hcju/isabelleps huggingface dataset:
left_idem: fixes a :: "'b"   and b :: "'b" shows "a \<sqinter> (a \<sqinter> b) = a \<sqinter> b"
mirror_\<alpha>_\<beta>: fixes p :: "fm"   and i :: "int"   and bs :: "int list" assumes "iszlfm p" shows "Inum (i # bs) ` set (\<alpha> p) = Inum (i # bs) ` set (\<beta> (mirror p))"
t_sift_l: shows "t_sift_down l x r \<le> height l + 1"
get_child_nodes_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "ptr |\<in>| object_ptr_kinds h"
wf: shows "wf {(cfg2, cfg1). \<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* cfg1 \<and> \<Gamma>\<turnstile> cfg1 \<rightarrow> cfg2}"
F_LUB: fixes S :: "nat \<Rightarrow> 'a process" assumes "chain S" shows "F (lim_proc (range S)) = \<Inter> (F ` range S)"
sublist_cons: fixes lo :: "nat"   and hi :: "nat"   and xs :: "'c list" assumes "lo < hi"   and "hi < length xs" shows "WB_Sort.sublist xs lo hi = xs ! lo # WB_Sort.sublist xs (Suc lo) hi"
tpick: fixes V :: "Preliminaries.var set"   and \<sigma> :: "'tp" assumes "tinfinite V" shows "tpick \<sigma> V \<in> V \<inter> tpOfV -` {\<sigma>}"
simps_1: fixes f :: "'b \<Rightarrow> ('a, 'c) state" shows "iterate_state f [] = State_Monad.return ()"
assms_2: assumes "\<alpha> = a[b]"   and "P = P'" shows "Prop (a[b]) P"
Levy_uniqueness: fixes M1 :: "real measure"   and M2 :: "real measure" assumes "real_distribution M1"   and "real_distribution M2"   and "char M1 = char M2" shows "M1 = M2"
ladder_n_bound: fixes u :: "nat" assumes "u < length L'" shows "ladder_n L' u \<le> length E"
joinable_non_Nil_1: fixes p :: "'a list"   and q :: "'a list" assumes "joinable p q" shows "p \<noteq> []"
min_divide_distrib_right: fixes x :: "'a"   and y :: "'a"   and p :: "'a" shows "min x y / p = (if (0::'a) \<le> p then min (x / p) (y / p) else max (x / p) (y / p))"
intros_4: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty"   and U :: "ty"   and u :: "trm" assumes "\<Gamma> \<turnstile> t : T \<rightarrow> U"   and "\<Gamma> \<turnstile> u : T" shows "\<Gamma> \<turnstile> t \<cdot> u : U"
in_Sup_multisetD: fixes x :: "'b"   and A :: "'b multiset set" assumes "x \<in># Sup A" shows "\<exists>X\<in>A. x \<in># X"
cInf_eq_minimum: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and X :: "'b set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq z x" shows "Inf X = z"
unlabel_Cons_inv: fixes A :: "('e strand_label \<times> 'd) list"   and b :: "'d"   and B :: "'d list" assumes "unlabel A = b # B" shows "\<exists>A'. (\<exists>n. A = (ln n, b) # A') \<or> A = (\<star>, b) # A'"
lt_1: shows "z \<otimes> x \<prec> z \<otimes> y"
substitutes_on_union: fixes a :: "'a"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set"   and A :: "'a set" assumes "a \<notin> f (B \<union> {a})"   and "substitutes_on (A \<union> B \<union> {a}) f"   and "finite A" shows "a \<notin> f (A \<union> B \<union> {a})"
SubsetSepTrans: fixes A :: "ZF"   and B :: "ZF"   and p :: "ZF \<Rightarrow> bool" assumes "HOLZF.subset A B" shows "HOLZF.subset (Sep A p) B"
simps_2: fixes n :: "nat"   and l :: "'b tree234"   and p :: "'b"   and r :: "'b tree234" shows "full\<^sub>i n (Up\<^sub>i l p r) = (full n l \<and> full n r)"
acc_point_closed: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "acc_point s \<in> carrier Zp"
