hcju/isabelleps huggingface dataset:
sds_lowering_strategyproof_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf"   and agents' :: "'a set"   and alts' :: "'b set" shows "sds_lowering_strategyproof agents alts sds agents' alts' \<equiv> strategyproof_sds agents alts sds \<and> sds_lowering agents alts sds agents' alts'"
Int_Union: fixes A :: "'d set"   and B :: "'d set set" shows "A \<inter> \<Union> B = \<Union> ((\<inter>) A ` B)"
asm: shows "\<forall>w\<in>ball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0 \<and> f w = h w * (w - z) powr of_int m \<and> h w \<noteq> 0"
principal_ideal: fixes a :: "'a" assumes "a \<in> carrier R" shows "ideal R (R \<diamondsuit>\<^sub>p a)"
not_alw: fixes \<phi> :: "'c stream \<Rightarrow> bool" shows "(\<lambda>xs. \<not> alw \<phi> xs) = ev (\<lambda>xs. \<not> \<phi> xs)"
cball_eq_ball_iff: fixes x :: "'a"   and d :: "real"   and y :: "'a"   and e :: "real" shows "(cball x d = ball y e) = (d < 0 \<and> e \<le> 0)"
F_subset_imp_T_subset: fixes P :: "'b process"   and Q :: "'b process" assumes "F P \<subseteq> F Q" shows "T P \<subseteq> T Q"
plus_oracle_split_asm: fixes P :: "(('a + 'b) \<times> 'c) spmf \<Rightarrow> bool"   and left :: "'c \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'c) spmf"   and right :: "'c \<Rightarrow> 'e \<Rightarrow> ('b \<times> 'c) spmf"   and s :: "'c"   and lr :: "'d + 'e" shows "P ((left \<oplus>\<^sub>O right) s lr) = (\<not> ((\<exists>x. lr = Inl x \<and> \<not> P (map_spmf (apfst Inl) (left s x))) \<or> (\<exists>y. lr = Inr y \<and> \<not> P (map_spmf (apfst Inr) (right s y)))))"
hyps_6: fixes ba :: "trm"   and bb :: "trm"   and b :: "name" assumes "ba \<mapsto> bb" shows " n[b::=ba] \<mapsto>\<^sup>* n[b::=bb] "
plus_game_zero_right: fixes G :: "game" shows "plus_game G zero_game = G"
normalize_pos: fixes m :: "'a common_primitive match_expr" assumes "normalized_nnf_match m"   and "\<not> has_disc_negated disc False m" shows "\<forall>m'\<in>set (normalize_pos m). normalized_n_primitive (disc, sel) (\<lambda>ps. case ps of L4Ports x pts \<Rightarrow> length pts \<le> 1) m'"
comp_null_2: fixes f :: "'a" shows "C f C.null = C.null"
set_valued_functor_axioms: shows "set_valued_functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) Cones.map"
pivot_tableau_code_def: shows "pivot_tableau_code = Pivot'.pivot_tableau eq_idx_for_lvar pivot_eq subst_var"
sum_vimage_proof: fixes f :: "'d \<Rightarrow> nat"   and bd :: "nat"   and S :: "'d set"   and g :: "nat \<Rightarrow> real" assumes "\<And>w. f w < bd"   and "finite S" shows "(\<Sum>w\<in>S. g (f w)) = (\<Sum>m = 0..<bd. real (card (f -` {m} \<inter> S)) * g m)"
CE_holds_def: fixes var :: "'sc"   and \<phi> :: "('sf, 'sc, 'sz) formula"   and \<psi> :: "('sf, 'sc, 'sz) formula" shows "CE_holds var \<phi> \<psi> \<equiv> valid (\<phi> \<leftrightarrow> \<psi>) \<longrightarrow> valid (InContext var \<phi> \<leftrightarrow> InContext var \<psi>)"
cross7_simps_16: fixes a :: "'b"   and c :: "'b"   and b :: "'b" shows "(a = c - b) = (a + b = c)"
associative: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and c :: "('a \<times> 'a) set" assumes "a \<in> carrier_local_ring_at"   and "b \<in> carrier_local_ring_at"   and "c \<in> carrier_local_ring_at" shows "add_local_ring_at (add_local_ring_at a b) c = add_local_ring_at a (add_local_ring_at b c)"
