hcju/isabelleps huggingface dataset:
divides_cong_l: fixes x :: "'a"   and x' :: "'a"   and y :: "'a" assumes "x \<sim> x'"   and "x' divides y"   and "x \<in> carrier G" shows "x divides y"
hyps_2: shows "u \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
simps_10: fixes i :: "int"   and c :: "int"   and e :: "MIR.num" shows "\<zeta> (NDvd i (CN 0 c e)) = c"
lex_3: shows "Counterclockwise_2D_Arbitrary.lex r s"
submoduleE_5: fixes H :: "(nat \<Rightarrow> 'a) set"   and aa :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a" assumes "submodule H R P"   and "aa \<in> H"   and "b \<in> H" shows "aa \<oplus>\<^bsub>P\<^esub> b \<in> H"
conv_terminating_path: fixes x :: "'a" shows "terminating_path x = terminating_path (x\<^sup>T)"
disj_def: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "P \<or> Q \<equiv> \<lambda>A. P A \<or> Q A"
simps_10: fixes p :: "nat"   and a :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (floatarith.Floor a) vs = approx_un p (\<lambda>ivl. Some (floor_float_interval ivl)) (approx_floatarith p a vs)"
nonzero_memE_1: fixes a :: "'a" assumes "a \<in> nonzero R" shows "a \<in> carrier R"
PiM_def: fixes I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" shows "Pi\<^sub>M I M = extend_measure (\<Pi>\<^sub>E i\<in>I. space (M i)) {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} (\<lambda>(J, X). prod_emb I M J (Pi\<^sub>E J X)) (\<lambda>(J, X). \<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
intro: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme" assumes "h \<in> ring_hom R S" shows "ring_hom_ring_axioms R S h"
abs_eq: fixes xa :: "'a"   and x :: "'b poly" shows "pm_of_poly xa x = Abs_poly_mapping (\<lambda>t. poly.coeff x (lookup t xa) when t \<in> .[{xa}])"
inverse_arrows_compose: fixes g :: "'b"   and f :: "'b"   and f' :: "'b"   and g' :: "'b" assumes "B.seq g f"   and "B.inverse_arrows f f'"   and "B.inverse_arrows g g'" shows "B.inverse_arrows (B g f) (B f' g')"
D: shows "distributed M lborel X (\<lambda>x. ennreal (std_normal_density x))"
E_wf_1: fixes G :: "'c graph" assumes "wf_graph G" shows "fst ` edges G \<subseteq> nodes G"
tauChainStepCons: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'" obtains P'' :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> \<tau>.P \<Longrightarrow>\<^sub>\<tau> P''" and "\<Psi> \<rhd> P' \<sim> P''"
clause_saturated_and_inference_saturated: fixes S :: "'a eclause set" assumes "clause_saturated S"   and "\<forall>x\<in>S. finite (cl_ecl x)" shows "inference_saturated S"
cf_set_def: fixes cf :: "nat \<times> nat \<Rightarrow> int"   and e :: "nat \<times> nat"   and cap :: "int" shows "cf_set cf e cap \<equiv> assert (case e of (u, v) \<Rightarrow> u \<in> V \<and> v \<in> V) \<bind> (\<lambda>_. return (cf(e := cap)))"
assms_3: fixes x :: "'a"   and y :: "'a" assumes "f x = f y"   and "g x = g y" shows "x = y"
resolution_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and P :: "'b set multiset multiset" assumes "incidence_system point_set block_collection" shows "incidence_system.resolution point_set block_collection P = (partition_on_mset block_collection P \<and> (\<forall>S\<in>#P. distinct_mset S \<and> incidence_system.resolution_class point_set block_collection (set_mset S)))"
add_minus: fixes V :: "'c set"   and x :: "'c" assumes "vectorspace V"   and "x \<in> V" shows "x + - x = (0::'c)"
mult_smult_right: fixes p :: "'c poly"   and a :: "'c"   and q :: "'c poly" shows "p * smult a q = smult a (p * q)"
assms_1: shows "\<langle>M;P\<rangle> \<turnstile>\<^sub>r m"
