hcju/isabelleps huggingface dataset:
Vagree_def: fixes \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec"   and \<nu>' :: "(real, 'c) vec \<times> (real, 'c) vec"   and V :: "('c + 'c) set" shows "Vagree \<nu> \<nu>' V \<equiv> (\<forall>i. Inl i \<in> V \<longrightarrow> fst \<nu> $ i = fst \<nu>' $ i) \<and> (\<forall>i. Inr i \<in> V \<longrightarrow> snd \<nu> $ i = snd \<nu>' $ i)"
Greatest_equality: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "P x"   and "\<And>y. P y \<Longrightarrow> y \<le> x" shows "Greatest P = x"
insort_left_comm: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.insort_key less_eq (\<lambda>x. x) x (linorder.insort_key less_eq (\<lambda>x. x) y xs) = linorder.insort_key less_eq (\<lambda>x. x) y (linorder.insort_key less_eq (\<lambda>x. x) x xs)"
thm_cont_propos_1: fixes v :: "i"   and p :: "\<o>" shows "[NonContingent p \<^bold>\<equiv> NonContingent (p\<^sup>-) in v]"
finite_range_map_of: fixes xys :: "('b \<times> 'a) list" shows "finite (range (table_of xys))"
abs_eq: fixes x :: "assertion" shows "Ribbon x = abs_interface (Ribbon_conc x)"
dom_char: fixes f :: "'c" shows "Cop.dom f = C.cod f"
no_disc: shows "\<forall>a. \<not> disc2 (C a)"
Collect_disj_eq: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "{x. P x \<or> Q x} = {x. P x} \<union> {x. Q x}"
inj_on_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and A :: "'b set" assumes "class.cancel_semigroup_add plus" shows "inj_on (plus a) A"
cut_ge_bound: fixes n :: "'a"   and i :: "'a set"   and t :: "'a" assumes "n \<in> i \<down>\<ge> t" shows "t \<le> n"
Exists_intro: fixes i :: "name"   and B :: "fm"   and e :: "(name, hf) finfun"   and A :: "fm" assumes "atom i \<sharp> B" shows "Exists \<lbrakk>\<guillemotleft>A IMP B\<guillemotright>\<rbrakk>e \<lbrakk>\<guillemotleft>SyntaxN.Ex i A IMP B\<guillemotright>\<rbrakk>e"
target_size_def: shows "target_size \<equiv> map_fun (map_fun id Word.rep) id (\<lambda>_. LENGTH('c))"
drop_append: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'b list" shows "drop n (xs \<cdot> ys) = drop n xs \<cdot> drop (n - \<^bold>|xs\<^bold>|) ys"
mod_mult_self3: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (times c b) a) b = modulo a b"
mult_nonneg_nonneg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "Rings.class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (times a b)"
full_componentE: fixes x :: "'a"   and i :: "nat"   and t :: "(nat, nat) pp" assumes "snd (rep_nat_term x) = i" obtains y :: "'a" where "rep_nat_term y = (t, i)"
assms: shows "sgnx (poly p) (at_right a) \<noteq> sgnx (poly p) at_top"
inject_6: fixes x6 :: "game"   and y6 :: "game" shows "(x6** = y6**) = (x6 = y6)"
add_missing_ret_unfoldings_alt: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list"   and s :: "state"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_missing_ret_unfoldings rs1 rs2, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match_MatchAnd_foldr (map (\<lambda>r. MatchNot (get_match r)) (filter (\<lambda>r. get_action r = Return) rs1)) rs2, s\<rangle> \<Rightarrow> t"
isSucc_pred_in: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r"   and "isSucc r i" shows "(pred r i, i) \<in> r"
det_iarrays_rings_def: fixes A :: "'c iarray iarray" shows "det_iarrays_rings A = (let A' = echelon_form_of_det_iarrays A euclid_ext2 in (1::'c) div fst A' * (\<Prod>i\<leftarrow>[0..<nrows_iarray A]. snd A' !! i !! i))"
