hcju/isabelleps huggingface dataset:
next_thread_eq_SomeD: fixes seed :: "natural \<times> natural"   and active :: "'s_t"   and t :: "'t"   and seed' :: "natural \<times> natural" assumes "next_thread seed active = \<lfloor>(t, seed')\<rfloor>"   and "thr'_invar active"   and "True" shows "t \<in> thr'_\<alpha> active"
trivial_init_filtrated_prob_space_axioms: shows "trivial_init_filtrated_prob_space M G"
ereal_divide_Infty_1: fixes x :: "ereal" shows "x / \<infinity> = 0"
Arccos_0: shows "Arccos 0 = complex_of_real (pi / 2)"
tm_comp_length: fixes A :: "(action \<times> nat) list"   and B :: "(action \<times> nat) list" shows "length (A |+| B) = length A + length B"
Rf_model: fixes I :: "'b set"   and N :: "'b literal multiset set" assumes "I \<Turnstile>s N - standard_redundancy_criterion.Rf N" shows "I \<Turnstile>s N"
affected_fixate_binary_circle: fixes f :: "'b perm"   and a :: "'b" assumes "order f a = 2" shows "affected (fixate a f) = affected f - {a, f \<langle>$\<rangle> a}"
MATCH1_matchers: fixes x :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)"   and y :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)" assumes "MATCH1 x y" shows "matchers_map (snd x) \<inter> matchers (set_mset (fst x)) = matchers_map (snd y) \<inter> matchers (set_mset (fst y))"
Red1FAcc: fixes h :: "'heap"   and a :: "'addr"   and D :: "String.literal"   and F :: "String.literal"   and v :: "'addr val"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and xs :: "'addr val list" assumes "heap_read h a (CField D F) v" shows "uf,P,t \<turnstile>1 \<langle>addr a\<bullet>F{D},(h, xs)\<rangle> -\<lbrace>ReadMem a (CField D F) v\<rbrace>\<rightarrow> \<langle>Val v,(h, xs)\<rangle>"
BIT_init_def: fixes init :: "'a list" shows "BIT_init init = map_pmf (\<lambda>l. (l, init)) (bv (length init))"
lambda_strict: shows "(\<lambda>x. \<bottom>) = \<bottom>"
EV_disj: fixes P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(EV P \<or> EV Q) = EV (\<lambda>n. P n \<or> Q n)"
monic_power: fixes p :: "'c poly"   and n :: "nat" assumes "monic p" shows "monic (p ^ n)"
cross7_components_6: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "(x \<times>\<^sub>7 y) $ 6 = x $ 1 * y $ 5 - x $ 5 * y $ 1 + x $ 7 * y $ 2 - x $ 2 * y $ 7 + x $ 3 * y $ 4 - x $ 4 * y $ 3"
shd_sset: fixes a :: "'a stream" shows "shd a \<in> i.R a"
neg_def: fixes \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" shows "\<not>\<^sup>c \<gamma> \<equiv> \<lambda>t n. \<not> \<gamma> t n"
start_point_zero_point: fixes x :: "'a" assumes "path x" shows "start_points x = 0 \<or> is_point (start_points x)"
ctr: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c stream"   and a :: "'b" shows "sscan f xs a = f (shd xs) a ## sscan f (stl xs) (f (shd xs) a)"
pq_ex: shows "(\<exists>x. Ifm (real_of_int x # bs) p) = (\<exists>x. Ifm (real_of_int x # bs) (fst (unit p)))"
continuous_on_translation_eq: fixes A :: "'b set"   and a :: "'c"   and g :: "'b \<Rightarrow> 'c" shows "continuous_on A ((+) a \<circ> g) = continuous_on A g"
rep_eq: fixes x :: "('a, 'b) ffun" shows "ffun_graph x = pfun_graph (pfun_of x)"
simps_1: fixes x1 :: "'b match_expr"   and x2 :: "action"   and y1 :: "'b match_expr"   and y2 :: "action" shows "(Rule x1 x2 = Rule y1 y2) = (x1 = y1 \<and> x2 = y2)"
