hcju/isabelleps huggingface dataset:
prod_topology_homeomorphic_space_right: fixes X :: "'d topology"   and a :: "'d"   and Y :: "'e topology" assumes "topspace X = {a}" shows "prod_topology X Y homeomorphic_space Y"
Scan_elem_in_Gen: fixes I :: "item set"   and k :: "nat"   and u :: "nat"   and T :: "(symbol \<times> character list) set"   and x :: "item" assumes "I \<subseteq> Gen (\<P> k u)"   and "k \<le> length Doc"   and "T \<subseteq> \<Z> k u"   and "x \<in> Scan T k I" shows "x \<in> Gen (\<P> k u)"
dist_fpxs_def: fixes f :: "'b fpxs"   and g :: "'b fpxs" shows "dist f g = (if f = g then 0 else 2 powr - real_of_rat (fpxs_val (f - g)))"
nstutinv_def: fixes P :: "(nat \<Rightarrow> 'c) \<Rightarrow> 'd" shows "NSTUTINV P \<equiv> \<forall>\<sigma> \<tau>. first \<sigma> = first \<tau> \<and> tail \<sigma> \<approx> tail \<tau> \<longrightarrow> P \<sigma> = P \<tau>"
priority_Min_priorities: fixes q :: "('c, 'd) pq" assumes "\<not> is_empty q" shows "priority q (PQ.min q) = Some (Min (set \<parallel>q\<parallel>))"
l2_partial_secrecy_def: shows "l2_partial_secrecy \<equiv> {s. synth (analz (ik s)) \<inter> secret s = {}}"
is_functor: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" assumes "identity_functor C" shows "functor C C (identity_functor.map C)"
A_B_C_D_2: fixes n :: "nat" shows "B n = (\<Prod>(a, i)\<in>as - UNIV \<times> {0..<n}. a)"
contour_integral_on_mirror_iff: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" assumes "valid_path g" shows "(f contour_integrable_on - g) = ((\<lambda>x. - f (- x)) contour_integrable_on g)"
n_mult_lb1: fixes x :: "'a"   and y :: "'a" shows "!x \<cdot> !y \<sqsubseteq> !x"
bounded_cbilinear_axioms: shows "bounded_cbilinear (*\<^sub>V)"
pathfinish_def: fixes g :: "real \<Rightarrow> 'b" shows "pathfinish g = g 1"
least_exists: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P n" shows "\<exists>n. is_least P n"
state_list_length_non_zero: fixes s :: "('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" shows "0 \<noteq> length (state_list s as)"
f5: shows "t \<otimes> (snd x' \<otimes> r \<ominus> s \<otimes> fst x') = \<zero>"
ValIsNoLoc_def: fixes v :: "Val" shows "ValIsNoLoc v = (v = RVal Nullref \<or> (\<exists>i. v = IVal i))"
hcis_mult: fixes a :: "real star"   and b :: "real star" shows "hcis a * hcis b = hcis (a + b)"
map_eq_appendE: fixes f :: "'c \<Rightarrow> 'b"   and ls :: "'c list"   and fl :: "'b list"   and fl' :: "'b list" assumes "map f ls = fl @ fl'" obtains l :: "'c list" and l' :: "'c list" where "ls = l @ l'" and "map f l = fl" and "map f l' = fl'"
wf_below_trancl: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "wf_below (r\<^sup>+) A = wf_below r A"
cases: fixes a1 :: "lam"   and a2 :: "lam" assumes " a1 \<longrightarrow>\<^sub>1 a2" obtains M :: "lam" where "a1 = M" and "a2 = M"   | t1 :: "lam" and t2 :: "lam" and s1 :: "lam" and s2 :: "lam" where "a1 = App t1a__ s1__" and "a2 = App t2a__ s2__" and " t1a__ \<longrightarrow>\<^sub>1 t2a__" and " s1__ \<longrightarrow>\<^sub>1 s2__"   | s1 :: "lam" and s2 :: "lam" and a :: "name" where "a1 = Lam [a__].s1a__" and "a2 = Lam [a__].s2a__" and " s1a__ \<longrightarrow>\<^sub>1 s2a__"   | a :: "name" and s1 :: "lam" and s2 :: "lam" and t1 :: "lam" and t2 :: "lam" where "a1 = App (Lam [a].t1) s1" and "a2 = t2[a::=s2]" and "a \<sharp> (s1, s2)" and " s1 \<longrightarrow>\<^sub>1 s2" and " t1 \<longrightarrow>\<^sub>1 t2"
