hcju/isabelleps huggingface dataset:
AnalBinds_delete_bot: fixes ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>"   and x :: "Vars.var"   and exp :: "exp \<Rightarrow> 'b \<rightarrow> 'c"   and \<Gamma> :: "(Vars.var \<times> exp) list" assumes "ae x = \<bottom>" shows "ExpAnalysis.AnalBinds exp (delete x \<Gamma>)\<cdot>ae = ExpAnalysis.AnalBinds exp \<Gamma>\<cdot>ae"
prob_space_distr: fixes f :: "ennreal \<Rightarrow> 'd"   and i :: "'b"   and M' :: "'d measure" assumes "Y.random_variable i M' f" shows "prob_space (distr (distr M borel (Y i)) M' f)"
restrict_indets_sum: fixes f :: "'e \<Rightarrow> ('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and A :: "'e set" shows "restrict_indets (sum f A) = (\<Sum>a\<in>A. restrict_indets (f a))"
dis: fixes C :: "'a set" assumes "C \<in> components (- S)" shows "\<not> bounded C"
Lit: fixes Env :: "env"   and B :: "lname set"   and v :: "val" shows "Env\<turnstile> B \<guillemotright>\<langle>Lit v\<rangle>\<guillemotright> \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
elims: fixes x :: "'b set set"   and xa :: "'b ltl"   and y :: "('b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) \<times> ('b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) \<times> ((('b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) \<times> 'b set \<times> 'b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set \<times> (('b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) \<times> 'b set \<times> 'b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set set) set" assumes "ltl_to_generalized_rabin_af x xa = y" obtains \<Sigma> :: "'b set set" and \<phi> :: "'b ltl" where "x = \<Sigma>" and "xa = \<phi>" and "y = ltl_to_rabin_base_def.ltl_to_generalized_rabin \<up>af \<up>af\<^sub>G Abs Abs M_fin \<Sigma> \<phi>"
fls_X_inv_power_times_comm: fixes n :: "nat"   and f :: "'a fls" shows "fls_X_inv ^ n * f = f * fls_X_inv ^ n"
spmf_of_set_def: fixes A :: "'a set" shows "spmf_of_set A = (if finite' A then spmf_of_pmf (pmf_of_set A) else return_pmf None)"
tail_invar_app_single: fixes bq :: "('c, 'd) SkewBinomialTree list"   and t' :: "('c, 'd) SkewBinomialTree" assumes "tail_invar bq"   and "\<forall>t\<in>set bq. rank t < rank t'"   and "tree_invar t'" shows "tail_invar (bq @ [t'])"
WT: shows "\<I>2, \<I>3 \<turnstile>\<^sub>C CNV callee s \<surd>"
simps_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and D :: "'c \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and intP :: "'b \<Rightarrow> 'c list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'c"   and r :: "'b"   and Tl :: "'a trm list" assumes "U.Struct wtFsym wtPsym arOf parOf D intF intP" shows "U.Struct.satA intF intP \<xi> (Pr r Tl) = intP r (map (U.Struct.int intF \<xi>) Tl)"
cinner_diff_left: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (minus x y) z = cinner x z - cinner y z"
os_sum_def: fixes p :: "int node ref option" shows "os_sum p \<equiv> os_it_init p \<bind> (\<lambda>it. os_sum' it 0)"
mem_Collect_eq: fixes a :: "'h"   and P :: "'h \<Rightarrow> bool" shows "(a \<in> Set.Collect P) = P a"
reversed_Arden: fixes A :: "'b list set"   and X :: "'b list set"   and B :: "'b list set" assumes "[] \<notin> A" shows "(X = X \<cdot> A \<union> B) = (X = B \<cdot> A\<star>)"
simps_3: fixes uv :: "nat" shows "enumerate_regs (V (I uv)) = {}"
b_not_in_no_head: fixes \<C> :: "'b list set"   and b :: "'b list" assumes "code \<C>"   and "b \<in> \<C>" shows "b \<notin> no_head \<C> b"