neq_if_length_neq: fixes xs :: "'a list"   and ys :: "'a list" assumes "length xs \<noteq> length ys" shows "xs = ys \<equiv> False"
cfs_monom_mult: fixes p :: "nat \<Rightarrow> 'a"   and a :: "'a"   and k :: "nat"   and n :: "nat" assumes "p \<in> carrier P"   and "a \<in> carrier R"   and "k < n" shows "(p \<otimes>\<^bsub>P\<^esub> monom P a n) k = \<zero>"
lfp_lowerbound: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and f :: "'g \<Rightarrow> 'g"   and A :: "'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq (f A) A" shows "less_eq (complete_lattice.lfp Inf less_eq f) A"
f_def: shows "f = (\<lambda>z. if z = 0 then 1 else z / (exp z - 1))"
Gen_Shleg_altdef: fixes k :: "nat"   and n :: "nat"   and x :: "real" assumes "k \<le> n" shows "Gen_Shleg n k x = (\<Sum>i\<le>k. (- 1) ^ (k - i) * real (k choose i) * real_of_int (int (pochhammer (n - i + 1) i * pochhammer (n - k + i + 1) (k - i))) * x ^ (n - i) * (1 - x) ^ (n - k + i))"
swap_nilpotent: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "Fun.swap a b (Fun.swap a b f) = f"
lem2: shows "((\<lambda>x. \<Sum>b\<in>basis. f ((g1 +++ g2) x) \<bullet> b * (vector_derivative (g1 +++ g2) (at x within {0..1}) \<bullet> b)) has_integral i2) {1 / 2..1}"
ls_rehash_def: fixes l :: "('a \<times> 'b) list list" shows "ls_rehash l = ls_copy (length l) l (replicate (2 * length l) [])"
inv_closed: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "field R"   and "x \<in> carrier R"   and "x \<noteq> \<zero>\<^bsub>R\<^esub>" shows "inv\<^bsub>R\<^esub> x \<in> carrier R"
interaction_bound_lift_spmf: fixes "consider" :: "'d \<Rightarrow> bool"   and p :: "'e spmf" shows "interaction_bound consider (lift_spmf p) = 0"
safety_closed_iff: fixes P :: "'a llist set" assumes "P \<subseteq> A\<^sup>\<infinity>" shows "safety A P = (P closed)"
factorize_polys_has_factors: fixes ps :: "rat poly list"   and ftrs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and p :: "rat poly"   and c :: "rat"   and fs :: "(rat poly \<times> nat) list" assumes "factorize_polys ps = (ftrs, data)"   and "p \<in> set ps"   and "factorize_rat_poly_monic p = (c, fs)" shows "set (map fst fs) \<subseteq> set ftrs"
units_eq: fixes e1 :: "'c"   and e2 :: "'c" assumes "e1 \<in> E"   and "e2 \<in> E"   and "D e1 e2" shows "e1 = e2"
inf_sup_ord_3: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y \<le> x"
restrict_space_sets_cong: fixes A :: "'c set"   and B :: "'c set"   and M :: "'c measure"   and N :: "'c measure" assumes "A = B"   and "sets M = sets N" shows "sets (restrict_space M A) = sets (restrict_space N B)"
Step: fixes A :: "('fun, 'var) extstrand_step list"   and x :: "('fun, 'var) strand_step" assumes "well_analyzed A" shows "well_analyzed (A @ [Step x])"
finite_label_def: fixes l :: "('c, 'd) label" shows "finite_label l \<equiv> case l of Skip \<Rightarrow> True | Assume e \<Rightarrow> finite (Bexp.vars e) | Assign x e \<Rightarrow> finite (Aexp.vars e)"
is_valid_problem_strips_operator_variable_sets_2: fixes \<Pi> :: "'a strips_problem"   and op :: "'a strips_operator" assumes "is_valid_problem_strips \<Pi>"   and "op \<in> set (\<Pi>\<^sub>\<O>)" shows "set (add_effects_of op) \<subseteq> set (\<Pi>\<^sub>\<V>)"