simps_4: fixes x :: "'b trm"   and y :: "'b trm"   and "next" :: "indices list"   and S :: "'b trm" shows "subterm (x \<cdot> y) (indices.Left # next) S = subterm x next S"
neg_imp: fixes Q :: "bool"   and P :: "bool" assumes "\<not> Q \<longrightarrow> \<not> P" shows "P \<longrightarrow> Q"
conseq2: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and t :: "Term.term"   and Q' :: "'b assn"   and Q :: "'b assn" assumes "G,A\<turnstile>{P} t\<succ> {Q'}"   and "Q' \<Rightarrow> Q" shows "G,A\<turnstile>{P} t\<succ> {Q}"
IH: shows "multiple_blocks n rep x = \<B> rep x * int n"
safe_reac_direct_def': fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'a \<Rightarrow> 'a \<times> instr list \<Rightarrow> bool"   and safe :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option) \<Rightarrow> bool"   and cfg :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)" shows "ini.safe_reach_direct program_step safe cfg \<equiv> \<forall>cfg'. ini.direct_concurrent_steps program_step cfg cfg' \<longrightarrow> safe cfg'"
weak_conv_imp_bdd_ae_continuous_conv: fixes f :: "real \<Rightarrow> 'b"   and B :: "real" assumes "emeasure M {x. \<not> isCont f x} = 0"   and "\<And>x. norm (f x) \<le> B"   and "f \<in> borel_measurable borel" shows "(\<lambda>n. integral\<^sup>L (\<mu> n) f) \<longlonglongrightarrow> integral\<^sup>L M f"
induct: fixes P :: "dB \<Rightarrow> bool"   and a0 :: "dB" assumes "\<And>n. P (Var n)"   and "\<And>n t. P t \<Longrightarrow> P (Var n \<degree> t)"   and "\<And>s1 s2 t. \<lbrakk>P (s1 \<degree> s2); P t\<rbrakk> \<Longrightarrow> P (s1 \<degree> s2 \<degree> t)"   and "\<And>u t. \<lbrakk>P u; P t\<rbrakk> \<Longrightarrow> P (Abs u \<degree> t)"   and "\<And>s. P s \<Longrightarrow> P (Abs s)" shows "P a0"
eq: shows "(\<exists>b\<in>set (filter (\<lambda>b. fst b \<noteq> 0) bs). punit.const_lt_component (fst b) = Some ()) = (\<exists>b\<in>set bs. fst b \<noteq> 0 \<and> punit.const_lt_component (fst b) = Some ())"
block_sat_fresh: fixes M :: "('d, 'e) model"   and g :: "'f \<Rightarrow> 'd"   and block :: "('e, 'f) fm list \<times> 'f"   and i :: "'f"   and v :: "'d" assumes "M, g \<Turnstile>\<^sub>B block"   and "i \<notin> block_nominals block" shows "M, g(i := v) \<Turnstile>\<^sub>B block"
nDAcharn: fixes p :: "('c, 'd) Combinators list" shows "noDenyAll p = (\<forall>r\<in>set p. \<not> member DenyAll r)"
path_join_path_ends: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "path (g1 +++ g2)"   and "path g2" shows "pathfinish g1 = pathstart g2"
normalize_right_idem: fixes a :: "'a"   and b :: "'a" shows "gcd a (normalize b) = gcd a b"
intros_4: fixes a :: "'b"   and xs :: "'b list" shows "a # a # xs \<approx>2 a # xs"
simps_2: fixes i :: "nat"   and xss :: "'a list list" shows "radix_sort (Suc i) xss = radix_sort i (sort_col i xss)"
