hcju/isabelleps huggingface dataset:
return_def: shows "return = COERCE(udom \<rightarrow> udom\<cdot>'c, 'b \<rightarrow> 'b\<cdot>'c)\<cdot>returnU"
SFAccInitDoneRed: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sh :: "sheap"   and sfs :: "sfields"   and h :: "heap"   and l :: "sfields" assumes "P \<turnstile> C has F,Static:t in D"   and "sh D = \<lfloor>(sfs, Done)\<rfloor>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},(h, l, sh),False\<rangle> \<rightarrow> \<langle>C\<bullet>\<^sub>sF{D},(h, l, sh),True\<rangle>"
size_mset_Fun_less: fixes ss :: "('c, 'd) Term.term list"   and ts :: "('e, 'f) Term.term list"   and f :: "'c"   and g :: "'e" shows "size_mset (mset (zip ss ts)) < size_mset {#(Fun f ss, Fun g ts)#}"
relcompEpair: fixes a :: "'c"   and c :: "'d"   and r :: "('c \<times> 'e) set"   and s :: "('e \<times> 'd) set" assumes "(a, c) \<in> r O s" obtains b :: "'e" where "(a, b) \<in> r" and "(b, c) \<in> s"
Suc_n_div_2_gt_zero: fixes n :: "nat" assumes "0 < n" shows "0 < Suc n div 2"
inject_2: fixes x2 :: "lam"   and x1 :: "lam"   and y2 :: "lam"   and y1 :: "lam" shows "(App x2 x1 = App y2 y1) = (x2 = y2 \<and> x1 = y1)"
mod_ring_plus: shows "(mod_ring_rel ===> mod_ring_rel ===> mod_ring_rel) (plus_p p) (+)"
span_span: fixes W :: "'a vec set" assumes "W \<subseteq> carrier_vec n" shows "local.span (local.span W) = local.span W"
CONSTRAINT_def: fixes P :: "'f \<Rightarrow> 'e"   and x :: "'f" shows "CONSTRAINT P x \<equiv> P x"
disj_nb: fixes p :: "fm"   and q :: "fm" assumes "bound0 p"   and "bound0 q" shows "bound0 (MIR.disj p q)"
choose_step_def: fixes X0 :: "'c set"   and h :: "real" shows "choose_step X0 h = method_spec \<bind> (\<lambda>mid. if mid = 2 then rk2_step X0 h else euler_step X0 h)"
statEqvt'': fixes p :: "(name \<times> name) list"   and M :: "'a"   and N :: "'a" shows "p \<bullet> M \<leftrightarrow> N = (p \<bullet> M) \<leftrightarrow> (p \<bullet> N)"
i_drop_the_conv: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Up> k = (THE g. \<exists>xs. length xs = k \<and> xs \<frown> g = f)"
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and tell :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_writer return bind tell" shows "monad return bind"
bounded_T: fixes t :: "symbol \<times> character list" assumes "t \<in> T" shows "length (chars_of_token t) \<le> l"
cross_mult_left: fixes c :: "real"   and x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "(c *\<^sub>R x) \<times> y = c *\<^sub>R x \<times> y"
Image_def: fixes r :: "('d \<times> 'c) set"   and s :: "'d set" shows "r `` s = {y. \<exists>x\<in>s. s2p r (x, y)}"
inf_aci_2: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) z = inf x (inf y z)"
MBC_def: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "MBC t n = (SOME b. b \<in> actHn (t n) \<and> pow_cond t n (length (bc (\<sigma>\<^bsub>b\<^esub>t n))))"
mult_zero_left: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "mbot \<odot> a = mbot"
subspace_Times: fixes A :: "'b set"   and B :: "'c set" assumes "vs1.subspace A"   and "vs2.subspace B" shows "p.subspace (A \<times> B)"
discI: fixes tree :: "'d tree"   and x21 :: "'d tree"   and x22 :: "'d"   and x23 :: "'d tree" assumes "tree = \<langle>x21, x22, x23\<rangle>" shows "tree \<noteq> \<langle>\<rangle>"
ipresIGFreshAll_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('f, 'g, 'varSort, 'h, 'i, 'var, 'd, 'e, 'j) model_scheme" shows "ipresIGFreshAll h hA termMOD MOD = ipresFreshAll h hA MOD"
