hcju/isabelleps huggingface dataset:
group_hom_axioms_def: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" shows "group_hom_axioms G H h \<equiv> h \<in> hom G H"
1: shows "\<not> (i < length cs \<longrightarrow> (\<forall>j<i. cs ! j \<noteq> cs ! i) \<longrightarrow> cs ! i \<noteq> x \<longrightarrow> cs ! i \<notin> set bs \<longrightarrow> x < cs ! i in s_TS init h \<sigma> (length (as @ [x] @ bs @ [x]) + i + 1))"
pcp_imp_pgp: fixes u :: "'v"   and vo :: "'v option"   and s0 :: "'s"   and s :: "'s" assumes "nofail (gds_init gds \<bind> WHILE gen_cond gen_step')"   and "nofail gen_dfs"   and "post_choose_pending u vo s0 s" shows "post_get_pending u vo s0 s"
generate_in_carrier: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set"   and h :: "'e" assumes "Group.group G"   and "H \<subseteq> carrier G"   and "h \<in> generate G H" shows "h \<in> carrier G"
snd_map_prod: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'a"   and x :: "'c \<times> 'd" shows "snd (map_prod f g x) = g (snd x)"
wf_digraph_axioms: shows "wf_digraph (with_proj (subdivide G e w))"
deep_def: fixes pr :: "('a, 'b) Digit"   and m :: "('a, 'b) FingerTreeStruc"   and sf :: "('a, 'b) Digit" shows "FingerTreeStruc.deep pr m sf = Deep (FingerTreeStruc.gmd pr + FingerTreeStruc.gmft m + FingerTreeStruc.gmd sf) pr m sf"
Nil: fixes G :: "prog"   and s0 :: "st" shows "G\<turnstile>Norm s0 \<midarrow>[]\<doteq>\<succ>[]\<rightarrow> Norm s0"
sl4: shows "- (ordinate C - ordinate D) \<in> radical_sqrt"
tree_eq_rule: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and v :: "'c"   and w :: "'c"   and u :: "'c" assumes "DFS_invar G param s"   and "(v, w) \<in> tree_edges s"   and "(u, w) \<in> tree_edges s" shows "v = u"
and_mask_eq_iff_le_mask: fixes w :: "'a word"   and n :: "nat" shows "(w AND mask n = w) = (w \<le> mask n)"
find_Some_iff: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list"   and x :: "'c" shows "(find P xs = Some x) = (\<exists>i<length xs. P (xs ! i) \<and> x = xs ! i \<and> (\<forall>j<i. \<not> P (xs ! j)))"
simps_2: fixes e :: "char list exp" shows "fin\<^sub>1 (throw e) = throw (fin\<^sub>1 e)"
ord_nat_code: fixes n :: "nat"   and a :: "nat" shows "ord n a = (if n = 0 then if a = 1 then 1 else 0 else if coprime n a then Min (Set.filter (\<lambda>k. [a ^ k = 1] (mod n)) {0<..n}) else 0)"
that_2: assumes "Q = (\<lambda>s. False)"   and "P = (\<lambda>s. False)" shows "thesis"
divided_differences_main: fixes k :: "nat"   and i :: "nat" assumes "k \<le> n"   and "i < k" shows "(\<Sum>j\<leftarrow>[0..<Suc k - i]. local.xij_f i (i + j) * local.pprod k i (i + j)) = (\<Sum>j\<leftarrow>[0..<Suc k - Suc i]. local.xij_f (Suc i) (Suc i + j) * local.pprod k (Suc i) (Suc i + j))"
noetherian_ring_def: fixes R :: "('c, 'd) ring_scheme" shows "noetherian_ring R \<equiv> ring R \<and> noetherian_ring_axioms R"
prefix_append: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" shows "prefix xs (ys @ zs) = (prefix xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefix us zs))"
assms_3: shows "\<^bold>|u\<^bold>| \<le> \<^bold>|p \<cdot> w\<^bold>|"
arr_cod_iff_arr: fixes f :: "'b" shows "D.arr (D.cod f) = D.arr f"
turn_true_simps_2: fixes \<A> :: "'b \<Rightarrow> bool"   and F :: "'b formula" assumes "\<not> \<A> \<Turnstile> F" shows "turn_true \<A> F = \<^bold>\<not> F"
bigo_plus_idemp: fixes f :: "'a \<Rightarrow> 'b" shows "O(f) + O(f) = O(f)"
vbrelation_vconverse: shows "vbrelation (vrat_uminus\<inverse>\<^sub>\<circ>)"