SUP_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'c \<times> 'd"   and D :: "'b \<Rightarrow> 'c \<times> 'd" assumes "A = B"   and "\<And>x. s2p B x \<Longrightarrow> C x = D x" shows "Sup_prod (\<P> C A) = Sup_prod (\<P> D B)"
inf_absorb2: fixes y :: "'b \<times> 'c"   and x :: "'b \<times> 'c" assumes "less_eq_prod y x" shows "inf_prod x y = y"
comp_arr_dom: fixes f :: "'b"   and a :: "'b" assumes "J.arr f"   and "J.dom f = a" shows "J.comp f a = f"
connected_connected_component: fixes S :: "'a set"   and x :: "'a" shows "connected (connected_component_set S x)"
asIMOD_igSwapAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c) semDom"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and A :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'c) \<Rightarrow> ('varSort, 'c) sAbs" shows "igSwapAbs (asIMOD SEM) zs z1 z2 A = (\<lambda>val. if sWlsVal SEM val then A (val ^[z1 \<and> z2]_zs) else undefined)"
sorted_Max: fixes l :: "'a list" assumes "l \<noteq> []"   and "sorted l" shows "Max (set l) = hd (rev l)"
mult_ac_1: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
quot_Eats: fixes x :: "tm"   and y :: "tm" shows "\<guillemotleft>Eats x y\<guillemotright> = HPair (HTuple 1) (HPair \<guillemotleft>x\<guillemotright> \<guillemotleft>y\<guillemotright>)"
inverse_eq_imp_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "inverse a = inverse b" shows "a = b"
LI_preserves_subst_wf_3: fixes S\<^sub>1 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>1 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S\<^sub>2 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>2 :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(S\<^sub>1, \<theta>\<^sub>1) \<leadsto>\<^sup>* (S\<^sub>2, \<theta>\<^sub>2)"   and "fv\<^sub>s\<^sub>t S\<^sub>1 \<inter> bvars\<^sub>s\<^sub>t S\<^sub>1 = {}"   and "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>\<^sub>1"   and "subst_domain \<theta>\<^sub>1 \<inter> vars\<^sub>s\<^sub>t S\<^sub>1 = {}"   and "range_vars \<theta>\<^sub>1 \<inter> bvars\<^sub>s\<^sub>t S\<^sub>1 = {}" shows "subst_domain \<theta>\<^sub>2 \<inter> vars\<^sub>s\<^sub>t S\<^sub>2 = {}"
IH: fixes x2a :: "('a, 'b) Term.term" assumes "x2a \<in> set X"   and "x2a \<cdot> \<theta> = x2a" shows "fv x2a \<inter> subst_domain \<theta> = {}"
critical_set_def: fixes A :: "ordinal set" shows "critical_set A = ordinal_rec0 A (\<lambda>p x. x \<inter> range (oDeriv (OrdinalVeblen.ordering x))) (\<lambda>f. \<Inter> (range f))"
broadcast_before_delivery: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and m :: "'c"   and i :: "nat" assumes "network history msg_id"   and "Deliver m \<in> set (history i)" shows "\<exists>j. node_histories.history_order history (Broadcast m) j (Deliver m)"
iteratei_rule_P: fixes m :: "'s"   and I :: "'u set \<Rightarrow> '\<sigma> \<Rightarrow> bool"   and \<sigma>0 :: "'\<sigma>"   and c :: "'\<sigma> \<Rightarrow> bool"   and f :: "'u \<times> 'v \<Rightarrow> '\<sigma> \<Rightarrow> '\<sigma>"   and P :: "'\<sigma> \<Rightarrow> bool" assumes "invar m"   and "I (dom (\<alpha> m)) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; \<alpha> m k = Some v; it \<subseteq> dom (\<alpha> m); I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom (\<alpha> m); it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iteratei m c f \<sigma>0)"