DeltaForall: fixes A :: "('a, 'b) form"   and n :: "'a"   and G :: "('a, 'b) form list" assumes "\<turnstile> A[App n []/0] # G"   and "news n (A # G)" shows "\<turnstile> Forall A # G"
m_lcancel: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a \<noteq> \<zero>\<^bsub>poly_ring R\<^esub>"   and "a \<in> carrier (poly_ring R)"   and "b \<in> carrier (poly_ring R)"   and "c \<in> carrier (poly_ring R)" shows "(a \<otimes>\<^bsub>poly_ring R\<^esub> b = a \<otimes>\<^bsub>poly_ring R\<^esub> c) = (b = c)"
assms_2: shows "t2 < t + 3 + k + d"
prefix_append: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "prefix xs (ys @ zs) = (prefix xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefix us zs))"
image_subset_iff: fixes f :: "'i \<Rightarrow> 'h"   and A :: "'i set"   and B :: "'h set" shows "(f ` A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
foldli_def: shows "foldli \<equiv> rec_list (\<lambda>c f \<sigma>. \<sigma>) (\<lambda>x xs xsa c f \<sigma>. if c \<sigma> then xsa c f (f x \<sigma>) else \<sigma>)"
ff_def: shows "ff = (\<lambda>x. if b \<le> x then f x else f b)"
HFail_HInv5_q2: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "p \<noteq> q"   and "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s q) < mbal (disk s d qq) \<and> \<not> hasRead s q d qq" shows "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s' q) < mbal (disk s' d qq) \<and> \<not> hasRead s' q d qq"
exhaust: fixes y :: "Enum.finite_3" obtains "y = a\<^sub>1"   | "y = a\<^sub>2"   | "y = a\<^sub>3"
initReachable_def: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and cfg :: "('a, 'c, 'b) configuration" shows "asynchronousSystem.initReachable trans sends start cfg \<equiv> \<exists>cfg0. asynchronousSystem.initial start cfg0 \<and> asynchronousSystem.reachable trans sends cfg0 cfg"
peek_and_and: fixes P :: "'b assn"   and p :: "state \<Rightarrow> bool" shows "(P \<and>. p \<and>. p) = (P \<and>. p)"
scc_root_unique_scc: fixes v :: "'v"   and scc :: "'v set"   and scc' :: "'v set" assumes "scc_root s' v scc"   and "scc_root s' v scc'" shows "scc = scc'"
ext_ids_def: fixes s :: "('c, 'd option) insert_message list" shows "ext_ids s = \<turnstile> # map (\<lambda>x. \<lbrakk>I x\<rbrakk>) s @ [\<stileturn>]"
dd: shows "dim ((+) (- a) ` S) < DIM('n)"
mapl_U_alt_def: fixes l1 :: "'n \<Rightarrow> 't" shows "mapl_U l1 = map_U l1 id id id id"
WR_def: fixes x :: "hf"   and y :: "hf" shows "WR x y \<equiv> \<exists>s. SeqWR s x y"
top_greatest: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and top :: "'g"   and a :: "'g" assumes "class.order_top less_eq less top" shows "less_eq a top"
S_pred_stream_abss_\<S>: fixes xs :: "('s \<times> ('c \<Rightarrow> real)) stream" shows "pred_stream (\<lambda>s. s \<in> S) xs = pred_stream (\<lambda>s. s \<in> \<S>) (smap abss xs)"
ring_hom_ring_def: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" shows "ring_hom_ring R S h \<equiv> ring R \<and> ring S \<and> ring_hom_ring_axioms R S h"
normal_form_def: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "normal_form F p = (SOME q. (punit.red (punit.reduced_GB F))\<^sup>*\<^sup>* p q \<and> \<not> punit.is_red (punit.reduced_GB F) q)"
get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('z, 'x, 'y, 'ag option) RShadowRoot_scheme \<Rightarrow> 'aj" assumes "l_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter"
