hcju/isabelleps huggingface dataset:
typing_induct: fixes x1 :: "binding list"   and x2 :: "trm"   and x3 :: "POPLmarkRecord.type"   and P1 :: "binding list \<Rightarrow> trm \<Rightarrow> POPLmarkRecord.type \<Rightarrow> bool" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> i U T. \<lbrakk>\<Gamma> \<turnstile>\<^sub>w\<^sub>f; \<Gamma>\<langle>i\<rangle> = \<lfloor>VarB U\<rfloor>; T = \<up>\<^sub>\<tau> (Suc i) 0 U\<rbrakk> \<Longrightarrow> P1 \<Gamma> (Var i) (\<up>\<^sub>\<tau> (Suc i) 0 U)"   and "\<And>T\<^sub>1 \<Gamma> t\<^sub>2 T\<^sub>2. \<lbrakk>VarB T\<^sub>1 \<Colon> \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2; P1 (VarB T\<^sub>1 \<Colon> \<Gamma>) t\<^sub>2 T\<^sub>2\<rbrakk> \<Longrightarrow> P1 \<Gamma> (\<lambda>:T\<^sub>1. t\<^sub>2) (T\<^sub>1 \<rightarrow> \<down>\<^sub>\<tau> 1 0 T\<^sub>2)"   and "\<And>\<Gamma> t\<^sub>1 T\<^sub>1\<^sub>1 T\<^sub>1\<^sub>2 t\<^sub>2. \<lbrakk>\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1\<^sub>1 \<rightarrow> T\<^sub>1\<^sub>2; P1 \<Gamma> t\<^sub>1 (T\<^sub>1\<^sub>1 \<rightarrow> T\<^sub>1\<^sub>2); \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>1\<^sub>1; P1 \<Gamma> t\<^sub>2 T\<^sub>1\<^sub>1\<rbrakk> \<Longrightarrow> P1 \<Gamma> (t\<^sub>1 \<bullet> t\<^sub>2) T\<^sub>1\<^sub>2"   and "\<And>T\<^sub>1 \<Gamma> t\<^sub>2 T\<^sub>2. \<lbrakk>TVarB T\<^sub>1 \<Colon> \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2; P1 (TVarB T\<^sub>1 \<Colon> \<Gamma>) t\<^sub>2 T\<^sub>2\<rbrakk> \<Longrightarrow> P1 \<Gamma> (\<lambda><:T\<^sub>1. t\<^sub>2) (\<forall><:T\<^sub>1. T\<^sub>2)"   and "\<And>\<Gamma> t\<^sub>1 T\<^sub>1\<^sub>1 T\<^sub>1\<^sub>2 T\<^sub>2. \<lbrakk>\<Gamma> \<turnstile> t\<^sub>1 : (\<forall><:T\<^sub>1\<^sub>1. T\<^sub>1\<^sub>2); P1 \<Gamma> t\<^sub>1 (\<forall><:T\<^sub>1\<^sub>1. T\<^sub>1\<^sub>2); \<Gamma> \<turnstile> T\<^sub>2 <: T\<^sub>1\<^sub>1\<rbrakk> \<Longrightarrow> P1 \<Gamma> (t\<^sub>1 \<bullet>\<^sub>\<tau> T\<^sub>2) (T\<^sub>1\<^sub>2[0 \<mapsto>\<^sub>\<tau> T\<^sub>2]\<^sub>\<tau>)"   and "\<And>\<Gamma> t S T. \<lbrakk>\<Gamma> \<turnstile> t : S; P1 \<Gamma> t S; \<Gamma> \<turnstile> S <: T\<rbrakk> \<Longrightarrow> P1 \<Gamma> t T"   and "\<And>\<Gamma> t\<^sub>1 T\<^sub>1 p \<Delta> t\<^sub>2 T\<^sub>2. \<lbrakk>\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1; P1 \<Gamma> t\<^sub>1 T\<^sub>1; \<turnstile> p : T\<^sub>1 \<Rightarrow> \<Delta>; \<Delta> @ \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2; P1 (\<Delta> @ \<Gamma>) t\<^sub>2 T\<^sub>2\<rbrakk> \<Longrightarrow> P1 \<Gamma> (LET p = t\<^sub>1 IN t\<^sub>2) (\<down>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> 0 T\<^sub>2)"   and "\<And>\<Gamma> fs fTs. \<Gamma> \<turnstile> fs [:] fTs \<Longrightarrow> P1 \<Gamma> (Rcd fs) (RcdT fTs)"   and "\<And>\<Gamma> t fTs l T. \<lbrakk>\<Gamma> \<turnstile> t : RcdT fTs; P1 \<Gamma> t (RcdT fTs); fTs\<langle>l\<rangle>\<^sub>? = \<lfloor>T\<rfloor>\<rbrakk> \<Longrightarrow> P1 \<Gamma> (t..l) T" shows "P1 x1 x2 x3"
lowdim_eq_hyperplane: fixes S :: "'a set" assumes "dim S = DIM('a) - 1" obtains a :: "'a" where "a \<noteq> (0::'a)" and "span S = {x. a \<bullet> x = 0}"
