hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "bool"   and a2 :: "v sem_env"   and a3 :: "'b state"   and a4 :: "top0"   and a5 :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_top a1 a2 a3 a4 a5" obtains ck :: "bool" and s1 :: "'b state" and s2 :: "'b state" and env :: "v sem_env" and d :: "dec" and new_env :: "v sem_env" where "a1 = ck__" and "a2 = enva__" and "a3 = s1__" and "a4 = Tdec d__" and "a5 = (s2__, Rval new_env__)" and "evaluate_dec ck__ [] enva__ s1__ d__ (s2__, Rval new_env__)"   | ck :: "bool" and s1 :: "'b state" and s2 :: "'b state" and env :: "v sem_env" and d :: "dec" and err :: "v error_result" where "a1 = cka__" and "a2 = envb__" and "a3 = s1a__" and "a4 = Tdec d" and "a5 = (s2a__, Rerr err__)" and "evaluate_dec cka__ [] envb__ s1a__ d (s2a__, Rerr err__)"   | ck :: "bool" and s1 :: "'b state" and s2 :: "'b state" and env :: "v sem_env" and ds :: "dec list" and mn :: "char list" and specs :: "spec list option" and new_env :: "v sem_env" where "a1 = ckb__" and "a2 = envc__" and "a3 = s1b__" and "a4 = Tmod mn__ specs__ ds__" and "a5 = (update_defined_mods (\<lambda>_. {[mn__]} \<union> defined_mods s2b__) s2b__, Rval (make_sem_env (nsLift mn__ (sem_env.v new_env)) (nsLift mn__ (c new_env))))" and "[mn__] \<notin> defined_mods s1b__ \<and> no_dup_types ds__ \<and> evaluate_decs ckb__ [mn__] envc__ s1b__ ds__ (s2b__, Rval new_env)"   | ck :: "bool" and s1 :: "'b state" and s2 :: "'b state" and env :: "v sem_env" and ds :: "dec list" and mn :: "char list" and specs :: "spec list option" and err :: "v error_result" where "a1 = ckc__" and "a2 = envd__" and "a3 = s1c__" and "a4 = Tmod mna__ specsa__ dsa__" and "a5 = (update_defined_mods (\<lambda>_. {[mna__]} \<union> defined_mods s2) s2, Rerr err)" and "[mna__] \<notin> defined_mods s1c__ \<and> no_dup_types dsa__ \<and> evaluate_decs ckc__ [mna__] envd__ s1c__ dsa__ (s2, Rerr err)"   | ck :: "bool" and s1 :: "'b state" and env :: "v sem_env" and ds :: "dec list" and mn :: "char list" and specs :: "spec list option" where "a1 = ckd__" and "a2 = enve__" and "a3 = s1" and "a4 = Tmod mnb__ specsb__ dsb__" and "a5 = (s1, Rerr (Rabort Rtype_error))" and "\<not> no_dup_types dsb__"   | ck :: "bool" and env :: "v sem_env" and s :: "'b state" and mn :: "char list" and specs :: "spec list option" and ds :: "dec list" where "a1 = ck" and "a2 = env" and "a3 = s" and "a4 = Tmod mn specs ds" and "a5 = (s, Rerr (Rabort Rtype_error))" and "[mn] \<in> defined_mods s"
fls_times_both_shifted_simp: fixes m :: "int"   and f :: "'b fls"   and n :: "int"   and g :: "'b fls" shows "fls_shift m f * fls_shift n g = fls_shift (m + n) (f * g)"
wf_supercls_distinct_app: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "char list"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'a mdecl list"   and Cs :: "char list list" assumes "wf_prog wf_md P"   and "C \<noteq> Object"   and "class P C = \<lfloor>(D, fs, ms)\<rfloor>"   and "supercls_lst P (C # Cs)"   and "distinct (C # Cs)" shows "distinct (D # C # Cs)"
r_normal_form_recfn: shows "recfn 2 r_normal_form"
cases: fixes a :: "'k"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "a \<in> MFMC_Network.SINK f" obtains x :: "'k" where "a = x" and "d_OUT f x = 0"
algebra_generator: shows "algebra (space (Pi\<^sub>M I M)) generator"
div_gcd_coprime: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a) \<or> b \<noteq> (0::'a)" shows "coprime (a div gcd a b) (b div gcd a b)"