ennreal_spmf_bind: fixes p :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf"   and x :: "'c" shows "ennreal (spmf (p \<bind> f) x) = \<integral>\<^sup>+ xa. ennreal (spmf (f xa) x) \<partial>measure_spmf p"
bottomD_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set"   and x :: "'b" assumes "ordering less_eq less"   and "ordering.has_bottom less_eq P"   and "x \<in> P" shows "less_eq (ordering.bottom less_eq P) x"
ortho_span_span: fixes W :: "'b vec set"   and n :: "nat"   and X :: "'b vec set"   and w :: "'b vec"   and x :: "'b vec" assumes "W \<subseteq> carrier_vec n"   and "X \<subseteq> carrier_vec n"   and "\<And>w x. \<lbrakk>w \<in> W; x \<in> X\<rbrakk> \<Longrightarrow> w \<bullet> x = (0::'b)"   and "w \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) W"   and "x \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) X" shows "w \<bullet> x = (0::'b)"
ipc_precondition_def: fixes tid :: "thread_id_t"   and dir :: "ipc_direction_t"   and partner :: "thread_id_t"   and page :: "page_t"   and s :: "state_t" shows "ipc_precondition tid dir partner page s \<equiv> let sender = case dir of SEND \<Rightarrow> tid | RECV \<Rightarrow> partner; receiver = case dir of SEND \<Rightarrow> partner | RECV \<Rightarrow> tid; local_access_mode = case dir of SEND \<Rightarrow> READ | RECV \<Rightarrow> WRITE in sp_impl_subj_subj s (Step.partition sender) (Step.partition receiver) \<and> sp_impl_subj_obj s (Step.partition tid) (PAGE page) local_access_mode"
l_1: shows "(w \<circ> a) \<longlonglongrightarrow> limsup w"
filterlim_norm_at_top_imp_at_infinity: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "LIM x F. norm (f x) :> at_top" shows "filterlim f at_infinity F"
rev_hull0: fixes x :: "'b list"   and G :: "'b list set" assumes "x \<in> rev ` \<langle>G\<rangle>" shows "x \<in> \<langle>rev ` G\<rangle>"
is_rcoset_replistD_distinct: fixes G :: "'a set"   and H :: "'a set"   and gs :: "'a list" assumes "Group G"   and "Group.is_rcoset_replist G H gs" shows "distinct (map (\<lambda>g. Group.rcoset_rel G H `` {g}) gs)"
nonneg: fixes ja__ :: "'n"   and j :: "'n" shows "0 \<le> A $h i $h ja__ * x $h j"
R_1: shows "new_last \<leadsto>R\<leadsto>\<^bsub>H\<^esub> v1"
sel_7: fixes x31 :: "'a"   and x32 :: "'c"   and x33 :: "'b" shows "new_id (Extension x31 x32 x33) = x31"
col_cong2_bet1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and D :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B D"   and "Bet A C B"   and "Cong A B C D"   and "Cong A C B D" shows "Bet C B D"
im_sheaf_morphisms_def: fixes U :: "'a set"   and V :: "'a set" shows "id.im_sheaf_morphisms U V \<equiv> \<rho> (identity S \<^sup>\<inverse> S U) (identity S \<^sup>\<inverse> S V)"
padic_a_inv_def: fixes p :: "int"   and f :: "nat \<Rightarrow> int" shows "padic_a_inv p f \<equiv> \<lambda>n. \<ominus>\<^bsub>residue_ring (p ^ n)\<^esub> f n"