nn_integral_return: fixes x :: "'c"   and M :: "'c measure"   and g :: "'c \<Rightarrow> ennreal" assumes "x \<in> space M"   and "g \<in> borel_measurable M" shows "integral\<^sup>N (return M x) g = g x"
\<beta>s: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau>' s $ Fv x \<tau>') (subst_bv (Fv x \<tau>') s)"
simps_1: fixes x :: "char list" shows "ground\<^sub>t (Var x) = False"
valid_def: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A \<equiv> \<forall>s t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<longrightarrow> s \<in> Normal ` P \<longrightarrow> t \<notin> Fault ` F \<longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A"
build_eq: fixes a\<^sub>1 :: "'b"   and w\<^sub>1 :: "nat \<Rightarrow> 'b"   and a\<^sub>2 :: "'b"   and w\<^sub>2 :: "nat \<Rightarrow> 'b" shows "(a\<^sub>1 ## w\<^sub>1 = a\<^sub>2 ## w\<^sub>2) = (a\<^sub>1 = a\<^sub>2 \<and> w\<^sub>1 = w\<^sub>2)"
hcomp_simps_1: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "B.hseq \<nu> \<mu>" shows "src\<^sub>B (\<nu> \<star>\<^sub>B \<mu>) = src\<^sub>B \<mu>"
iso: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "isomorphic_categories A B" shows "\<exists>F G. inverse_functors A B F G"
hyps_1: shows "\<forall>n. \<Gamma>,\<Theta>_\<Turnstile>n:\<^bsub>/F_\<^esub> P_ c_ Q_,A_"
globala_def: fixes P :: "(nat \<Rightarrow> 'd) \<times> 'f \<times> (nat \<Rightarrow> 'd) \<Rightarrow> bool" shows "globala P \<equiv> \<lambda>((\<sigma>, uu_), a, \<sigma>', uu_). P (\<sigma>, a, \<sigma>')"
Interp_imp_INTERP: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and N :: "'a literal multiset set" assumes "C \<le> D"   and "Interp N D \<Turnstile> C" shows "INTERP N \<Turnstile> C"
collapse_2: fixes sum :: "'a + 'b" assumes "\<not> isl sum" shows "Inr (projr sum) = sum"
fold_map_idx_len: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b \<times> 'a"   and i :: "nat"   and y :: "'b"   and xs :: "'c list" shows "length (snd (fold_map_idx f i y xs)) = length xs"
mem_subring_mem_ring: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and x :: "'e" assumes "Ring R"   and "Subring R S"   and "x \<in> carrier S" shows "x \<in> carrier R"
ordertype_VWF_finite_nat: fixes A :: "V set" assumes "finite A" shows "ordertype A VWF = ord_of_nat (card A)"
simps_1: fixes uu :: "(real, 'b) vec" shows "min_height (Leaf uu) = 0"
ncoplanar_perm_5: 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 C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet A D C B"
lindep_span: fixes U :: "'c set" assumes "U \<subseteq> carrier V"   and "finite U" shows "lin_dep U = (\<exists>u\<in>U. u \<in> local.module.span (U - {u}))"
least_power_def: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "least_power f x = (LEAST n. (f ^^ n) x = x \<and> 0 < n)"
GroupHomSet_def: fixes G :: "'c set"   and H :: "'d set" shows "GroupHomSet G H \<equiv> {T. GroupHom G T} \<inter> {T. T ` G \<subseteq> H}"
iD: shows "isFDERIV DIM('a) [0..<DIM('a)] fas (list_of_eucl x @ params)"
discovered_eq_finished_un_stack: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme" assumes "DFS_invar G param s" shows "dom (discovered s) = dom (finished s) \<union> set (stack s)"
induct: fixes x :: "('a, 'b) ODE"   and P :: "('a, 'b) ODE \<Rightarrow> bool" assumes "osafe x"   and "\<And>c. P (OVar c)"   and "\<And>\<theta> x. dfree \<theta> \<Longrightarrow> P (OSing x \<theta>)"   and "\<And>ODE1 ODE2. \<lbrakk>osafe ODE1; P ODE1; osafe ODE2; P ODE2; ODE_dom ODE1 \<inter> ODE_dom ODE2 = {}\<rbrakk> \<Longrightarrow> P (OProd ODE1 ODE2)" shows "P x"
