hcju/isabelleps huggingface dataset:
fst_corec_prod: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and s :: "'e" shows "fst (corec_prod f g s) = f s"
JF1rel_F1rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d JF1"   and b :: "'e JF1" shows "JF1rel R a b = F1rel R (JF1rel R) (JF2rel R) (dtor1 a) (dtor1 b)"
octo_eqI: fixes x :: "octo"   and y :: "octo" assumes "Ree x = Ree y"   and "Im1 x = Im1 y"   and "Im2 x = Im2 y"   and "Im3 x = Im3 y"   and "Im4 x = Im4 y"   and "Im5 x = Im5 y"   and "Im6 x = Im6 y"   and "Im7 x = Im7 y" shows "x = y"
set_mset_tree: fixes t :: "'b tree" shows "set_mset (mset_tree t) = set_tree t"
commute: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" shows "ordered_powerprod_lin.max a b = ordered_powerprod_lin.max b a"
simps_2: fixes p :: "int \<times> 'a" shows "del None (Branch2 Empty p Empty) = Some (p, True, Empty)"
ord_of_minus_1: fixes n :: "nat" assumes "0 < n" shows "ord_of_nat n = ZFC_in_HOL.succ (ord_of_nat (n - 1))"
elapsed_ge: fixes u :: "'c \<Rightarrow> 'a"   and u' :: "'c \<Rightarrow> 'a"   and c :: "'c" assumes "eq_elapsed u u'"   and "c \<in> \<X>" shows "u' c - u c \<le> elapsed u u'"
simps_1: fixes x1 :: "atom_sort"   and x2 :: "nat"   and y1 :: "atom_sort"   and y2 :: "nat" shows "(Atom x1 x2 = Atom y1 y2) = (x1 = y1 \<and> x2 = y2)"
prj_simps_13: shows "C.dom f_src_src.Prj\<^sub>1 = Chn (f \<star> src f \<star> src f)"
dvd_trans: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a b"   and "dvd.dvd times b c" shows "dvd.dvd times a c"
octo_unfold: fixes q :: "octo" shows "q = Ree q *\<^sub>R octo_e0 + Im1 q *\<^sub>R octo_e1 + Im2 q *\<^sub>R octo_e2 + Im3 q *\<^sub>R octo_e3 + Im4 q *\<^sub>R octo_e4 + Im5 q *\<^sub>R octo_e5 + Im6 q *\<^sub>R octo_e6 + Im7 q *\<^sub>R octo_e7"
induct: fixes P :: "'a prog"   and x1 :: "char list"   and x2 :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and Pa :: "char list \<Rightarrow> ((char list \<times> char list) \<times> staticb \<times> ty) list \<Rightarrow> bool" assumes "P \<turnstile> x1 has_fields x2"   and "\<And>C D fs ms FDTs FDTs'. \<lbrakk>class P C = \<lfloor>(D, fs, ms)\<rfloor>; C \<noteq> Object; P \<turnstile> D has_fields FDTs; Pa D FDTs; FDTs' = map (\<lambda>(F, b, T). ((F, C), b, T)) fs @ FDTs\<rbrakk> \<Longrightarrow> Pa C FDTs'"   and "\<And>D fs ms FDTs. \<lbrakk>class P Object = \<lfloor>(D, fs, ms)\<rfloor>; FDTs = map (\<lambda>(F, b, T). ((F, Object), b, T)) fs\<rbrakk> \<Longrightarrow> Pa Object FDTs" shows "Pa x1 x2"
add_independentS_append_reduce_right: fixes As :: "'a set list"   and Bs :: "'a set list" assumes "add_independentS (As @ Bs)" shows "add_independentS Bs"
norm_eq_square: fixes x :: "'a"   and a :: "real" shows "(norm x = a) = (0 \<le> a \<and> inner x x = a\<^sup>2)"
k_frontier_def: fixes a :: "'a" shows "k_frontier a \<equiv> map (simInit a \<circ> envObs a) envInit"
Suc: shows "fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n"
Init_Join: fixes F :: "'c program"   and G :: "'c program" shows "Init (F \<squnion> G) = Init F \<inter> Init G"
simps_2: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and ys :: "'g list" shows "sorted_wrt P (x # ys) = (Ball (set ys) (P x) \<and> sorted_wrt P ys)"