edges_of_walk_edge: fixes xs :: "'a list"   and v :: "'a"   and w :: "'a" assumes "walk xs"   and "(v, w) \<in> edges_of_walk xs" shows "v \<rightarrow> w"
continuous_Pair: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. (f x, g x))"
wf_fmla_atom_alt: fixes D :: "ast_domain"   and ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and \<phi> :: "'a atom formula" shows "ast_domain.wf_fmla_atom D ty_ent \<phi> = (is_predAtom \<phi> \<and> ast_domain.wf_fmla D ty_ent \<phi>)"
insert_Nil: fixes x :: "'b" shows "List.insert x \<epsilon> = [x]"
total_to_partial: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,{}\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,{}\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
spr_trans_def: fixes kbp :: "('b, 'd, 'e) GC list"   and envAction :: "'c \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'e) \<Rightarrow> 'c \<Rightarrow> 'c"   and val :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "SPRViewSingle.spr_trans kbp envAction envTrans val \<equiv> \<lambda>a X. concat (map (\<lambda>s. concat (map (\<lambda>eact. map (\<lambda>aact. envTrans eact (\<lambda>a'. aact) s) (SPRViewSingle.spr_simAction kbp val a X)) (envAction s))) (toList X))"
get_component_def: fixes ptr :: "(_) object_ptr" shows "get_component ptr = Heap_Error_Monad.bind (get_root_node ptr) to_tree_order"
wf_mdeclD1: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "methd" assumes "wf_mdecl G C (sig, m)" shows "wf_mhead G (pid C) sig (mhead m) \<and> unique (lcls (mbody m)) \<and> (\<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None) \<and> (\<forall>(vn, T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T)"
that: fixes M' :: "nat \<Rightarrow> nat \<Rightarrow> 'a DBMEntry" assumes "[M]\<^bsub>v,n\<^esub> = [M']\<^bsub>v,n\<^esub> \<and> dbm_int M' n \<and> (\<forall>i\<le>n. M' i i = \<one>)" shows "thesis"
cyclic_group_def: fixes G :: "('e, 'f) monoid_scheme" shows "cyclic_group G \<equiv> \<exists>x\<in>carrier G. subgroup_generated G {x} = G"
rderivs_replicate_mult_rest: fixes i :: "nat"   and as :: "'a"   and r :: "'b rexp"   and k :: "nat"   and m :: "nat" assumes "\<guillemotleft>rderivs (replicate i as) r\<guillemotright> = \<guillemotleft>r\<guillemotright>"   and "k < i" shows "\<guillemotleft>rderivs (replicate (m * i + k) as) r\<guillemotright> = \<guillemotleft>rderivs (replicate k as) r\<guillemotright>"
tensor_domain_wall_additivity: fixes xs :: "wall"   and ys :: "wall" shows "domain_wall (xs \<otimes> ys) = domain_wall xs + domain_wall ys"
h21: shows "real_of_int \<parallel>h\<parallel>\<^sup>2 \<le> (real_of_int (norm1 h))\<^sup>2"
simps_2: fixes n :: "'b"   and l :: "'b tree"   and v :: "'b"   and va :: "'b tree"   and vb :: "'b tree"   and vc :: "nat"   and h :: "nat" shows "rightmost_item (MKT n l (MKT v va vb vc) h) = rightmost_item (MKT v va vb vc)"
MGU_Const1: fixes c :: "'c"   and d :: "'c" shows "MGU Var (Fun c []) (Fun d []) = (c = d)"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "x \<in> Domain R"