uniform_measure_def: fixes M :: "'c measure"   and A :: "'c set" shows "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
N_def: shows "N = map_minsky (\<lambda>s. if s = 1 then 2 else if s = 2 then 1 else s) id M"
unitary_def: fixes A :: "'c mat" shows "unitary A = (A \<in> carrier_mat (dim_row A) (dim_row A) \<and> inverts_mat A (adjoint A))"
implies_not_eq: fixes y :: "'a"   and x :: "'a" assumes "y \<sqsubset> x"   and "x \<in> X" shows "x \<noteq> y"
wf_inst_atom: fixes a :: "PDDL_STRIPS_Semantics.term atom" assumes "wf_atom (ty_term Q constT) a" shows "wf_atom objT (map_atom (subst_term f) a)"
negation_type_to_match_expr_def: fixes m :: "'b negation_type" shows "negation_type_to_match_expr m \<equiv> negation_type_to_match_expr_f id m"
permutationNotEmpty: fixes l :: "'d list"   and n :: "nat" assumes "l \<noteq> []" shows "perm2 l n \<noteq> []"
toSet_def: fixes X :: "'c odlist" shows "toSet X = set (toList X)"
c\<^sub>5: shows "stack c\<^sub>5 = stack c\<^sub>1"
R1: fixes w :: "'b \<Rightarrow> 'c"   and v :: "'b \<Rightarrow> 'd"   and F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "|~ Unchanged w \<longrightarrow> Unchanged v" shows "\<turnstile> \<box>[F]_w \<longrightarrow> \<box>[F]_v"
elims: fixes x :: "ereal"   and y :: "real" assumes "real_of_ereal x = y" obtains r :: "real" where "x = ereal r" and "y = r"   | "x = \<infinity>" and "y = 0"   | "x = - \<infinity>" and "y = 0"
primeness_condition: fixes p :: "'a" assumes "p \<in> carrier R" shows "ring_irreducible p = ring_prime p"
redT_thread_not_disappear: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t' :: "'thread_id" assumes "P \<turnstile> s -t\<triangleright>ta\<rightarrow>\<^bsub>jvmd\<^esub> s'"   and "thr s' t' = None" shows "thr s t' = None"
continuous_map_on_empty: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "topspace X = {}" shows "continuous_map X Y f"
image_Collect: fixes f :: "'d \<Rightarrow> 'c"   and P :: "'d \<Rightarrow> bool" shows "f ` {x. P x} = {f x |x. P x}"
rfb_inj_iff: fixes R :: "('d \<times> 'c) set"   and S :: "('d \<times> 'c) set" shows "(fb\<^sub>\<R> R = fb\<^sub>\<R> S) = (R = S)"
unit_is_iso: shows "local.iso (local.inv \<epsilon>\<^sub>0)"
mult_two_impl2: fixes a :: "enat"   and b :: "enat" assumes "a * 2 = 1 + 2 * b" shows "a = \<infinity> \<and> b = \<infinity>"
nth_root_nat_unique: fixes m :: "nat"   and k :: "nat"   and n :: "nat" assumes "m ^ k \<le> n"   and "n < (m + 1) ^ k" shows "nth_root_nat k n = m"
Union_supports_multiset: fixes M :: "'a multiset" shows "\<Union> {supp x |x. x \<in># M} supports M"
G_def: shows "G \<equiv> \<lambda>t. C + K * integral {t..b} g"
negligible_def: fixes s :: "'b set" shows "negligible s = (\<forall>a b. (indicat_real s has_integral 0) (cbox a b))"
sup_bot_left: fixes x :: "'b multiset" shows "\<Empt> \<union># x = x"
val_type_eq_PRODUCT: fixes x :: "val"   and t1 :: "pdf_type"   and t2 :: "pdf_type" shows "(val_type x = PRODUCT t1 t2) = (\<exists>a b. val_type a = t1 \<and> val_type b = t2 \<and> x = <|a, b|>)"
det_singleton: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and A :: "'b mat" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "A \<in> carrier_mat 1 1" shows "Determinant.det A = A $$ (0, 0)"
\<tau>exec_move_\<tau>exec_1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and body :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and D :: "String.literal"   and frs :: "'addr frame list" assumes "\<tau>exec_move ci P t body h (stk, loc, pc, xcp) (stk', loc', pc', xcp')"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = \<lfloor>body\<rfloor> in D" shows "\<tau>exec_1 (compP2 P) t (xcp, h, (stk, loc, C, M, pc) # frs) (xcp', h, (stk', loc', C, M, pc') # frs)"