subset_image_inj: fixes S :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and T :: "'c set" shows "order.greater_eq (f ` T) S = (\<exists>U. order.greater_eq T U \<and> inj_on f U \<and> S = f ` U)"
hyps_3: assumes "\<And>a. a \<in> A \<Longrightarrow> f a < \<top>" shows "enn2real (sum f A) = (\<Sum>a\<in>A. enn2real (f a))"
card_Nset_im: fixes f :: "nat \<Rightarrow> 'e"   and n :: "nat"   and A :: "'e set" assumes "f \<in> {j. j \<le> n} \<rightarrow> A" shows "Suc 0 \<le> card (f ` {j. j \<le> n})"
lprefix_lnullD: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lprefix xs ys"   and "lnull ys" shows "lnull xs"
ide_implies_finite_set_1: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a" shows "finite (local.set a)"
Gauss_Jordan_upt_k_iarrays_PA_def: fixes A :: "'b iarray iarray"   and k :: "nat" shows "Gauss_Jordan_upt_k_iarrays_PA A k = (let foldl = foldl Gauss_Jordan_column_k_iarrays_PA (mat_iarray (1::'b) (nrows_iarray A), 0, A) [0..<Suc k] in (fst foldl, snd (snd foldl)))"
pickEff: fixes r :: "'rule"   and s :: "'state" assumes "enabled r s" shows "eff r s (pickEff r s)"
h_frag: fixes x :: "'a" assumes "x \<in> S" shows "h (frag_of x) = (frag_of \<circ> f) x"
Abs_def: fixes xs :: "'varSort"   and x :: "'var"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set" shows "Abs xs x X \<equiv> asAbs (qAbs xs x (pick X))"
Least_equality: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "P x"   and "\<And>y. P y \<Longrightarrow> x \<le> y" shows "Least P = x"
q: fixes t :: "real" assumes "0 \<le> t"   and "t \<le> 1" shows "q (\<lambda>x. k (t, x))"
s_Tr_3: shows "\<lless>\<pi>\<^sub>2, eV\<^sub>2\<ggreater> V\<rightarrow> \<lless>\<pi>\<^sub>3, eV\<^sub>3\<ggreater>"
simps_1: fixes vo :: "'d option"   and ts :: "('c \<times> ('c, 'd) trie) list" shows "delete_trie [] (Trie vo ts) = Trie None ts"
Main_no_return_source: fixes a :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a"   and "knd a = Q'\<hookleftarrow>\<^bsub>Main\<^esub>f'" shows "False"
aff_dim_sing: fixes a :: "'b" shows "aff_dim {a} = 0"
writerT_eq_iff: fixes a :: "'b\<cdot>('c, 'd) writerT"   and b :: "'b\<cdot>('c, 'd) writerT" shows "(a = b) = (runWriterT\<cdot>a = runWriterT\<cdot>b)"
h_def: shows "h \<equiv> fps_X ^ (subdegree g - subdegree f)"
runit_naturality: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "\<r>[S.cod \<mu>] \<cdot>\<^sub>S R \<mu> = \<mu> \<cdot>\<^sub>S \<r>[S.dom \<mu>]"
min_set_finite: fixes X :: "'b set" assumes "finite X" shows "finite (min_set X)"
l_get_disconnected_nodes_axioms: shows "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"
to_contain_is_to_divide: fixes a :: "('c multiset \<Rightarrow> 'a) list"   and b :: "('c multiset \<Rightarrow> 'a) list" assumes "a \<in> carrier (poly_ring L)"   and "b \<in> carrier (poly_ring L)" shows "(PIdl\<^bsub>poly_ring L\<^esub> b \<subseteq> PIdl\<^bsub>poly_ring L\<^esub> a) = a divides\<^bsub>poly_ring L\<^esub> b"
assms_4: shows "Map f \<in> Hom (Dom f) (Cod f)"
replace_f: fixes m :: "'a match_expr" assumes "normalized_nnf_match m" shows "matches \<gamma> (replace_primitive_matchexpr disc_sel replace_f m) a p = matches \<gamma> m a p"
type_II_chain_vert_bound_valid: shows "\<forall>(k, \<gamma>)\<in>two_chain_vertical_boundary two_chain. valid_path \<gamma>"