intros_4: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and G :: "'b formula" assumes "F, \<Gamma> \<Rightarrow>\<^sub>n"   and "G, \<Gamma> \<Rightarrow>\<^sub>n" shows "F \<^bold>\<or> G, \<Gamma> \<Rightarrow>\<^sub>n"
algebraic_uminus_iff: fixes x :: "'a" shows "algebraic (- x) = algebraic x"
that: fixes ns :: "'node list" assumes "g \<turnstile> defNode g q-ns\<rightarrow>defNode g p"   and "1 < length ns"   and "\<forall>n\<in>set (butlast ns). \<exists>p q m ns'. r p q \<and> g \<turnstile> defNode g q-ns'\<rightarrow>m \<and> defNode g q \<notin> set (tl ns') \<and> q \<in> phiUses g m \<and> m \<in> set (predecessors g (defNode g p)) \<and> n \<in> set ns' \<and> set ns' \<subseteq> set ns \<and> defNode g p \<in> set ns" shows "thesis"
uncountable_open_segment: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "uncountable (open_segment a b)"
one: fixes U :: "'a set"   and t0 :: "'a"   and t1 :: "'a" assumes "open U"   and "t0 \<in> S"   and "t0 \<in> U"   and "t1 \<in> S - U" shows "\<exists>V. open V \<and> t0 \<in> V \<and> S \<inter> V \<subseteq> U \<and> (\<forall>e>0. \<exists>f\<in>R. f ` S \<subseteq> {0..1} \<and> (\<forall>t\<in>S \<inter> V. f t < e) \<and> (\<forall>t\<in>S - U. 1 - e < f t))"
ord_isom_le: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and a :: "'a"   and b :: "'a" assumes "Order E"   and "ord_isom D E f"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq> b = f a \<preceq>\<^bsub>E\<^esub> f b"
SeqAuto_EmptySet: shows "({SOME x. True}, SOME x. True, {}, {}) \<in> {(S, I, L, D) |S I L D. SeqAuto S I L D}"
scalar_vec_one: fixes v :: "'b vec" shows "(1::'b) \<cdot>\<^sub>v v = v"
\<psi>\<phi>: shows "\<psi> \<circ> \<phi> = J.map"
lprefix_code_1: fixes ys :: "'b llist" shows "lprefix <> ys = True"
arc_lp: shows "arc (linepath (a - complex_of_real e) (a + complex_of_real e))"
TRACES_def: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" shows "TRACES P = {tr. \<exists>a. a \<in> FAILURES P \<and> tr = fst a}"
two_block_non_interfering_row_comp2: fixes i :: "nat"   and A :: "'a mat"   and B :: "'a mat" assumes "i < dim_row (two_block_non_interfering A B)"   and "dim_row A \<le> i" shows "row (two_block_non_interfering A B) i = 0\<^sub>v (dim_col A) @\<^sub>v row B (i - dim_row A)"
nsum_in_Vr: fixes v :: "'a \<Rightarrow> ant"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "valuation K v"   and "\<forall>j\<le>n. f j \<in> carrier K"   and "\<forall>j\<le>n. 0 \<le> v (f j)" shows "\<Sigma>\<^sub>e K f n \<in> carrier (Vr K v)"
AddSA_CompFun_ran_not_mem: fixes SA2 :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto"   and SA1 :: "('d, 'e, 'f) seqauto"   and S :: "'d" assumes "States SA2 \<inter> (HAStates A \<union> States SA1) = {}"   and "States SA1 \<inter> HAStates A = {}"   and "S \<in> HAStates A" shows "{SA2} \<notin> ran (CompFun A [f+] (S, SA1))"
wf_treeOf: fixes ct :: "('sequent, 'rule, 'link) ctree" assumes "cwf ct" shows "local.wf (treeOf pointsTo ct)"
C\<^sub>2s: shows "\<^bold>\<forall>x xa xb. E ((x \<cdot> xa) \<cdot> xb) \<^bold>\<and> \<^bold>\<not> (\<^bold>\<not> (E (x \<cdot> (xa \<cdot> xb))) \<^bold>\<or> (x \<cdot> xa) \<cdot> xb \<noteq> x \<cdot> (xa \<cdot> xb)) \<^bold>\<leftarrow> E (x \<cdot> xa) \<^bold>\<and> E (xa \<cdot> xb)"