set_map_of_compr: fixes xs :: "('c \<times> 'd) list" assumes "distinct (map fst xs)" shows "set xs = {(k, v). table_of xs k = Some v}"
obj_class_Arr1: fixes obj :: "obj"   and T :: "ty"   and k :: "int" assumes "tag obj = Arr T k" shows "obj_class obj = Object"
simps_1: fixes s :: "nat \<Rightarrow> 'd list" shows "inf_last_ti s 0 = s 0"
HSem_reorder: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> Value" assumes "map_of \<Gamma> = map_of \<Delta>" shows "\<lbrace>\<Gamma>\<rbrace>\<rho> = \<lbrace>\<Delta>\<rbrace>\<rho>"
post_delete: fixes t :: "(('c \<times> 'd) \<times> nat) tree"   and x :: "'c" assumes "invar t" shows "post_del t (AA_Map.delete x t)"
linear_independent_extend_subspace: fixes B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "vs1.independent B" shows "\<exists>g. linear (*a) (*b) g \<and> (\<forall>x\<in>B. g x = f x) \<and> range g = vs2.span (f ` B)"
abs_eq: fixes x :: "('a \<times> 'b) list" shows "fmap_of_list x = Abs_fmap (map_of x)"
combine_neq_Leaf_if_both_non_Leaf: fixes a :: "'b tree"   and z :: "'b tree" assumes "a \<noteq> Leaf"   and "z \<noteq> Leaf" shows "combine a z \<noteq> Leaf"
product_aforms_def: fixes x :: "(real \<times> real pdevs) list"   and y :: "(real \<times> real pdevs) list" shows "product_aforms x y = x @ msum_aforms (degree_aforms x) (replicate (length y) (0, zero_pdevs)) y"
Spy_see_private_Key: fixes evs :: "event list"   and b :: "bool"   and A :: "agent" assumes "evs \<in> set_mr" shows "(Key (invKey (publicKey b A)) \<in> parts (knows Spy evs)) = (A \<in> bad)"
b_def: shows "b \<equiv> ln (1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / pi"
mat_append_col_dims: fixes A :: "'a mat"   and b :: "'a vec" shows "mat_append_col A b \<in> carrier_mat (dim_row A) (dim_col A + 1)"
cfun_arg_cong: fixes x :: "'r"   and y :: "'r"   and f :: "'r \<rightarrow> 's" assumes "x = y" shows "f\<cdot>x = f\<cdot>y"
axioms_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool" assumes "subcategory C Arr" shows "category C"
no_spoofing_iface: fixes rs :: "'a common_primitive rule list"   and ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option" assumes "simple_ruleset (rs::'a common_primitive rule list)"   and "\<forall>r::'a common_primitive rule\<in>set (rs::'a common_primitive rule list). normalized_nnf_match (get_match r)"   and "\<forall>iface::iface\<in>dom (ipassmt::iface \<Rightarrow> ('a word \<times> nat) list option). no_spoofing_iface iface ipassmt (rs::'a common_primitive rule list)" shows "no_spoofing TYPE('b) (ipassmt::iface \<Rightarrow> ('a word \<times> nat) list option) (rs::'a common_primitive rule list)"
trans: fixes b :: "('d, 'e) Term.term"   and a :: "('d, 'e) Term.term"   and c :: "('d, 'e) Term.term" assumes "b \<unrhd> a"   and "c \<unrhd> b" shows "c \<unrhd> a"
adapt_bexp_def: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and s :: "'c \<Rightarrow> nat" shows "adapt_bexp e s = (\<lambda>\<sigma>. e (\<lambda>x. \<sigma> (symvar x s)))"
sub: fixes i :: "name"   and t1 :: "tm"   and t2 :: "tm"   and t3 :: "tm"   and t4 :: "tm"   and x :: "tm" assumes "atom i \<sharp> tms" shows "(P t1 t2 t3 t4)(i::=x) = P (subst i x t1) (subst i x t2) (subst i x t3) (subst i x t4)"
rule_sub_lem_1_d: fixes v :: "i"   and \<psi> :: "'a \<Rightarrow> \<o>"   and \<chi> :: "'a \<Rightarrow> \<o>" assumes "\<And>x. [\<^bold>\<box>(\<psi> x \<^bold>\<equiv> \<chi> x) in v]" shows "[(\<^bold>\<forall>\<alpha>. \<psi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<chi> \<alpha>) in v]"