t_find_closest_bound: fixes p :: "int \<times> int"   and ps :: "(int \<times> int) list"   and \<delta> :: "real"   and ps\<^sub>L :: "(int \<times> int) set"   and ps\<^sub>R :: "(int \<times> int) set"   and l :: "int" assumes "distinct (p # ps)"   and "sorted_snd (p # ps)"   and "0 \<le> \<delta>"   and "set (p # ps) = ps\<^sub>L \<union> ps\<^sub>R"   and "\<forall>p'\<in>set (p # ps). real_of_int l - \<delta> < real_of_int (fst p') \<and> real_of_int (fst p') < real_of_int l + \<delta>"   and "\<forall>p\<in>ps\<^sub>L. fst p \<le> l"   and "\<forall>p\<in>ps\<^sub>R. l \<le> fst p"   and "sparse \<delta> ps\<^sub>L"   and "sparse \<delta> ps\<^sub>R" shows "t_find_closest p \<delta> ps \<le> 8"
nth_prime_gt_zero: fixes n :: "nat" shows "0 < nth_prime n"
fst_rec2_pair: fixes ga :: "('d, 'd IF1 \<times> 'f, 'd IF2 \<times> 'e) F1 \<Rightarrow> 'f"   and g :: "('d, 'd IF1 \<times> 'f, 'd IF2 \<times> 'e) F2 \<Rightarrow> 'e" shows "fst \<circ> fold2 <ctor1 \<circ> F1map id fst fst , ga> <ctor2 \<circ> F2map id fst fst , g> = id"
map_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and v :: "'d tree" shows "map_tree g (map_tree f v) = map_tree (g \<circ> f) v"
rassocl_inverse: shows "lassocr\<^sub>C \<odot> rassocl\<^sub>C = 1\<^sub>C"
a_comm: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "x \<oplus>\<^bsub>P\<^esub> y = y \<oplus>\<^bsub>P\<^esub> x"
mat_of_row_dim_2: fixes y :: "'c vec" shows "dim_col (mat_of_row y) = dim_vec y"
lprod_2_4: fixes a :: "'b"   and b :: "'b"   and R :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> R" shows "([c, a], [c, b]) \<in> lprod R"
zip_option_intros_1: fixes ys :: "'a list"   and xs :: "'b list"   and zs :: "('b \<times> 'a) list" assumes "length ys = length xs"   and "length zs = length xs"   and "zs = zip xs ys" shows "zip_option xs ys = Some zs"
that: fixes v :: "'t"   and s :: "'a" assumes "v \<in> keys r"   and "s \<in> Keys (set fs)"   and "t = pp_of_term v + s" shows "thesis"
deg_pp_eq_0_iff: fixes s :: "('b, 'a) pp" shows "(deg_pp s = (0::'a)) = (s = 0)"
halt_lemma2: fixes LE :: "('a \<times> 'a) set"   and Q :: "'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "wf LE"   and "Q (f 0)"   and "\<not> P (f 0)"   and "\<forall>n. \<not> P (f n) \<and> Q (f n) \<longrightarrow> Q (f (Suc n)) \<and> (f (Suc n), f n) \<in> LE" shows "\<exists>n. P (f n) \<and> Q (f n)"
eq_commute: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (b = a)"
nn_integral_bij_count_space: fixes g :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and f :: "'d \<Rightarrow> ennreal" assumes "bij_betw g A B" shows "\<integral>\<^sup>+ x. f (g x) \<partial>count_space A = integral\<^sup>N (count_space B) f"
ordering2_def: fixes f :: "nat \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a set" shows "ordering2 f ord X \<equiv> (\<forall>n. (finite X \<longrightarrow> n < card X) \<longrightarrow> f n \<in> X) \<and> (\<forall>x\<in>X. \<exists>n. (finite X \<longrightarrow> n < card X) \<and> f n = x) \<and> (\<forall>n n' n''. (finite X \<longrightarrow> n'' < card X) \<and> Suc n = n' \<and> Suc n' = n'' \<longrightarrow> ord (f n) (f n') (f n''))"