exE_realizer': fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d \<times> 'c" assumes "P (snd p) (fst p)" obtains x :: "'d" and y :: "'c" where "P y x"
one: shows "R (arith_ops_record.one ops) (1::'a)"
dist_s_A: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list"   and n :: "nat" assumes "MTF_Off as rs s0" shows "distinct (MTF_Off.s_A as rs s0 n)"
assms_4: shows "diff_fun k (charts_submanifold W) f'"
inject: fixes x1 :: "'c"   and x2 :: "'c stream"   and y1 :: "'c"   and y2 :: "'c stream" shows "(x1 ## x2 = y1 ## y2) = (x1 = y1 \<and> x2 = y2)"
fsafe_Geq: fixes t1 :: "('a, 'b) trm"   and t2 :: "('a, 'b) trm" assumes "dsafe t1"   and "dsafe t2" shows "fsafe (Geq t1 t2)"
extendFalsum: fixes S :: "sequent"   and \<Gamma> :: "form multiset"   and \<Delta> :: "form multiset" assumes "extend S ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "ff \<in># \<Gamma>"
equals_flowI: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t0 :: "real"   and T' :: "real set"   and x0 :: "'b"   and z :: "real \<Rightarrow> 'b"   and t :: "real" assumes "ll_on_open_it T f X"   and "t0 \<in> T'"   and "is_interval T'"   and "T' \<subseteq> ll_on_open.existence_ivl T f X t0 x0"   and "(z solves_ode f) T' X"   and "z t0 = ll_on_open.flow T f X t0 x0 t0"   and "t \<in> T'" shows "z t = ll_on_open.flow T f X t0 x0 t"
set2_FGl_def: fixes x :: "(('q, 'p, 'r, 's, 't, 'u, 'v) G, 'q, 'w, 'r, 'x, 'y, 't, 'z, 'aa, 'ab) F" shows "set2_FGl x = \<Union> (set2_G ` set1_F x)"
accept\<^sub>G\<^sub>R_def: fixes R :: "('b \<Rightarrow> 'c \<Rightarrow> 'b) \<times> 'b \<times> (('b \<times> 'c \<times> 'b) set \<times> ('b \<times> 'c \<times> 'b) set set) set"   and w :: "nat \<Rightarrow> 'c" shows "accept\<^sub>G\<^sub>R R w \<equiv> \<exists>(Fin, Inf)\<in>snd (snd R). accepting_pair\<^sub>G\<^sub>R (fst R) (fst (snd R)) (Fin, Inf) w"
simps_9: fixes v :: "linear_poly"   and va :: "linear_poly" shows "nonstrict_constr (LTPP v va) = False"
quot_domain_imp_primeideal: fixes P :: "'a set" assumes "ideal P R"   and "domain (R Quot P)" shows "primeideal P R"
solves_Vagree_trans: fixes F :: "real \<Rightarrow> variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and U :: "variable set"   and I :: "interp"   and x :: "char"   and \<theta> :: "trm"   and \<zeta> :: "real" assumes "Uvariation (F 0) \<omega> U"   and "solves_ODE I F x \<theta>" shows "Uvariation (F \<zeta>) \<omega> (U \<union> {RVar x, DVar x})"
trivial_group_subset: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.group G" shows "trivial_group G = (carrier G \<subseteq> {\<one>\<^bsub>G\<^esub>})"
ide_compE: fixes g :: "'s"   and f :: "'s" assumes "ide (g \<cdot> f)" obtains "seq g f" and "seq f g" and "g \<cdot> f = local.dom f" and "g \<cdot> f = cod g"
underS_underS_trans: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "trans r"   and "antisym r"   and "a \<in> underS r b"   and "b \<in> underS r c" shows "a \<in> underS r c"
apply_ncrit_subset': fixes crit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "(bool \<times> bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "snd ` set (apply_ncrit crit data gs bs hs ps) \<subseteq> snd ` snd ` set ps"
rpr_complete: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "rpr A r"   and "x \<in> A"   and "y \<in> A" shows "x \<^bsub>r\<^esub>\<preceq> y \<or> y \<^bsub>r\<^esub>\<preceq> x"