l: shows "(v \<circ> h) \<longlonglongrightarrow> l"
infTp: fixes \<sigma> :: "'tp"   and intT :: "'tp \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'fsym \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'psym \<Rightarrow> univ list \<Rightarrow> bool" assumes "infTp \<sigma>"   and "Ik.MModel intT intF intP" shows "infinite {a. intT \<sigma> a}"
geven_def: shows "PROP geven \<equiv> (\<And>n kvs t k v kvs'. \<lbrakk>0 < n; n \<le> Suc (length kvs); Q n kvs; rbtreeify_g n kvs = (t, (k, v) # kvs'); n \<le> Suc (length kvs'); Q n kvs'\<rbrakk> \<Longrightarrow> Q (2 * n) kvs)"
assms_1: fixes j :: "nat" assumes "j < dim_vec v"   and "j \<noteq> i" shows "v $ j = (0::'a)"
distinct_removeAll: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs" shows "distinct (removeAll x xs)"
simps_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "monoidal_language C" shows "monoidal_language.Diagonalize C \<^bold>\<I> = \<^bold>\<I>"
uy_frees: shows "\<forall>i\<in>frees (unlift_vars n y'). n \<le> i"
prog_step_preserves_separated_tmps: fixes i :: "nat"   and ts :: "((stmt \<times> nat) \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> (stmt \<times> nat) memref list \<times> 'a \<times> 'b) list"   and p :: "stmt \<times> nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'a"   and \<O> :: "'b"   and p' :: "stmt \<times> nat"   and is' :: "instr list" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>)"   and "\<theta>\<turnstile> p \<rightarrow>\<^sub>s (p', is')"   and "separated_tmps ts" shows "separated_tmps (ts[i := (p', is @ is', \<theta>, sb @ [Prog\<^sub>s\<^sub>b p p' is'], \<D>, \<O>)])"
FullModel_axioms: shows "FFullModel InfModel.intTF local.M.intFF local.M.intPF"
simps_1: fixes I :: "\<I>"   and nt :: "nat" shows "eval_until I nt [] = ([], [])"
f_2: fixes x :: "'a" shows "0 \<le> f x"
Dom_comp: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.arr f"   and "Maps.arr g"   and "Maps.Dom g = Maps.Cod f" shows "Maps.Dom (g \<odot> f) = Maps.Dom f"
valid_augment_Faults: 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"   and F' :: "'f set" assumes "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"   and "F \<subseteq> F'" shows "\<Gamma>\<Turnstile>\<^bsub>/F'\<^esub> P c Q,A"
Card_order_Times1: fixes r :: "('c \<times> 'c) set"   and B :: "'d set" assumes "Card_order r"   and "B \<noteq> {}" shows "r \<le>o |Field r \<times> B|"
satC_e: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "('fsym, 'psym) lit list" shows "U.satC \<xi> c = M.satC \<xi> c"
Value_InitStatus_HAInitValue: fixes A :: "('e, 'f, 'd) hierauto" shows "Value (InitStatus A) = HAInitValue A"
leg_simps_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data" assumes "span_in_category C S" shows "partial_magma.dom C (Leg0 S) = span_in_category.apex C S"
tendsto_at_right_continuous_on_avoidI: fixes i :: "'c"   and b :: "'c"   and I :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on ({i..b} - I) g"   and "\<And>x. \<lbrakk>i < x; x < b\<rbrakk> \<Longrightarrow> g x = f x"   and "i \<notin> I"   and "finite I"   and "i < b" shows "(f \<longlongrightarrow> g i) (at_right i)"
t_2: shows "(v \<circ> r \<circ> s \<circ> t) \<longlonglongrightarrow> liminf (v \<circ> r \<circ> s)"