sub: fixes x :: "'a"   and y :: "'a" assumes "x \<in> space M"   and "y \<in> space M"   and "g x = g y" shows "f x = f y"
f_def: shows "f \<equiv> snd (normalize_quot (a, d))"
standard_simplex_mono: fixes p :: "nat"   and q :: "nat" assumes "p \<le> q" shows "standard_simplex p \<subseteq> standard_simplex q"
Not_eq_iff: fixes P :: "bool"   and Q :: "bool" shows "((\<not> P) = (\<not> Q)) = (P = Q)"
less_trans: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.preorder less_eq less"   and "less x y"   and "less y z" shows "less x z"
tranclp_trancl_eq: fixes r :: "('b \<times> 'b) set" shows "(\<lambda>x xa. (x, xa) \<in> r)\<^sup>+\<^sup>+ = (\<lambda>x xa. (x, xa) \<in> r\<^sup>+)"
mod_star_trueE': fixes P :: "assn"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> P * true" obtains h' :: "heap \<times> nat set" where "fst h' = fst h" and "snd h' \<subseteq> snd h" and "h' \<Turnstile> P"
foo: fixes a :: "'a"   and b :: "'a" shows "let (p, q, u, v, gcd_a_b) = bezout a b in p * a + q * b = gcd_a_b \<and> gcd_a_b dvd a \<and> gcd_a_b dvd b \<and> (\<forall>d'. d' dvd a \<and> d' dvd b \<longrightarrow> d' dvd gcd_a_b) \<and> gcd_a_b * u = - b \<and> gcd_a_b * v = a"
r: fixes x :: "'a" assumes "x \<in> s" shows "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (top_of_set s) u \<and> compact v"
\<mu>'_via_\<sigma>: fixes j :: "nat"   and i :: "nat" assumes "j \<le> i"   and "i < length (map of_int_hom.vec_hom fs)" shows "gs.\<mu>' i j = (if j = 0 then map of_int_hom.vec_hom fs ! i \<bullet> map of_int_hom.vec_hom fs ! j else gs.\<mu>' (j - 1) (j - 1) * (map of_int_hom.vec_hom fs ! i \<bullet> map of_int_hom.vec_hom fs ! j) - gs.\<sigma> j i j)"
coeffs_Poly: fixes as :: "'b list" shows "coeffs (Poly as) = strip_while ((=) (0::'b)) as"
\<delta>: shows "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"
subst_fm_commute: fixes j :: "name"   and A :: "fm"   and i :: "name"   and t :: "tm"   and u :: "tm" assumes "atom j \<sharp> A" shows "(A(i::=t))(j::=u) = A(i::=subst j u t)"
Nil_in_shuffles: fixes xs :: "'b list"   and ys :: "'b list" shows "([] \<in> shuffles xs ys) = (xs = [] \<and> ys = [])"
nu_witness: shows "seq (nu_witness + nu_witness) = seq nu_witness"
add_match_rot_add_missing_ret_unfoldings: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m (add_missing_ret_unfoldings rs1 rs2), Undecided\<rangle> \<Rightarrow> Undecided = \<Gamma>,\<gamma>,p\<turnstile> \<langle>add_missing_ret_unfoldings rs1 (add_match m rs2), Undecided\<rangle> \<Rightarrow> Undecided"
uniformity_transE: fixes E :: "'a \<times> 'a \<Rightarrow> bool" assumes "eventually E uniformity" obtains D :: "'a \<times> 'a \<Rightarrow> bool" where "eventually D uniformity" and "\<And>x y z. \<lbrakk>D (x, y); D (y, z)\<rbrakk> \<Longrightarrow> E (x, z)"
rank_col_rank: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A = col_rank A"
theFT_Int_theNFT: fixes cl :: "('d, 'e, 'f) cmd list" shows "theFT cl \<inter> theNFT cl = {}"
min_eq_find_min_None: fixes xs :: "('c, 'd) bintree option list" shows "(Binomial_Queue.min xs = None) = (find_min xs = None)"
x: shows "x \<in> S \<and> x \<in> affine hull (S - {x})"