full_gb_def: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "full_gb bs = map (\<lambda>k. (monomial (1::'d) (term_of_pair (0::'a, k)), 0, default)) (remdups (map component_of_term (Keys_to_list (map fst bs))))"
staticSecret_parts_agent: fixes m :: "msg"   and C :: "agent"   and evs :: "event list"   and A :: "agent" assumes "m \<in> parts (knows C evs)"   and "m \<in> staticSecret A" shows "A = C \<or> (\<exists>D E X. Says D E X \<in> set evs \<and> m \<in> parts {X}) \<or> (\<exists>Y. Notes C Y \<in> set evs \<and> m \<in> parts {Y})"
ereal_Lim_uminus: fixes f :: "'b \<Rightarrow> ereal"   and f0 :: "ereal"   and net :: "'b filter" shows "(f \<longlongrightarrow> f0) net = ((\<lambda>x. - f x) \<longlongrightarrow> - f0) net"
G_def: shows "G \<equiv> {C \<in> components (U - S). C \<inter> K = {}}"
\<chi>\<sigma>z: shows "subst \<chi>\<sigma> (Var z) inp = exi zz (cnj (subst (subst \<sigma> (Var z) inp) (Var zz) out) (subst \<chi> (Var zz) inp))"
Lim_UN1: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> ('f, 'g) Old_Datatype.node set" shows "Old_Datatype.Lim (\<lambda>x. \<Union> (range (f x))) = (\<Union>y. Old_Datatype.Lim (\<lambda>x. f x y))"
Re_prod_Reals: fixes A :: "'a set"   and f :: "'a \<Rightarrow> complex" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<real>" shows "Re (prod f A) = (\<Prod>x\<in>A. Re (f x))"
list_of_eucl_image_lv_rel_inverse: fixes x :: "'a set" shows "(x, list_of_eucl ` x) \<in> \<langle>lv_rel\<inverse>\<rangle>set_rel"
blocksEval: fixes ps :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   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 e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option" assumes "length ps = length Ts"   and "length ps = length vs"   and "P,E \<turnstile> \<langle>blocks (ps, Ts, vs, e),(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>" shows "\<exists>l'' vs'. P,E(ps [\<mapsto>] Ts) \<turnstile> \<langle>e,(h, l(ps [\<mapsto>] vs'))\<rangle> \<Rightarrow> \<langle>e',(h', l'')\<rangle> \<and> P \<turnstile> Ts Casts vs to vs'  \<and> length vs' = length vs"
degree_mod_less': fixes b :: "'a poly"   and a :: "'a poly" assumes "b \<noteq> 0"   and "a mod b \<noteq> 0" shows "degree (a mod b) < degree b"
euler_err_fas_def: fixes X0 :: "'d list"   and h :: "floatarith"   and CX :: "floatarith list" shows "euler_err_fas X0 h CX = map (euler_err_fas_nth X0 h CX) [0..<length X0]"
closedin_closed: fixes U :: "'a set"   and S :: "'a set" shows "closedin (top_of_set U) S = (\<exists>T. closed T \<and> S = U \<inter> T)"
Y_\<nu>: shows "Y \<subseteq> subformulas\<^sub>\<nu> \<phi>"
qGood_qTerm_induct: fixes phi :: "('d, 'e, 'varSort, 'var, 'f) qTerm \<Rightarrow> bool"   and phiAbs :: "('d, 'e, 'varSort, 'var, 'f) qAbs \<Rightarrow> bool"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "\<And>xs x. phi (qVar xs x)"   and "\<And>delta inp binp. \<lbrakk>|{i. inp i \<noteq> None}| <o |UNIV|; |{i. binp i \<noteq> None}| <o |UNIV|; liftAll (\<lambda>X. qGood X \<and> phi X) inp; liftAll (\<lambda>A. qGoodAbs A \<and> phiAbs A) binp\<rbrakk> \<Longrightarrow> phi (qOp delta inp binp)"   and "\<And>xs x X. \<lbrakk>qGood X; \<And>Y. qGood Y \<and> qSkel Y = qSkel X \<Longrightarrow> phi Y; \<And>Y. (X, Y) \<in> qSwapped \<Longrightarrow> phi Y\<rbrakk> \<Longrightarrow> phiAbs (qAbs xs x X)" shows "(qGood X \<longrightarrow> phi X) \<and> (qGoodAbs A \<longrightarrow> phiAbs A)"