cases: fixes a1 :: "nat"   and a2 :: "('c, 'd) bintree list" assumes "is_bintree_list a1 a2" obtains "a1 = 0" and "a2 = []"   | l :: "nat" and ts :: "('c, 'd) bintree list" and t :: "('c, 'd) bintree" where "a1 = Suc l" and "a2 = t # ts" and "is_bintree_list l ts" and "is_bintree_list l (children t)"
finite_extension_carrier: fixes xs :: "'a list" assumes "set xs \<subseteq> carrier R" shows "finite_extension (carrier R) xs = carrier R"
distinct_14: fixes x51 :: "pdf_type"   and x52 :: "pdf_type" shows "PRODUCT x51 x52 \<noteq> BOOL"
cases: fixes a :: "hmultiset" assumes "local.well_base\<^sub>h a" obtains M :: "hmultiset" where "a = M" and "Multiset.Ball (hmsetmset M) local.well_base\<^sub>h" and "local.well_base (hmsetmset M)"
open_Collect_conj: fixes "open" :: "'c set \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "class.topological_space open"   and "open {x. P x}"   and "open {x. Q x}" shows "open {x. P x \<and> Q x}"
valid_as_valid_exec: fixes as :: "(('c, bool) fmap \<times> ('c, bool) fmap) list"   and PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap" assumes "as \<in> valid_plans PROB"   and "s \<in> valid_states PROB" shows "exec_plan s as \<in> valid_states PROB"
effect_lookupI: fixes h' :: "heap"   and h :: "heap"   and x :: "'a"   and r :: "'a ref" assumes "h' = h"   and "x = Ref.get h r" shows "effect (!r) h h' x"
assms_3: fixes n :: "nat"   and a :: "'a" assumes "(n, a) \<in> set n_as"   and "norm a = 1" shows "n \<le> N"
get_child_nodes_def: shows "Shadow_DOM.get_child_nodes \<equiv> l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes"
cblinfun_apply_inverse: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "CBlinfun ((*\<^sub>V) x) = x"
0: fixes z :: "complex" assumes "z \<in> S" shows "f z \<noteq> 0"
evalCPS_def: fixes l :: "lambda" shows "\<PR> l = (let ve = Map.empty; \<beta> = Map.empty; f = \<A> (L l) \<beta> ve in \<F>\<cdot>(Discr (f, [Stop], ve, \<binit>)))"
surjective: fixes r :: "'a bdd_scheme" shows "r = \<lparr>dpm = dpm r, dcl = dcl r, \<dots> = bdd.more r\<rparr>"
moebius_similarity_lines_to_lines: fixes a :: "complex"   and b :: "complex"   and H :: "circline" assumes "a \<noteq> 0" shows "(\<infinity>\<^sub>h \<in> circline_set (moebius_circline (moebius_similarity a b) H)) = (\<infinity>\<^sub>h \<in> circline_set H)"
last_quotient: assumes "1 < length \<HH>" shows "last comp\<HH>.quotients = G Mod \<HH> ! (length \<HH> - 1 - 1)"
fresh_psubst: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs :: "'varSort"   and z :: "'var" assumes "good X"   and "goodEnv rho" shows "fresh zs z X #[rho] = (\<forall>ys y. fresh ys y X \<or> freshImEnvAt zs z rho ys y)"
zero_order_3: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and n :: "'e" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
leg1_uniquely_isomorphic_1: shows "s\<^sub>1 \<cong> u\<^sub>1 \<star> \<chi>.chine"
set_hyp: shows "sign \<in> set (characterize_consistent_signs_at_roots p qs)"
RGraph_Impl_def: fixes c :: "nat \<times> nat \<Rightarrow> int"   and s :: "nat"   and t :: "nat"   and N :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> int" shows "RGraph_Impl c s t N cf \<equiv> RGraph c s t cf \<and> Network_Impl c s t N"
isolate_variable_sparse_def: shows "isolate_variable_sparse \<equiv> map_fun id (map_fun id (map_fun id MPoly)) (\<lambda>mp x d. \<Sum>m | m \<in> monomials mp \<and> lookup m x = d. monomial (MPoly_Type.coeff mp m) (Poly_Mapping.update x 0 m))"
