d :: "nat"   and str :: "nat \<Rightarrow> 'a"   and t :: "nat"   and i :: "nat"   and k :: "nat"   and x :: "'a" assumes "\<forall>j\<le>2 * d. str (t + j + Suc (i + k)) = x" shows "\<forall>j\<le>d. str (t + i + k + d + Suc j) = x"
quot_dbtm_inject_lemma: fixes e :: "(name, hf) finfun"   and t :: "dbtm"   and u :: "dbtm" shows "(\<lbrakk>quot_dbtm t\<rbrakk>e = \<lbrakk>quot_dbtm u\<rbrakk>e) = (t = u)"
normalize_ports_generic_nnf: fixes m :: "'b common_primitive match_expr"   and m' :: "'b common_primitive match_expr"   and normalize_pos :: "'b common_primitive match_expr \<Rightarrow> 'b common_primitive match_expr list"   and rewrite_neg :: "'b common_primitive match_expr \<Rightarrow> 'b common_primitive match_expr"   and disc :: "'b common_primitive \<Rightarrow> bool" assumes "normalized_nnf_match m"   and "m' \<in> set (normalize_ports_generic normalize_pos rewrite_neg m)"   and "\<not> has_disc_negated disc False (rewrite_neg m)"   and "\<And>m m'. \<lbrakk>normalized_nnf_match m; \<not> has_disc_negated disc False m; m' \<in> set (normalize_pos m)\<rbrakk> \<Longrightarrow> normalized_nnf_match m'" shows "normalized_nnf_match m'"
E: shows "take (Suc n) ys = take n ys @ [ys ! n]"
top_join: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "weak_bounded_lattice L"   and "x \<in> carrier L" shows "\<top>\<^bsub>L\<^esub> \<squnion>\<^bsub>L\<^esub> x .=\<^bsub>L\<^esub> \<top>\<^bsub>L\<^esub>"
inner_eq_zero_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(inner x x = 0) = (x = zero)"
foldri_def: fixes l :: "'b list" shows "foldri l = foldli (rev l)"
final_Power_dfa: shows "dfa.final Power_dfa = {HF (epsclo Q) |Q. Q \<subseteq> nfa.states M \<and> Q \<inter> nfa.final M \<noteq> {}}"
part_Un_singl2: fixes K01 :: "'a set"   and I0 :: "'a set"   and K1 :: "'a set"   and P :: "'a set set" assumes "K01 = I0 \<union> K1"   and "part K1 P"   and "\<And>I. I \<in> P \<Longrightarrow> I0 \<inter> I = {}" shows "part K01 ({I0} \<union> P)"
finfun_ext: fixes f :: "('a, 'b) finfun"   and g :: "('a, 'b) finfun" assumes "\<And>a. finfun_apply f a = finfun_apply g a" shows "f = g"
introduce_parallel: fixes i :: "'a::refinement_lattice"   and j :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (i::'a::refinement_lattice)"   and "chaos \<sqsubseteq> (j::'a::refinement_lattice)" shows "(c::'a::refinement_lattice) \<iinter> (d::'a::refinement_lattice) \<sqsubseteq> (j::'a::refinement_lattice) \<iinter> c // (i::'a::refinement_lattice) \<parallel> i \<iinter> d // j"
is_spanning_tree_def: fixes G :: "'a ugraph"   and T :: "'a ugraph" shows "is_spanning_tree G T \<equiv> tree T \<and> nodes T = nodes G \<and> edges T \<subseteq> edges G"
complement_bibd_index: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2" shows "int (\<B>\<^sup>C index ps) = \<b> + \<Lambda> - 2 * (\<Lambda> * (des.\<v> - 1) div (\<k> - 1))"
f2: shows "4 \<le> rk {A, B, P, R, a}"
AbstTermP_graph_aux_def: shows "AbstTermP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>AbstTermP_aux s v i t u k. x1 = (v, i, t, u) \<and> x2 = OrdP i AND SyntaxN.Ex s (SyntaxN.Ex k (SeqStTermP v (Q_Ind i) t u (Var s) (Var k))) \<and> atom s \<sharp> (v, i, t, u, k) \<and> atom k \<sharp> (v, i, t, u))"
Abs': fixes T :: "POPLmarkRecord.type"   and t :: "trm" assumes "x = (\<lambda>:T. t)" shows "thesis"
ineqs_2: shows "C\<^sub>1 / (1 - a - c) \<le> C\<^sub>3"
gt_halfspace_def: fixes sctn :: "'b sctn"   and x :: "'b" shows "gt_halfspace sctn x = (pstn sctn < x \<bullet> normal sctn)"
feana: shows "OF_priority_match_ana \<gamma> ft p = Action fe"
LPDs_Test_3: fixes \<phi> :: "nat" shows "LPDs (MTestNeg \<phi>) \<subseteq> {MTestNeg \<phi>}"