p_strict_mono: fixes a :: "real"   and v :: "real"   and x :: "real"   and y :: "real"   and s0 :: "real" assumes "braking_movement a v"   and "x < y"   and "y \<le> movement.t_stop a v" shows "movement.p a v s0 x < movement.p a v s0 y"
ideal_view_1_def: fixes x :: "'grp \<times> 'grp"   and y :: "bool"   and z :: "'e"   and S :: "(('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'd spmf) \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp) \<times> 'f) spmf) \<times> (('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'd spmf) \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> unit \<Rightarrow> 'f \<Rightarrow> 'd spmf)"   and \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'd spmf)" shows "mal_def.ideal_view_1 x y z S \<A> = (let (S1, S2) = S in mal_def.ideal_game_1 x y z (S1 \<A>, S2 \<A>))"
is_sublist_notlast: fixes vs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "distinct vs"   and "x = last vs" shows "\<not> is_sublist [x, y] vs"
bezout_matrix_row_def: fixes A :: "(('h, 'i) vec, 'g) vec"   and a :: "'g"   and b :: "'g"   and j :: "'i"   and bezout :: "'h \<Rightarrow> 'h \<Rightarrow> 'f \<times> 'f \<times> 'f \<times> 'f \<times> 'j"   and x :: "'g" shows "bezout_matrix_row A a b j bezout x = (let (p, q, u, v, d) = bezout (A $ a $ j) (A $ b $ j) in \<chi>y. if x = a \<and> y = a then p else if x = a \<and> y = b then q else if x = b \<and> y = a then u else if x = b \<and> y = b then v else if x = y then 1::'f else (0::'f))"
that: fixes s :: "real" assumes "is_Arg z s"   and "s \<in> R" shows "thesis"
degree_neville_aitken_interpolation_poly: fixes xs_ys :: "('a \<times> 'a) list" shows "degree (neville_aitken_interpolation_poly xs_ys) \<le> length xs_ys - 1"
subst_compose1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y1 :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y2 :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y1"   and "good Y2" shows "X #[Y1 / y]_ys #[Y2 / y]_ys = X #[Y1 #[Y2 / y]_ys / y]_ys"
ts_b2': fixes x :: "nat"   and y :: "nat"   and qs :: "nat list"   and h :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "qs \<in> lang (seq [Atom x, Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom y, Atom y])"   and "(\<exists>hs. h = [x, x] @ hs) \<or> h = []" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs = length qs - 3 \<and> Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs = ([y, x], rev qs @ h) \<and> (\<exists>hs. rev qs @ h = [y, y] @ hs)"
freshEnv_getEnv: fixes ys :: "'d"   and y :: "'e"   and rho :: "'d \<Rightarrow> 'e \<Rightarrow> ('f, 'g, 'd, 'e, 'h) qTerm set option" assumes "freshEnv ys y rho" shows "rho ys y = None"
poly_rel_uminus: shows "(poly_rel ===> poly_rel) (uminus_poly_i ops) uminus_class.uminus"
complex_cnj_cancel_iff: fixes x :: "complex"   and y :: "complex" shows "(cnj x = cnj y) = (x = y)"
simps_2: fixes a :: "String.literal"   and Ts :: "typ list"   and b :: "String.literal"   and Us :: "typ list"   and subs :: "((variable \<times> String.literal set) \<times> typ) list" shows "assoc_match (Ty a Ts) (Ty b Us) subs = (if a = b \<and> length Ts = length Us then fold (\<lambda>(T, U) subs. Option.bind subs (assoc_match T U)) (zip Ts Us) (Some subs) else None)"
dist_blinfun_def: fixes a :: "'c \<Rightarrow>\<^sub>L 'd"   and b :: "'c \<Rightarrow>\<^sub>L 'd" shows "dist a b = norm (a - b)"
bit_horner_sum_bit_iff: fixes bs :: "bool list"   and n :: "nat" shows "bit (horner_sum of_bool (2::'a) bs) n = (n < length bs \<and> bs ! n)"