intros: fixes F :: "'a"   and basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real" assumes "is_expansion F basis"   and "\<forall>\<^sub>F x in at_top. eval F x = f x" shows "(f expands_to F) basis"
vec_cnj_mult_vm: fixes v :: "complex \<times> complex"   and A :: "complex \<times> complex \<times> complex \<times> complex" shows "vec_cnj (v *\<^sub>v\<^sub>m A) = vec_cnj v *\<^sub>v\<^sub>m mat_cnj A"
equiv_Int: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and B :: "'b set"   and s :: "('b \<times> 'b) set" assumes "equiv A r"   and "equiv B s" shows "equiv (A \<inter> B) (r \<inter> s)"
less_imp_of_nat_less: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "m < n" shows "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
lsided_prop: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" assumes "prod.lsd x" shows "less_eq_prod (times_prod' y x) x"
actionScopeDest: fixes \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi"   and \<beta> :: "'d action"   and x :: "name"   and Q :: "('d, 'e, 'f) psi" assumes "\<alpha> \<prec> P = \<beta> \<prec> \<lparr>\<nu>x\<rparr>Q"   and "x \<sharp> bn \<alpha>"   and "x \<sharp> bn \<beta>" obtains R :: "('d, 'e, 'f) psi" where "P = \<lparr>\<nu>x\<rparr>R" and "\<alpha> \<prec> R = \<beta> \<prec> Q"
assms_1: shows "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' y f"
ipt_iprange_compress_def: shows "ipt_iprange_compress = wi_2_cidr_ipt_iprange_list \<circ> ipt_iprange_negation_type_to_br_intersect"
dist_scaleR: fixes x :: "real"   and a :: "'a"   and y :: "real" shows "dist (x *\<^sub>R a) (y *\<^sub>R a) = \<bar>x - y\<bar> * norm a"
LowLink_eqI: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and s' :: "('c, 'e) state_scheme"   and w :: "'c"   and X :: "'c set" assumes "DFS_invar G param s"   and "DFS_invar G param s'"   and "discovered s \<subseteq>\<^sub>m discovered s'"   and "param_DFS_defs.lowlink_set G s w \<subseteq> param_DFS_defs.lowlink_set G s' w"   and "param_DFS_defs.lowlink_set G s' w \<subseteq> param_DFS_defs.lowlink_set G s w \<union> X"   and "w \<in> dom (discovered s)"   and "\<And>x. \<lbrakk>x \<in> X; x \<in> param_DFS_defs.lowlink_set G s' w\<rbrakk> \<Longrightarrow> Min (the_discovered s ` param_DFS_defs.lowlink_set G s w) \<le> the_discovered s' x" shows "Min (the_discovered s ` param_DFS_defs.lowlink_set G s w) = Min (the_discovered s' ` param_DFS_defs.lowlink_set G s' w)"
affine_parallel_expl: fixes S :: "'a set"   and T :: "'a set" shows "affine_parallel S T = (\<exists>a. \<forall>x. (x \<in> S) = (a + x \<in> T))"
from_nat_neq_rows: fixes i :: "nat"   and A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and k :: "nat" assumes "i < nrows A"   and "k < nrows A"   and "i \<noteq> k" shows "from_nat i \<noteq> from_nat k"
sum_list'_sum_list_le: fixes y :: "'a"   and xs :: "'a list"   and p :: "nat" assumes "y \<le> sum_list xs" shows "y \<le> sum_list' p xs"
continuous_on_id': fixes s :: "'b set" shows "continuous_on s id"
projection_on_subset2: fixes l :: "'a list"   and L :: "'a set"   and X' :: "'a set"   and X :: "'a set" assumes "set l \<subseteq> L"   and "l \<upharpoonleft> X' = []"   and "X \<inter> L \<subseteq> X'" shows "l \<upharpoonleft> X = []"
merge_exists: fixes as :: "nat list list"   and bs :: "nat list list" assumes "strict_sorted (concat as)"   and "strict_sorted (concat bs)"   and "as \<in> lists (- {[]})"   and "bs \<in> lists (- {[]})"   and "hd as < hd bs"   and "as \<noteq> []"   and "bs \<noteq> []"   and "\<And>a b. \<lbrakk>a \<in> list.set as; b \<in> list.set bs\<rbrakk> \<Longrightarrow> a < b \<or> b < a" shows "\<exists>us vs. merge as bs us vs"