finite_subset_image: fixes B :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "finite B"   and "B \<subseteq> f ` A" shows "\<exists>C\<subseteq>A. finite C \<and> B = f ` C"
minus_unique: fixes y :: "'e"   and x :: "'e"   and y' :: "'e" assumes "y \<oplus>\<^bsub>N\<^esub> x = \<zero>\<^bsub>N\<^esub>"   and "x \<oplus>\<^bsub>N\<^esub> y' = \<zero>\<^bsub>N\<^esub>"   and "x \<in> carrier N"   and "y \<in> carrier N"   and "y' \<in> carrier N" shows "y = y'"
sq_norm_as_sq_abs: fixes a :: "'b" shows "sq_norm_conjugate a = \<bar>a\<bar>\<^sup>2"
basic_subtype_ObjectType_x: fixes \<C> :: "'a"   and \<sigma> :: "'a basic_type" assumes "\<langle>\<C>\<rangle>\<^sub>\<T> \<sqsubset>\<^sub>B \<sigma>" obtains \<D> :: "'a" where "\<sigma> = \<langle>\<D>\<rangle>\<^sub>\<T>" and "\<C> < \<D>"   | "\<sigma> = OclAny"
lincomb_card_bound: fixes X :: "'a vec set"   and Bnd :: "'a"   and c :: "'a vec \<Rightarrow> 'a"   and k :: "nat" assumes "X \<subseteq> Bounded_vec Bnd"   and "X \<subseteq> carrier_vec n"   and "(0::'a) \<le> Bnd"   and "\<And>x. x \<in> X \<Longrightarrow> \<bar>c x\<bar> \<le> (1::'a)"   and "card X \<le> k" shows "lincomb c X \<in> Bounded_vec (of_nat k * Bnd)"
of_int_1: shows "of_int 1 = (1::'a)"
n1_type: fixes t :: "'a bro"   and h :: "nat" assumes "t \<in> Bp h" shows "n1 t \<in> T (Suc h)"
IH: fixes ys :: "'a list" shows "cnt x (remove_cycles xs x ys) \<le> max 1 (cnt x ys)"
eval_fds_power: fixes f :: "'a fds"   and s :: "'a"   and n :: "nat" assumes "fds_abs_converges f s" shows "eval_fds (f ^ n) s = eval_fds f s ^ n"
abs_of_neg: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less a zero" shows "abs a = uminus a"
increasing_nat2Nat: fixes n :: "nat" assumes "0 < n" shows "Elem (nat2Nat (n - 1)) (nat2Nat n)"
SN: shows "0 < length (snd (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M)))) \<and> fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! 0 = 0 \<and> snd (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! 0 = 0"
enum_ind_def: fixes x :: "'b" shows "enum_ind x = first_ind enum_class.enum x 0"
pvs_def: shows "pvs = xs @ ys @ y # bppvs"
ennreal_ineq_diff_add: fixes b :: "ennreal"   and a :: "ennreal" assumes "b \<le> a" shows "a = b + (a - b)"
simps_2: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd"   and c :: "tcolor"   and uu :: "'c \<times> 'd"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" shows "invc \<langle>l, (a, c, uu), r\<rangle> = (invc l \<and> invc r \<and> (c = Red \<longrightarrow> color l = Black \<and> color r = Black))"
dist_le_zero_iff: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.metric_space dist uniformity open" shows "(dist x y \<le> 0) = (x = y)"
eq: shows "S = (\<Union>n. {x \<in> S. norm x \<le> real n \<and> onorm (f' x) \<le> real n})"
derived_set_subset_gen: fixes X :: "'c topology"   and S :: "'c set" shows "(X derived_set_of S \<subseteq> S) = closedin X (topspace X \<inter> S)"
foldl_echelon_form_column_k_eq_3: fixes k :: "nat"   and A :: "(('c, 'd) vec, 'e) vec"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" assumes "k < ncols A" shows "snd (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k]) \<le> nrows A"
arr_cod_iff_arr: fixes f :: "'d" shows "D.arr (D.cod f) = D.arr f"
iter_tensor_of_H_times_\<psi>\<^sub>2_is_\<psi>\<^sub>3: shows "(Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)) \<Otimes> Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) = Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then \<Sum>k<2 ^ n. (- 1) ^ (f k + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else \<Sum>k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1))))"