wf_sub_rel: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" assumes "wf_sub_rel_of g subs" shows "wf_sub_rel subs"
abs_float: fixes x :: "real" assumes "x \<in> float" shows "\<bar>x\<bar> \<in> float"
dist: shows "distinct (C' # clinit_classes ((stk, loc, C, M, pc, Throwing Cs a) # frs))"
sol_dim_in_max_lp: fixes x :: "'a vec"   and c :: "'a vec"   and A :: "'a mat"   and b :: "'a vec" assumes "x \<in> max_lp A b c" shows "dim_vec x = dim_col A"
field_def: fixes P :: "'b prog"   and C :: "char list"   and F :: "char list" shows "field P C F \<equiv> THE (D, b, T). P \<turnstile> C sees F,b:T in D"
not_dom_eq_not_sdom: fixes G :: "nat graph_rec"   and n1 :: "nat"   and n2 :: "nat" assumes "cfg_doms G"   and "\<not> cfg_doms.dominate G n1 n2" shows "\<not> cfg_doms.strict_dominate G n1 n2"
max_sls_def: shows "max_sls \<equiv> Abs_semilattice_set max"
homotopic_paths_trans: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a"   and r :: "real \<Rightarrow> 'a" assumes "homotopic_paths s p q"   and "homotopic_paths s q r" shows "homotopic_paths s p r"
Vset_Transset: fixes m :: "hf" assumes "Ord m" shows "Transset (Vset m)"
Equiv: fixes I :: "('a, 'b, 'c, 'd) Interpretation"   and A :: "'a"   and e1 :: "('a, 'b) Expression"   and e2 :: "('a, 'b) Expression"   and B :: "'a" assumes "Interpretation I"   and "L\<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IBool True" shows "\<exists>g. L\<lbrakk>Vx : A \<turnstile> e1 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g \<and> L\<lbrakk>Vx : A \<turnstile> e2 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g"
compact_translation: fixes s :: "'a set"   and a :: "'a" assumes "compact s" shows "compact ((+) a ` s)"
sfilter_strict: fixes p :: "'a \<rightarrow> tr" shows "sfilter\<cdot>p\<cdot>\<bottom> = \<bottom>"
nth_prime_exists1: fixes n :: "nat" shows "\<exists>!p. prime p \<and> card {q. prime q \<and> q < p} = n"
diff_self_eq_0: fixes m :: "nat" shows "m - m = 0"
idem: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "semilattice f" shows "f a a = a"
inner_Basis_def: fixes a :: "'a"   and b :: "'a" shows "local.inner_Basis a b = (\<Sum>i\<in>B. representation B a i * representation B b i)"
assms_2: shows "crdt_ops (ys @ [(oid, oper)] @ zs) deps"
assms_3: shows "\<iota>\<^sub>m k = Some q\<^sub>0"
cInput: fixes C :: "'a"   and a :: "name"   and x :: "name"   and P :: "pi" shows "F C (a<x>.P) a x P"
SUP_sup_distrib: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'a" shows "sup (Sup (f ` A)) (Sup (g ` A)) = (SUP a\<in>A. sup (f a) (g a))"
preservation1_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y \<le> x \<cdot> y \<cdot> x"   and "y \<cdot> x \<le> y" shows "(x \<cdot> y)\<^sup>\<dagger> \<cdot> x = x \<cdot> y\<^sup>\<dagger>"
thetaSeqW_def: shows "thetaSeqW \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. c1 \<approx>wT d1 \<and> c2 \<approx>w d2}"
anormal_aneg: fixes a :: "'a" assumes "anormal a" shows "\<forall>b\<in>atoms (aneg a). anormal b"
