hcju/isabelleps huggingface dataset:
interleave_s_assoc1: fixes w :: "'b list"   and w1 :: "'b list"   and ws :: "'b list"   and w2 :: "'b list"   and w3 :: "'b list" assumes "w \<in> w1 \<otimes> ws \<and> ws \<in> w2 \<otimes> w3" shows "\<exists>ws'\<in>w1 \<otimes> w2. w \<in> ws' \<otimes> w3"
hyps_1: fixes s :: "'a"   and c :: "('a, 'b, 'c) com" assumes "\<Gamma>\<turnstile>c1' \<down> Normal s"   and "c \<subseteq>\<^sub>g c1'"   and "\<Gamma>\<turnstile> \<langle>c1',Normal s\<rangle> \<Rightarrow>\<notin>range Fault" shows "\<Gamma>\<turnstile>c \<down> Normal s"
endswith: shows "endswith\<cdot>pat = p oo sfoldl\<cdot>op\<cdot>z"
uniform_limit_set_lebesgue_integral_at_top: fixes A :: "'a set"   and a :: "real"   and f :: "'a \<Rightarrow> real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> real"   and M :: "real measure" assumes "\<And>x y. \<lbrakk>x \<in> A; a \<le> y\<rbrakk> \<Longrightarrow> norm (f x y) \<le> g y"   and "set_integrable M {a..} g"   and "\<And>x. x \<in> A \<Longrightarrow> set_borel_measurable M {a..} (f x)"   and "sets borel \<subseteq> sets M" shows "uniform_limit A (\<lambda>b x. set_lebesgue_integral M {a..b} (f x)) (\<lambda>x. set_lebesgue_integral M {a..} (f x)) at_top"
hvzk: fixes x0 :: "'g"   and x1 :: "'a"   and w0 :: "'h"   and w1 :: "'b"   and e :: "'e" assumes "((x0, x1), w0, w1) \<in> Rel_AND"   and "e \<in> challenge_space" shows "\<Sigma>_AND.R (x0, x1) (w0, w1) e = \<Sigma>_AND.S (x0, x1) e"
homeomorphic_eq_injective_perfect_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "homeomorphic_map X Y f = (perfect_map X Y f \<and> inj_on f (topspace X))"
atomize_all: fixes P :: "'b \<Rightarrow> bool" shows "(\<And>x. P x) \<equiv> \<forall>x. P x"
widen_Class: fixes P :: "'a prog"   and T :: "ty"   and C :: "String.literal" shows "P \<turnstile> T \<le> Class C = (T = NT \<or> (\<exists>D. T = Class D \<and> P \<turnstile> D \<preceq>\<^sup>* C) \<or> C = Object \<and> (\<exists>A. T = A\<lfloor>\<rceil>))"
module_with_axioms_def: fixes plus\<^sub>M :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "module_with_axioms plus\<^sub>M scale \<equiv> ((\<forall>a x y. scale a (plus\<^sub>M x y) = plus\<^sub>M (scale a x) (scale a y)) \<and> (\<forall>a b x. scale (a + b) x = plus\<^sub>M (scale a x) (scale b x))) \<and> (\<forall>a b x. scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. scale (1::'e) x = x)"
M: shows "\<forall>m\<in>M. homogeneous\<^sub>l\<^sub>s\<^sub>t m \<A> Sec \<and> m \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
greyI_1: fixes r :: "'e"   and s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state"   and p :: "'g process_name" assumes "r \<in> ghost_honorary_grey (s p)" shows "grey r s"
case_Some: fixes m :: "'a simple_match" assumes "Some m = simple_match_and m1 m2" shows "(simple_matches m1 p \<and> simple_matches m2 p) = simple_matches m p"
Notes_TTP_imp_Says_A: fixes A :: "agent"   and B :: "agent"   and L :: "nat"   and K :: "nat"   and con_K :: "msg"   and evs :: "event list"   and sub_K :: "msg" assumes "Notes TTP \<lbrace>Number f_con, Agent A, Agent B, Nonce L, Key K, con_K\<rbrace> \<in> set evs"   and "sub_K = Crypt (priEK A) \<lbrace>Number f_sub, Agent B, Nonce L, Key K\<rbrace>"   and "A \<notin> broken"   and "evs \<in> zg" shows "Says A TTP \<lbrace>Number f_sub, Agent B, Nonce L, Key K, sub_K\<rbrace> \<in> set evs"
PPs_insertI: fixes tx :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and t :: "'b \<Rightarrow>\<^sub>0 nat"   and x :: "'b"   and e :: "nat" assumes "tx \<in> .[X]"   and "t = monomial e x + tx" shows "t \<in> .[insert x X]"
x_split_var: fixes x :: "'a" shows "x \<sqinter> nc + x \<cdot> (0::'a) = x"
CAS1Throw: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr1"   and e3 :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Throw a\<bullet>compareAndSwap(D\<bullet>F, e2, e3),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