eq_commute: fixes a :: "'e"   and b :: "'e" shows "(a = b) = (b = a)"
dist_eqs_2: fixes a :: "'a"   and e' :: "'b" shows "Ok\<cdot>a \<noteq> Err\<cdot>e'"
lhopital: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and F :: "real filter" assumes "f \<midarrow>x\<rightarrow> 0"   and "g \<midarrow>x\<rightarrow> 0"   and "\<forall>\<^sub>F x in at x. g x \<noteq> 0"   and "\<forall>\<^sub>F x in at x. g' x \<noteq> 0"   and "\<forall>\<^sub>F x in at x. (f has_real_derivative f' x) (at x)"   and "\<forall>\<^sub>F x in at x. (g has_real_derivative g' x) (at x)"   and "LIM x at x. f' x / g' x :> F" shows "LIM x at x. f x / g x :> F"
splitFace_f21_new_vertices: fixes f12 :: "face"   and f21 :: "face"   and newGraph :: "graph"   and g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVs :: "nat list"   and v :: "nat" assumes "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF newVs"   and "v \<in> set newVs" shows "v \<in> \<V> f21"
card_stabiliser: shows "card (tetrahedral.stabiliser A) = 3"
domain_invol: fixes x :: "('a \<times> 'a) set" shows "rel_ad (rel_ad (rel_ad (rel_ad x))) = rel_ad (rel_ad x)"
csm: shows "cs\<^bsup>\<pi>\<^esup> m = [\<pi> m]"
perm_simps_1: fixes p :: "perm"   and trm1 :: "trm"   and trm2 :: "trm" shows "p \<bullet> trm1 EQ trm2 = (p \<bullet> trm1) EQ (p \<bullet> trm2)"
alphaInp_sym: fixes qinp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and qinp' :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" assumes "(qinp, qinp') \<in> alphaInp" shows "(qinp', qinp) \<in> alphaInp"
phi_in_int: fixes x :: "real" assumes "x \<in> {0..1}" shows "\<phi> x \<in> {0..1}"
simps_6: fixes r :: "'c rexp"   and s :: "'c rexp" shows "\<guillemotleft>Times r s\<guillemotright> = Times \<guillemotleft>r\<guillemotright> \<guillemotleft>s\<guillemotright>"
IH: fixes n :: "nat" assumes "n < array_length a \<Longrightarrow> P (Suc n)" shows "P n"
alg_MapOps_empty_1: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'g"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'h"   and jviewInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'g"   and jviewIncr :: "'b \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'g"   and simf :: "'e Trace \<Rightarrow> 'i"   and simRels :: "'b \<Rightarrow> ('i \<times> 'i) set"   and simVal :: "'i \<Rightarrow> 'c \<Rightarrow> bool"   and simAbs :: "'j \<Rightarrow> 'i set"   and simObs :: "'b \<Rightarrow> 'j \<Rightarrow> 'h"   and simInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'j"   and simTrans :: "'b \<Rightarrow> 'j \<Rightarrow> 'j list"   and simAction :: "'b \<Rightarrow> 'j \<Rightarrow> 'd list"   and aOps :: "('k, 'j, 'd list) MapOps"   and tOps :: "('l, 'j \<times> 'h, 'j) MapOps"   and a :: "'b"   and ec :: "'j" assumes "AlgorithmForAgent jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction aOps tOps"   and "AlgorithmForAgent.k_isNode jkbp envInit envAction envTrans envVal jview simf simAbs a ec" shows "lookup aOps (MapOps.empty aOps) ec = None"
OV_inv3_obsv_round: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and D :: "process \<Rightarrow> val option"   and Ob :: "process set" shows "{OV_inv3 \<inter> OV_inv2} obsv_round r S v D Ob {> OV_inv3}"