simps_6: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and \<nu> :: "'a set" shows "af_letter (\<phi> or \<psi>) \<nu> = af_letter \<phi> \<nu> or af_letter \<psi> \<nu>"
neg_line_def: shows "neg_line = linepath (complex_of_real (- R)) (complex_of_real (- r))"
substitute_only_empty_both: fixes f :: "'b \<Rightarrow> 'b ttree"   and t' :: "'b ttree"   and T :: "'b set"   and t :: "'b ttree" assumes "const_on f (carrier t') TTree.empty" shows "substitute f T (t \<otimes>\<otimes> t') = substitute f T t \<otimes>\<otimes> t'"
measure_of_subset: fixes M :: "'b set set"   and \<Omega> :: "'b set"   and M' :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal"   and \<mu>' :: "'b set \<Rightarrow> ennreal" assumes "M \<subseteq> Pow \<Omega>"   and "M' \<subseteq> M" shows "sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
transcendental: fixes x :: "real"   and p :: "nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> int" assumes "liouville x p q" shows "\<not> algebraic x"
cases: fixes a :: "fm" assumes "a \<in> boolean_axioms" obtains A :: "fm" where "a = Aa__ IMP Aa__"   | A :: "fm" and B :: "fm" where "a = Ab__ IMP Ab__ OR Ba__"   | A :: "fm" where "a = Ac__ OR Ac__ IMP Ac__"   | A :: "fm" and B :: "fm" and C :: "fm" where "a = Ad__ OR Bb__ OR C__ IMP (Ad__ OR Bb__) OR C__"   | C :: "fm" and A :: "fm" and B :: "fm" where "a = C OR A IMP (C IMP B) IMP A OR B"
aux_abs_int: fixes c :: "int"   and x :: "int" assumes "c \<noteq> 0" shows "\<bar>x\<bar> \<le> \<bar>x * c\<bar>"
lincomb_0coeffs: fixes ms :: "'m list"   and rs :: "'r list" assumes "set ms \<subseteq> M"   and "\<forall>s\<in>set rs. s = (0::'r)" shows "rs \<bullet>\<cdot> ms = (0::'m)"
simps_4: fixes \<tau> :: "ty"   and \<rho> :: "val set list" shows "T (Forall \<tau>) \<rho> = {v. (\<exists>v'. v = Abs (Some v') \<and> (\<forall>V. v' \<in> T \<tau> (V # \<rho>))) \<or> v = Abs None}"
one_closed: shows "\<one>\<^bsub>G\<^esub> \<in> H"
ch2ch_fun: fixes S :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and x :: "'c" assumes "chain S" shows "chain (\<lambda>i. S i x)"
intro: fixes x :: "'b \<Longrightarrow> 'c" assumes "weak_lens x"   and "pbij_lens_axioms x" shows "pbij_lens x"
choose_poly_bound: fixes i :: "nat"   and d :: "nat"   and r :: "nat" assumes "i \<le> d" shows "r choose i \<le> max 1 (r ^ d)"
map_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "\<alpha>.map f = (if CCC.arr f then \<alpha> (CCC.cod f) \<cdot> T.ToTC f else null)"
hyps: fixes xs :: "'a llist" assumes "the_enat (llength xs) < the_enat (llength xsa__)"   and "lfinite xs" shows "P xs"
SystemDICut_def: shows "SystemDICut = ((TT \<rightarrow> [[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) TT]]Geq (Differential (Var vid1)) (Differential (Const 0))) \<rightarrow> (TT \<rightarrow> Geq (Var vid1) (Const 0)) \<rightarrow> [[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) TT]]Geq (Var vid1) (Const 0))"
remdups_into_doubles: fixes xs :: "'b list" shows "double\<^sup>*\<^sup>* (remdups xs) xs"
integrable_distr_eq: fixes g :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and f :: "'d \<Rightarrow> 'e" assumes "g \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> borel_measurable N" shows "integrable (distr M N g) f = integrable M (\<lambda>x. f (g x))"
zmax_ge_l: fixes x :: "int"   and y :: "int" shows "x \<le> zmax x y"
that: fixes C :: "V" assumes "C \<subseteq>\<^sub>\<circ> A"   and "B = r `\<^sub>\<circ> C" shows "thesis"