atm_of_lit_in_atms_of: fixes L :: "'b literal"   and C :: "'b literal multiset" assumes "L \<in># C" shows "atm_of L \<in> atms_of C"
tautology_def: fixes C :: "'a Literal set" shows "tautology C \<equiv> \<exists>A. Pos A \<in> C \<and> Neg A \<in> C"
intra_proc_matched: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and a' :: "'a" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "a' \<in> get_return_edges a" shows "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses (CFG_node (targetnode a)) [CFG_node (targetnode a)] (CFG_node (sourcenode a'))"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "monoid_add_hom hom" shows "group_add_hom hom"
simps_3: fixes v :: "'c"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree" shows "is_heap (T v E (T va vb vc)) = (val (T va vb vc) \<le> v \<and> is_heap (T va vb vc))"
card_atLeastAtMost_int: fixes l :: "int"   and u :: "int" shows "card {l..u} = nat (u - l + 1)"
drop_eq_nths: fixes n :: "nat"   and xs :: "'g list" shows "drop n xs = nths xs {i. n \<le> i}"
BNTr6: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and W1 :: "'a set"   and W2 :: "'a set" assumes "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x"   and "W1 \<in> WWa D f a"   and "W2 \<in> WWa D f a"   and "W1 \<subset> W2" shows "\<exists>b\<in>carrier (Iod D W2). ord_equiv (Iod D W1) (Iod D (segment (Iod D W2) b))"
atLeastAtMost_iff: fixes i :: "'x \<Rightarrow>\<^sub>0 nat"   and l :: "'x \<Rightarrow>\<^sub>0 nat"   and u :: "'x \<Rightarrow>\<^sub>0 nat" shows "(i \<in> ordered_powerprod_lin.atLeastAtMost l u) = (l \<preceq> i \<and> i \<preceq> u)"
digraph_isomorphism_invI: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "aG.digraph_isomorphism hom" shows "pre_digraph.digraph_isomorphism (app_iso hom (app_iso hom G)) (aG.inv_iso hom)"
S_def: fixes Rel :: "('a \<times> 'b) set"   and valid_pub :: "'a set"   and S_raw :: "'a \<Rightarrow> 'd \<Rightarrow> ('c \<times> 'e) spmf"   and h :: "'a"   and e :: "'d" assumes "\<Sigma>_protocols_base Rel valid_pub" shows "\<Sigma>_protocols_base.S S_raw h e = map_spmf (\<lambda>(a, z). (a, e, z)) (S_raw h e)"
hyps_3: assumes "atom x \<sharp> (\<Delta> @ \<Gamma>, body)" shows "atom x \<sharp> (\<Theta>, z) \<or> x \<in> domA \<Theta> - set L"
boxInit: fixes F :: "'a \<Rightarrow> bool" shows "\<box>F \<equiv> \<box>Init F"
project_preserves_I: fixes G :: "'c program"   and v :: "'a \<Rightarrow> 'd"   and C :: "'c set" assumes "G \<in> preserves (v \<circ> f)" shows "project h C G \<in> preserves v"
union_disjoint: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "prod g (A \<union> B) = prod g A * prod g B"
negative_cycle_dest_diag: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'd"   and n :: "nat" assumes "\<not> cycle_free M n" shows "\<exists>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<and> len M i i xs < \<one>"
vconverse_vunion: fixes r :: "V"   and s :: "V" shows "(r \<union>\<^sub>\<circ> s)\<inverse>\<^sub>\<circ> = r\<inverse>\<^sub>\<circ> \<union>\<^sub>\<circ> s\<inverse>\<^sub>\<circ>"
not_le: fixes x :: "'c word"   and y :: "'c word" shows "(\<not> x \<le>s y) = (y <s x)"
goodInp_imp_qGoodInp_pickInp: fixes inp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option" assumes "goodInp inp" shows "qGoodInp (pickInp inp)"
try_gpv_Fail: fixes gpv' :: "('d, 'e, 'f) gpv" shows "TRY Fail ELSE gpv' = gpv'"
WhileBodyThrow: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and c :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>c,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
map_cong: fixes xs :: "'d list"   and ys :: "'d list"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "map f xs = map g ys"