convex_sum: fixes S :: "'b set"   and C :: "'c set"   and a :: "'b \<Rightarrow> real"   and y :: "'b \<Rightarrow> 'c" assumes "finite S"   and "convex C"   and "sum a S = 1"   and "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i"   and "\<And>i. i \<in> S \<Longrightarrow> y i \<in> C" shows "(\<Sum>j\<in>S. a j *\<^sub>R y j) \<in> C"
set_of_graph_rulesD_3: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and R :: "('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph" assumes "set_of_graph_rules Rs"   and "R \<in> Rs" shows "subgraph (fst R) (snd R)"
assms: shows "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>"
mk_conn_def: fixes H1 :: "'a hrel \<Rightarrow> 'a hrel"   and \<H>\<^sub>1 :: "'b hrel \<Rightarrow> 'a hrel"   and \<H>\<^sub>2 :: "'a hrel \<Rightarrow> 'b hrel"   and H2 :: "'b hrel \<Rightarrow> 'b hrel" shows "H1 \<Leftarrow>\<langle>\<H>\<^sub>1,\<H>\<^sub>2\<rangle>\<Rightarrow> H2 \<equiv> \<lparr>orderA = utp_order H1, orderB = utp_order H2, lower = \<H>\<^sub>2, upper = \<H>\<^sub>1\<rparr>"
powr_strict: fixes a :: "real"   and b :: "real" assumes "p powr a < p powr b" shows "a < b"
degreen_npolyhCN: fixes c :: "poly"   and n :: "nat"   and p :: "poly"   and n0 :: "nat" assumes "isnpolyh (CN c n p) n0" shows "degreen c n = 0"
nemp_len: fixes u :: "'b list" assumes "u \<noteq> \<epsilon>" shows "\<^bold>|u\<^bold>| \<noteq> 0"
set_def: fixes al :: "('b, 'c) assoc_list" shows "Assoc_List.set al = set (assoc_list.impl_of al)"
reaches1_reaches_iff1: fixes x :: "'a"   and y :: "'a" shows "(x \<rightarrow>\<^sup>+ y) = (\<exists>z. x \<rightarrow> z \<and> z \<rightarrow>* y)"
obind_def: fixes f :: "'b \<Rightarrow> 'd option"   and g :: "'d \<Rightarrow> 'b \<Rightarrow> 'c option" shows "f |>> g \<equiv> \<lambda>s. case f s of None \<Rightarrow> None | Some x \<Rightarrow> g x s"
derivative_def: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and a :: "'c" shows "derivative R f a = taylor_expansion R a f 1"
orthogonal_clauses_5: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner"   and "real_inner.orthogonal inner a x"   and "real_inner.orthogonal inner a y" shows "real_inner.orthogonal inner a (minus x y)"
hom_power: fixes x :: "real poly"   and n :: "nat" shows "map_poly complex_of_real (x ^ n) = map_poly complex_of_real x ^ n"
alt_seq_def: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" shows "alt_seq s = (\<lambda>k. if even k then s k else res_lim s)"
CallRedsObj: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and Copt :: "char list option"   and M :: "char list"   and es :: "expr list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>Call e Copt M es,s\<rangle> \<rightarrow>* \<langle>Call e' Copt M es,s'\<rangle>"
f_join_f_shrink_iMODb: fixes k :: "nat"   and xs :: "'b message_af list"   and n :: "nat"   and d :: "nat" assumes "0 < k" shows "xs \<longmapsto> k \<Join>\<^sub> [ n * k + (k - 1), mod k, d ] = xs \<div> k \<Join>\<^sub> [n\<dots>,d]"
deny: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and a :: "action"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p"   and "a = Drop \<or> a = Reject" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
simps_1: fixes uu :: "typ"   and t :: "Core.term" shows "depth (Abs uu t) = depth t + 1"