even_mask_div_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (divide (minus (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m) one) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)) = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n = zero \<or> m \<le> n)"
simps_2: shows "\<guillemotleft>One\<guillemotright> = One"
Lp_D_4: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "0 < p"   and "f \<in> space\<^sub>N (\<LL> (ennreal p) M)" shows "eNorm (\<LL> (ennreal p) M) f = ennreal ((LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p))"
wp_seq_r: fixes P :: "('b, 'd) urel"   and Q :: "('d, 'c) urel"   and r :: "'c upred" shows "P ;; Q wp r = P wp (Q wp r)"
inject_2: fixes x21 :: "'c net_state"   and x22 :: "'c net_state"   and y21 :: "'c net_state"   and y22 :: "'c net_state" shows "(SubnetS x21 x22 = SubnetS y21 y22) = (x21 = y21 \<and> x22 = y22)"
distinct_upto: fixes i :: "int"   and j :: "int" shows "distinct [i..j]"
sel_7: fixes x41 :: "typ"   and x42 :: "Core.term" shows "un_Abs2 (Abs x41 x42) = x42"
arcosh_mono: fixes x :: "real"   and y :: "real" assumes "1 \<le> x"   and "1 \<le> y" shows "(arcosh y \<le> arcosh x) = (y \<le> x)"
T_spaceM_stable_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "qmpt M T"   and "x \<in> space M" shows "T x \<in> space M"
nonregular_event: fixes ev :: "('d, 'e, 'f) event" shows "regular_event ev \<noteq> (isSnapshot ev \<or> isRecvMarker ev)"
indices_state_def: fixes s :: "('c, 'd) state" shows "indices_state s = {i. \<exists>x b. Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x = Some (i, b) \<or> Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x = Some (i, b)}"
g: fixes F :: "real filter" shows "\<forall>\<^sub>F x in F. g x = exp (f x)"
coincidence_dterm: fixes \<theta> :: "('d, 'e) trm"   and \<nu> :: "(real, 'e) vec \<times> (real, 'e) vec"   and \<nu>' :: "(real, 'e) vec \<times> (real, 'e) vec"   and I :: "('d, 'f, 'e) interp" assumes "dsafe \<theta>"   and "Vagree \<nu> \<nu>' (FVT \<theta>)" shows "dterm_sem I \<theta> \<nu> = dterm_sem I \<theta> \<nu>'"
rewrite_X_enat_infinity_invariant: fixes \<phi> :: "'a ltln" shows "(snd (rewrite_X_enat \<phi>) = \<infinity>) = is_constant (fst (rewrite_X_enat \<phi>))"
piecewise_C1_differentiable_on_components: fixes t :: "real set"   and f :: "real \<Rightarrow> 'a"   and s :: "real set" assumes "finite t"   and "\<And>i. i \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> i) C1_differentiable_on s - t"   and "\<And>i. i \<in> Basis \<Longrightarrow> continuous_on s (\<lambda>x. f x \<bullet> i)" shows "f piecewise_C1_differentiable_on s"
tailrec_transform2: fixes m :: "'d nres"   and a :: "'d \<Rightarrow> 'd nres"   and b :: "'d \<Rightarrow> 'c nres" shows "lfp (\<lambda>x. sup m (x \<bind> a)) \<bind> b \<le> m \<bind> REC (\<lambda>D s. sup (a s \<bind> D) (b s))"
gcsort_add_inv: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and t' :: "nat \<times> nat list \<times> 'c list"   and p :: "nat"   and t :: "nat \<times> nat list \<times> 'c list"   and n :: "nat" assumes "index_less index key"   and "t' \<in> gcsort_set index key p t"   and "add_inv n t"   and "n \<le> p" shows "add_inv n t'"
map_lasso_simps_4: fixes f :: "'f \<Rightarrow> 'e"   and L :: "('f, 'g) lasso_scheme" shows "lasso_v0 (map_lasso f L) = f (lasso_v0 L)"
list_seq_nth': fixes seq :: "nat \<Rightarrow> 'c list"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "\<And>i. \<exists>x. seq (Suc i) = x # seq i"   and "i < length (seq j)"   and "i < length (seq k)" shows "rev (seq k) ! i = rev (seq j) ! i"