sH_cond_iff: fixes P :: "'a \<Rightarrow> bool"   and T :: "'a \<Rightarrow> bool"   and X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'a) set"   and Q :: "'a \<Rightarrow> bool" shows "PRE P (IF T THEN X ELSE Y FI) POST Q = (PRE P \<sqinter> T X POST Q \<and> PRE P \<sqinter> - T Y POST Q)"
bell11_def: shows "|\<beta>\<^sub>1\<^sub>1\<rangle> \<equiv> complex_of_real (1 / sqrt 2) \<cdot>\<^sub>m |vec 4 (\<lambda>i. if i = 1 then 1 else if i = 2 then - 1 else 0)\<rangle>"
non_succ_LimitI: fixes i :: "V" assumes "i \<noteq> 0"   and "Ord i"   and "\<And>y. ZFC_in_HOL.succ y \<noteq> i" shows "Limit i"
case_2: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "int \<Rightarrow> 'a"   and f5 :: "CTypeId \<Rightarrow> nat \<Rightarrow> 'a"   and f6 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and f7 :: "'a"   and x2 :: "int" shows "(case intgV x2 of boolV x \<Rightarrow> f1 x | intgV x \<Rightarrow> f2 x | shortV x \<Rightarrow> f3 x | byteV x \<Rightarrow> f4 x | objV x xa \<Rightarrow> f5 x xa | arrV x xa \<Rightarrow> f6 x xa | nullV \<Rightarrow> f7) = f2 x2"
HPow_iff: fixes u :: "hf"   and x :: "hf" shows "(u \<^bold>\<in> HPow x) = (u \<le> x)"
simps_2: fixes Chn :: "'b"   and Dom :: "'b span_data"   and Cod :: "'b span_data"   and more :: "'c" shows "Dom \<lparr>Chn = Chn, Dom = Dom, Cod = Cod, \<dots> = more\<rparr> = Dom"
l9_19_R2: fixes X :: "'p"   and Y :: "'p"   and P :: "'p"   and A :: "'p"   and B :: "'p" assumes "Col X Y P"   and "P Out A B"   and "\<not> Col X Y A" shows "X Y OS A B"
rep_eq: fixes x :: "real" shows "enn2ereal (ennreal x) = sup 0 (ereal x)"
square_part_uminus: fixes x :: "'b" shows "square_part (- x) = square_part x"
induct: fixes P :: "(dynamic, nat, bool) unboxed list \<Rightarrow> bool"   and a0 :: "(dynamic, nat, bool) unboxed list" assumes "\<And>x y. P [OpUbx1 x, OpUbx1 y]"   and "P []"   and "\<And>vb va. P (OpDyn vb # va)"   and "\<And>vb va. P (OpUbx2 vb # va)"   and "\<And>v. P [v]"   and "\<And>v vd vc. P (v # OpDyn vd # vc)"   and "\<And>v vd vc. P (v # OpUbx2 vd # vc)"   and "\<And>v vb vd ve. P (v # vb # vd # ve)" shows "P a0"
basis_row_space_iarrays_eq: fixes A :: "(('b, 'c) vec, 'd) vec" shows "basis_row_space_iarrays (matrix_to_iarray A) = set (map (\<lambda>i. row_iarray i (Gauss_Jordan_iarrays (matrix_to_iarray A))) [0..<rank_iarray (matrix_to_iarray A)])"
WT_restrict_gpv: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" shows "\<I> \<turnstile>g restrict_gpv \<I> gpv \<surd>"
neg_zmset_pos: fixes M :: "'a multiset" shows "mset_neg (zmset_of M) = {#}"
adhoc_norm_triangle: fixes a :: "real"   and y :: "'b"   and b :: "real"   and x :: "'b" assumes "a + norm y \<le> b"   and "norm x \<le> a" shows "norm (x + y) \<le> b"
ipurge_tr_append: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list"   and ys :: "'c list" shows "ipurge_tr I D u (xs @ ys) = ipurge_tr I D u xs @ ipurge_tr_aux I D (insert u (sinks I D u xs)) ys"
return: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> Rule m Return # rs \<Rightarrow>\<^sub>r Undecided"
extends_imp_wf_sub_rel: fixes subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and sub :: "('d \<times> nat) \<times> 'd \<times> nat"   and g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs' :: "(('d \<times> nat) \<times> 'd \<times> nat) set" assumes "wf_sub_rel subs"   and "SubExt.extends g subs sub subs'" shows "wf_sub_rel subs'"