strict_implies_order: fixes b :: "'loc \<times> 't"   and a :: "'loc \<times> 't" assumes "b<\<^sub>pa" shows "b\<le>\<^sub>pa"
ereal_le_real_iff: fixes x :: "real"   and y :: "ereal" shows "(x \<le> real_of_ereal y) = ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0))"
aux: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('d, 'e) \<I>"   and s :: "('a, 'b, 'd, 'e) converter" assumes "\<I> \<turnstile>g gpv \<surd>"   and "\<I>, \<I>' \<turnstile>\<^sub>C s \<surd>" shows "set_spmf (inline1 run_converter gpv s) \<subseteq> {Inr (out, callee', rpv') |out callee' rpv'. \<exists>call\<in>outs_\<I> \<I>. \<exists>s. \<I>, \<I>' \<turnstile>\<^sub>C s \<surd> \<and> (\<forall>x\<in>responses_\<I> \<I>' out. callee' x \<in> sub_gpvs \<I>' (run_converter s call))} \<union> {Inl (x, s') |x s'. x \<in> results_gpv \<I> gpv \<and> \<I>, \<I>' \<turnstile>\<^sub>C s' \<surd>}"
is_type_nt: fixes P :: "'a prog" shows "is_type P NT = True"
Units_l_cancel: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> Units H"   and "y \<in> carrier H"   and "z \<in> carrier H" shows "(x \<otimes>\<^bsub>H\<^esub> y = x \<otimes>\<^bsub>H\<^esub> z) = (y = z)"
infinite_eqpoll_lists: fixes A :: "'a set" assumes "infinite A" shows "lists A \<approx> A"
keys_def: shows "RBT.keys \<equiv> map_fun rbt.impl_of id RBT_Impl.keys"
AbGroup_inner_dirsum_el_decomp_ex1_double: fixes G :: "'a set"   and H :: "'a set"   and x :: "'a" assumes "AbGroup G"   and "AbGroup H"   and "add_independentS [G, H]"   and "x \<in> G \<oplus> H" shows "\<exists>!gh. fst gh \<in> G \<and> snd gh \<in> H \<and> x = fst gh + snd gh"
monadic_filter_rev_aux_rule: fixes l :: "'b list"   and P :: "'b \<Rightarrow> bool nres"   and Q :: "'b \<Rightarrow> bool"   and a :: "'b list" assumes "\<And>x. x \<in> set l \<Longrightarrow> P x \<le> (spec r. r = Q x)" shows "monadic_filter_rev_aux a P l \<le> (spec r. r = filter_rev_aux a Q l)"
monoid_comm_monoidI: assumes "\<And>x y. \<lbrakk>x \<in> UNIV; y \<in> UNIV\<rbrakk> \<Longrightarrow> x * y = y * x" shows "Group.comm_monoid class_ring"
insert_eqvt: fixes pi :: "('b \<times> 'b) list"   and x :: "'a"   and X :: "'a set" shows "pi \<bullet> insert x X = insert (pi \<bullet> x) (pi \<bullet> X)"
simps_4: fixes F :: "'b formula"   and G :: "'b formula" shows "depth (F \<^bold>\<and> G) = Suc (max (depth F) (depth G))"
dynC_prop: shows "G,statT \<turnstile> dynC valid_lookup_cls_for is_static emh"
fiter_unfold: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<star> = nil \<sqinter> c ; c\<^sup>\<star>"
unit_preserving_hom_axioms: shows "unit_preserving_hom (\<lambda>p. p \<circ>\<^sub>p x_y)"
red_rtrancl_uminus: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "(punit.red F)\<^sup>*\<^sup>* p q" shows "(punit.red F)\<^sup>*\<^sup>* (- p) (- q)"
strict_trans1: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b"   and "b \<subset># c" shows "a \<subset># c"
simps_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(In1 x1 = In1 y1) = (x1 = y1)"
finite_minimal_antichain_path_weightp: fixes l1 :: "'vtx"   and l2 :: "'vtx" shows "finite (minimal_antichain {x. path_weightp l1 l2 x})"
symmetric_conv: fixes G :: "('a, 'b) pre_digraph" shows "symmetric G = (\<forall>e1\<in>arcs G. \<exists>e2\<in>arcs G. tail G e1 = head G e2 \<and> head G e1 = tail G e2)"