Union_rel_def: fixes L :: "('d \<times> 'f set) set"   and S :: "('f \<times> 'e set) set" shows "\<langle>L, S\<rangle>Union_rel = L O \<langle>S\<rangle>set_rel O br \<Union> top"
prv_neg_neg_exi: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "bprv (neg \<phi>)" shows "bprv (neg (exi x \<phi>))"
inv_msog_def: fixes M :: "((accent \<times> 'b) \<times> (accent \<times> 'b) list) multiset" shows "inv_msog M = {#(inv_letter x, inv_greek t). (x, t) \<in># M#}"
monoid_axioms: shows "Group_Theory.monoid G (\<cdot>) \<one>"
pivotandupdate_satisfies_tableau: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "PivotAndUpdate eq_idx_for_lvar pivot_and_update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i)" shows "\<langle>\<V> s\<rangle> \<Turnstile>\<^sub>t \<T> s \<longrightarrow> \<langle>\<V> (pivot_and_update x\<^sub>i x\<^sub>j c s)\<rangle> \<Turnstile>\<^sub>t \<T> s"
Rep_Node: fixes x :: "('g, 'f) Old_Datatype.node" shows "Rep_Node x \<in> Old_Datatype.Node"
that: fixes A :: "fm" assumes "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<and> {} \<turnstile> A" shows "thesis"
hd_append2: fixes xs :: "'d list"   and ys :: "'d list" assumes "xs \<noteq> []" shows "hd (xs @ ys) = hd xs"
star_rtc_2: fixes y :: "'a"   and x :: "'a" assumes "(1::'a) + y \<cdot> y + x \<le> y" shows "x\<^sup>\<star> \<le> y"
ideal_npow_closed: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and x :: "'c"   and n :: "nat" assumes "Ring R"   and "ideal R I"   and "x \<in> I"   and "0 < n" shows "x^\<^bsup>R n\<^esup> \<in> I"
monomial_0_iff: fixes t :: "'e"   and c :: "'f" shows "(monomial c t = 0) = (c = (0::'f))"
S_def: shows "S = (\<Sum>(c, \<alpha>)\<leftarrow>xs. c * exp \<alpha>)"
intros_2: fixes a :: "('e \<times> 'f) \<times> 'd"   and A :: "('e, 'f, 'd, 'g) FSM_scheme"   and p :: "'d"   and r :: "(('e \<times> 'f) \<times> 'd) list" assumes "snd a \<in> succ A (fst a) p"   and "path A r (snd a)" shows "path A (a # r) p"
carrier_vec_conjugate: fixes m :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "m \<in> carrier_mat nr nc" shows "conjugate m \<in> carrier_mat nr nc"
bind_returns_result_E: fixes h :: "'x"   and f :: "('x, 'y, 'aa) prog"   and g :: "'aa \<Rightarrow> ('x, 'y, 'z) prog"   and y :: "'z" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y" obtains x :: "'aa" and h' :: "'x" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>r y"
submod: fixes K :: "('g, 'h) ring_scheme"   and W :: "'i set"   and V :: "('g, 'i, 'j) module_scheme" assumes "subspace K W V" shows "LinearCombinations.submodule K W V"
AE_convergence_imp_convergence_in_prob: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> real"   and \<epsilon> :: "real" assumes "\<And>i. random_variable borel (X i)"   and "random_variable borel Y"   and "AE x in M. (\<lambda>i. X i x) \<longlonglongrightarrow> Y x"   and "0 < \<epsilon>" shows "(\<lambda>i. prob {x \<in> space M. \<epsilon> < \<bar>X i x - Y x\<bar>}) \<longlonglongrightarrow> 0"
sees_field_def: fixes P :: "'a prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" shows "P \<turnstile> C sees F:T (fm) in D \<equiv> \<exists>FDTs. P \<turnstile> C has_fields FDTs \<and> map_of (map (\<lambda>((F, D), Tm). (F, D, Tm)) FDTs) F = \<lfloor>(D, T, fm)\<rfloor>"