not_eq_order_implies_strict: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "ordering less_eq less"   and "a \<noteq> b"   and "less_eq a b" shows "less a b"
x_eq: shows "s1 \<cdot> s \<cdot> fst x = s1 \<cdot> snd x \<cdot> r"
RBT_Impl_rbt_ex_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and rbt :: "('c, 'd) rbt" shows "RBT_Impl_rbt_ex P rbt = (\<exists>(k, v)\<in>set (RBT_Impl.entries rbt). P k v)"
order_refl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.preorder (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) x"
adv_P1_def: fixes x :: "'a \<times> 'a"   and y :: "bool"   and z :: "'e"   and S :: "(('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'e \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'f \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'g spmf) \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'e \<Rightarrow> (('a \<times> 'a) \<times> 'f) spmf) \<times> (('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'e \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'f \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'g spmf) \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'e \<Rightarrow> unit \<Rightarrow> 'f \<Rightarrow> 'g spmf)"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'e \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'f \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'g spmf)"   and D :: "'g \<times> 'a \<Rightarrow> bool spmf" shows "mal_def.adv_P1 x y z S \<A> D = \<bar>spmf (P1_real_model x y z \<A> \<bind> D) True - spmf (mal_def.ideal_view_1 x y z S \<A> \<bind> D) True\<bar>"
open_Un: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and T :: "'a set" assumes "class.topological_space open"   and "open S"   and "open T" shows "open (S \<union> T)"
surjective: fixes r :: "'a point_scheme" shows "r = \<lparr>Xcoord = Xcoord r, Ycoord = Ycoord r, \<dots> = more r\<rparr>"
elims: fixes x :: "'b heap list"   and y :: "'b heap" assumes "merge_pairs x = y" obtains "x = []" and "y = heap.Empty"   | h :: "'b heap" where "x = [h]" and "y = h"   | h1 :: "'b heap" and h2 :: "'b heap" and hs :: "'b heap list" where "x = h1 # h2 # hs" and "y = merge (merge h1 h2) (merge_pairs hs)"
Torder_def: fixes D :: "('c, 'd) Order_scheme" shows "Torder D \<equiv> Order D \<and> Torder_axioms D"
dist_fps_def: fixes a :: "'a fps"   and b :: "'a fps" shows "dist a b = (if a = b then 0 else inverse (2 ^ subdegree (a - b)))"
supp_finfun_const: fixes b :: "'b" shows "supp (finfun_const b) = supp b"
n_left_distrib_var: fixes p :: "'a"   and x :: "'a"   and y :: "'a" assumes "test p" shows "p \<cdot> (x + y) = p \<cdot> x + p \<cdot> y"
le_supI: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and x :: "'d"   and b :: "'d" assumes "class.semilattice_sup sup less_eq less"   and "less_eq a x"   and "less_eq b x" shows "less_eq (sup a b) x"
fbox_simp: fixes x :: "'a"   and y :: "'a" shows "antidomain_semiringl.fbox a (\<cdot>) x y = antidomain_semiringl.fbox a (\<cdot>) x (d y)"
Carmichael_odd_prime_power: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "odd p"   and "0 < k" shows "Carmichael (p ^ k) = p ^ (k - 1) * (p - 1)"
components_are_iso: fixes a :: "'a arrow_of_spans_data" assumes "ide a" shows "local.iso (src a)"
simps_2: fixes l :: "linear_poly"   and r :: "QDelta"   and v :: "nat" shows "qdelta_constraint_to_atom (GEQ_ns l r) v = (if is_monom l then monom_to_atom (GEQ_ns l r) else Geq v r)"