pred_set: fixes P :: "'a \<Rightarrow> bool" shows "list_all P = (\<lambda>x. Ball (set x) P)"
list_update_swap: fixes i :: "nat"   and i' :: "nat"   and xs :: "'d list"   and x :: "'d"   and x' :: "'d" assumes "i \<noteq> i'" shows "xs[i := x, i' := x'] = xs[i' := x', i := x]"
map_simp: fixes f :: "'b" assumes "J.arr f" shows "J.map f = f"
IF2rel_F2rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "('d, 'd IF1, 'd IF2) F2"   and b :: "('e, 'e IF1, 'e IF2) F2" shows "IF2rel R (ctor2 a) (ctor2 b) = F2rel R (IF1rel R) (IF2rel R) a b"
S: shows "S \<subseteq> {x. a \<bullet> x \<le> b}"
ortho: fixes w :: "'a vec"   and x :: "'a vec" assumes "w \<in> W"   and "x \<in> X" shows "x \<bullet> w = (0::'a)"
K_B: fixes n :: "nat" shows "K n \<subseteq> B n"
base_le: fixes a :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "a \<in> s" shows "b j \<le> a j"
zip_eq_conv: fixes xs :: "'c list"   and ys :: "'d list"   and zs :: "('c \<times> 'd) list" assumes "length xs = length ys" shows "(zip xs ys = zs) = (map fst zs = xs \<and> map snd zs = ys)"
sum_upto_moebius_mu_over_n_eq: fixes x :: "real" assumes "x < 2" shows "M x = (if 1 \<le> x then 1 else 0)"
indRelRTPO_impl_TRel_is_strong_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "\<forall>P Q P'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q')" shows "strong_reduction_simulation (TRel\<^sup>+) Target"
split_sel: fixes P :: "'a \<Rightarrow> bool"   and f :: "exesignature \<Rightarrow> Core.term list \<Rightarrow> 'a"   and exetheory :: "exetheory" shows "P (case exetheory of ExeTheory x xa \<Rightarrow> f x xa) = (exetheory = ExeTheory (exesig exetheory) (exeaxioms_of exetheory) \<longrightarrow> P (f (exesig exetheory) (exeaxioms_of exetheory)))"
H_elem_map_carrier: fixes H1 :: "'a set" assumes "H1 \<in> rcosets H" shows "(SOME x. x \<in> carrier G \<and> H #> x = H1) \<in> carrier G"
induct: fixes P :: "'c \<Rightarrow> (('c \<times> 'd) \<times> color) tree \<Rightarrow> bool"   and a0 :: "'c"   and a1 :: "(('c \<times> 'd) \<times> color) tree" assumes "\<And>x. P x \<langle>\<rangle>"   and "\<And>x l a b c r. \<lbrakk>\<lbrakk>cmp x a = LT; l \<noteq> \<langle>\<rangle> \<and> color l = Black\<rbrakk> \<Longrightarrow> P x l; \<lbrakk>cmp x a = LT; \<not> (l \<noteq> \<langle>\<rangle> \<and> color l = Black)\<rbrakk> \<Longrightarrow> P x l; \<lbrakk>cmp x a = GT; r \<noteq> \<langle>\<rangle> \<and> color r = Black\<rbrakk> \<Longrightarrow> P x r; \<lbrakk>cmp x a = GT; \<not> (r \<noteq> \<langle>\<rangle> \<and> color r = Black)\<rbrakk> \<Longrightarrow> P x r\<rbrakk> \<Longrightarrow> P x \<langle>l, ((a, b), c), r\<rangle>" shows "P a0 a1"
assms: shows "\<forall>S\<subseteq>topspace X. f ` (X closure_of S) \<subseteq> Y closure_of f ` S"
induct: fixes P :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and a0 :: "'a option"   and a1 :: "'a option" assumes "\<And>x y. P (Some x) (Some y)"   and "\<And>y. P None y"   and "\<And>uu_. P (Some uu_) None" shows "P a0 a1"
dirichlet_prod_cong: fixes f :: "nat \<Rightarrow> 'a"   and f' :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and g' :: "nat \<Rightarrow> 'a" assumes "\<And>n. 0 < n \<Longrightarrow> f n = f' n"   and "\<And>n. 0 < n \<Longrightarrow> g n = g' n" shows "dirichlet_prod f g = dirichlet_prod f' g'"