cases: fixes a :: "'a"   and g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)" assumes "a \<in> terminates_on g" obtains s :: "'a" where "a = s__" and "\<not> fst g s__"   | s :: "'a" where "a = s" and "fst g s" and "snd (snd g s) \<in> terminates_on g"
distinct_34: fixes x71 :: "trm"   and x72 :: "char list"   and x31 :: "POPLmarkRecord.type"   and x32 :: "trm" shows "x71..x72 \<noteq> (\<lambda><:x31. x32)"
non_dom_eval_eq: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd option" shows "(x \<notin> dom m) = (m x = None)"
mult_isor: fixes x :: "('a \<times> ('b \<times> 'a) list) set"   and y :: "('a \<times> ('b \<times> 'a) list) set"   and z :: "('a \<times> ('b \<times> 'a) list) set" assumes "x \<subseteq> y" shows "t_prod x z \<subseteq> t_prod y z"
comm_monoid_axioms_def: fixes G :: "('e, 'f) monoid_scheme" shows "Group.comm_monoid_axioms G \<equiv> \<forall>x y. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
the_lcg_edges: fixes Rules :: "(('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph) set"   and sel :: "nat \<times> ('b \<times> nat \<times> nat) set \<Rightarrow> ((('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph) \<times> ('c \<times> nat) set) option"   and L :: "'b set"   and G :: "nat \<times> ('b \<times> nat \<times> nat) set" assumes "valid_selector Rules sel"   and "fst ` \<Union> ((edges \<circ> snd) ` Rules) \<subseteq> L"   and "fst ` snd G \<subseteq> L" shows "fst ` edges (the_lcg sel Rules G) \<subseteq> L"
simps_3: fixes x :: "'b" shows "extB.Val x + extB.Bot = extB.Val x"
Ssegment_inc_psub: fixes D :: "'c Order"   and x :: "'c"   and a :: "'c" assumes "Order D"   and "x \<in> Ssegment D a" shows "Ssegment D x \<subset> Ssegment D a"
rel_mono_strong: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a pmf"   and y :: "'b pmf"   and Ra :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_pmf R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_pmf x; yb \<in> set_pmf y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_pmf Ra x y"
assms: shows "low_equal s1 s2 \<and> t1 = cpu_reg_mod w cr s1 \<and> t2 = cpu_reg_mod w cr s2"
length_transfer_kind_slice_kind: fixes a :: "'edge"   and s\<^sub>1 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s\<^sub>2 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s\<^sub>1' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and S :: "'node SDG_node set"   and s\<^sub>2' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "valid_edge a"   and "length s\<^sub>1 = length s\<^sub>2"   and "transfer (kind a) s\<^sub>1 = s\<^sub>1'"   and "transfer (slice_kind S a) s\<^sub>2 = s\<^sub>2'" shows "length s\<^sub>1' = length s\<^sub>2'"
max2_equals2: fixes r :: "('d \<times> 'd) set"   and a :: "'d"   and b :: "'d" assumes "wo_rel r"   and "a \<in> Field r"   and "b \<in> Field r" shows "(max2 r a b = b) = ((a, b) \<in> r)"
orderI: fixes a :: "'b"   and b :: "'b" assumes "a = sup a b" shows "b \<le> a"
list_ex_iff: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "list_ex P xs = Bex (set xs) P"
nth_Cons: fixes x :: "'d"   and xs :: "'d list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
lexer_correctness_2: fixes r :: "'b rexp"   and s :: "'b list" shows "(lexer r s = None) = (\<nexists>v. s \<in> r \<rightarrow> v)"
fup1: fixes f :: "'c \<rightarrow> 'b" shows "fup\<cdot>f\<cdot>\<bottom> = \<bottom>"
mem_carrier: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "H \<le> G"   and "x \<in> H" shows "x \<in> carrier G"
one_dim_iso_is_of_complex: shows "one_dim_iso = of_complex"
ladder_\<alpha>_full: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and a :: "symbol list"   and D' :: "(nat \<times> symbol \<times> symbol list) list"   and L' :: "(nat \<times> nat \<times> nat) list" assumes "is_ladder D L"   and "ladder_last_n L = length D" shows "ladder_\<alpha> a (D @ D') (L @ L') (length L) = Derive a D"