fold_Cons_commute: fixes x :: "'a"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>a b. \<lbrakk>a \<in> set (x # xs); b \<in> set (x # xs)\<rbrakk> \<Longrightarrow> (f a \<circ>\<circ> f) b = (f b \<circ>\<circ> f) a" shows "fold f (x # xs) = (f x \<circ>\<circ>\<circ> fold) f xs"
fps_nth_inject: fixes x :: "'a fps"   and y :: "'a fps" shows "(fps_nth x = fps_nth y) = (x = y)"
open_vimage_ProjD: fixes T :: "'b proj_space set" assumes "open (Proj -` T)" shows "open T"
LI_completeness: fixes S\<^sub>1 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>1 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>1 \<theta>\<^sub>1"   and "\<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>1,\<theta>\<^sub>1\<rangle>" shows "\<exists>S\<^sub>2 \<theta>\<^sub>2. (S\<^sub>1, \<theta>\<^sub>1) \<leadsto>\<^sup>* (S\<^sub>2, \<theta>\<^sub>2) \<and> simple S\<^sub>2 \<and> \<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>2,\<theta>\<^sub>2\<rangle>"
hausdorff_distanceI: fixes A :: "'b set"   and B :: "'b set"   and D :: "real" assumes "\<And>x. x \<in> A \<Longrightarrow> infdist x B \<le> D"   and "\<And>x. x \<in> B \<Longrightarrow> infdist x A \<le> D"   and "0 \<le> D" shows "hausdorff_distance A B \<le> D"
image_def: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "f ` A = {y. \<exists>x\<in>A. y = f x}"
wf_ci_ins_jump_ok: fixes ci :: "'a instr \<Rightarrow> (nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> 'b \<Rightarrow> 'a val list \<Rightarrow> 'a val list \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> nat \<Rightarrow> 'a frame list \<Rightarrow> bool"   and ins :: "'a instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'b"   and stk :: "'a val list"   and loc :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'a frame list"   and pc' :: "nat" assumes "wf_ci ci"   and "ci ins P h stk loc C M pc frs"   and "pc \<le> pc'" shows "ins_jump_ok ins pc'"
intros_4: fixes s :: "Core.term"   and t :: "Core.term"   and T :: "typ" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "Abs T s \<rightarrow>\<^sub>\<beta> Abs T t"
If: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and n :: "nat"   and b :: "val"   and s1 :: "state"   and c1 :: "stmt"   and c2 :: "stmt"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>(if the_Bool b then c1 else c2)\<midarrow>n\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>If(e) c1 Else c2\<midarrow>n\<rightarrow> s2"
rel_cases: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c sctn"   and b :: "'d sctn" assumes "rel_sctn R a b" obtains x1a :: "'c" and x2a :: "real" and y1a :: "'d" and y2a :: "real" where "a = Sctn x1a x2a" and "b = Sctn y1a y2a" and "R x1a y1a" and "x2a = y2a"
tau_c: shows "\<tau> 1\<^sub>\<pi> = 1\<^sub>\<pi>"
sum_coprime_conv_dirichlet_prod_moebius_mu: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> complex"   and n :: "nat" assumes "\<And>a b d. \<lbrakk>d dvd a; d dvd b\<rbrakk> \<Longrightarrow> f (a div d) (b div d) = f a b" shows "(\<Sum>k | k \<in> {1..n} \<and> coprime k n. f k n) = dirichlet_prod moebius_mu (\<lambda>n. \<Sum>k = 1..n. f k n) n"
Init: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and A :: "'b triple set"   and P :: "'b assn"   and Q :: "'b assn"   and R :: "'b assn" assumes "the (class G C) = c"   and "G,A\<turnstile>{Normal (P \<and>. Not \<circ> initd C ;. supd (init_class_obj G C))} .(if C = Object then Skip else Init (super c)). {Q}"   and "\<forall>l. G,A\<turnstile>{Q \<and>. (\<lambda>s. l = locals (snd s)) ;. set_lvars Map.empty} .init c. {set_lvars l .; R}" shows "G,A\<turnstile>{Normal (P \<and>. Not \<circ> initd C)} .Init C. {R}"