multpw_listE: fixes X :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set" assumes "(X, Y) \<in> multpw ns" obtains xs :: "'a list" and ys :: "'a list" where "length xs = length ys" and "X = mset xs" and "Y = mset ys" and "\<forall>i<length ys. (xs ! i, ys ! i) \<in> ns"
in_polychain_of_ccw: fixes e :: "real \<times> real \<Rightarrow> real"   and qs :: "(real \<times> real) list"   and seg :: "(real \<times> real) \<times> real \<times> real"   and Pc :: "real \<times> real" assumes "e \<in> UNIV \<rightarrow> {0<..<1}"   and "ccw'.sortedP 0 qs"   and "length qs \<noteq> 1"   and "seg \<in> set (polychain_of Pc qs)" shows "ccw' (fst seg) (snd seg) (Pc + (\<Sum>P\<in>set qs. e P *\<^sub>R P))"
gallery_map: fixes Cs :: "'a set list" assumes "folding.gallery Cs" shows "folding.gallery (f \<Turnstile> Cs)"
P2R_def: fixes \<phi> :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "P2R \<phi> \<equiv> {(x, y). \<phi> x y}"
simps_3: fixes z :: "quat" shows "Im2 (Quaternions.cnj z) = - Im2 z"
iterates_increasing: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool"   and i :: "nat" assumes "healthy (wp body)" shows "le_trans (iterates body G i) (iterates body G (Suc i))"
empty_Collect_eq: fixes P :: "'g \<Rightarrow> bool" shows "({} = Collect P) = (\<forall>x. \<not> P x)"
well_formed_def: shows "well_formed \<equiv> length astI = numVars \<and> (\<forall>x<numVars. astI ! x < numVals x) \<and> wf_partial_state astG \<and> distinct (map fst ast\<delta>) \<and> Ball (set ast\<delta>) wf_operator"
Set_insert_code_2: fixes x :: "'b"   and xs :: "'b list" shows "insert x (Set_Monad xs) = Set_Monad (x # xs)"
Abs_hierauto_inverse: fixes y :: "'d data \<times> ('e, 'f, 'd) seqauto set \<times> 'f set \<times> ('e \<Rightarrow> ('e, 'f, 'd) seqauto set option)" assumes "y \<in> hierauto" shows "Rep_hierauto (Abs_hierauto y) = y"
s_prod_isor: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<cdot> z \<le> y \<cdot> z"
mono: fixes x :: "nat \<times> nat"   and y :: "nat" assumes "P x y" shows "fst x \<le> y + c \<and> snd x \<le> y + d"
fst_divmod_abs: fixes k :: "integer"   and l :: "integer" shows "fst (Code_Numeral.divmod_abs k l) = \<bar>k\<bar> div \<bar>l\<bar>"
image_inv_f_f: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj f" shows "inv f ` f ` A = A"
fps_shift_times_fps_X'': fixes n :: "nat"   and f :: "'a fps" assumes "1 \<le> n" shows "fps_shift n (f * fps_X) = fps_shift (n - 1) f"
ordinal_modI: fixes x :: "ordinal"   and y :: "ordinal"   and q :: "ordinal"   and r :: "ordinal" assumes "x = y * q + r"   and "r < y" shows "x mod y = r"
pre_list: shows "wf_until_auxlist \<sigma> V n R pos \<phi> I \<psi> auxlist ne"
mahler_measure_dvd: fixes f :: "int poly"   and h :: "int poly" assumes "f \<noteq> 0"   and "h dvd f" shows "mahler_measure h \<le> mahler_measure f"
Abs_fin_string_cases: fixes x :: "fin_string" obtains y :: "char list" where "x = Abs_fin_string y" and "y \<in> {s. length s \<le> MAX_STR}"
Liminf_bounded: fixes C :: "'d"   and X :: "'b \<Rightarrow> 'd"   and F :: "'b filter" assumes "\<forall>\<^sub>F n in F. C \<le> X n" shows "C \<le> Liminf F X"