get\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap" shows "get shadow_root_ptr h = Option.bind (get (cast shadow_root_ptr) h) cast"
aform_err_uminus_aform: fixes e :: "nat \<Rightarrow> real"   and x :: "real"   and X :: "real \<times> real pdevs"   and ba :: "real" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "x \<in> aform_err e (X, ba)" shows "- x \<in> aform_err e (uminus_aform X, ba)"
fc_equiv_def: fixes a :: "'b"   and b :: "'b" shows "H.fc_equiv a b \<equiv> a \<in> H.face_cycle_set b"
a1: fixes m :: "nat" assumes "m \<le> n" shows "geom_proc m x = geom_proc m y"
intro: fixes p :: "real" assumes "0 \<le> p"   and "p \<le> 1" shows "edge_space p"
Dp: shows "DIM_precond TYPE((real, 'n) vec) CARD('n)"
bangUnfoldLeft: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "Id \<subseteq> Rel" shows "P \<parallel> !P \<leadsto>[Rel] !P"
magQ_plus: fixes x :: "'e['f, 'g]"   and y :: "'e['f, 'g]" shows "\<lbrakk>x + y\<rbrakk>\<^sub>Q = \<lbrakk>x\<rbrakk>\<^sub>Q + \<lbrakk>y\<rbrakk>\<^sub>Q"
arr_simps_1: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "src \<mu> = a"
A7': fixes a :: "'a"   and p :: "'a"   and c :: "'a"   and b :: "'a"   and q :: "'a" assumes "B a p c"   and "B b q c" shows "\<exists>x. B p x b \<and> B q x a"
obj_eval_Obj: fixes t :: "'c local.term" assumes "Obj t" shows "obj \<lbrace>t\<rbrace>"
assms: fixes xs :: "'a"   and x :: "'b" shows "rho xs x = rho' xs x"
reconstruct_lasso_def: fixes Vr :: "'q set"   and Vl :: "'q set" shows "reconstruct_lasso Vr Vl \<equiv> reconstruct_reach Vr Vl \<bind> (\<lambda>(pr, va). let cS_full = {0..<num_acc}; E = E \<inter> UNIV \<times> Vl in WHILE\<^sub>T\<^bsup>rec_loop_invar Vl va\<^esup> (\<lambda>(uu_, uu_, cS). cS \<noteq> cS_full) (\<lambda>(v, p, cS). ASSERT (\<exists>v'. (v, v') \<in> E\<^sup>* \<and> \<not> acc v' \<subseteq> cS) \<bind> (\<lambda>_. find_path E {v} (\<lambda>v. \<not> acc v \<subseteq> cS) \<bind> (\<lambda>sr. ASSERT (sr \<noteq> None) \<bind> (\<lambda>_. let (p_seg, v) = the sr in RETURN (v, p @ p_seg, cS \<union> acc v))))) (va, [], acc va) \<bind> (\<lambda>(vd, p, uu_). (if p = [] then find_path1 E vd ((=) va) else find_path E {vd} ((=) va)) \<bind> (\<lambda>p_close_r. ASSERT (p_close_r \<noteq> None) \<bind> (\<lambda>_. let (p_close, uu_) = the p_close_r in RETURN (pr, p @ p_close)))))"
prems_3: shows "j < output_size' (Pool m1 m2)"
sumh: shows "MPoly_Type.degree (Var x ^ i * Var x) x = MPoly_Type.degree (Var x ^ i) x + MPoly_Type.degree (Var x) x"
neq_V_not_region: shows "- V \<notin> \<R>"
le_def: fixes r\<^sub>A :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "Product.le r\<^sub>A r\<^sub>B = (\<lambda>(a\<^sub>1, b\<^sub>1) (a\<^sub>2, b\<^sub>2). a\<^sub>1 \<sqsubseteq>\<^bsub>r\<^sub>A\<^esub> a\<^sub>2 \<and> b\<^sub>1 \<sqsubseteq>\<^bsub>r\<^sub>B\<^esub> b\<^sub>2)"
allNotDSource_NotSource: fixes S :: "CSet"   and i :: "AbstrLevelsID" assumes "\<forall>C. S \<notin> DSources i C" shows "\<forall>Z. S \<notin> Sources i Z"
ipresSubstAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'g, 'h, 'var, 'c, 'd, 'i) Iteration.model_scheme" shows "ipresSubstAbs h hA MOD \<equiv> \<forall>ys Y y us s A. wls (asSort ys) Y \<and> wlsAbs (us, s) A \<longrightarrow> hA (A $[Y / y]_ys) = igSubstAbs MOD ys (h Y) y (hA A)"