pure_assn_def: fixes b :: "bool" shows "\<up> b \<equiv> Abs_assn (pure_assn_raw b)"
fresh_nat: fixes a :: "'a"   and n :: "nat" shows "a \<sharp> n"
op_set_pick_remove_def: fixes X :: "'c set" shows "op_set_pick_remove X \<equiv> SPEC (\<lambda>(x, X'). x \<in> X \<and> X' = X - {x})"
surj_def: fixes f :: "'d \<Rightarrow> 'c" shows "surj f = (\<forall>y. \<exists>x. y = f x)"
semiring_norm_7: fixes m :: "num"   and n :: "num" shows "num.Bit0 m + num.Bit1 n = num.Bit1 (m + n)"
star_subdist_var_1: fixes x :: "'a"   and y :: "'a" shows "x \<le> (x + y)\<^sup>\<star>"
simps_2: fixes m :: "'a \<Rightarrow> nat"   and h :: "nat"   and X :: "char list set" assumes "Measure1_fun m h" shows "Measure1_fun.m_o m h None X = h * card X + 1"
perms_inv: fixes f :: "'b \<Rightarrow> 'b" assumes "f \<in> perms" shows "inv f \<in> perms"
uint_bl: fixes w :: "'b word" shows "to_bl w = bin_to_bl (size w) (uint w)"
ipresSwapAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'f, 'g, 'var, 'b, 'c, 'h) model_scheme" shows "ipresSwapAll h hA MOD \<equiv> ipresSwap h MOD \<and> ipresSwapAbs hA MOD"
d_IN_alt_def: fixes y :: "'a"   and G :: "('a, 'b) graph_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "\<And>x. (x, y) \<notin> \<^bold>E\<^bsub>G\<^esub> \<Longrightarrow> g (x, y) = 0" shows "d_IN g y = (\<Sum>\<^sup>+ x\<in>\<^bold>I\<^bold>N\<^bsub>G\<^esub> y. g (x, y))"
rep_eq: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xa :: "('a, 'b) RBT.rbt"   and xb :: "('a, 'b) RBT.rbt" shows "impl_of (RBT.combine x xa xb) = rbt_union_with x (impl_of xa) (impl_of xb)"
good: shows "rel_fun (\<lambda>x y. bad1 x = bad2 y \<and> (\<not> bad2 y \<longrightarrow> A x = B y)) (=) (\<lambda>x. A x \<and> \<not> bad1 x) (\<lambda>y. B y \<and> \<not> bad2 y)"
prems_5: shows "\<forall>e\<in>E - (set accu \<union> set (e # Es)). \<not> all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = E, flows_state = {e} \<union> set accu\<rparr>)"
root_summation_inv: fixes k :: "nat"   and n :: "nat" assumes "0 < k"   and "k < n" shows "sum ((^) ((1 / FFT.root n) ^ k)) {0..<n} = 0"
cbig: shows "(\<sigma>', coll') \<in> cbig P \<sigma>"
isGlb_def: fixes S :: "'b set"   and po :: "'b potype" shows "isGlb S po = (\<lambda>G. G \<in> pset po \<and> (\<forall>y\<in>S. (G, y) \<in> order po) \<and> (\<forall>z\<in>pset po. (\<forall>y\<in>S. (z, y) \<in> order po) \<longrightarrow> (z, G) \<in> order po))"
perm_2: fixes pi :: "(name \<times> name) list"   and x1 :: "name"   and x2 :: "('a, 'b, 'c) input" shows "pi \<bullet> \<nu>x1x2 = \<nu>(pi \<bullet> x1)(pi \<bullet> x2)"
pos_def: fixes rs :: "'a stream"   and r :: "'a" shows "pos rs r \<equiv> LEAST n. shd (sdrop n rs) = r"
assms_3: shows "fmrel R (xm(i \<mapsto>\<^sub>f x)) ym"
assms_21: shows "InvariantVarsM (getM state) F0 Vbl"
geq: fixes x :: "'a"   and n :: "nat" assumes "x \<in> - E n" shows "g n x = indicat_real S x"
select_convs_2: fixes ext :: "real_int"   and lan :: "nat_int"   and own :: "cars"   and more :: "'a" shows "lan \<lparr>ext = ext, lan = lan, own = own, \<dots> = more\<rparr> = lan"
stake_snth': fixes n :: "nat"   and w :: "'b stream"   and x :: "'b stream" assumes "\<And>i. Suc i \<le> n \<Longrightarrow> w !! i = x !! i" shows "stake n w = stake n x"