IH: shows "card {i. n < i \<and> i \<le> n' \<and> mining i} \<le> bc n' - bc n"
scalar_product_def: fixes ns :: "nat list"   and "is" :: "int list" shows "scalar_product ns is = sum_list (map_index (\<lambda>i. (*) (if i < length ns then map int ns ! i else 0)) is)"
design_H2: fixes P :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool"   and Q :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool" assumes "\<forall>a b. P (a, b\<lparr>ok := True\<rparr>) \<longrightarrow> P (a, b\<lparr>ok := False\<rparr>)" shows "(P \<turnstile> Q) is H2 healthy"
d_dist: fixes x' :: "'a"   and x :: "'a"   and e :: "real"   and f :: "'a \<Rightarrow> 'b" assumes "dist x' x < d x e"   and "x \<in> S"   and "x' \<in> S"   and "0 < e"   and "f \<in> \<F>" shows "dist (f x') (f x) < e"
eq_iff: fixes x :: "'a"   and d :: "nat"   and y :: "'a" shows "(Polynomial.monom x d = Polynomial.monom y d) = (x = y)"
step_is_path: fixes c :: "'b"   and c' :: "'b"   and astep :: "'b \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool" assumes "(c, c') \<in> (asystem.step astep)\<^sup>*" shows "\<exists>p. LTS.path astep c p c'"
assms: shows "\<exists>i. k \<in> dom (map_of (l ! i)) \<and> i < length l"
lead_coeff_monom: fixes c :: "'b"   and n :: "nat" shows "lead_coeff (monom c n) = c"
vpairs_vsubset_vdomain_vrange: fixes r :: "V" shows "vpairs r \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<times>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
locally_empty: fixes P :: "'b set \<Rightarrow> bool" shows "locally P {}"
insort_append_first: fixes ys :: "'a list"   and x :: "'a"   and xs :: "'a list" assumes "\<forall>y\<in>set ys. x \<le> y" shows "insort x (xs @ ys) = insort x xs @ ys"
eq: shows "(\<lambda>x. restrict x J) -` A \<inter> Pi\<^sub>E I S \<subseteq> (\<lambda>x. restrict x J) -` B \<inter> Pi\<^sub>E I S"
ldrop_lappend: fixes n :: "enat"   and xs :: "'c llist"   and ys :: "'c llist" shows "ldrop n (lappend xs ys) = (if n < llength xs then lappend (ldrop n xs) ys else ldrop (n - llength xs) ys)"
suffix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "suffix xs ys = (\<exists>zs. ys = zs @ xs)"
sel_8: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "'b option"   and x24 :: "'b tree"   and x25 :: "'b tree" shows "lchild (Node x21 x22 x23 x24 x25) = x24"
otnaux1: fixes x :: "'b"   and y :: "'b" shows "{x, y, x, y} = {x, y}"
eexp_inj: fixes x :: "ereal"   and y :: "ereal" shows "(eexp x = eexp y) = (x = y)"
typ_of1_split_App: fixes Ts :: "typ list"   and t :: "Core.term"   and u :: "Core.term"   and ty :: "typ" assumes "typ_of1 Ts (t $ u) = Some ty" shows "\<exists>uty. typ_of1 Ts t = Some (uty \<rightarrow> ty) \<and> typ_of1 Ts u = Some uty"
that: fixes B' :: "'a set set" assumes "B' \<subseteq> B"   and "X = \<Union> B'" shows "thesis"
plus_fds_def: fixes f :: "'b fds"   and g :: "'b fds" shows "f + g = fds (\<lambda>n. fds_nth f n + fds_nth g n)"
FL_valid_Act_fresh: fixes \<alpha> :: "'act"   and f :: "'effect"   and P :: "'state"   and x :: "('a, 'pred, 'act, 'effect) formula" assumes "bn \<alpha> \<sharp>* \<langle>f\<rangle>P" shows "P \<Turnstile> Act f \<alpha> x = (\<exists>P'. \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>,P'\<rangle> \<and> P' \<Turnstile> x)"