less_SUP_iff: 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 bot :: "'b"   and top :: "'b"   and a :: "'b"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less a (Sup (f ` A)) = (\<exists>x\<in>A. less a (f x))"
length_pos_if_in_set: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "0 < |xs|"
assms_4: shows "(r, Store e mo r) \<in> set (node_deliver_messages xs)"
execn_elim_cases_10: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>While b c,s\<rangle> =n\<Rightarrow> t" obtains f :: "'f" where "sb__ = Fault f" and "t = Fault f"   | s :: "'e" and s' :: "('e, 'f) xstate" where "sb__ = Normal sc__" and "sc__ \<in> b" and "\<Gamma>\<turnstile> \<langle>c,Normal sc__\<rangle> =n\<Rightarrow> s'" and "\<Gamma>\<turnstile> \<langle>While b c,s'\<rangle> =n\<Rightarrow> t"   | s :: "'e" where "sb__ = Normal sd__" and "t = Normal sd__" and "sd__ \<notin> b"   | "sb__ = Stuck" and "t = Stuck"   | s :: "'e" where "sb__ = Abrupt s" and "t = Abrupt s"
decE_length: fixes n :: "nat"   and k :: "nat"   and \<Gamma> :: "binding list" shows "\<parallel>\<down>\<^sub>e n k \<Gamma>\<parallel> = \<parallel>\<Gamma>\<parallel>"
rel_gpv''_mono_strong: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and gpv :: "('d, 'f, 'h) gpv"   and gpv' :: "('e, 'g, 'i) gpv"   and R' :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and A' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C' :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "rel_gpv'' A C R gpv gpv'"   and "\<And>x y. \<lbrakk>x \<in> results_gpv (\<I>_uniform UNIV {x. Domainp R' x}) gpv; y \<in> results_gpv (\<I>_uniform UNIV {x. Rangep R' x}) gpv'; A x y\<rbrakk> \<Longrightarrow> A' x y"   and "\<And>x y. \<lbrakk>x \<in> outs_gpv (\<I>_uniform UNIV {x. Domainp R' x}) gpv; y \<in> outs_gpv (\<I>_uniform UNIV {x. Rangep R' x}) gpv'; C x y\<rbrakk> \<Longrightarrow> C' x y"   and "R' \<le> R" shows "rel_gpv'' A' C' R' gpv gpv'"
rreq_oip_kD: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:3..PRreq-:28} \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>))"
osum_Refl: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Refl r"   and "Refl r'" shows "Refl (r +o r')"
wf_mbufn_notin_set: fixes i :: "nat"   and js :: "nat list"   and Ps :: "(nat \<Rightarrow> event_data option list set \<Rightarrow> bool) list"   and buf :: "event_data option list set list list"   and j :: "nat" assumes "wf_mbufn i js Ps buf"   and "[] \<notin> set buf"   and "j \<in> set js" shows "i < j"
well_typed: shows "\<turnstile> \<Gamma> {c} \<Gamma>'"
broadcast_only_valid_msgs: fixes pre :: "('a \<times> ('a, 'b) operation) event list"   and m :: "'a \<times> ('a, 'b) operation"   and i :: "nat" assumes "pre @ [Broadcast m] prefix of i" shows "\<exists>state. apply_operations pre = Some state \<and> valid_rga_msg state m"
polynomial_p_def: shows "polynomial_p = (SOME p. vars p \<subseteq> {..<weight_space_dim} \<and> (\<forall>x. insertion x p = polynomial_f x))"
set_nzrows: fixes A :: "'c mat" shows "set (nzrows A) = set (rows A) - {0\<^sub>v (dim_col A)}"