simps_2: fixes d :: "nat"   and f :: "real" shows "get_intv_val (Intv d) f = real d + f"
simps_2: fixes s :: "'b \<Rightarrow> ('b, 'c) pform"   and n :: "'c"   and ps :: "('b, 'c) pform list" shows "subst s (Fun n ps) = Fun n (map (subst s) ps)"
Imp_Imp_commute: fixes H :: "fm set"   and B :: "fm"   and A :: "fm"   and C :: "fm" assumes "H \<turnstile> B IMP A IMP C" shows "H \<turnstile> A IMP B IMP C"
prems: shows "higher_differentiable_on X f (Suc (Suc i))"
image_transfer: fixes A :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_fun (rel_fun A B) (rel_fun (rel_set A) (rel_set B)) (`) (`)"
apply_inj: fixes f :: "'a perm"   and a :: "'a"   and b :: "'a" shows "(f \<langle>$\<rangle> a = f \<langle>$\<rangle> b) = (a = b)"
multi_self_add_other_not_self: fixes M :: "'k multiset"   and x :: "'k" shows "(M = add_mset x M) = False"
rules_singleton_rev_E: fixes m :: "'c match_expr"   and a :: "action"   and rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list"   and P :: "'c match_expr \<Rightarrow> action \<Rightarrow> bool" assumes "[Rule m a] = rs\<^sub>1 @ rs\<^sub>2"   and "\<lbrakk>rs\<^sub>1 = [Rule m a]; rs\<^sub>2 = []\<rbrakk> \<Longrightarrow> P m a"   and "\<lbrakk>rs\<^sub>1 = []; rs\<^sub>2 = [Rule m a]\<rbrakk> \<Longrightarrow> P m a" shows "P m a"
never_blocked: fixes l :: "nat"   and s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" shows "A.can_step l s = (l < N)"
induct: fixes x :: "'node SDG_node"   and S :: "'node SDG_node set"   and P :: "'node SDG_node \<Rightarrow> bool" assumes "x \<in> combine_SDG_slices S"   and "\<And>n. n \<in> S \<Longrightarrow> P n"   and "\<And>n' n'' p n. \<lbrakk>n' \<in> S; n'' s-p\<rightarrow>\<^bsub>ret\<^esub> CFG_node (parent_node n'); n \<in> sum_SDG_slice2 n'\<rbrakk> \<Longrightarrow> P n" shows "P x"
noFault: shows "\<Gamma>\<turnstile> \<langle>Catch c1' c2',Normal s\<rangle> \<Rightarrow>\<notin>range Fault"
bounded_by_def: fixes b :: "real"   and P :: "'c \<Rightarrow> real" shows "bounded_by b P \<equiv> \<forall>x. P x \<le> b"
assms_2: shows "u \<le>s t \<^sup>@ k"
inverts_2: fixes discr :: "bool discr"   and discr' :: "bool discr" shows "(B\<cdot>discr \<sqsubseteq> B\<cdot>discr') = (discr \<sqsubseteq> discr')"
filter_def: fixes P :: "'a \<Rightarrow> bool"   and A :: "'a set" shows "Set.filter P A = {a \<in> A. P a}"
genby_lcoset_closed: fixes a :: "'b"   and S :: "'b set"   and A :: "'b set" assumes "a \<in> \<langle>S\<rangle>"   and "order.greater_eq \<langle>S\<rangle> A" shows "order.greater_eq \<langle>S\<rangle> (a +o A)"
cimage_cempty: fixes f :: "'e \<Rightarrow> 'd" shows "cimage f cempty = cempty"
I_r_closed: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme"   and a :: "'a"   and x :: "'a" assumes "ideal I R"   and "a \<in> I"   and "x \<in> carrier R" shows "a \<otimes>\<^bsub>R\<^esub> x \<in> I"
equalityI: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater_eq B A"   and "order.greater_eq A B" shows "A = B"
Nonce_compromise: fixes evs :: "event list"   and KK :: "nat set"   and N :: "nat" assumes "evs \<in> set_pur"   and "\<forall>K\<in>KK. K \<notin> range priEK" shows "(Nonce N \<in> analz (Key ` KK \<union> knows Spy evs)) = (Nonce N \<in> analz (knows Spy evs))"
rep_transfer: shows "rel_fun cr_uint32 (=) (\<lambda>x. x) Rep_uint32"