reduction_ample: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and init :: "'b \<Rightarrow> bool"   and int :: "'b \<Rightarrow> 'c"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and src :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ren :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and q :: "'b" assumes "ample_abstract ex en init int ind src ren"   and "q \<in> transition_system_initial.nodes ex en init" shows "ample_base.ample_set ex en int ind src q {a. ren a q}"
cases: fixes a1 :: "('a \<times> 'a) set"   and a2 :: "'a list" assumes "sorted_on a1 a2" obtains r :: "('a \<times> 'a) set" where "a1 = r__" and "a2 = []"   | x :: "'a" and r :: "('a \<times> 'a) set" and xs :: "'a list" where "a1 = r" and "a2 = x # xs" and "x \<in> Field r" and "\<forall>y\<in>set xs. (x, y) \<in> r" and "sorted_on r xs"
con_rews: fixes int :: "int" shows "MkI\<cdot>int \<noteq> \<bottom>"
essentially_equalI: fixes G :: "('c, 'd) monoid_scheme"   and fs1 :: "'c list"   and fs1' :: "'c list"   and fs2 :: "'c list" assumes "Group.monoid G"   and "fs1 <~~> fs1'"   and "fs1' [\<sim>]\<^bsub>G\<^esub> fs2" shows "essentially_equal G fs1 fs2"
taut_shAll_intro: fixes P :: "'a \<Rightarrow> 'b upred" assumes "\<forall>x. `P x`" shows "`ushAll P`"
UNION_disjoint: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and g :: "'d \<Rightarrow> 'e" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" shows "prod g (\<Union> (A ` I)) = (\<Prod>x\<in>I. prod g (A x))"
bounded_ball: fixes x :: "'a"   and e :: "real" shows "bounded (ball x e)"
assms_3: fixes d :: "nat"   and a :: "nat"   and b :: "nat" assumes "d dvd a"   and "d dvd b" shows "f (a div d) (b div d) = f a b"
restricted_algebra: fixes A :: "'a set" assumes "A \<in> M" shows "algebra A (restricted_space A)"
trss_er_path_proc_const: fixes p :: "'p"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and q :: "'p"   and c' :: "'n list multiset" assumes "(([entry fg p], c), w, [return fg q], c') \<in> trcl (trss fg)" shows "p = q"
r_inv_ex: fixes x :: "'b" assumes "x \<in> carrier (G i)" shows "\<exists>y\<in>carrier (G i). x \<otimes>\<^bsub>G i\<^esub> y = \<one>\<^bsub>G i\<^esub>"
stop_no_eval: fixes mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and c' :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds' :: "Mode \<Rightarrow> var\<^sub>C set"   and mem' :: "var\<^sub>C \<Rightarrow> nat" shows "(\<langle>Stop, mds, mem\<rangle>\<^sub>C, \<langle>c', mds', mem'\<rangle>\<^sub>C) \<notin> C.eval\<^sub>w"
differing_empty_eq: fixes mem :: "'Var \<Rightarrow> 'Val"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "differing_vars mem mem' = {}" shows "mem = mem'"
ext_inject: fixes nodesL :: "'b list"   and edgesL :: "('b \<times> 'b) list"   and more :: "'c"   and nodesL' :: "'b list"   and edgesL' :: "('b \<times> 'b) list"   and more' :: "'c" shows "(\<lparr>nodesL = nodesL, edgesL = edgesL, \<dots> = more\<rparr> = \<lparr>nodesL = nodesL', edgesL = edgesL', \<dots> = more'\<rparr>) = (nodesL = nodesL' \<and> edgesL = edgesL' \<and> more = more')"
posterior_separate_def: fixes a :: "nat"   and g :: "vname gexp list"   and u :: "(nat \<times> vname aexp) list"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option" shows "posterior_separate a g u i r = (if can_take a g i r then Some (apply_updates u (join_ir i r) r) else None)"
absorb_iff2: fixes b :: "'g multiset"   and a :: "'g multiset" shows "(b \<subseteq># a) = (a \<inter># b = b)"