update_cons_val: fixes x :: "'h"   and xs :: "'h list"   and opt :: "('f, 'g, 'h) env option"   and a :: "'f" shows "update (x # xs) opt (Val a) = Val a"
pelims: fixes x :: "complex \<times> complex"   and xa :: "complex \<times> complex"   and y :: "complex" assumes "x *\<^sub>v\<^sub>v xa = y"   and "mult_vv_dom (x, xa)" obtains xb :: "complex" and ya :: "complex" and a :: "complex" and b :: "complex" where "x = (xb, ya)" and "xa = (a, b)" and "y = xb * a + ya * b" and "mult_vv_dom ((xb, ya), a, b)"
goalTrue: shows "\<forall>ea\<in>E - (set (e # accu) \<union> set Es). \<not> all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = E, flows_state = {ea} \<union> set (e # accu)\<rparr>)"
remove_max_multiset': fixes l :: "'b"   and m :: "'a"   and l' :: "'b" assumes "\<not> is_empty l"   and "(m, l') = remove_max l" shows "add_mset m (multiset l') = multiset l"
cut_le_mem_iff: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" shows "(x \<in> I \<down>\<le> t) = (x \<in> I \<and> x \<le> t)"
vV: shows "v = #(#v + Polynomial.smult (of_nat q) (#v'))"
the_wall_betw_nempty_2: fixes C :: "'a set"   and D :: "'a set" assumes "THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>}}) \<and> separated_by H C D) \<noteq> {}" shows "separated_by (THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>}}) \<and> separated_by H C D)) C D"
inverse_prod_list: fixes xs :: "'b list" shows "inverse (prod_list xs) = prod_list (map inverse xs)"
oalist_inv_rawI: fixes xs :: "('a \<times> 'd) list" assumes "(0::'d) \<notin> snd ` set xs"   and "sorted_wrt lt (map fst xs)" shows "oalist_inv_raw xs"
backtrack_s_def: shows "backtrack_s = Incremental_State_Ops_Simplex.backtrack_s"
pw_ASSERT_2: fixes \<Phi> :: "bool"   and x :: "unit" shows "inres (ASSERT \<Phi>) x"
check_FSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "FSbImorph check check (errMOD MOD) MOD"
assms: shows "pp_pm.is_GB_cofactor_bound (Poly_Mapping.map_key PP ` set fs) b"
simps_86: fixes f1 :: "char list \<Rightarrow> 'c trm list \<Rightarrow> 'b"   and f2 :: "'c trm \<Rightarrow> 'c trm \<Rightarrow> 'b"   and f3 :: "'c formula \<Rightarrow> 'b"   and f4 :: "'c formula \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f5 :: "'c formula \<Rightarrow> 'b"   and f6 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f7 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f8 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f9 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and x5 :: "'c formula" shows "(case Exists x5 of Pred x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa | Neg x \<Rightarrow> f3 x | Or x xa \<Rightarrow> f4 x xa | Exists x \<Rightarrow> f5 x | Prev x xa \<Rightarrow> f6 x xa | Next x xa \<Rightarrow> f7 x xa | Since x xa xb \<Rightarrow> f8 x xa xb | Until x xa xb \<Rightarrow> f9 x xa xb) = f5 x5"
simps_2: fixes ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" shows "nonempty_lasttreebal (Node ts t) = ((\<exists>ls tsub tsep. ts = ls @ [(tsub, tsep)] \<and> height tsub = height t) \<and> nonempty_lasttreebal t)"