mcont_def: fixes luba :: "'a set \<Rightarrow> 'a"   and orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lubb :: "'b set \<Rightarrow> 'b"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" shows "mcont luba orda lubb ordb f = (monotone orda ordb f \<and> cont luba orda lubb ordb f)"
generatePolygonTame_def: fixes n :: "nat"   and v :: "nat"   and f :: "face"   and g :: "graph" shows "generatePolygonTame n v f g \<equiv> let enumeration = Enumerator.enum n |vertices f|; enumeration = filter (\<lambda>is. \<not> containsDuplicateEdge g f v is) enumeration; vertexLists = map (indexToVertexList f v) enumeration in filter (\<lambda>g'. \<not> notame g') (map (subdivFace g f) vertexLists)"
\<eta>_map_simp: fixes y :: "'d" assumes "D.ide y" shows "transformation_by_components.map (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) D.map \<phi>\<psi>.\<eta>o y = \<phi> y (F y)"
get_min: fixes h :: "'b heap" assumes "pheap h"   and "h \<noteq> Empty" shows "get_min h = Min_mset (mset_heap h)"
r_triangle_prim: shows "prim_recfn 1 r_triangle"
maps_all_maps_list_def: fixes G_list :: "'b list \<times> ('b \<times> 'b) list" shows "maps_all_maps_list G_list = map (to_map (set (snd G_list)) \<circ> lists_succ) (all_maps_list G_list)"
try_gpv_Done: fixes x :: "'d"   and gpv' :: "('d, 'e, 'f) gpv" shows "TRY Generative_Probabilistic_Value.Done x ELSE gpv' = Generative_Probabilistic_Value.Done x"
mat_comm_monoid: fixes nr :: "nat"   and nc :: "nat"   and b :: "'d" shows "Group.comm_monoid (mat_monoid nr nc b)"
subdiv_props_3: shows "\<forall>(k, \<gamma>)\<in>subdiv. valid_path \<gamma>"
cases: fixes a :: "'a stream" assumes "A.run a" obtains x :: "'a" and y :: "'a" and xs :: "'a stream" where "a = x ## y ## xs" and "A x y" and "A.run (y ## xs)"
primepow_def: fixes n :: "'a" shows "primepow n = (\<exists>p k. prime p \<and> 0 < k \<and> n = p ^ k)"
lowlink_set_not_empty: 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" assumes "DFS_invar G param s"   and "v \<in> dom (discovered s)" shows "param_DFS_defs.lowlink_set G s v \<noteq> {}"
simps: fixes l :: "name"   and s :: "tm"   and k :: "tm"   and v :: "tm"   and sli :: "name"   and sl :: "name"   and sl' :: "name"   and m :: "name"   and n :: "name"   and smi :: "name"   and sm :: "name"   and sm' :: "name"   and sni :: "name"   and sn :: "name"   and sn' :: "name"   and i :: "tm"   and x :: "tm"   and x' :: "tm" assumes "atom l \<sharp> (s, k, v, sli, sl, sl', m, n, smi, sm, sm', sni, sn, sn')"   and "atom sli \<sharp> (s, v, sl, sl', m, n, smi, sm, sm', sni, sn, sn')"   and "atom sl \<sharp> (s, v, sl', m, n, smi, sm, sm', sni, sn, sn')"   and "atom sl' \<sharp> (s, v, m, n, smi, sm, sm', sni, sn, sn')"   and "atom m \<sharp> (s, n, smi, sm, sm', sni, sn, sn')"   and "atom n \<sharp> (s, smi, sm, sm', sni, sn, sn')"   and "atom smi \<sharp> (s, sm, sm', sni, sn, sn')"   and "atom sm \<sharp> (s, sm', sni, sn, sn')"   and "atom sm' \<sharp> (s, sni, sn, sn')"   and "atom sni \<sharp> (s, sn, sn')"   and "atom sn \<sharp> (s, sn')"   and "atom sn' \<sharp> s" shows "SeqAbstFormP v i x x' s k = LstSeqP s k (HPair i (HPair x x')) AND All2 l (SUCC k) (SyntaxN.Ex sli (SyntaxN.Ex sl (SyntaxN.Ex sl' (HPair (Var l) (HPair (Var sli) (HPair (Var sl) (Var sl'))) IN s AND (AbstAtomicP