power_increasing_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and b :: "'c"   and x :: "nat"   and y :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one b" shows "less_eq (power.power one times b x) (power.power one times b y) = (x \<le> y)"
len_cut: fixes P :: "'v"   and n :: "nat" assumes "len P \<le> n" shows "cut n P = P"
uminus_epi_iff: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(f \<circ> \<partial> = g \<circ> \<partial>) = (f = g)"
Await_sound: fixes pre :: "'b set"   and rely :: "('b \<times> 'b) set"   and post :: "'b set"   and P :: "'b com"   and b :: "'b set"   and guar :: "('b \<times> 'b) set" assumes "stable pre rely"   and "stable post rely"   and "\<forall>V. \<turnstile> P sat [pre \<inter> b \<inter> {s. s = V}, {(s, t). s = t}, UNIV, {s. (V, s) \<in> guar} \<inter> post] \<and> \<Turnstile> P sat [pre \<inter> b \<inter> {s. s = V}, {(s, t). s = t}, UNIV, {s. (V, s) \<in> guar} \<inter> post]" shows "\<Turnstile> Await b P sat [pre, rely, guar, post]"
simps_22: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "char list \<Rightarrow> 'a" shows "(case Void of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f1"
take_0: fixes xs :: "'b list" shows "take 0 xs = []"
sup_inf_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (x \<squnion> y) = x"
rwof_step': fixes step :: "'a \<Rightarrow> 'a nres"   and s :: "'a"   and m0 :: "'a nres"   and cond :: "'a \<Rightarrow> bool" assumes "nofail (step s)"   and "rwof m0 cond step s"   and "cond s" shows "step s \<le> SPEC (rwof m0 cond step)"
SUP_bot_conv_1: 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 B :: "'d \<Rightarrow> 'b"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup (B ` A) = bot) = (\<forall>x\<in>A. B x = bot)"
while_def: shows "local.while \<equiv> spmf.fixp_fun (\<lambda>while s. if s \<notin> B then spmf_of_set A \<bind> while else return_spmf s)"
elliptic_isometry_inv_iff: fixes f :: "'b \<Rightarrow> 'b" assumes "bij f" shows "elliptic_isometry (inv f) = elliptic_isometry f"
simps_2: fixes r :: "rat" shows "invariant_2 (Rational r) = True"
assms_3: fixes i :: "'c" assumes "i \<in> B - A" shows "h i = \<one>"
m_comm: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)"   and "y \<in> carrier (DirProds G I)" shows "x \<otimes>\<^bsub>DirProds G I\<^esub> y = y \<otimes>\<^bsub>DirProds G I\<^esub> x"
nontrans2: fixes u :: "'b"   and r :: "'b" assumes "u \<parallel> r" shows "\<nexists>t. r \<parallel> t \<and> t \<parallel> u"
sublinear_wp_repeat: fixes p :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "sublinear (wp p)"   and "healthy (wp p)" shows "sublinear (wp (repeat n p))"
rel_distinct_2: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and y21 :: "'g"   and y22 :: "'g list" shows "\<not> list_all2 R (y21 # y22) []"
poly_no_roots_less_less: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "(\<forall>x. a < x \<and> x < b \<longrightarrow> poly p x \<noteq> 0) = (b \<le> a \<or> p \<noteq> 0 \<and> count_roots_between p a b = (if poly p b = 0 then 1 else 0))"
insert_commute: fixes x :: "'b::type"   and y :: "'b::type"   and A :: "'b::type set" shows "insert (x::'b::type) (insert (y::'b::type) (A::'b::type set)) = insert y (insert x A)"
inv_mult_group: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus> (x \<oplus> y) = \<ominus> y \<oplus> \<ominus> x"
image_union_split: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and B :: "'e set"   and g :: "'f \<Rightarrow> 'd"   and C :: "'f set" assumes "f ` (A \<union> B) = g ` C" obtains D :: "'f set" and E :: "'f set" where "f ` A = g ` D" and "f ` B = g ` E" and "D \<subseteq> C" and "E \<subseteq> C"
HInfinite_HFinite_not_Infinitesimal_mult2: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> HInfinite"   and "y \<in> HFinite - Infinitesimal" shows "y * x \<in> HInfinite"