map_of_set_simps_1: fixes s :: "'a set" shows "dom (map_of_set s) = s"
h: shows "h \<in> K \<rightarrow>\<^sub>M stock_measure (PRODUCT t1 t2)"
lifted_map_def: shows "lifted_map \<equiv> \<Lambda> a. \<mu> uu. lifted_abs oo u_map\<cdot>a oo lifted_rep"
subprob_space_def: fixes M :: "'b measure" shows "subprob_space M \<equiv> finite_measure M \<and> subprob_space_axioms M"
gim: shows "g ` rel_interior T \<subseteq> rel_interior S"
rel_mset_Plus: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e"   and M :: "'d multiset"   and N :: "'e multiset" assumes "R a b"   and "rel_mset R M N" shows "rel_mset R (add_mset a M) (add_mset b N)"
induct: fixes P :: "Ferrack.num \<Rightarrow> bool"   and a0 :: "Ferrack.num" assumes "\<And>i. P (C i)"   and "\<And>n c t. (\<And>x. P t) \<Longrightarrow> P (CN n c t)"   and "\<And>v. P (Bound v)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)" shows "P a0"
edge_rev_in_arcs: fixes a :: "'b" shows "(rev_H a \<in> arcs H) = (a \<in> arcs H)"
precondition_ind_def: fixes s :: "'state_t"   and execs :: "'dom_t \<Rightarrow> 'action_t list list" shows "precondition_ind s execs \<equiv> invariant s \<and> (\<forall>d. fst (control s d (execs d)) \<rightharpoonup> AS_precondition s d)"
exhaust: fixes y :: "Literal" obtains x1 :: "nat" where "y = Pos x1"   | x2 :: "nat" where "y = Neg x2"
ll_on_open_axioms_def: fixes T :: "real set"   and f :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and X :: "'c set" shows "ll_on_open_axioms T f X \<equiv> (local_lipschitz T X f \<and> (\<forall>x. x \<in> X \<longrightarrow> continuous_on T (\<lambda>t. f t x))) \<and> open T \<and> open X"
Or_ltlp_def: fixes \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "\<phi> or\<^sub>p \<psi> \<equiv> not\<^sub>p \<phi> implies\<^sub>p \<psi>"
right_cancel: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(y + x = z + x) = (y = z)"
calc_atm_1: fixes a :: "name"   and b :: "name"   and pi :: "(name \<times> name) list"   and x :: "name" shows "((a, b) # pi) \<bullet> x = swap (a, b) (pi \<bullet> x)"
map_le_trans: fixes m1 :: "'c \<Rightarrow> 'd option"   and m2 :: "'c \<Rightarrow> 'd option"   and m3 :: "'c \<Rightarrow> 'd option" assumes "m1 \<subseteq>\<^sub>m m2"   and "m2 \<subseteq>\<^sub>m m3" shows "m1 \<subseteq>\<^sub>m m3"
valid_def: fixes \<phi> :: "('sf, 'sc, 'sz) formula" shows "valid \<phi> \<equiv> \<forall>I \<nu>. is_interp I \<longrightarrow> \<nu> \<in> fml_sem I \<phi>"
subclsEval: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog"   and c :: "class" assumes "G\<turnstile>C\<prec>\<^sub>C D"   and "ws_prog G"   and "class G C = Some c"   and "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object\<rbrakk> \<Longrightarrow> \<exists>sc. class G (super c) = Some sc" shows "D \<in> superclasses G C"
trace_id: fixes n :: "nat" shows "trace (1\<^sub>m n) = int n"
assms: shows "aEvalUni (LeqUni (a, b, c)) x \<noteq> aEvalUni (LeqUni (a, b, c)) y \<and> x < y"
A6': fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" assumes "B\<^sub>\<real> a b a" shows "a = b"
lset_induct: fixes x :: "'a"   and xs :: "'a llist"   and P :: "'a llist \<Rightarrow> bool" assumes "x \<in> lset xs"   and "\<And>xs. P (LCons x xs)"   and "\<And>x' xs. \<lbrakk>x \<in> lset xs; x \<noteq> x'; P xs\<rbrakk> \<Longrightarrow> P (LCons x' xs)" shows "P xs"