v (Var sli) (Var sl) (Var sl') OR OrdP (Var sli) AND SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex smi (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sni (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (HPair (Var smi) (HPair (Var sm) (Var sm'))) IN s AND HPair (Var n) (HPair (Var sni) (HPair (Var sn) (Var sn'))) IN s AND (Var sli EQ Var smi AND Var sli EQ Var sni AND Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR Var sli EQ Var smi AND Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR SUCC (Var sli) EQ Var smi AND Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm')))))))))))))))"
2: shows "f x \<sqinter> y \<le> f (x \<sqinter> - - g y)"
Int_UN_distrib2: fixes A :: "'c \<Rightarrow> 'b set"   and I :: "'c set"   and B :: "'d \<Rightarrow> 'b set"   and J :: "'d set" shows "\<Union> (A ` I) \<inter> \<Union> (B ` J) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
exhaust: fixes y :: "trieP" obtains "y = LfP"   | x21 :: "bool list" and x22 :: "bool" and x23 :: "trieP \<times> trieP" where "y = NdP x21 x22 x23"
PAC_Format_induct_split: fixes V :: "nat set"   and A :: "int mpoly multiset"   and V' :: "nat set"   and A' :: "int mpoly multiset"   and P :: "nat set \<Rightarrow> int mpoly multiset \<Rightarrow> nat set \<Rightarrow> int mpoly multiset \<Rightarrow> bool" assumes "PAC_Format (V, A) (V', A')"   and "\<And>p A q p' \<V>. \<lbrakk>p \<in># A; q \<in># A; p + q - p' \<in> More_Modules.ideal polynomial_bool; vars p' \<subseteq> \<V>\<rbrakk> \<Longrightarrow> P \<V> A \<V> (add_mset p' A)"   and "\<And>p A q p' \<V>. \<lbrakk>p \<in># A; p * q - p' \<in> More_Modules.ideal polynomial_bool; vars p' \<subseteq> \<V>; vars q \<subseteq> \<V>\<rbrakk> \<Longrightarrow> P \<V> A \<V> (add_mset p' A)"   and "\<And>p A \<V>. p \<in># A \<Longrightarrow> P \<V> A \<V> (A - {#p#})"   and "\<And>p' \<V> x A. \<lbrakk>p'\<^sup>2 - p' \<in> More_Modules.ideal polynomial_bool; vars p' \<subseteq> \<V>; x \<notin> \<V>\<rbrakk> \<Longrightarrow> P \<V> A (\<V> \<union> {x' \<in> vars (- Var x + p'). x' \<notin> \<V>}) (add_mset (- Var x + p') A)" shows "P V A V' A'"
elims: fixes x :: "trilean"   and xa :: "trilean"   and y :: "trilean" assumes "x \<or>? xa = y" obtains uu_ :: "trilean" where "x = invalid" and "xa = uu_" and "y = invalid"   | "x = true" and "xa = invalid" and "y = invalid"   | "x = false" and "xa = invalid" and "y = invalid"   | "x = true" and "xa = true" and "y = true"   | "x = true" and "xa = false" and "y = true"   | "x = false" and "xa = true" and "y = true"   | "x = false" and "xa = false" and "y = false"
assms_2: shows "valid ds (\<lambda>ds fp. R ds fp \<and> P ds fp \<and> Q ds fp) Q"
map_spmf_of_pmf: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d pmf" shows "map_spmf f (spmf_of_pmf p) = spmf_of_pmf (map_pmf f p)"
left_one: fixes x :: "'a" shows "one \<odot> x = x"
trm_strong_induct: fixes P :: "'a set \<Rightarrow> 'a trm \<Rightarrow> bool"   and S :: "'a set"   and M :: "'a trm" assumes "P S Unit"   and "\<And>x. P S (Var x)"   and "\<And>A B. \<lbrakk>P S A; P S B\<rbrakk> \<Longrightarrow> P S (App A B)"   and "\<And>x T A. \<lbrakk>x \<notin> S; P S A\<rbrakk> \<Longrightarrow> P S (Fn x T A)"   and "\<And>A B. \<lbrakk>P S A; P S B\<rbrakk> \<Longrightarrow> P S (SimplyTyped.Pair A B)"   and "\<And>A. P S A \<Longrightarrow> P S (Fst A)"   and "\<And>A. P S A \<Longrightarrow> P S (Snd A)"   and "finite S" shows "P S M"