inject_2: fixes x1 :: "name"   and x2 :: "('a, 'b, 'c) input"   and y1 :: "name"   and y2 :: "('a, 'b, 'c) input" shows "(\<nu>x1x2 = \<nu>y1y2) = ([x1].x2 = [y1].y2)"
bsums: fixes x :: "'a" assumes "norm (x - \<xi>) < r"   and "x \<noteq> \<xi>" shows "(\<lambda>n. b n * (x - \<xi>) ^ n) sums (f x / (a m * (x - \<xi>) ^ m))"
dvd_diffD: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k dvd m - n"   and "k dvd n"   and "n \<le> m" shows "k dvd m"
seq: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and t :: "state"   and rs\<^sub>2 :: "'c rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'"   and "no_matching_Goto \<gamma> p rs\<^sub>1" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1 @ rs\<^sub>2, Undecided\<rangle> \<Rightarrow> t'"
m_ac_2: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y = y \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x"
arc_implies_awalk: fixes e :: "'b" assumes "e \<in> arcs G" shows "awalk (tail G e) [e] (head G e)"
psubset_fset: fixes xs :: "'b fset"   and ys :: "'b fset" shows "(xs |\<subset>| ys) = (fset xs \<subset> fset ys)"
lift_def: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> (nat \<times> int) list \<Rightarrow> ((nat \<times> int) list \<Rightarrow> real) \<Rightarrow> (nat \<times> int) list \<Rightarrow> real"   and dm :: "nat"   and lm :: "nat"   and d :: "nat" shows "lift f dm lm d = foldr (\<lambda>p. f d (lm - level p) p) (gridgen (start dm) ({0..<dm} - {d}) lm)"
drop_update_swap: fixes m :: "nat"   and n :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "m \<le> n" shows "drop m (xs[n := x]) = (drop m xs)[n - m := x]"
int_pow_pow: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and n :: "int"   and m :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "(x [^]\<^bsub>G\<^esub> n) [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> (n * m)"
midpoint_between: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "(b = midpoint a c) = (between (a, c) b \<and> dist a b = dist b c)"
assms_5: shows "c \<in> hyp2 \<union> S"
Knaster_Tarski: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" assumes "weak_complete_lattice L"   and "f \<in> carrier L \<rightarrow> carrier L"   and "Mono\<^bsub>L\<^esub> f" shows "weak_complete_lattice (fpl L f)"
assms_2: fixes x :: "real"   and y :: "real" shows "f (x + y + 1) = f x + f y"
spec\<^sub>0: shows "vars spec \<subseteq> \<V>\<^sub>0"
simps_1: fixes l2 :: "'a list" shows "merge [] l2 = l2"
ex: shows "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> P x = P1"
set_less_aux_finiteD: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" assumes "ord.set_less_aux less_eq A B" shows "finite A \<and> B \<in> infinite_complement_partition"
mult_1s_1: fixes a :: "'c" shows "Numeral1 * a = a"
lexord_take_index_conv: fixes x :: "'g list"   and y :: "'g list"   and r :: "('g \<times> 'g) set" shows "((x, y) \<in> lexord r) = (length x < length y \<and> take (length x) y = x \<or> (\<exists>i<min (length x) (length y). take i x = take i y \<and> (x ! i, y ! i) \<in> r))"
bv_extend_longer: fixes n :: "nat"   and w :: "bit list"   and b :: "bit" assumes "n \<le> length w" shows "bv_extend n b w = w"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "hf_of (x # xs) = \<langle>hf_of x, hf_of xs\<rangle>"
moebius_inv: fixes M :: "moebius" shows "moebius_pt (moebius_inv M) = inv (moebius_pt M)"
u_dvdm_res: shows "u dvdm [:resultant f g:]"
get_curr_win_privilege: fixes s' :: "'b sparc_state"   and s :: "'b sparc_state" assumes "s' = snd (fst (get_curr_win () s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