from_hma\<^sub>v_diff: fixes v :: "('d, 'e) Finite_Cartesian_Product.vec"   and w :: "('d, 'e) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>v v - Mod_Type_Connect.from_hma\<^sub>v w = Mod_Type_Connect.from_hma\<^sub>v (v - w)"
filterlim_at_top_mult_at_top: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "filterlim f at_top F"   and "filterlim g at_top F" shows "LIM x F. f x * g x :> at_top"
M_def: shows "M \<equiv> sum_upto moebius_mu"
eq: shows "\<rho>\<^sub>Y V W q = \<rho>\<^sub>Y V' W q'"
word_plus_and_or: fixes x :: "'a word"   and y :: "'a word" shows "(x AND y) + (x OR y) = x + y"
opt_mru_round_def: fixes r :: "nat"   and Q :: "process set"   and S :: "process set"   and v :: "val"   and r_decisions :: "process \<Rightarrow> val option" shows "opt_mru_round r Q S v r_decisions = {(s, s'). r = opt_mru_state.next_round s \<and> (S \<noteq> {} \<longrightarrow> opt_mru_guard (mru_vote s) Q v) \<and> d_guard r_decisions (const_map v S) \<and> s' = s\<lparr>mru_vote := mru_vote s ++ const_map (r, v) S, opt_mru_state.next_round := Suc r, opt_mru_state.decisions := opt_mru_state.decisions s ++ r_decisions\<rparr>}"
bij_prop1: fixes g :: "'a"   and y :: "'c" assumes "g \<in> carrier G"   and "y \<in> E" shows "\<exists>!x. x \<in> E \<and> \<phi> g x = y"
eql_subst_fmla_Iff: fixes t :: "trm"   and u :: "trm"   and H :: "fmla set"   and A :: "fmla"   and i :: "name" shows "insert (t EQ u) H \<turnstile> A(i::=t) IFF A(i::=u)"
set_inner_verts: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.apath u p v" shows "set (S.inner_verts p) = set (pawalk_verts u p) - {u, v}"
join_wf_tuple: fixes x :: "'b option list"   and X :: "'b option list set"   and b :: "bool"   and Y :: "'b option list set"   and n :: "nat"   and A :: "nat set"   and B :: "nat set"   and C :: "nat set" assumes "x \<in> join X b Y"   and "\<forall>v\<in>X. wf_tuple n A v"   and "\<forall>v\<in>Y. wf_tuple n B v"   and "\<not> b \<Longrightarrow> B \<subseteq> A"   and "A \<union> B = C" shows "wf_tuple n C x"
eq: shows "sets (completion (distr M N X)) = sets (distr M (completion N) X)"
l_inv_add_rng_of_frac: fixes r :: "'a"   and s :: "'a" assumes "(r, s) \<in> carrier rel" shows "(\<ominus> r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r |\<^bsub>rel\<^esub> s) = \<zero>\<^bsub>rec_rng_of_frac\<^esub>"
length_Cons: fixes x :: "'a"   and xs :: "'a list" shows "\<parallel>x \<Colon> xs\<parallel> = Suc \<parallel>xs\<parallel>"
lim_explicit: fixes f :: "nat \<Rightarrow> 'a"   and f0 :: "'a" shows "f \<longlonglongrightarrow> f0 = (\<forall>S. open S \<longrightarrow> f0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> S))"
subst_img_comp_subset_const': fixes \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and \<tau> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and x :: "'b"   and c :: "'a" assumes "(\<delta> \<circ>\<^sub>s \<tau>) x = Fun c []" shows "\<delta> x = Fun c [] \<or> (\<exists>z. \<delta> x = Var z \<and> \<tau> z = Fun c [])"
add_eq_0_iff_both_eq_0: fixes x :: "'a multiset"   and y :: "'a multiset" shows "(x + y = {#}) = (x = {#} \<and> y = {#})"