is_max: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and \<aa> :: "'c set" assumes "max_lideal I R addition multiplication zero unit"   and "lideal \<aa> R addition multiplication zero unit"   and "\<aa> \<noteq> R"   and "I \<subseteq> \<aa>" shows "I = \<aa>"
borel_measurable_add': fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "f + g \<in> borel_measurable M"
countable_vector: fixes B :: "'a \<Rightarrow> 'b set" assumes "\<And>i. countable (B i)" shows "countable {V. \<forall>i. V $ i \<in> B i}"
lookup_delete: fixes k :: "'b"   and m :: "('b, 'a) mapping"   and k' :: "'b" shows "Mapping.lookup (Mapping.delete k m) k' = (if k = k' then None else Mapping.lookup m k')"
lr: shows "(c::'a::refinement_lattice)\<^sup>\<star> ; c\<^sup>\<star> \<sqsubseteq> c\<^sup>\<star>"
Shift: fixes G :: "('a, 'b) form list" assumes "\<turnstile> rotate1 G" shows "\<turnstile> G"
split_part: fixes P :: "bool"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(\<lambda>(a, b). P \<and> Q a b) = (\<lambda>ab. P \<and> (case ab of (x, xa) \<Rightarrow> Q x xa))"
prod_list_map_filter: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "prod_list (map g (filter f xs)) * prod_list (map g (filter (\<lambda>x. \<not> f x) xs)) = prod_list (map g xs)"
ceiling_log_nat_eq_powr_iff: fixes b :: "nat"   and k :: "nat"   and n :: "nat" assumes "2 \<le> b"   and "0 < k" shows "(\<lceil>log (real b) (real k)\<rceil> = int n + 1) = (b ^ n < k \<and> k \<le> b ^ (n + 1))"
drop_tl: fixes n :: "nat"   and xs :: "'g list" shows "drop n (tl xs) = tl (drop n xs)"
post_top: fixes p :: "'b" shows "post_fun p p = \<top>"
minus_mult_distrib_mat_vec: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and v :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "(A - B) *\<^sub>v v = A *\<^sub>v v - B *\<^sub>v v"
coeff_a_p2: shows "local.coeff (map ((\<otimes>) a) p2 @ replicate (degree (a # p1)) \<zero>) = (\<lambda>i. \<Oplus>k\<in>{..i}. (if k = length p1 then a else \<zero>) \<otimes> local.coeff p2 (i - k))"
Call: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set" assumes "\<forall>(P, c, Q)\<in>C. \<exists>p. c = CALL p"   and "C \<tturnstile> {(P, b, Q). \<exists>p. (P, CALL p, Q) \<in> C \<and> b = body p}" shows "{} \<tturnstile> C"
commutative_square_def: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and h :: "'a \<times> 'a"   and k :: "'a \<times> 'a" shows "B.VV.commutative_square f g h k \<equiv> B.VV.cospan f g \<and> B.VV.span h k \<and> B.VV.dom f = B.VV.cod h \<and> B.VV.comp f h = B.VV.comp g k"
vrat_fpair_in_fproduct_vrat: fixes a :: "rat"   and b :: "rat" shows "[a\<^sub>\<rat>, b\<^sub>\<rat>]\<^sub>\<circ> \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>"
min_absorb2: fixes y :: "'g"   and x :: "'g" assumes "y \<le> x" shows "min x y = y"
assms_2: shows "d = {d. d \<noteq> [] \<and> hd d \<in> ev ` A \<and> (\<exists>a. ev a = hd d \<and> tl d \<in> D (P a))}"
below_rep_def: fixes p :: "int extended \<times> int extended" shows "below_rep p = (if is_empty_rep p then empty_rep else let (l, h) = p in (-\<infinity>, h))"
f3: shows "rk {a, c, A, C, \<beta>} \<le> 3"
star_left_slide: fixes x :: "'a"   and y :: "'a" shows "(x * y)\<^sup>\<star> * x \<le> x * (y * x)\<^sup>\<star>"
differentiable_on_def: fixes f :: "'c \<Rightarrow> 'd"   and s :: "'c set" shows "(f differentiable_on s) = (\<forall>x\<in>s. f differentiable at x within s)"
inf_bot_left: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and x :: "'e" assumes "class.bounded_lattice_bot inf less_eq less sup bot" shows "inf bot x = bot"