T_BIT_absch: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat" assumes "BIT_Off acts qs init"   and "n \<le> length qs" shows "BIT_Off.T_BIT qs init n \<le> 7 / 4 * real_of_int (BIT_Off.T_A' acts qs init n) - 3 / 4 * real n"
image_set: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "f ` set xs = set (map f xs)"
rt_fresher_single: fixes ip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2" shows "the (rt1 ip) \<sqsubseteq> the (rt2 ip)"
valid_Act_strong: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and X :: "'d"   and P :: "'a"   and \<alpha> :: "'c"   and x :: "('e, 'b, 'c) formula" assumes "nominal_ts satisfies transition"   and "finite (supp X)" shows "nominal_ts.valid satisfies transition P (Act \<alpha> x) = (\<exists>\<alpha>' x' P'. Act \<alpha> x = Act \<alpha>' x' \<and> transition P \<langle>\<alpha>',P'\<rangle> \<and> nominal_ts.valid satisfies transition P' x' \<and> bn \<alpha>' \<sharp>* X)"
p: fixes x :: "real" assumes "x \<in> {real_of_int i' / 2 ^ (l' + 1)..(real_of_int i' + 1) / 2 ^ (l' + 1)}" shows "(\<Phi>_p has_real_derivative \<phi> (l', i') x * \<phi> (l, i) x) (at x)"
rhs: shows "sample_uniform_units q = spmf_of_set ({..<q} - {0})"
aezfun_setspanD: fixes x :: "('b, 'c) aezfun"   and A :: "'c set" assumes "x \<in> aezfun_setspan A" shows "supp (aezfun x) \<subseteq> A"
inc_inj: fixes n :: "Arity"   and n' :: "Arity" shows "(inc\<cdot>n = inc\<cdot>n') = (n = n')"
valid_folded: shows "valid_mmuaux' args cur nt aux auxlist"
not_square: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'b" shows "\<turnstile> (\<not> [A]_v) = <\<not> A>_v"
amod_def: fixes x :: "ant"   and y :: "ant" shows "x amod y = ant (tna x mod tna y)"
nt: shows "nt = \<tau> \<sigma> (ne + length aux)"
wt: shows "P,E,h,sh(C \<mapsto> (sfs, Error)) \<turnstile> e : T"
rep_eq: fixes x :: "(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 xa :: "char list \<Rightarrow> ty option"   and xb :: "expr list"   and xc :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list, val) mapping) set) option) \<times> (char list \<Rightarrow> val option)"   and xd :: "expr list"   and xe :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list, val) mapping) set) option) \<times> (char list \<Rightarrow> val option)" shows "x,xa \<turnstile> \<langle>xb,xc\<rangle> [\<Rightarrow>'] \<langle>xd,xe\<rangle> = x,xa \<turnstile> \<langle>xb,map_prod (map_fun id (map_option (map_prod id ((`) (map_prod id mapping.rep))))) id xc\<rangle> [\<Rightarrow>] \<langle>xd,map_prod (map_fun id (map_option (map_prod id ((`) (map_prod id mapping.rep))))) id xe\<rangle>"
newman_def: fixes s :: "complex" shows "newman s = 1 / (s - 1) * (pre_newman s - deriv zeta s / zeta s)"
p_def: fixes n :: "nat" shows "p n = real n powr (\<epsilon> - 1)"
wt_elim_cases_19: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and es :: "expr list"   and Ts :: "tys" assumes "E,dt\<Turnstile>In3 (e # es)\<Colon>Ts" obtains T :: "ty" and Tsa :: "ty list" where "Ts = Inr (T # Tsa)" and "E,dt\<Turnstile>e\<Colon>-T" and "E,dt\<Turnstile>es\<Colon>\<doteq>Tsa"