xin: shows "x\<^sub>C \<in> - range var\<^sub>C_of"
that_1: fixes c :: "'form" assumes "c \<in> set conclusions"   and "v = (c, [])" shows "thesis"
cbox_interval: fixes a :: "'c"   and b :: "'c" shows "cbox a b = {a..b}"
Domain_fst: fixes r :: "('b \<times> 'c) set" shows "Domain r = fst ` r"
size_eq_0_iff_empty: fixes M :: "'a multiset" shows "(size M = 0) = (M = \<Empt>)"
trail_def: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" shows "G.trail u p v \<equiv> G.awalk u p v \<and> distinct p"
assms: shows "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"
nonneg_cexpr_Mult: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr"   and V :: "nat set" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : REAL"   and "free_vars e1 \<subseteq> V"   and "free_vars e2 \<subseteq> V"   and "nonneg_cexpr V \<Gamma> e1"   and "nonneg_cexpr V \<Gamma> e2" shows "nonneg_cexpr V \<Gamma> (e1 *\<^sub>c e2)"
sr_spectral_radius: shows "sr = HMA_Connect.spectral_radius cA"
sup_inf_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
assms: shows "a 0 = 0 \<or> (\<exists>i\<in>{1..n}. a i \<noteq> 0)"
countable_vimage: fixes B :: "'b set"   and f :: "'c \<Rightarrow> 'b" assumes "B \<subseteq> range f"   and "countable (f -` B)" shows "countable B"
prod_eqI: fixes p :: "'b \<times> 'c"   and q :: "'b \<times> 'c" assumes "fst p = fst q"   and "snd p = snd q" shows "p = q"
bet__le2313: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C" shows "B C Le A C"
getData_def: shows "getData \<equiv> fst \<circ> snd"
finfun_rec_def: fixes cnst :: "'b \<Rightarrow> 'a"   and upd :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "('c, 'b) finfun" shows "finfun_rec cnst upd f \<equiv> let b = finfun_default f; g = THE g. f = Abs_finfun (map_default b g) \<and> finite (dom g) \<and> b \<notin> ran g in Finite_Set.fold (\<lambda>a. upd a (map_default b g a)) (cnst b) (dom g)"
row_add_iterate_A: shows "row_add_iterate A n i j $ a $ b = A $ a $ b"
treehead_in_multiset: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "t \<in> set bq" shows "(SkewBinomialHeapStruc.val t, SkewBinomialHeapStruc.prio t) \<in># SkewBinomialHeapStruc.queue_to_multiset bq"
vangle_commute: fixes u :: "'b"   and v :: "'b" shows "vangle u v = vangle v u"
a_Inj1: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> e, eP, eV : \<tau>\<^sub>1" shows "\<Gamma> \<turnstile> Inj1 e, Inj1 eP, Inj1 eV : Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2"
Resolution_useless_infinite: fixes S :: "'b literal set set"   and R :: "'b literal set" assumes "S \<turnstile> R"   and "finite R" shows "\<exists>S'\<subseteq>S. Ball S' finite \<and> finite S' \<and> (S' \<turnstile> R)"
g0: shows "g 0 = f (\<lambda>j. if j = 0 then 1 else 0)"
elims_3: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b list" assumes "\<not> successive x xa" obtains P :: "'b \<Rightarrow> bool" and xb :: "'b" and y :: "'b" and xs :: "'b list" where "x = P" and "xa = xb # y # xs" and "\<not> (\<not> P y \<and> successive P xs \<or> \<not> P xb \<and> successive P (y # xs))"