distinct_sorted_list_of_set: fixes A :: "'t set" shows "distinct (ord_term_lin.sorted_list_of_set A)"
of_real_mangoldt: fixes n :: "nat" shows "of_real (mangoldt n) = mangoldt n"
relabel_effect_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" shows "relabel_effect f l u \<equiv> l(u := Min {l v |v. (u, v) \<in> cfE_of f} + 1)"
rel_fun_mono_strong: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and A' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B' :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "rel_fun A B f g"   and "A' \<le> A"   and "\<And>x y. \<lbrakk>x \<in> f ` {x. Domainp A' x}; y \<in> g ` {x. Rangep A' x}; B x y\<rbrakk> \<Longrightarrow> B' x y" shows "rel_fun A' B' f g"
discr_ZObis: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and d :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "proper c"   and "proper d"   and "PL_Indis.discr aval tval cval indis c"   and "PL_Indis.discr aval tval cval indis d" shows "(c, d) \<in> PL_Indis.ZObis aval tval cval indis"
g: shows "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>"
snd_map_prod: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'e"   and x :: "'g \<times> 'h" shows "snd (map_prod f g x) = g (snd x)"
bi_unique_rel_12_23: assumes "three TYPE('d)" shows "bi_unique three.rel_12_23"
det_int_def: fixes A :: "int mat" shows "det_int A = det_code (select_min (\<lambda>x. nat \<bar>x\<bar>)) gcd_mute_fun A"
weakCongUnfold: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<sigma> :: "(name list \<times> 'a list) list" assumes "P \<doteq>\<^sub>c Q"   and "wellFormedSubst \<sigma>" shows "P[<\<sigma>>] \<doteq>\<^sub>c Q[<\<sigma>>]"
while_k_def: fixes uu :: "nat" shows "while_k uu \<equiv> while_C"
u: shows "(r \<cdot> s) \<^sup>@ Suc i = u"
elims: fixes x :: "primfun"   and y :: "real \<Rightarrow> real" assumes "eval_primfun' x = y" obtains n :: "nat" where "x = LnChain n" and "y = ln ^^ n"
sub_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (sub_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
right_invertible_transpose: fixes A :: "(('c, 'd) vec, 'e) vec" shows "(\<exists>B. Finite_Cartesian_Product.transpose A ** B = mat (1::'c)) = (\<exists>B. B ** A = mat (1::'c))"
set_filter: fixes P :: "'k \<Rightarrow> bool"   and xs :: "'k list" shows "set (filter P xs) = {x \<in> set xs. P x}"
tortoise_hare_def: shows "tortoise_hare \<equiv> find_nu ;; find_mu ;; find_lambda"
elims: fixes x :: "('a, 'b, 'c, 'op, 'opinl) Inca.instr"   and y :: "('a, 'b, 'c, 'op) Std.instr" assumes "norm_instr x = y" obtains d :: "'a" where "x = Inca.instr.IPush d" and "y = Std.instr.IPush d"   | "x = Inca.instr.IPop" and "y = Std.instr.IPop"   | xa :: "'b" where "x = Inca.instr.ILoad xa__" and "y = Std.instr.ILoad xa__"   | xa :: "'b" where "x = Inca.instr.IStore xaa__" and "y = Std.instr.IStore xaa__"   | op :: "'op" where "x = Inca.instr.IOp op" and "y = Std.instr.IOp op"   | opinl :: "'opinl" where "x = IOpInl opinl" and "y = Std.instr.IOp (\<DD>\<ee>\<II>\<nn>\<ll> opinl)"   | n :: "nat" where "x = Inca.instr.ICJump n" and "y = Std.instr.ICJump n"   | xa :: "'c" where "x = Inca.instr.ICall xa" and "y = Std.instr.ICall xa"