order_tendstoD_2: fixes f :: "'b \<Rightarrow> 'a"   and y :: "'a"   and F :: "'b filter"   and a :: "'a" assumes "(f \<longlongrightarrow> y) F"   and "y < a" shows "\<forall>\<^sub>F x in F. f x < a"
complex_of_real_integrable_eq: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" shows "complex_integrable M (\<lambda>x. complex_of_real (f x)) = integrable M f"
intro: fixes T :: "'a \<Rightarrow> 'b"   and M :: "'a set"   and N :: "'b set" assumes "bij_betw T M N" shows "RModuleIso_axioms M T N"
mono_body_fixp: fixes F :: "('c \<Rightarrow> 'd llist) \<Rightarrow> 'c \<Rightarrow> 'd llist" assumes "\<And>x. mono_llist (\<lambda>f. F f x)" shows "llist.fixp_fun F = F (llist.fixp_fun F)"
shows_prec_ratfps_def: fixes p :: "nat"   and x :: "'a ratfps" shows "shows_prec p x = showsp_ratfps p x"
rep_eq: fixes x :: "real_alg"   and xa :: "real_alg" shows "compare x xa = compare_3 (rep_real_alg x) (rep_real_alg xa)"
invertible_right_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "invertible x"   and "x \<in> G"   and "y \<in> G"   and "z \<in> G" shows "(y \<cdot> x = z \<cdot> x) = (y = z)"
word_succ_p1: fixes a :: "'a word" shows "word_succ a = a + 1"
mcont2mcont: fixes lub' :: "'d set \<Rightarrow> 'd"   and ord' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e spmf"   and lub :: "'f set \<Rightarrow> 'f"   and ord :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and t :: "'f \<Rightarrow> 'd" assumes "mcont lub' ord' lub_spmf (ord_spmf (=)) f"   and "mcont lub ord lub' ord' t" shows "mcont lub ord lub_spmf (ord_spmf (=)) (\<lambda>x. f (t x))"
cos_of_real: fixes x :: "real" shows "cos (of_real x) = of_real (cos x)"
MS_def: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" shows "MS M l s ll t = (\<exists>k. (M, l, s, k, ll, t) \<in> MStep)"
odd_succ_div_two: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "divide (plus a one) (numeral.numeral one plus (num.Bit0 num.One)) = plus (divide a (numeral.numeral one plus (num.Bit0 num.One))) one"
on': shows "higher_differentiable_on (S \<inter> f -` {..<0}) (\<lambda>x. - (1::'b)) n"
neqE: fixes x :: "'a Word.word"   and y :: "'a Word.word" assumes "x \<noteq> y" obtains "x <s y"   | "y <s x"
assms: shows "lineA = \<lparr>basepoint = basepoint lineB, direction = direction lineB\<rparr>"
f_approx_nonneg: fixes x :: "real" assumes "x\<^sub>0 \<le> x" shows "0 \<le> f_approx x"
exp_Reals_eq: fixes z :: "complex" assumes "z \<in> \<real>" shows "exp z = complex_of_real (exp (Re z))"
semiring_norm_58: fixes a :: "'a" shows "a - (0::'a) = a"
nsum_ideal_incTr: fixes A :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A" shows "(\<forall>i\<le>n. f i \<in> A) \<longrightarrow> \<Sigma>\<^sub>e R f n \<in> A"
grey_def: fixes r :: "'g" shows "grey r = (\<lambda>s. \<exists>x. r \<in> WL x s)"
finite_dimensionE': fixes K :: "'a list set"   and E :: "'a list set" assumes "UP.finite_dimension K E" obtains n :: "nat" where "UP.dimension n K E"
ex_cong: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>x. P x \<Longrightarrow> Q x"   and "\<And>x. Q x \<Longrightarrow> P (f x)" shows "(\<exists>x. P x) = (\<exists>x. Q x)"
essential_mono: fixes G :: "('k, 'l) graph_scheme"   and B :: "'k set"   and S :: "'k set"   and x :: "'k"   and S' :: "'k set" assumes "essential G B S x"   and "S' \<subseteq> S" shows "essential G B S' x"