inject_5: fixes x5 :: "msg"   and y5 :: "msg" shows "(Hash x5 = Hash y5) = (x5 = y5)"
bisim1Throw1: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,throw e,h \<turnstile> (throw e', xs) \<leftrightarrow> (stk, loc, pc, xcp)"
A81_NSources_L1: shows "\<forall>C\<in>AbstrLevel level1. C \<noteq> sA92 \<and> C \<noteq> sA91 \<and> C \<noteq> sA81 \<longrightarrow> sA81 \<notin> Sources level1 C"
downset_iso_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.preorder less_eq less" shows "(ord.downset less_eq x \<subseteq> ord.downset less_eq y) = less_eq x y"
restrict_map_self: fixes m :: "'d \<Rightarrow> 'e option" shows "m |` dom m = m"
numeral_eq_enat: fixes k :: "num" shows "numeral k = enat (numeral k)"
assms_3: shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'"
degree_isovarspar: fixes p :: "real mpoly"   and x :: "nat"   and i :: "nat" shows "MPoly_Type.degree (isolate_variable_sparse p x i) x = 0"
substEnv_vsubstEnv_compose2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Z :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and y1 :: "'var" assumes "goodEnv rho"   and "good Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "rho &[y1 // y]_ys &[Z / z]_zs = rho &[Z / z]_zs &[Var ys y1 #[Z / z]_zs / y]_ys"
wf_eq_minimal: fixes r :: "('b \<times> 'b) set" shows "wf r = (\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q))"
A_2: shows "\<forall>k\<le>n. M k k \<le> (0::'a)"
dvd_imp_le_int: fixes i :: "int"   and d :: "int" assumes "i \<noteq> 0"   and "d dvd i" shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
hyp2_rep_abs_examples_4: shows "hyp2_rep (hyp2_abs (vector [1 / 4, 1 / 4])) = vector [1 / 4, 1 / 4]"
tuple_iso_tuple: shows "isomorphic_tuple Record.tuple_iso_tuple"
LHS: shows "Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) (Cring_Poly.compose R q r) = a \<odot>\<^bsub>P\<^esub> Cring_Poly.compose R q r [^]\<^bsub>P\<^esub> n"
simps: fixes i :: "char list"   and p_iface :: "char list" shows "match_iface (Iface i) p_iface = internal_iface_name_match i p_iface"
filter_ASSUME_def: fixes c :: "'b \<Rightarrow> bool"   and m :: "'b nres" shows "filter_ASSUME c m \<equiv> m \<bind> (\<lambda>x. ASSUME (c x) \<bind> (\<lambda>_. RETURN x))"
wls_Abs_ainj_all: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and s' :: "'sort"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "wls s X"   and "wls s' X'" shows "(Abs xs x X = Abs xs' x' X') = (xs = xs' \<and> (\<forall>y. (y = x \<or> fresh xs y X) \<and> (y = x' \<or> fresh xs y X') \<longrightarrow> X #[y \<and> x]_xs = X' #[y \<and> x']_xs))"
n_gr_0: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "1 < card X"   and "finite F"   and "F \<subseteq> P[X]"   and "f \<in> F"   and "\<And>f'. f' \<in> F \<Longrightarrow> homogeneous f'"   and "\<And>f'. f' \<in> F \<Longrightarrow> poly_deg f' \<le> poly_deg f"   and "0 < poly_deg f"   and "ideal {f} \<noteq> ideal F" shows "0 < card X"
let\<^sub>1: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "stack_elem list"   and e :: "exp" assumes "atom ` domA \<Delta> \<sharp>* \<Gamma>"   and "atom ` domA \<Delta> \<sharp>* S" shows "(\<Gamma>, Let \<Delta> e, S) \<Rightarrow> (\<Delta> @ \<Gamma>, e, S)"
Skip: shows "\<turnstile>\<^sub>3\<^sub>' {$ 1} SKIP { $ 0}"
valid_eqvt: fixes P :: "'state"   and x :: "('a, unit, ('act, 'pred) S_action) formula"   and p :: "perm" assumes "P \<Turnstile>\<^sub>S x" shows "p \<bullet> P \<Turnstile>\<^sub>S p \<bullet> x"