lm13: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "inj_on f A"   and "inj_on f B" shows "inj_on f (A \<union> B) = (f ` (A - B) \<inter> f ` (B - A) = {})"
SFAssNonStatic: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "P \<turnstile> C has F,NonStatic:t in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW IncompatibleClassChangeError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"
dropWhile_append3: fixes P :: "'a \<Rightarrow> bool"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "\<not> P y" shows "dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
pred_sexp_def: shows "pred_sexp = (\<Union>M\<in>sexp. \<Union>N\<in>sexp. {(M, Scons M N), (N, Scons M N)})"
Newman: fixes r :: "('a \<times> 'a) set" assumes "SN r"   and "WCR r" shows "CR r"
simps_3: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat" shows "index (x # xs) (Suc n) = index xs n"
assms_2: fixes S :: "'a set" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> P (S - {x})" shows "P S"
periodic_continuous_onI: fixes f :: "real \<Rightarrow> real"   and p :: "real"   and a :: "real" assumes "\<And>x. f (x + p) = f x"   and "0 < p"   and "continuous_on {a..a + p} f" shows "continuous_on UNIV f"
sgnx_eq: shows "sgnx g (at_left x) = sgnx g (at_right x)"
sinh_complex: fixes z :: "complex" shows "(exp z - inverse (exp z)) / 2 = - \<i> * sin (\<i> * z)"
sets_interval_measure: fixes F :: "real \<Rightarrow> real" shows "sets (interval_measure F) = sets borel"
Fvars_trm_simps_2: fixes a :: "name" shows "Fvars (Var a) = {a}"
reachable_closed_reachable_pnet: fixes s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and n :: "net_tree" assumes "s \<in> reachable (closed (pnet np n)) TT" shows "s \<in> reachable (pnet np n) TT"
boundedByAdd1: fixes N :: "nat"   and B :: "(nat \<times> 'b) set"   and M :: "nat" assumes "boundedBy N B" shows "boundedBy (N + M) B"
ce_rel_lm_32: fixes r :: "(nat \<times> nat) set"   and s :: "(nat \<times> nat) set" shows "ce_rel_to_set (r \<union> s) = ce_rel_to_set r \<union> ce_rel_to_set s"
schroeder_6_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<le> - z) = (z[[x]] \<le> - y\<^sup>T)"
subgraph_trans: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph"   and I :: "('c, 'd) pre_digraph" assumes "subgraph G H"   and "subgraph H I" shows "subgraph G I"
gtchr: shows "(\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 1}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) = rat_of_int (int (card {x. poly p x = 0 \<and> (\<forall>p\<in>set (retrieve_polys qs (fst I)). poly p x = 0) \<and> 0 < poly (prod_list (retrieve_polys qs (snd I))) x}))"
separate: fixes a :: "(real mpoly \<times> nat) list"   and b :: "(real mpoly \<times> nat) list"   and v :: "nat"   and p :: "real mpoly"   and f :: "nat \<Rightarrow> real" assumes "(a, b) = chophelper (to_list v p) []" shows "insertion f p = (\<Sum>(term, i)\<leftarrow>a. insertion f term * f v ^ i) + (\<Sum>(term, i)\<leftarrow>b. insertion f term * f v ^ i)"
adjoin_id: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := adjoin G H a\<rparr>)"   and "a \<in> H" shows "adjoin (G\<lparr>carrier := adjoin G H a\<rparr>) H a = H"
a_group_hom: shows "group_hom (add_monoid R) (add_monoid S) h"
I_minWeightBasis_sorted: fixes T :: "'a set"   and E :: "'a set" assumes "I_minWeightBasis (T, E)" shows "\<forall>x\<in>T. \<forall>y\<in>E. weight x \<le> weight y"
obj_simps_1: fixes a :: "'b" assumes "D.obj a" shows "D.arr a"
indexed_var_def: fixes i :: "'c" shows "M.indexed_var i = M.indexed_pmult (M.indexed_const \<one>\<^bsub>M\<^esub>) i"
br'_rcm_aux: fixes Q :: "'d set"   and W :: "'d set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<delta> :: "('d, 'e) ta_rule set"   and q :: "'d" assumes "(Q, W, rcm) \<in> br'_invar \<delta>"   and "q \<in> W" shows "{lhs r |r. r \<in> \<delta> \<and> q \<in> set (rhsq r) \<and> the (rcm r) \<le> Suc 0} = {lhs r |r. r \<in> \<delta> \<and> q \<in> set (rhsq r) \<and> set (rhsq r) \<subseteq> Q - (W - {q})}"