vimage_Int: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set"   and B :: "'d set" shows "f -` (A \<inter> B) = f -` A \<inter> f -` B"
td_a: shows "type_definition Rep_a Abs_a (Collect (Domainp A))"
closure_of_bool_cases: fixes f :: "unit \<Rightarrow> bool"   and P :: "(unit \<Rightarrow> bool) \<Rightarrow> bool" assumes "f = (\<lambda>u. False) \<Longrightarrow> P f"   and "f = (\<lambda>u. True) \<Longrightarrow> P f" shows "P f"
lfinite_inf_llist: fixes f :: "nat \<Rightarrow> 'a" shows "\<not> lfinite (inf_llist f)"
PPs_Int: fixes X :: "'b set"   and Y :: "'b set" shows ".[X \<inter> Y] = .[X] \<inter> .[Y]"
S_def: fixes i :: "nat" shows "S i = {x \<in> T. 1 / real (Suc i) < norm (\<mu> x)}"
Start_nsys_xcpts1_5: shows "Start \<noteq> IncompatibleClassChangeError"
prime_divisor_exists_strong_nat: fixes m :: "nat" assumes "1 < m"   and "\<not> prime m" shows "\<exists>p k. m = p * k \<and> 1 < p \<and> p < m \<and> 1 < k \<and> k < m \<and> prime p"
vec_of_poly_rev_shifted_scalar_prod: fixes p :: "'b poly"   and m :: "nat"   and v :: "'b vec"   and n :: "nat"   and j :: "nat" assumes "degree p = m"   and "dim_vec v = n"   and "j < m + n" shows "Resultant.vec_of_poly_rev_shifted p n (n + m - Suc j) \<bullet> v = coeff (p * poly_of_vec v) j"
count_mset: fixes xs :: "'b list"   and x :: "'b" shows "count (mset xs) x = length (filter ((=) x) xs)"
fold_A_im_min_gallery': fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set"   and Cs :: "'b set list" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z"   and "SimplicialComplex.min_maxsimpchain ((ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) \<turnstile> A) (C # Cs)" shows "SimplicialComplex.min_maxsimpchain A (C # Cs)"
singleton_list_cong_eq_iff: fixes xs :: "'a list"   and ys :: "'a list" shows "({xs} // {\<cong>} = {ys} // {\<cong>}) = xs \<cong> ys"
dg_prod_3_Cod_app: fixes f :: "V"   and f' :: "V"   and f'' :: "V" assumes "[f, f', f'']\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Arr\<rparr>" shows "(\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Cod\<rparr> \<lparr>f, f', f''\<rparr>\<^sub>\<bullet> = [\<AA>\<lparr>Cod\<rparr>\<lparr>f\<rparr>, \<BB>\<lparr>Cod\<rparr>\<lparr>f'\<rparr>, \<CC>\<lparr>Cod\<rparr>\<lparr>f''\<rparr>]\<^sub>\<circ>"
Int_left_absorb: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> (A \<inter> B) = A \<inter> B"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "min (min a b) c = min a (min b c)"
HInfinite_ge_HInfinite: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HInfinite"   and "x \<le> y"   and "0 \<le> x" shows "y \<in> HInfinite"
z_idem: shows "bot_homomorphism ((\<squnion>) bot)"
dropWhile_append2: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and ys :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "dropWhile P (xs @ ys) = dropWhile P ys"
arcs_app_iso: fixes hom :: "('e, 'f, 'd, 'c) digraph_isomorphism"   and G :: "('e, 'f) pre_digraph" shows "arcs (app_iso hom G) = iso_arcs hom ` arcs G"
periodic_set_union: fixes S :: "real set"   and \<delta> :: "real"   and T :: "real set" assumes "periodic_set S \<delta>"   and "periodic_set T \<delta>" shows "periodic_set (S \<union> T) \<delta>"
induct: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "enat" assumes "interaction_bounded_by consider gpv n" obtains "interaction_bound consider gpv \<le> n"
l: shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l' x) < e"