vsv_axioms: shows "vsv (\<CC>\<lparr>Cod\<rparr>)"
fimage_fimage: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c fset" shows "f |`| g |`| A = (\<lambda>x. f (g x)) |`| A"
SV_inv3_def: shows "SV_inv3 = {s. \<forall>r a b v w. votes s r a = Some v \<and> votes s r b = Some w \<longrightarrow> v = w}"
Domain_Collect_case_prod: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "Domain {(x, y). P x y} = {x. \<exists>y. P x y}"
ultrafilters_def: shows "pseudo_hoop_algebra.ultrafilters (\<lambda>a b. b * a) (\<le>) = pseudo_hoop_algebra.properfilters (\<lambda>a b. b * a) (\<le>) \<inter> pseudo_hoop_algebra.maximalfilters (\<lambda>a b. b * a) (\<le>)"
unwinding_theorem_SD: fixes SES :: "('a, 'b) SES_rec"   and \<V> :: "'b V_rec"   and \<V>' :: "'b V_rec"   and ur :: "('a \<times> 'a) set" assumes "Unwinding SES \<V>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>"   and "Unwinding.lrf SES \<V>' ur"   and "Unwinding.osc SES \<V>' ur" shows "SD \<V> Tr\<^bsub>(induceES SES)\<^esub>"
less_append_same_iff: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs < xs @ ys) = ([] < ys)"
span_def: fixes S :: "'b set" shows "ideal S = ideal.subspace hull S"
uminus_set_def: fixes A :: "'a set" shows "- A = Collect (- (\<lambda>x. x \<in> A))"
geodesic_segment_param_in_geodesic_spaces_6: fixes t :: "real"   and x :: "'a"   and y :: "'a" assumes "t \<in> {0..dist x y}" shows "dist x (geodesic_segment_param {x--y} x t) = t"
M_def: shows "M \<equiv> \<lambda>m. real (card {(n, d). n \<in> {0<..m} \<and> d dvd n}) / real (card {0<..m})"
1: shows "(\<lambda>x. eval\<Pi>\<^sub>1 F x dj) = (\<lambda>x. eval\<Pi>\<^sub>1 G x dj)"
wt_elim_cases_14: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e0 :: "expr"   and e1 :: "expr"   and e2 :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (e0 ? e1 : e2)\<Colon>T" obtains T1 :: "ty" and T2 :: "ty" and T :: "ty" where "Ta__ = Inl T" and "E,dt\<Turnstile>e0\<Colon>-PrimT Boolean" and "E,dt\<Turnstile>e1\<Colon>-T1" and "E,dt\<Turnstile>e2\<Colon>-T2" and "prg E\<turnstile>T1\<preceq>T2 \<and> T = T2 \<or> prg E\<turnstile>T2\<preceq>T1 \<and> T = T1"
inject: fixes x1 :: "'a"   and y1 :: "'a" shows "(ev x1 = ev y1) = (x1 = y1)"
assms_1: shows "irreflp p\<^sup>+\<^sup>+"
assms_2: fixes x :: "'a" assumes "x \<in> A" shows "f x \<noteq> {}"
injective_axioms: fixes d :: "nat" shows "injective (\<lambda>x. monom x d)"
pp_dist_sup: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "\<ominus> \<ominus> (x \<oplus> y) = \<ominus> \<ominus> x \<oplus> \<ominus> \<ominus> y"
hyps_2: shows "P \<turnstile> \<langle>e1,s1,b1\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>"
components_are_iso: fixes a :: "'a" assumes "hom'.ide a" shows "hom'.iso (FG.map a)"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> nat \<Rightarrow> 'a"   and tm :: "'b tm" shows "P (case tm of TM x xa \<Rightarrow> f x xa) = (\<forall>x1 x2. tm = TM x1 x2 \<longrightarrow> P (f x1 x2))"
set_Pi_pmf_subset: fixes A :: "'b set"   and dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf" assumes "finite A" shows "set_pmf (Pi_pmf A dflt p) \<subseteq> {f. \<forall>x. x \<notin> A \<longrightarrow> f x = dflt}"
elims: fixes x :: "nat"   and xa :: "'c list list"   and y :: "'c list list" assumes "Matrix_Legacy.transpose x xa = y" obtains nr :: "nat" where "x = nr__" and "xa = []" and "y = replicate nr__ []"   | nr :: "nat" and v :: "'c list" and m :: "'c list list" where "x = nr" and "xa = v # m" and "y = map2 (#) v (Matrix_Legacy.transpose nr m)"