closedin_closed: fixes U :: "'c set"   and S :: "'c set" shows "closedin (top_of_set U) S = (\<exists>T. closed T \<and> S = U \<inter> T)"
bounded_linear: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_antilinear f" shows "bounded_linear f"
final_threadE: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "final_thread s t" obtains x :: "'x" where "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" and "final x" and "wset s t = None"
wf12: shows "mat nr nc (mat_multI ze pl ti nr m1 m2)"
union_diff2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "comm_monoid_set f z"   and "finite A"   and "finite B" shows "comm_monoid_set.F f z g (A \<union> B) = f (f (comm_monoid_set.F f z g (A - B)) (comm_monoid_set.F f z g (B - A))) (comm_monoid_set.F f z g (A \<inter> B))"
dom_simp_3: shows "partial_magma.dom (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) parallel_pair.j0 = quiver.Zero"
assms_3: shows "\<forall>s t. s \<le> t \<longrightarrow> subalgebra (G t) (G s)"
induct: fixes x1 :: "com"   and x2 :: "loc \<Rightarrow> nat"   and x3 :: "loc \<Rightarrow> nat"   and eval :: "((exp \<times> (loc \<Rightarrow> nat)) \<times> nat \<times> (loc \<Rightarrow> nat)) set"   and P :: "com \<Rightarrow> (loc \<Rightarrow> nat) \<Rightarrow> (loc \<Rightarrow> nat) \<Rightarrow> bool" assumes "(x1, x2) -[eval]-> x3"   and "\<And>s. P SKIP s s"   and "\<And>e s v s' x. ((e, s), v, s') \<in> eval \<Longrightarrow> P (x := e) s (s'(x := v))"   and "\<And>c0 s s2 c1 s1. \<lbrakk>(c0, s) -[eval]-> s2; P c0 s s2; (c1, s2) -[eval]-> s1; P c1 s2 s1\<rbrakk> \<Longrightarrow> P (c0;;c1) s s1"   and "\<And>e s s' c0 s1 c1. \<lbrakk>((e, s), 0, s') \<in> eval; (c0, s') -[eval]-> s1; P c0 s' s1\<rbrakk> \<Longrightarrow> P (IF e THEN c0 ELSE c1) s s1"   and "\<And>e s s' c1 s1 c0. \<lbrakk>((e, s), Suc 0, s') \<in> eval; (c1, s') -[eval]-> s1; P c1 s' s1\<rbrakk> \<Longrightarrow> P (IF e THEN c0 ELSE c1) s s1"   and "\<And>e s s1 c. ((e, s), Suc 0, s1) \<in> eval \<Longrightarrow> P (WHILE e DO c) s s1"   and "\<And>e s s1 c s2 s3. \<lbrakk>((e, s), 0, s1) \<in> eval; (c, s1) -[eval]-> s2; P c s1 s2; (WHILE e DO c, s2) -[eval]-> s3; P (WHILE e DO c) s2 s3\<rbrakk> \<Longrightarrow> P (WHILE e DO c) s s3" shows "P x1 x2 x3"
False: shows "\<not> {x. poly p x = 0 \<and> a < x \<and> x < b} \<noteq> {}"
cut: fixes \<Gamma>' :: "'a formula multiset"   and A :: "'a formula"   and \<Delta>' :: "'a formula multiset" assumes "\<Gamma>' \<Rightarrow> A, \<Delta>'"   and "A, \<Gamma>' \<Rightarrow> \<Delta>'" shows "\<Gamma>' \<Rightarrow> \<Delta>'"
spmf_map_pmf_Some: fixes p :: "'a pmf"   and i :: "'a" shows "spmf (map_pmf Some p) i = pmf p i"
cospan_\<nu>\<pi>: shows "C.cospan \<nu>.leg0 \<pi>\<rho>.leg1"
asymp_equiv_sandwich': fixes l :: "real"   and c :: "real"   and g :: "'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and u :: "real" assumes "\<And>c'. c' \<in> {l<..<c} \<Longrightarrow> \<forall>\<^sub>F x in F. c' * g x \<le> f x"   and "\<And>c'. c' \<in> {c<..<u} \<Longrightarrow> \<forall>\<^sub>F x in F. f x \<le> c' * g x"   and "l < c"   and "c < u"   and "c \<noteq> 0" shows "f \<sim>[F] (\<lambda>x. c * g x)"
cosets_finite: fixes G :: "('c, 'd) monoid_scheme"   and c :: "'c set"   and H :: "'c set" assumes "Group.group G"   and "c \<in> rcosets\<^bsub>G\<^esub> H"   and "H \<subseteq> carrier G"   and "finite (carrier G)" shows "finite c"
split_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and err :: "'c err" shows "P (case err of Err \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = (\<not> (err = Err \<and> \<not> P f1 \<or> (\<exists>x2. err = OK x2 \<and> \<not> P (f2 x2))))"
ksimplex_replace_0: fixes p :: "nat"   and n :: "nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "ksimplex p n s"   and "a \<in> s"   and "j < n"   and "\<forall>x\<in>s - {a}. x j = 0" shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1"