WhilePostdomination_aux: fixes prog :: "cmd" shows "Postdomination sourcenode targetnode kind (valid_edge prog) (_Entry_) (_Exit_)"
ballpE: fixes A :: "('a \<times> 'b) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "\<forall>(x, y)\<in>A. P x y" obtains "P x y"   | "(x, y) \<notin> A"
proj_in_set_iff_2: fixes d :: "'e"   and i :: "'d"   and D :: "('d strand_label \<times> 'e) list" shows "((\<star>, d) \<in> set (proj i D)) = ((\<star>, d) \<in> set D)"
tranclp_tranclp_to_tranclp_r: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<And>x y z. \<lbrakk>R\<^sup>+\<^sup>+ x y; R y z; P x; P z\<rbrakk> \<Longrightarrow> P y"   and "R\<^sup>+\<^sup>+ x y"   and "R\<^sup>+\<^sup>+ y z"   and "P x"   and "P z" shows "P y"
floor_add2: fixes x :: "'d"   and y :: "'d" assumes "x \<in> \<int> \<or> y \<in> \<int>" shows "\<lfloor>x + y\<rfloor> = \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>"
HomFtorMor': fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<in> mor\<^bsub>C\<^esub>" shows "Hom\<^bsub>C\<^esub>[X,f] maps\<^bsub>SET'\<^esub> Hom\<^bsub>C\<^esub> X dom\<^bsub>C\<^esub> f to Hom\<^bsub>C\<^esub> X cod\<^bsub>C\<^esub> f"
A4: shows "[\<lambda>w. \<forall>x. (P x m\<rightarrow> \<box> (P x)) w]"
env_delete_restr: fixes x :: "'a"   and m :: "'a \<Rightarrow> 'b" shows "env_delete x m = m f|` (- {x})"
assms: shows "strategy_attracts_via p \<sigma> v0 A W"
dns5: fixes x :: "'a"   and y :: "'a" shows "ans_d x \<cdot> (ans_d x + ans_d y) = ans_d x"
ring_distribs_4: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b - c) = a * b - a * c"
subgame_V: fixes V' :: "'a set" shows "V\<^bsub>G'.subgame V'\<^esub> \<subseteq> V\<^bsub>G'\<^esub>"
c_2: shows "\<forall>\<^sub>F x in F. R (norm (f x)) (c * norm (g x))"
q0: fixes q :: "int poly"   and ri :: "root_info" assumes "(q, ri) \<in> set polys" shows "q \<noteq> 0"
rel_refl: fixes Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'b \<rightharpoonup>\<^sub>f 'a" assumes "\<And>x. Ra x x" shows "fmrel Ra x x"
assms: shows "C\<cdot>r \<sqsubseteq> demand f"
Gcd_factorial_eq_0_iff: fixes A :: "'a set" shows "(Gcd_factorial A = (0::'a)) = (A \<subseteq> {0::'a})"
outputFreshChainDerivative_1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and yvec :: "name list" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "xvec \<sharp>* M"   and "distinct xvec"   and "yvec \<sharp>* P"   and "yvec \<sharp>* xvec" shows "yvec \<sharp>* N"
embed2_axioms: shows "embed2 \<Sigma> wf_atom project lookup embed singleton"
commutative: fixes x :: "'a"   and y :: "'a" assumes "x \<in> R"   and "y \<in> R" shows "x + y = y + x"
cvalidt_def: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A \<equiv> (\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P Call p Q,A) \<longrightarrow> \<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
singleton_table_def: fixes n :: "nat"   and i :: "nat"   and x :: "'a" shows "singleton_table n i x = {tabulate (\<lambda>j. if i = j then Some x else None) 0 n}"
c: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" shows "length (a @ b @ c) = length a + length b + length c"