imp_igSwapBinpIPresIGWlsBinp: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" assumes "igSwapAbsIPresIGWlsAbs MOD" shows "igSwapBinpIPresIGWlsBinp MOD"
D: shows "D \<in> folding.\<C> - f \<turnstile> folding.\<C>"
subprob_space_axioms_def: fixes M :: "'b measure" shows "subprob_space_axioms M \<equiv> emeasure M (space M) \<le> 1 \<and> space M \<noteq> {}"
preserves_ide: fixes a :: "'b" assumes "B.ide a" shows "C.ide (G a)"
cases: fixes x :: "('a \<times> 'b) \<times> 'a \<times> 'c" obtains v1 :: "'a" and p1 :: "'b" and v2 :: "'a" and p2 :: "'c" where "x = ((v1, p1), v2, p2)"
fam_eq: fixes U :: "'a set"   and x :: "'b" assumes "is_open U"   and "x \<in> \<FF> U" shows "fam_morphisms' U x = fam_morphisms U x"
f: shows "\<forall>i\<in>Field r. f i \<in> As (succ r i) - As i"
Const\<^sub>0_numeral: fixes x :: "num" shows "Const\<^sub>0 (numeral x) = numeral x"
recursive_compile_correct: fixes recf :: "recf"   and args :: "nat list"   and ap :: "abc_inst list"   and arity :: "nat"   and fp :: "nat"   and anything :: "nat list" assumes "terminate recf args"   and "rec_ci recf = (ap, arity, fp)" shows "{\<lambda>nl. nl = args @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = args @ rec_exec recf args # 0 \<up> (fp - Suc arity) @ anything}"
intro: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" assumes "func f S T"   and "continuous_axioms f S T" shows "continuous f S T"
jatt': shows "jAction (SPR.MCn (tLength t)) t a = jAction (SPR.MCn (tLength t')) t' a"
value_h: fixes i :: "nat" shows "value_flow \<Delta> (fn.h i) = value_flow \<Delta> (f_i i)"
nth_map_zip: fixes i :: "nat"   and xs :: "'b list"   and ys :: "'c list"   and f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "i < length xs"   and "i < length ys" shows "map f (zip xs ys) ! i = f (xs ! i, ys ! i)"
fst_snd_eq: fixes a :: "'b \<times> 'c"   and x :: "'b"   and y :: "'c" assumes "fst a = x"   and "snd a = y" shows "(x, y) = a"
4: shows "x\<^sup>T \<le> x\<^sup>\<star> ; r"
chop_position_of: fixes s :: "('c, 'd) tm" assumes "is_App s" shows "position_of s (replicate (num_args (fun s)) dir.Left @ [dir.Right])"
conseq_no_aux: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P' :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q' :: "'e set"   and A' :: "'e set"   and P :: "'e set"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c Q',A'"   and "\<forall>s. s \<in> P \<longrightarrow> s \<in> P' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
isomorphicI: fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.isomorphic (VV.dom f) (VV.cod f)"
permutes_inv_o_1: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "p permutes S" shows "p \<circ> inv p = id"
sum_list_map_eq_sum_count: fixes f :: "'a \<Rightarrow> nat"   and xs :: "'a list" shows "sum_list (map f xs) = (\<Sum>x\<in>set xs. count_list xs x * f x)"
bij_\<phi>: shows "bij_betw \<phi> (elts \<alpha>) A"
Diff_eq_empty_iff: fixes A :: "'f set"   and B :: "'f set" shows "(A - B = {}) = (A \<subseteq> B)"
canonical_sampled: fixes A :: "('b \<Rightarrow> bool) set"   and \<sigma> :: "nat \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> nat" assumes "canonical_sequence A \<sigma>" shows "canonical_sequence A (\<sigma> \<circ> f)"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
tiny_digraph_op: shows "tiny_digraph \<alpha> (op_dg \<BB>)"
lprefix_conv_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" shows "lprefix xs ys = (\<exists>zs. ys = lappend xs zs)"