lossless_fdr_step: fixes v :: "nat"   and c :: "nat" shows "lossless_spmf (local.fdr_step v c) = (0 < v)"
length_stake: fixes n :: "nat"   and s :: "'b stream" shows "length (stake n s) = n"
euler_liar_cong: fixes a :: "int"   and b :: "int"   and p :: "nat" assumes "[a = b] (mod int p)" shows "euler_liar a p = euler_liar b p"
assms: shows "s' = snd (fst (add_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
fequalityI: fixes A :: "'a fset"   and B :: "'a fset" assumes "A |\<subseteq>| B"   and "B |\<subseteq>| A" shows "A = B"
appL: fixes s :: "Core.term"   and t :: "Core.term"   and u :: "Core.term" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "s $ u \<rightarrow>\<^sub>\<eta> t $ u"
ennreal_of_enat_eSuc: fixes x :: "enat" shows "ennreal_of_enat (eSuc x) = 1 + ennreal_of_enat x"
holomorphic_on_extend: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and \<xi> :: "complex" assumes "f holomorphic_on S - {\<xi>}"   and "\<xi> \<in> interior S"   and "f \<in> O[at \<xi>](\<lambda>_. 1)" shows "\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S - {\<xi>}. g z = f z)"
impCE: fixes P :: "bool"   and Q :: "bool" assumes "P \<longrightarrow> Q" obtains "\<not> P"   | "Q"
mult2_s_locally_one_step': fixes ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and X :: "'b multiset"   and Y :: "'b multiset" assumes "ns O s \<subseteq> s"   and "refl ns"   and "locally_irrefl s X"   and "locally_irrefl s Y"   and "trans s" shows "((X, Y) \<in> mult2_s ns s) = (\<exists>X1 X2 Y1 Y2. X = X1 + X2 \<and> Y = Y1 + Y2 \<and> (X1, Y1) \<in> multpw ns \<and> (X2, Y2) \<in> mult s)"
valid_validn: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and K :: "nat" assumes " \<Turnstile> \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I" shows "\<Turnstile>\<^sub>K \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I "
r_coset_hom_Mod: fixes H :: "'b set"   and G :: "('b, 'c) monoid_scheme" assumes "H \<lhd> G" shows "(#>\<^bsub>G\<^esub>) H \<in> hom G (G Mod H)"
elims_2: fixes x :: "symbol list"   and xa :: "(nat \<times> symbol \<times> symbol list) list"   and xb :: "symbol list" assumes "Derivation x xa xb" obtains a :: "symbol list" and b :: "symbol list" where "x = aa__" and "xa = []" and "xb = ba__" and "aa__ = ba__"   | a :: "symbol list" and d :: "nat \<times> symbol \<times> symbol list" and D :: "(nat \<times> symbol \<times> symbol list) list" and b :: "symbol list" where "x = a" and "xa = d # D" and "xb = b" and "\<exists>x. Derives1 a (fst d) (snd d) x \<and> Derivation x D b"
simps_3: fixes M :: "('b, 'c) kripke"   and s :: "'c"   and p :: "'b fm"   and q :: "'b fm" shows "(M, s \<Turnstile> p \<^bold>\<or> q) = (M, s \<Turnstile> p \<or> M, s \<Turnstile> q)"
less_name_def: fixes xs :: "name"   and ys :: "name" shows "(xs < ys) = ((as_string xs, as_string ys) \<in> lexord {(u, v). of_char u < of_char v})"
dims_Aw'_2: shows "dim_col Aw' = prod_list (nths (dims Aw) {n. odd n})"
cong_exp_iff_simps_3: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and n :: "num"   and q :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit1 n)) (numeral.numeral one plus (num.Bit0 q)) = zero) = False"
2: shows "exec_plan s (as1 @ as3) = exec_plan s (as1 @ as2 @ as3)"
minus_zero: fixes R :: "('d, 'e) ring_scheme" assumes "ring R" shows "\<ominus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
cnf_\<alpha>_struct_1: shows "cnf_\<alpha> [] = {}"
exactly_def: fixes s :: "char list"   and x :: "char list" shows "exactly s x = ??.Parser_Monad.exactly_aux s x s x"
lprefix_not_lnullD: fixes xs :: "'f llist"   and ys :: "'f llist" assumes "lprefix xs ys"   and "\<not> lnull xs" shows "\<not> lnull ys"
stake_append: fixes n :: "nat"   and u :: "'b list"   and s :: "'b stream" shows "stake n (u @- s) = take (min (length u) n) u @ stake (n - length u) s"