hta_states_correct_2: shows "hs.invar (hta_states H)"
NCisSD2aux: fixes a :: "('b set set, 'c) Combinators"   and aa :: "('b set set, 'c) Combinators" assumes "onlyTwoNets a"   and "onlyTwoNets aa"   and "first_bothNet a \<noteq> first_bothNet aa"   and "\<not> member DenyAll a"   and "\<not> member DenyAll aa"   and "allNetsDistinct [a, aa]" shows "disjSD_2 a aa"
simps_2: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and ta :: "('f, 'i, 'j) new_thread_action"   and tas :: "('f, 'i, 'j) new_thread_action list" shows "redT_updTs' ts (ta # tas) = redT_updTs' (redT_updT' ts ta) tas"
UN_E: fixes b :: "'b"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" assumes "b \<in> \<Union> (B ` A)" obtains x :: "'c" where "x \<in> A" and "b \<in> B x"
set_mult_card_eq_impl_empty_inter: fixes H :: "'a set"   and J :: "'a set"   and a :: "'a"   and b :: "'a" assumes "finite H"   and "finite J"   and "H \<subseteq> carrier G"   and "J \<subseteq> carrier G"   and "card (H <#> J) = card H * card J"   and "a \<in> H"   and "b \<in> H"   and "a \<noteq> b" shows "(\<otimes>) a ` J \<inter> (\<otimes>) b ` J = {}"
reduce_specs_reduce_spec: fixes d :: "nat" assumes "d = DIM('c)" shows "(reduce_specs d, reduce_spec) \<in> unit_rel \<rightarrow> \<langle>lv_rel\<rangle>set_rel \<rightarrow> \<langle>\<langle>lv_rel\<rangle>set_rel\<rangle>nres_rel"
ceiling_le_iff: fixes x :: "'d"   and z :: "int" shows "(\<lceil>x\<rceil> \<le> z) = (x \<le> of_int z)"
assms: shows "eval r_result [pdec1 y, pdec12 y, pdec22 y] \<down>= 0 \<or> pdec2 y \<noteq> prod_encode (i, x)"
simps_3: fixes x21 :: "('a, 'b) rbt"   and x22 :: "color"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt" shows "Node x21 x22 x23 x24 x25 \<noteq> Leaf"
list_of_eucl_eq_iff: fixes x :: "'a"   and y :: "'b" shows "(list_of_eucl x = list_of_eucl y) = (DIM('a) = DIM('b) \<and> (\<forall>i<DIM('b). x \<bullet> Basis_list ! i = y \<bullet> Basis_list ! i))"
SUP_UN_eq: fixes r :: "'b \<Rightarrow> 'a set"   and S :: "'b set" shows "(SUP i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> \<Union> (r ` S))"
inner_product_space_axioms_def: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> 'a" shows "inner_product_space_axioms scale inner \<equiv> ((\<forall>x y. inner x y = cnj_class.cnj (inner y x)) \<and> (\<forall>x y z. inner (x + y) z = inner x z + inner y z) \<and> (\<forall>r x y. inner (scale r x) y = r * inner x y)) \<and> ((\<forall>x. 0 \<le> real_of (inner x x)) \<and> (\<forall>x. (inner x x = (0::'a)) = (x = (0::'b)))) \<and> (\<forall>x A. real_of (inner x x) *\<^sub>R A = inner x x * A) \<and> (\<forall>x. (0 < real_of (inner x x)) = (x \<noteq> (0::'b)))"
HR_imp_eq_H: fixes x :: "hf"   and z :: "hf"   and e :: "(name, hf) finfun" assumes "HR x z" shows "z = \<lbrakk>Goedel_I.HF x\<rbrakk>e"
seq: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b"   and rs\<^sub>1 :: "'a rule list"   and t :: "state"   and rs\<^sub>2 :: "'a rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ rs\<^sub>2, Undecided\<rangle> \<Rightarrow> t'"
simps_2: fixes a :: "'d \<times> ('e, 'f) fmap"   and as :: "('d \<times> ('e, 'f) fmap) list" shows "no_effectless_act (a # as) = (fmdom' (snd a) \<noteq> {} \<and> no_effectless_act as)"
dom_def: fixes f :: "('o, 'a) arr" shows "local.dom f = (if domains f \<noteq> {} then SOME a. a \<in> domains f else null)"