sorted_hyp: fixes p :: "nat"   and q :: "nat" assumes "p < length (pivot_positions A)"   and "q < length (pivot_positions A)"   and "p < q" shows "fst (pivot_positions A ! p) < fst (pivot_positions A ! q)"
otherassms_5: shows "l' \<in> labels \<Gamma> q"
SeqWRP_Zero_E: fixes y :: "tm"   and H :: "fm set"   and A :: "fm"   and k :: "tm"   and s :: "tm" assumes "insert (y EQ Zero) H \<turnstile> A"   and "H \<turnstile> k EQ Zero" shows "insert (SeqWRP s k y) H \<turnstile> A"
restrict_idle_include: fixes n :: "nat"   and A :: "nat set"   and v :: "'b option list"   and I :: "nat set" assumes "wf_tuple n A v"   and "A \<subseteq> I" shows "restrict I v = v"
deg_pm_plus: fixes s :: "'e \<Rightarrow>\<^sub>0 'd"   and t :: "'e \<Rightarrow>\<^sub>0 'd" shows "deg_pm (s + t) = deg_pm s + deg_pm t"
berlekamp_mat_closed_2: fixes u :: "'a mod_ring poly" shows "dim_row (local.berlekamp_mat u) = degree u"
block_def: fixes i :: "nat"   and t :: "nat" shows "block i t \<equiv> pdec1 (the (eval r_state [t, i]))"
lin_polies_to_mat_def: fixes lst :: "linear_poly list" shows "lin_polies_to_mat lst = mat (length lst) (max_dim_poly_list lst) (\<lambda>(x, y). Abstract_Linear_Poly.coeff (lst ! x) y)"
mk_vec_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "mk_vec n f \<equiv> \<lambda>i. if i < n then f i else undef_vec (i - n)"
sum_impl_def: fixes g :: "'d \<Rightarrow> 'c nres"   and S :: "'d set" shows "sum_impl g S \<equiv> FOREACH S (\<lambda>x a. g x \<bind> (\<lambda>b. RETURN (a + b))) (0::'c)"
rel_spmf_bindI: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b spmf"   and q :: "'c spmf"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd spmf"   and g :: "'c \<Rightarrow> 'e spmf" assumes "rel_spmf R p q"   and "\<And>x y. R x y \<Longrightarrow> rel_spmf P (f x) (g y)" shows "rel_spmf P (p \<bind> f) (q \<bind> g)"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat list"   and y :: "real mat convnet" assumes "witness x xa xb = y" obtains Y :: "nat" and r :: "nat" and rs :: "nat list" where "x = Y" and "xa = r" and "xb = rs" and "y = Conv ((if length rs = 0 then id_matrix else if length rs = 1 then all1_matrix else copy_first_matrix) Y r) (witness' r rs)"
drop_map: fixes n :: "nat"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "drop n (map f xs) = map f (drop n xs)"
nth_def: shows "nth = Cn 4 r_nth [recf.Id 4 3, recf.Id 4 0]"
row_space_is_preserved: fixes P :: "'a mat"   and m :: "nat"   and A :: "'a mat"   and n :: "nat" assumes "invertible_mat P"   and "P \<in> carrier_mat m m"   and "A \<in> carrier_mat m n" shows "vec_space.row_space n (P * A) = vec_space.row_space n A"
continuous_on_cases_local_open: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool" assumes "openin (top_of_set (S \<union> T)) S"   and "openin (top_of_set (S \<union> T)) T"   and "continuous_on S f"   and "continuous_on T g"   and "\<And>x. x \<in> S \<and> \<not> P x \<or> x \<in> T \<and> P x \<Longrightarrow> f x = g x" shows "continuous_on (S \<union> T) (\<lambda>x. if P x then f x else g x)"
mkVarChannel_def: fixes v :: "varDecl"   and upd :: "((String.literal, variable) assoc_list \<Rightarrow> (String.literal, variable) assoc_list) \<Rightarrow> 'a gState_scheme \<times> pState \<Rightarrow> 'a gState_scheme \<times> pState"   and g :: "'a gState_scheme"   and p :: "pState" shows "mkVarChannel v upd g p = (let (k, v, cs) = toVariable g p v; (g', p') = upd (lm.update k v) (g, p) in mkChannels g' p' cs)"