subst_fresh_simp: fixes x :: "name"   and \<theta> :: "(name \<times> trm) list" assumes "x \<sharp> \<theta>" shows "\<theta><Var x> = Var x"
preserves_reflects_arr: fixes f :: "'a" shows "\<sigma>.B.arr (\<sigma> f) = \<sigma>.A.arr f"
assms_3: shows "R\<^sup>*\<^sup>* (f xm) (f ym)"
all_fields_in_has_fields: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and D :: "char list"   and D' :: "char list"   and fs :: "fdecl list"   and ms :: "'a mdecl list"   and F :: "char list"   and b :: "staticb"   and T :: "ty" assumes "P \<turnstile> C has_fields FDTs"   and "P \<turnstile> C \<preceq>\<^sup>* D"   and "class P D = \<lfloor>(D', fs, ms)\<rfloor>"   and "(F, b, T) \<in> set fs" shows "((F, D), b, T) \<in> set FDTs"
g_def: shows "g \<equiv> \<lambda>n. if odd n then - \<i> * \<i> ^ n / of_nat n else 0"
ChamberSubcomplex_def: fixes Y :: "'b set set" shows "codomain.ChamberSubcomplex Y \<equiv> order.greater_eq Y Y \<and> ChamberComplex Y \<and> (\<forall>C. SimplicialComplex.maxsimp Y C \<longrightarrow> codomain.chamber C)"
1: shows "meta_adjunction C D F F.G F.\<phi> F.\<psi>"
5: shows "injective (prim_P w v e\<^sup>T)"
bot_annil: fixes x :: "'a" shows "\<bottom> \<cdot> x = \<bottom>"
BB_PROTECT_def: shows "BB_PROTECT \<equiv> \<lambda>a. a"
sum_upto_0: fixes f :: "nat \<Rightarrow> 'b" shows "sum_upto f 0 = (0::'b)"
segment_a_a'_e: shows "segment G.face_cycle_succ a a' = {}"
A_def2: shows "A = {x. \<exists>y. q x y = 0}"
bounded_antilinear_cblinfun_matrix: fixes j :: "'a"   and i :: "'b" shows "bounded_antilinear (\<lambda>x. cblinfun_apply x j \<bullet>\<^sub>C i)"
assms: fixes gpv :: "('a, 'out, 'in) gpv" shows "\<I> \<turnstile>g gpv \<surd>"
Ch_range': fixes x :: "'x"   and h :: "'h"   and X :: "'x set" assumes "x \<in> Ch h X" shows "x \<in> X \<and> Xh x = h"
bounded_linear_compose: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c" assumes "bounded_linear f"   and "bounded_linear g" shows "bounded_linear (\<lambda>x. f (g x))"
SC: fixes r :: "real" shows "S r \<subseteq> C r"
array_is_sorted_def: fixes cmp :: "int \<Rightarrow> int \<Rightarrow> bool"   and a :: "int \<Rightarrow> int"   and l :: "int"   and r :: "int" shows "array_is_sorted cmp a l r \<equiv> \<forall>i j. bounded l r i \<longrightarrow> bounded l r j \<longrightarrow> i < j \<longrightarrow> cmp (a i) (a j)"
Pi_mem: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and x :: "'a" assumes "f \<in> Pi A B"   and "x \<in> A" shows "f x \<in> B x"
assms_1: shows "s' \<le> s \<cdot> z"
comm_neutral: fixes a :: "'c" shows "a + (0::'c) = a"
vDec: shows "initReachable c \<and> <\<bottom>, outM v> \<in># msgs c"
build_def: fixes n :: "nat"   and seed :: "natural \<times> natural" shows "local.build n seed = (let bound = of_nat n * 2 in local.gen_remove bound n (local.gen_build bound n (empty (), seed)))"
Data_select: fixes D :: "'b data" shows "Data (DataValue D) (Data.DataSpace D)"
dupe_monic_def: fixes D :: "'a poly"   and H :: "'a poly"   and S :: "'a poly"   and T :: "'a poly"   and U :: "'a poly" shows "Hensel_Lifting.dupe_monic D H S T U = (case pdivmod_monic (T * U) D of (q, r) \<Rightarrow> (S * U + H * q, r))"