cod_simp: fixes C1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<times> 'c" assumes "product_category C1 C2"   and "partial_magma.arr (product_category.comp C1 C2) f" shows "partial_magma.cod (product_category.comp C1 C2) f = (partial_magma.cod C1 (fst f), partial_magma.cod C2 (snd f))"
plus_length1: fixes k :: "nat"   and ls :: "'c list" shows "rotate (k + |ls|) ls = rotate k ls"
observable_behaviour_refinement: fixes R :: "('e \<times> 'f) set"   and pi :: "'g \<Rightarrow> 'h"   and Sa :: "('e, 'h) spec"   and Sc :: "('f, 'g) spec"   and bc :: "'g list" assumes "refines R pi Sa Sc"   and "bc \<in> obeh Sc" shows "map pi bc \<in> obeh Sa"
is_slim_def: fixes G :: "'b pair_pre_digraph"   and V :: "'b set" shows "pair_pre_digraph.is_slim G V \<equiv> (\<forall>v\<in>pverts G. v \<in> V \<or> in_degree (with_proj G) v \<le> 2 \<and> (\<exists>x p y. pre_digraph.gen_iapath (with_proj G) V x p y \<and> v \<in> set (pawalk_verts x p))) \<and> (\<forall>e\<in>parcs G. fst e \<noteq> snd e \<and> (\<exists>x p y. pre_digraph.gen_iapath (with_proj G) V x p y \<and> e \<in> set p)) \<and> (\<forall>u v p q. pre_digraph.gen_iapath (with_proj G) V u p v \<and> pre_digraph.gen_iapath (with_proj G) V u q v \<longrightarrow> p = q) \<and> V \<subseteq> pverts G"
TA: fixes f :: "nat \<Rightarrow> 'o"   and n :: "nat" shows "(\<Sum>i<n. T (f i) (state f i)) \<le> sum (A f) {..<n}"
of_nat_add: fixes m :: "nat"   and n :: "nat" shows "of_nat (m + n) = of_nat m + of_nat n"
a_aux_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a_aux a b c = a \<sqinter> e_aux a b c \<squnion> d_aux a b c"
map_of_zip_is_None: fixes xs :: "'b list"   and ys :: "'c list"   and x :: "'b" assumes "length xs = length ys" shows "(map_of (zip xs ys) x = None) = (x \<notin> set xs)"
wf_eq_minimal2: fixes r :: "('b \<times> 'b) set" shows "wf r = (\<forall>A. A \<subseteq> Field r \<and> A \<noteq> {} \<longrightarrow> (\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r))"
ereal_limsup_lim_add: fixes u :: "nat \<Rightarrow> ereal"   and a :: "ereal"   and v :: "nat \<Rightarrow> ereal" assumes "u \<longlonglongrightarrow> a"   and "\<bar>a\<bar> \<noteq> \<infinity>" shows "limsup (\<lambda>n. u n + v n) = a + limsup v"
iterative_override_on: shows "(\<mu> \<rho>'. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>')(x := Abs_cfun (ESem_syn e)\<cdot>\<rho>')) = (\<mu> \<rho>'. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<mu> \<rho>''. \<rho>' ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>''))(x := Abs_cfun (ESem_syn e)\<cdot>\<rho>'))"
point_mult_closed: fixes a :: "'b"   and b :: "'b"   and p :: "'b point"   and n :: "nat" assumes "on_curve a b p" shows "on_curve a b (point_mult a n p)"
U_sub_W: fixes x :: "'a"   and i :: "nat" assumes "x \<in> U i" shows "x \<in> W i"
IfTrue: fixes e :: "exp"   and s :: "loc \<Rightarrow> nat"   and s' :: "loc \<Rightarrow> nat"   and eval :: "((exp \<times> (loc \<Rightarrow> nat)) \<times> nat \<times> (loc \<Rightarrow> nat)) set"   and c0 :: "com"   and s1 :: "loc \<Rightarrow> nat"   and c1 :: "com" assumes "((e, s), 0, s') \<in> eval"   and "(c0, s') -[eval]-> s1" shows "(IF e THEN c0 ELSE c1, s) -[eval]-> s1"
has_integral_is_0: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> S \<Longrightarrow> f x = (0::'d)" shows "(f has_integral (0::'d)) S"
Min_eq_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and m :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(linorder.Min less_eq A = m) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq m a))"