mat0_mult_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and nc :: "nat"   and ncc :: "nat"   and m :: "'c list list"   and nr :: "nat" assumes "class.semiring_0 plus zero times"   and "mat nc ncc m" shows "mat_multI zero plus times nr (mat0I zero nr nc) m = mat0I zero nr ncc"
f12: shows "(f\<^sub>1, f\<^sub>2) = split_face f u v vs"
fun_upd_finfun: fixes y :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'b" shows "(y(a := b) \<in> finfun) = (y \<in> finfun)"
Minimal_Truth_Soundness_HBL1iff_Compl_Pf_axioms: shows "Minimal_Truth_Soundness_HBL1iff_Compl_Pf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv N isTrue Pf"
blocksEval: fixes ps :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap" assumes "length ps = length Ts"   and "length ps = length vs"   and "P \<turnstile> \<langle>blocks (ps, Ts, vs, e),(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',(h', l', sh')\<rangle>" shows "\<exists>l''. P \<turnstile> \<langle>e,(h, l(ps [\<mapsto>] vs), sh)\<rangle> \<Rightarrow> \<langle>e',(h', l'', sh')\<rangle>"
ccNeighbors_join: fixes x :: "Vars.var"   and G :: "CoCalls"   and G' :: "CoCalls" shows "ccNeighbors x (G \<squnion> G') = ccNeighbors x G \<union> ccNeighbors x G'"
PChain: shows "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"
ordLeq_transitive: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set"   and r'' :: "('d \<times> 'd) set" assumes "r \<le>o r'"   and "r' \<le>o r''" shows "r \<le>o r''"
assms: shows "z *\<^sub>h w = \<infinity>\<^sub>h"
\<beta>_zero: fixes i :: "nat" assumes "i < length fs" shows "\<beta> fs i \<noteq> (0::'a)"
less_fin_string_def: shows "(<) \<equiv> map_fun Rep_fin_string (map_fun Rep_fin_string id) less_charlist"
rSim: fixes R :: "pi"   and S :: "pi" assumes "(R, S) \<in> X" shows "R \<leadsto><(weakBisim O (X \<union> weakBisim) O bisim)> S"
P1_real_ddh0_\<sigma>_false: fixes \<sigma> :: "bool"   and M :: "'a \<times> 'a"   and z :: "'f"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('g \<Rightarrow> 'e spmf)"   and D :: "'e \<times> 'a \<Rightarrow> bool spmf" assumes "\<sigma> = False" shows "P1_real_model M \<sigma> z \<A> \<bind> D = local.ddh.DDH0 (P1_DDH_mal_adv_\<sigma>_false M z \<A> D)"
wfp_onE_min: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and x :: "'d"   and Q :: "'d set" assumes "wfp_on r A"   and "x \<in> Q"   and "Q \<subseteq> A" obtains z :: "'d" where "z \<in> Q" and "\<And>y. r y z \<Longrightarrow> y \<notin> Q"
weakSimAux_def: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" shows "weakSimAux P Rel Q \<equiv> (\<forall>Q' a x. Q \<longmapsto> a<\<nu>x> \<prec> Q' \<and> x \<sharp> P \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel)) \<and> (\<forall>Q' a x. Q \<longmapsto> a<x> \<prec> Q' \<and> x \<sharp> P \<longrightarrow> (\<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel)) \<and> (\<forall>Q' \<alpha>. Q \<longmapsto> \<alpha> \<prec> Q' \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel))"
FSwImorph_igWlsBinp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'f option"   and h :: "'e \<Rightarrow> 'h"   and hA :: "'f \<Rightarrow> 'i"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'h, 'i, 'j) model_scheme" assumes "igWlsBinp MOD delta binp"   and "FSwImorph h hA MOD MOD'" shows "igWlsBinp MOD' delta (lift hA binp)"
shows_prec_rat_def: fixes d :: "nat"   and x :: "rat" shows "shows_prec d x = (case quotient_of x of (den, num) \<Rightarrow> if num = 1 then shows den else shows den \<circ> shows CHR ''/'' \<circ> shows num)"