funion_commute: fixes A :: "'c fset"   and B :: "'c fset" shows "A |\<union>| B = B |\<union>| A"
omega_le_1: fixes x :: "'a"   and y :: "'a" shows "\<Omega> x y \<le> d y"
prj_simps_2: shows "C.dom trg_f.prj\<^sub>1 = trg.leg0 \<down>\<down> f.leg1"
Ch_gfp_F_XH_card: fixes h :: "'c" shows "card (Ch h (snd (gfp_F ds))) \<le> card (Ch h (snd XD_XH))"
not_summable_outside_conv_radius: fixes f :: "nat \<Rightarrow> 'b"   and z :: "'b" assumes "conv_radius f < ereal (norm z)" shows "\<not> summable (\<lambda>n. f n * z ^ n)"
integrable_norm_iff: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "integrable M (\<lambda>x. norm (f x)) = integrable M f"
valid_prefix_implies_ltl: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and i :: "nat" assumes "af \<phi> (prefix i w) \<sim>\<^sub>L true\<^sub>n" shows "w \<Turnstile>\<^sub>n \<phi>"
fromQ_cases: fixes y :: "('a, sdim, 'b) Measurement_System" assumes "y \<in> {x. Quantity.dim x = QD('c)}" obtains x :: "'a['c, 'b]" where "y = fromQ x"
multiplicative_sum_divisors: shows "multiplicative_function (\<lambda>n. \<Sum>d | d dvd n. \<chi> d)"
components_are_iso: fixes a :: "'a" assumes "\<phi>.A.ide a" shows "\<phi>.B.iso (F a)"
facet_unique_other_chamber: fixes B :: "'a set"   and z :: "'a set"   and C :: "'a set"   and D :: "'a set" assumes "folding_g.chamber B"   and "z \<lhd> B"   and "folding_g.chamber C"   and "z \<lhd> C"   and "folding_g.chamber D"   and "z \<lhd> D"   and "C \<noteq> B"   and "D \<noteq> B" shows "C = D"
indep_in_subI: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set"   and \<E>' :: "'b set" assumes "indep_system carrier indep"   and "indep_system.indep_in indep \<E> X" shows "indep_system.indep_in indep \<E>' (X \<inter> \<E>')"
Apps_snoc: fixes f :: "expr"   and xs :: "expr list"   and x :: "expr" shows "f \<bullet> (xs @ [x]) = f \<bullet> xs \<cdot> x"
is_Abs_def: fixes prot_fun :: "('a, 'b, 'c) prot_fun" shows "is_Abs prot_fun = (\<exists>x4. prot_fun = Abs x4)"
integrableI_bounded: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "f \<in> borel_measurable M"   and "\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>" shows "integrable M f"
has_limits_of_shape_def: fixes J :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "J'.has_limits_of_shape J \<equiv> \<forall>D. diagram J J' D \<longrightarrow> (\<exists>a \<chi>. limit_cone J J' D a \<chi>)"
simps_3: fixes r :: "real" shows "\<infinity> * ereal r = (if r = 0 then 0 else if 0 < r then \<infinity> else - \<infinity>)"
lincomb_vec_diff_add: fixes A :: "'a vec set"   and B :: "'a vec set"   and f :: "'a vec \<Rightarrow> 'a" assumes "A \<subseteq> carrier_vec n"   and "B \<subseteq> A"   and "finite A"   and "f \<in> A \<rightarrow> UNIV" shows "lincomb f A = lincomb f (A - B) + lincomb f B"
finite_sum_legendre_aux: fixes p :: "nat"   and x :: "real" assumes "prime p" shows "finite {m. 0 < m \<and> real (p ^ m) \<le> x}"
A_1: shows "val_type (expr_sem_rf \<sigma> e1) = t1"