clopenin_eq_frontier_of: fixes X :: "'c topology"   and S :: "'c set" shows "(closedin X S \<and> openin X S) = (S \<subseteq> topspace X \<and> X frontier_of S = {})"
Quo_div: fixes x :: "nat"   and y :: "nat" shows "Quo x y = y div x"
not_frequently: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter" shows "(\<not> frequently P F) = (\<forall>\<^sub>F x in F. \<not> P x)"
simps: fixes x :: "Vars.var"   and y :: "Vars.var"   and z :: "Vars.var" shows "x[y::v=z] = (if x = y then z else x)"
path_join_is_path: fixes G :: "graph"   and p :: "nat list"   and q :: "nat list" assumes "joinable G p q" shows "is_path G (path_join G p q)"
elims: fixes step :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and x :: "('a \<Rightarrow> 'b) \<times> ('a \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'b)"   and xa :: "'a \<times> 'b"   and xb :: "'c list"   and y :: "'a \<times> 'b" assumes "On_Off.config' step x xa xb = y" obtains A :: "('a \<Rightarrow> 'b) \<times> ('a \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'b)" and s :: "'a \<times> 'b" where "x = A__" and "xa = s__" and "xb = []" and "y = s__"   | A :: "('a \<Rightarrow> 'b) \<times> ('a \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'b)" and s :: "'a \<times> 'b" and r :: "'c" and rs :: "'c list" where "x = A" and "xa = s" and "xb = r # rs" and "y = On_Off.config' step A (On_Off.Step step A s r) rs"
analz_insert_freshK: fixes evs :: "event list"   and KAB :: "nat"   and K :: "nat" assumes "evs \<in> bankerberos"   and "KAB \<notin> range shrK" shows "(Key K \<in> analz (insert (Key KAB) (knows Spy evs))) = (K = KAB \<or> Key K \<in> analz (knows Spy evs))"
exhaust_sel: fixes stream :: "'a stream" obtains "stream = shd stream ## stl stream"
nonintSfmla_iff_nonint: shows "nonintSfmla [] = nonint"
distinct_16: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Skip"
copr_pderiv: shows "algebraic_semidom_class.coprime f (pderiv (prod_list fss))"
\<delta>: shows "\<delta> = \<eta> + \<delta>a"
mul_add_right: fixes M :: "'a multiset"   and N :: "'a multiset"   and r :: "('a \<times> 'a) set"   and P :: "'a multiset" assumes "(M, N) \<in> mul r" shows "(M, N + P) \<in> mul r"
sel_2: fixes x1 :: "(char list, char list, 'b) namespace"   and x2 :: "(char list, char list, nat \<times> tid_or_exn) namespace" shows "c (make_sem_env x1 x2) = x2"
inject_2: fixes x21 :: "real"   and x22 :: "real list"   and y21 :: "real"   and y22 :: "real list" shows "(Eq x21 x22 = Eq y21 y22) = (x21 = y21 \<and> x22 = y22)"
bn_heapToAssn: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "bn (heapToAssn \<Gamma>) = map (\<lambda>x. atom (fst x)) \<Gamma>"
diff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> affine hull S"   and "y \<in> affine hull S" shows "x - y = (f x - f y) *\<^sub>R z"
cc_SDG_path_ics_SDG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n cc-ns\<rightarrow>\<^sub>d* n'" shows "n ics-ns\<rightarrow>\<^sub>d* n'"
foldl_conv_fold: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and s :: "'a"   and xs :: "'b list" shows "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
Outside_def: fixes R :: "('b \<times> 'c) set"   and X :: "'b set" shows "Outside' X R = R - X \<times> Range R"
APP_def: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c" shows "f $ a \<equiv> f a"
ScopeExt: fixes y :: "name"   and R :: "pi"   and z :: "name"   and S :: "pi" assumes "y \<sharp> R" shows "(<\<nu>z>(R \<parallel> <\<nu>y>S), <\<nu>y><\<nu>z>(R \<parallel> S)) \<in> Rel"