eval_elim: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" obtains c :: "('Var, 'AExp, 'BExp) Stmt" and c'a :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "cb__ = cxt_to_stmt E__ cc__" and "c' = cxt_to_stmt E__ c'a__" and "mds' = mds" and "eval\<^sub>w_simple_abv (cc__, mem) (c'a__, mem')"   | c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" where "cb__ = c\<^sub>1 ;; c\<^sub>2" and "c' = c\<^sub>1' ;; c\<^sub>2" and "\<langle>c\<^sub>1, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem'\<rangle>"   | c :: "('Var, 'AExp, 'BExp) Stmt" and mu :: "'Var ModeUpd" and c'a :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "cb__ = cxt_to_stmt E (c@[mu])" and "c' = cxt_to_stmt E c'a" and "\<langle>c, update_modes mu mds, mem\<rangle> \<leadsto> \<langle>c'a, mds', mem'\<rangle>"
SUP_sup_distrib: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'a" shows "Sup (f ` A) \<squnion> Sup (g ` A) = (SUP a\<in>A. f a \<squnion> g a)"
B_def: fixes m :: "'a option"   and \<alpha> :: "'a \<Rightarrow> bool" shows "m \<rightharpoonup> \<alpha> \<equiv> case m of None \<Rightarrow> True | Some a \<Rightarrow> \<alpha> a"
EQ_imp_SUBS2: fixes u :: "tm"   and t :: "tm"   and H :: "fm set" shows "insert (u EQ t) H \<turnstile> t SUBS u"
pmod_def: fixes x :: "'a mpoly"   and y :: "'a mpoly" shows "x pmod y = (THE r. \<exists>a q. pseudo_divmod_rel a x y q r)"
leg_simps_1: shows "C.arr \<nu>.dom.leg0"
4: shows "\<exists>D. prefix D A \<and> C \<in> set (tr\<^sub>p\<^sub>c D [])"
assms: fixes i :: "nat" assumes "i \<le> n" shows "to_fract x ^ i / to_fract y ^ (i - 1) \<in> range to_fract"
monomial_P_mod_mod: fixes I :: "'a set"   and c :: "'a"   and p :: "'a"   and d :: "nat" assumes "ideal S I"   and "c \<in> carrier S"   and "p = c \<cdot>\<^sub>r X^\<^bsup>R d\<^esup>" shows "(c \<in> I) = P_mod R S X I p"
Cons_path2: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and n' :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "n' \<in> set (old.predecessors g n)" shows "g \<turnstile> n'-n' # ns\<rightarrow>m"
image_mset_add_mset: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and M :: "'d multiset" shows "f `# add_mset a M = add_mset (f a) (f `# M)"
mult_cancel_right: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.semiring_no_zero_divisors_cancel plus zero times" shows "(times a c = times b c) = (c = zero \<or> a = b)"
f_bm: shows "f \<in> borel_measurable (lebesgue_on S)"
sets_D: fixes K :: "'c \<Rightarrow> 'c pmf" shows "sets (stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x)))) = sets (stream_space (Pi\<^sub>M UNIV (\<lambda>s. count_space UNIV)))"
assms_2: shows "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> z"
t1: shows "\<forall>t\<in>mbal_update (\<lambda>_. mbal (dblock s' p)) ` allBlocksRead s p. Inv2a_innermost s p t"
key_compare_uminus: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "key_compare (- ko) x y = key_compare ko y x"
add_le_same_cancel1: fixes b :: "'d multiset"   and a :: "'d multiset" shows "(b + a \<subseteq># b) = (a \<subseteq># {#})"
confluent_distinct: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q1 :: "'t \<Rightarrow>\<^sub>0 'c"   and f1 :: "'t \<Rightarrow>\<^sub>0 'c"   and t1 :: "'a"   and q2 :: "'t \<Rightarrow>\<^sub>0 'c"   and f2 :: "'t \<Rightarrow>\<^sub>0 'c"   and t2 :: "'a"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "red_single p q1 f1 t1"   and "red_single p q2 f2 t2"   and "t1 \<oplus> lt f1 \<noteq> t2 \<oplus> lt f2"   and "f1 \<in> F"   and "f2 \<in> F" obtains s :: "'t \<Rightarrow>\<^sub>0 'c" where "(red F)\<^sup>*\<^sup>* q1 s" and "(red F)\<^sup>*\<^sup>* q2 s"