in_lset_conv_lnth: fixes x :: "'a"   and xs :: "'a llist" shows "(x \<in> lset xs) = (\<exists>n. enat n < llength xs \<and> lnth xs n = x)"
emeasure_pair_measure_Times: fixes A :: "'b set"   and N :: "'b measure"   and B :: "('i \<Rightarrow> 'a) set" assumes "A \<in> sets N"   and "B \<in> sets (Pi\<^sub>M J M)" shows "emeasure (N \<Otimes>\<^sub>M Pi\<^sub>M J M) (A \<times> B) = emeasure N A * emeasure (Pi\<^sub>M J M) B"
add: fixes f :: "'e \<Rightarrow> 'f"   and x :: "'e"   and y :: "'e" assumes "additive f" shows "f (x + y) = f x + f y"
comp_ide_self: fixes a :: "'a" assumes "Go\<tau>.A.ide a" shows "A a a = a"
Abs_inverse: fixes y :: "int" assumes "y \<in> uints LENGTH('b)" shows "uint (word_of_int y) = y"
B: shows "Maps.Map (\<lbrakk>\<lbrakk>u\<^sub>1\<rbrakk>\<rbrakk> \<odot> \<lbrakk>\<lbrakk>\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>) = Maps.Comp \<lbrakk>u\<^sub>1\<rbrakk> \<lbrakk>\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk>"
ntrunc_Atom: fixes k :: "nat"   and a :: "'a + nat" shows "ntrunc (Suc k) (Atom a) = Atom a"
r_b_def: shows "r_b = {(0, z) |z. z \<in> B}"
Try\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and C :: "char list"   and i :: "nat"   and e\<^sub>2 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>try e\<^sub>1 catch(C i) e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"
nths_zip: fixes xs :: "'b list"   and ys :: "'c list"   and I :: "nat set" shows "nths (zip xs ys) I = zip (nths xs I) (nths ys I)"
pw_ords_iff_1: fixes S :: "'b nres"   and S' :: "'b nres" shows "(S \<le> S') = (nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> inres S' x))"
affine_dependent_explicit_finite: fixes S :: "'c set" assumes "finite S" shows "affine_dependent S = (\<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = (0::'c))"
that: fixes h :: "'t \<Rightarrow>\<^sub>0 'b" assumes "h \<in> F"   and "h \<noteq> 0"   and "lt h adds\<^sub>t lt f" shows "thesis"
Collect_split_Suc_2: fixes P :: "nat \<Rightarrow> bool" assumes "P 0" shows "{i. P i} = {0} \<union> Suc ` {i. P (Suc i)}"
open_closed_segment: fixes u :: "'b"   and w :: "'b"   and z :: "'b" assumes "u \<in> {w<--<z}" shows "u \<in> {w--z}"
set_of_rel_inverse: fixes x :: "'a Executable_Relation.rel" shows "rel_of_set (set_of_rel x) = x"
max_subgraph_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and H :: "('e, 'f) pre_digraph"   and P :: "('e, 'f) pre_digraph \<Rightarrow> bool" assumes "digraph_isomorphism hom"   and "subgraph H (app_iso hom G)" shows "pre_digraph.max_subgraph (app_iso hom G) P H = max_subgraph (P \<circ> app_iso hom) (app_iso (inv_iso hom) H)"
naturality: fixes f :: "'a" assumes "arr f" shows "\<ll>' (cod f) \<cdot> local.map f = (\<I> \<otimes> f) \<cdot> \<ll>' (local.dom f)"
hom_power: fixes x :: "int poly"   and n :: "nat" shows "Ring_Hom_Poly.of_int_poly (x ^ n) = Ring_Hom_Poly.of_int_poly x ^ n"
unit_simps_4: fixes a :: "'a" assumes "C.obj a" shows "D.dom (G.unit a) = G.map\<^sub>0 a"
hom_boundary1: shows "\<exists>d. \<forall>p X S. d p X S \<in> hom (relative_homology_group (int p) X S) (homology_group (int (p - Suc 0)) (subtopology X S)) \<and> (\<forall>c. singular_relcycle p X S c \<longrightarrow> d p X S (homologous_rel_set p X S c) = homologous_rel_set (p - Suc 0) (subtopology X S) {} (chain_boundary p c))"