IH: assumes "\<exists>!x. x \<in> set xs \<and> P x" shows "(THE x. x \<in> set xs \<and> P x) = hd (filter P xs)"
Abinds_env_cong: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae' :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "\<And>x. x \<in> domA \<Delta> \<Longrightarrow> ae x = ae' x" shows "ArityAnalysis.ABinds Aexp \<Delta>\<cdot>ae = ArityAnalysis.ABinds Aexp \<Delta>\<cdot>ae'"
closed_Collect_le: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on UNIV f"   and "continuous_on UNIV g" shows "closed {x. f x \<le> g x}"
min_int_poly_represents: fixes x :: "'b" assumes "algebraic x" shows "min_int_poly x represents x"
sqrt_aform_err: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and p :: "nat"   and Y :: "(real \<times> real pdevs) \<times> real" assumes "x \<in> aform_err e X"   and "sqrt_aform_err p X = Some Y"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "sqrt x \<in> aform_err e Y"
E: fixes n :: "nat" shows "measurable_envelope M (A n) (E n)"
eq_paths: fixes P :: "'a set"   and Q :: "'a set"   and a :: "'a"   and b :: "'a" assumes "P \<in> \<P>"   and "Q \<in> \<P>"   and "a \<in> P"   and "b \<in> P"   and "a \<in> Q"   and "b \<in> Q"   and "a \<noteq> b" shows "P = Q"
composed_prot_def: fixes \<P>\<^sub>i :: "'a \<Rightarrow> ('a strand_label \<times> 'b) list set" shows "composed_prot \<P>\<^sub>i \<equiv> {\<A>. \<forall>n. proj n \<A> \<in> \<P>\<^sub>i n}"
locally_sound_modes_invariant: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and c :: "'c"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "'c"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.locally_sound_mode_use dma \<C>_vars eval ((c, mds), mem)"   and "(((c, mds), mem), (c', mds'), mem') \<in> eval" shows "sifum_security_init.locally_sound_mode_use dma \<C>_vars eval ((c', mds'), mem')"
pt_freshs_freshs: fixes pi :: "('e \<times> 'e) list"   and Xs :: "'e set"   and Ys :: "'e set"   and x :: "'d" assumes "pt TYPE('d) TYPE('e)"   and "at TYPE('e)"   and "set pi \<subseteq> Xs \<times> Ys"   and "Xs \<sharp>* x"   and "Ys \<sharp>* x" shows "pi \<bullet> x = x"
det_fold_set_def: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g \<Rightarrow> 'g"   and \<sigma> :: "'g"   and result :: "'f set \<Rightarrow> 'g" shows "det_fold_set R c f \<sigma> result \<equiv> \<forall>l. distinct l \<and> sorted_wrt R l \<longrightarrow> foldli l c f \<sigma> = result (set l)"
eq_id_iff: fixes f :: "'c \<Rightarrow> 'c" shows "(\<forall>x. f x = x) = (f = id)"
coboundedI: fixes A :: "'c DBMEntry set"   and a :: "'c DBMEntry" assumes "finite A"   and "a \<in> A" shows "linordered_monoid.Min A \<preceq> a"
group_iso_betw_hom: fixes G1 :: "('c, 'd) monoid_scheme"   and G2 :: "('e, 'f) monoid_scheme"   and i :: "'c \<Rightarrow> 'e"   and H :: "('g, 'h) monoid_scheme" assumes "Group.group G1"   and "Group.group G2"   and "i \<in> Group.iso G1 G2" shows "\<exists>f. bij_betw f (homr G2 H) (homr G1 H)"
cases: fixes a :: "enat" assumes "enat_setp a" obtains "a = 0"   | n :: "enat" where "a = eSuc n" and "enat_setp n"