cSup_finite_ex: fixes X :: "'b set" assumes "finite X"   and "X \<noteq> {}" shows "\<exists>x\<in>X. Sup X = x"
complete_game_return_true: fixes h :: "'g"   and w :: "'h"   and e :: "'e" assumes "(h, w) \<in> Rel0"   and "\<Sigma>0.completeness"   and "\<forall>h w. lossless_spmf (init0 h w)"   and "\<forall>r w e. lossless_spmf (response0 r w e)"   and "e \<in> carrier L" shows "\<Sigma>0.completeness_game h w e = return_spmf True"
Z_def: shows "Z = (\<lambda>t n x. Max {u i x |i. i < s} + (\<Sum>i<n. \<bar>u s ((T ^^ (i * s + t)) x)\<bar>) + birkhoff_sum F (n * s + t) x - d / 2 * real (card (V x \<inter> {1..n * s + t})))"
assms_2: shows "x \<in> space\<^sub>N N"
obtain_v1_v2_3: shows "disjnt (\<Union> (set lvs)) vs"
lQuot_epsilon: fixes a :: "'b" shows "lQuot a {[]} = {}"
iterate_Suc2: fixes n :: "nat"   and F :: "'c \<rightarrow> 'c"   and x :: "'c" shows "F\<^bsup>Suc n\<^esup>\<cdot>x = F\<^bsup>n\<^esup>\<cdot>(F\<cdot>x)"
list_ordered_it_correct: fixes m :: "'s" assumes "invar m" shows "map_iterator_linord (list_ordered_it m) (\<alpha> m)"
\<R>_def: shows "\<R> \<equiv> {region X I r |I r. valid_region X k I r}"
lesse2: shows "lnorm (lebesgue_on S) p (f - h) < e / 2"
inverse: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'b)"   and "f \<in> o[F](g)" shows "(\<lambda>x. inverse (g x)) \<in> o[F](\<lambda>x. inverse (f x))"
CAs_p: shows "ground_resolution_with_selection.ord_resolve S CAs DA AAs As E"
applicableUnitPropagate_def: fixes state :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" shows "applicableUnitPropagate state F0 decisionVars \<equiv> \<exists>state'. appliedUnitPropagate state state' F0 decisionVars"
get_owner_document_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (get_owner_document ptr) h"
prod_prime_factors': fixes n :: "nat" assumes "0 < n" shows "f n = (\<Prod>p\<in>prime_factors n. f_prime_power p (multiplicity p n))"
le_infI: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x a"   and "less_eq x b" shows "less_eq x (inf a b)"
solutions_sound: shows "set (solutions a b) \<subseteq> local.Minimal_Solutions"
le_int_of_uint32: fixes x :: "uint32"   and y :: "uint32" shows "(x \<le> y) = (int_of_uint32 x \<le> int_of_uint32 y)"
25: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
proper_it'I: fixes it :: "'f \<Rightarrow> ('g \<Rightarrow> bool) \<Rightarrow> ('h \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and it' :: "'f \<Rightarrow> ('i \<Rightarrow> bool) \<Rightarrow> ('h \<Rightarrow> 'i \<Rightarrow> 'i) \<Rightarrow> 'i \<Rightarrow> 'i" assumes "\<And>s. proper_it (it s) (it' s)" shows "proper_it' it it'"
prems_1: shows "0 < degree (prod_list (x # P))"
os_ts__sams: fixes B :: "'p"   and P :: "'p"   and A :: "'p"   and C :: "'p" assumes "B P TS A C"   and "A B OS P C" shows "SAMS A B P P B C"
dvd_triv_right: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a (times b a)"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G"   and "finite_group_axioms G" shows "finite_group G"
topspace_def: fixes T :: "'a topology" shows "topspace T = \<Union> {S. openin T S}"
sorted_ins_list: fixes xs :: "'a list"   and x :: "'a" assumes "Sorted_Less.sorted xs" shows "Sorted_Less.sorted (ins_list x xs)"
g'_def: fixes \<G> :: "'a cyclic_group"   and x :: "nat" assumes "okamoto_base \<G>" shows "okamoto_base.g' \<G> x = \<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> x"
r2s_ad_hom: fixes R :: "('b \<times> 'b) set" assumes "R \<subseteq> Id" shows "r2s (rel_n R) = - r2s R"