sorted_list_of_fset_sort: fixes l :: "'b list" shows "sorted_list_of_fset (fset_of_list l) = sort (remdups l)"
lift_valid_node: fixes prog :: "cmd"   and n :: "w_node" assumes "While_CFG.valid_node prog n" shows "CFG.valid_node src trg (lift_valid_edge (valid_edge prog) sourcenode targetnode kind (_Entry_) (_Exit_)) (LDCFG_node.Node n)"
neg_less_0_iff_less: fixes 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 plus zero minus uminus less_eq less" shows "less (uminus a) zero = less zero a"
disjunctive_def: fixes sup_b :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and sup_c :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and times_abc :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "disjunctive.disjunctive sup_b sup_c times_abc = {x. \<forall>y z. times_abc x (sup_b y z) = sup_c (times_abc x y) (times_abc x z)}"
mat_eqI: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'b list list"   and m2 :: "'b list list" assumes "mat nr nc m1"   and "mat nr nc m2"   and "\<And>i j. \<lbrakk>i < nc; j < nr\<rbrakk> \<Longrightarrow> m1 ! i ! j = m2 ! i ! j" shows "m1 = m2"
term_variants_pred_inv_Var_2: fixes P :: "'c \<Rightarrow> 'c list"   and t :: "('c, 'd) Term.term"   and x :: "'d" shows "term_variants_pred P t (Var x) = (t = Var x)"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "semigroup_add_hom hom"   and "injective hom" shows "inj_semigroup_add_hom hom"
simps_22: fixes x8 :: "('a, 'b, 'c) expr"   and x1 :: "('a, 'b, 'c) val" shows "Rjoin x8 \<noteq> VE x1"
scaleR_half_double: fixes a :: "'b" shows "(1 / 2) *\<^sub>R (a + a) = a"
ObjectC_def: shows "ObjectC = (Object, \<lparr>access = Public, cfields = [], methods = Object_mdecls, init = Skip, super = undefined, superIfs = []\<rparr>)"
openin: fixes x :: "'c topology" shows "openin x \<in> {L. istopology L}"
LiberalConseq_noguards_nothrows_sound: 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' :: "'d \<Rightarrow> 'f set"   and c :: "('f, 'e, 'g) com"   and Q' :: "'d \<Rightarrow> 'f set"   and A' :: "'d \<Rightarrow> 'f set"   and P :: "'f set"   and Q :: "'f set"   and n :: "nat"   and A :: "'f set" assumes "\<forall>Z n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> (P' Z) c (Q' Z),(A' Z)"   and "\<forall>s t. (\<forall>Z. s \<in> P' Z \<longrightarrow> t \<in> Q' Z) \<longrightarrow> s \<in> P \<longrightarrow> t \<in> Q"   and "noguards c"   and "\<forall>p\<in>dom \<Gamma>. noguards (the (\<Gamma> p))"   and "nothrows c"   and "\<forall>p\<in>dom \<Gamma>. nothrows (the (\<Gamma> p))"   and "noSpec c"   and "\<forall>p\<in>dom \<Gamma>. noSpec (the (\<Gamma> p))"   and "procs c \<subseteq> dom \<Gamma>"   and "\<forall>p\<in>dom \<Gamma>. procs (the (\<Gamma> p)) \<subseteq> dom \<Gamma>" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
p'_zero: fixes a :: "real"   and v :: "real" shows "movement.p' a v 0 = v"
set_replicate_conv_if: fixes n :: "nat"   and x :: "'b" shows "set (replicate n x) = (if n = 0 then {} else {x})"
comp_assoc: fixes h :: "'c local.term set"   and g :: "'c local.term set"   and f :: "'c local.term set" shows "(h \<cdot> g) \<cdot> f = h \<cdot> g \<cdot> f"