HMA_diagonal_to_Smith_aux_PQ_JNF: fixes xs :: "nat list" assumes "\<forall>j\<in>set xs. j < min CARD('b) CARD('c)" shows "((=) ===> HMA_M3 ===> HMA_M3) (diagonal_to_Smith_aux_PQ_JNF xs) (diagonal_to_Smith_aux_PQ xs)"
that: fixes T :: "'a set" assumes "open T \<and> x0 \<in> T \<and> (\<forall>x'\<in>T. f x' \<le> f x0 \<longrightarrow> f x' \<in> S)" shows "thesis"
BoxI: fixes \<phi> :: "\<o>"   and v :: "i" assumes "\<And>v. [\<phi> in v]" shows "[\<^bold>\<box>\<phi> in v]"
horiz_vector_helper_pos_R: fixes p :: "real poly"   and x :: "real"   and sign :: "rat list"   and qs :: "real poly list"   and I :: "nat list \<times> nat list" assumes "p \<noteq> 0"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec qs x"   and "list_constr (fst I) (length qs)"   and "list_constr (snd I) (length qs)" shows "((\<forall>p\<in>set (retrieve_polys qs (fst I)). poly p x = 0) \<and> 0 < poly (prod_list (retrieve_polys qs (snd I))) x) = (z_R I sign = 1)"
l7_22_aux_R1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A1 :: "'a"   and C :: "'a"   and B1 :: "'a"   and B2 :: "'a"   and M1 :: "'a"   and M2 :: "'a"   and A2 :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A1 C C"   and "Bet B1 C B2"   and "Cong C A1 C B1"   and "Cong C C C B2"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M1 A1 B1"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M2 A2 B2"   and "Tarski_neutral_dimensionless.Le Bet Cong C A1 C C" shows "Bet M1 C M2"
strict_iff_order: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" shows "lex_pm_strict a b = (lex_pm a b \<and> a \<noteq> b)"
supp_of_finite_sets: fixes S :: "'e set" assumes "finite S" shows "supp S = \<Union> (supp ` S)"
Graph_def: fixes f :: "'c \<Rightarrow> 'd" shows "Graph f = {(x, f x) |x. True}"
word_ao_equiv: fixes w :: "'b word"   and w' :: "'b word" shows "(w = w || w') = (w' = w && w')"
rel_def: shows "rel = measures [\<lambda>(t, i). N - i, \<lambda>(t, i). length t]"
IH1': shows "frechet I t1 (fst \<nu>) (snd \<nu>) = frechet I t1 (fst \<nu>') (snd \<nu>')"
tyenv_wellformed_def: fixes mds :: "Mode \<Rightarrow> 'Var set"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" shows "tyenv_wellformed mds \<Gamma> \<S> P \<equiv> mds_consistent mds \<Gamma> \<S> P \<and> types_wellformed \<Gamma> \<and> types_stable \<Gamma> \<S>"
FOREACHoi_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<Phi> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'a set" shows "FOREACH\<^sub>O\<^bsup>R,\<Phi>\<^esup> S \<equiv> FOREACH\<^sub>O\<^sub>C\<^bsup>R,\<Phi>\<^esup> S (\<lambda>_. True)"
simps_2: fixes n :: "'a itrm"   and x :: "dB" shows "norm_nn n (Pure x) = norm_pn (\<T> \<degree> x) n"
3: fixes l :: "'lbl" shows "GSMP (N2 (proj_unl l B)) \<subseteq> GSMP (N2 (proj_unl l A))"
wz_def: shows "wz = geodesic_segment_param {e--z} e t"
min_pm: fixes a :: "nat"   and b :: "nat" shows "min a b + (a - b) = a"
mon_w_single: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and e :: "('h, 'k) label" shows "mon_w fg [e] = mon_e fg e"
Seq_def: fixes s :: "hf"   and k :: "hf" shows "Seq s k = (hrelation s \<and> hfunction s \<and> k \<le> hdomain s)"
distinct_group_sort: fixes xs :: "(primfun \<times> real) list" shows "distinct (map fst (groupsort_primfun.group_sort xs))"
alist_of_push: fixes k :: "'c"   and p :: "'d"   and q :: "('c, 'd) pq" shows "alist_of (push k p q) = (if k \<notin> set |q| then insort_key snd (k, p) (alist_of q) else alist_of q)"