hcju/isabelleps huggingface dataset:
induct: fixes P1 :: "'a exp \<Rightarrow> bool"   and P2 :: "'a exp list \<Rightarrow> bool"   and exp :: "'a exp" assumes "\<And>x. P1 (new x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Cast x1 x2)"   and "\<And>x. P1 (Val x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 \<guillemotleft>x2\<guillemotright> x3)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (x1:=x2)"   and "\<And>x1 x2 x3. P1 x1 \<Longrightarrow> P1 (x1\<bullet>x2{x3})"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2{x3} := x4)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P2 x3\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2(x3))"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 {x1:x2; x3}"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (if (x1) x2 else x3)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (while (x1) x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (try x1 catch(x2 x3) x4)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 exp"
FGModuleIso_axioms: shows "FGModuleIso G (\<cdot>) V (\<star>) T W"
A_def: shows "A = Inf {u n / real n |n. 0 < n}"
cospan_diagram_axioms_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f0 :: "'b"   and f1 :: "'b" shows "cospan_diagram_axioms C f0 f1 \<equiv> partial_magma.arr C f0 \<and> partial_magma.arr C f1 \<and> partial_magma.cod C f0 = partial_magma.cod C f1"
progress_0_gen: fixes P :: "String.literal \<Rightarrow> nat option"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<phi> :: "Formula.formula" assumes "pred_mapping (\<lambda>x. x = 0) P" shows "Monitor.progress \<sigma> P \<phi> 0 = 0"
distinct_remdups_id: fixes xs :: "'g list" assumes "distinct xs" shows "remdups xs = xs"
axioms_2: fixes H :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "subring H R" shows "submonoid H R"
elims_2: fixes x :: "'b ext0"   and xa :: "'b ext0" assumes "x \<le> xa" obtains uu_ :: "'b ext0" where "xa__ = ext0.Bot" and "xa = uu_"   | x :: "'b" and y :: "'b" where "xa__ = ext0.Val x" and "xa = ext0.Val y" and "x \<le> y"   | uw_ :: "'b" where "xa__ = ext0.Val uw_" and "xa = ext0.Top"   | "xa__ = ext0.Top" and "xa = ext0.Top"
no_head_gen_def: shows "no_head_gen \<equiv> \<lambda>C b. {x. no_head_genp (\<lambda>x. x \<in> C) b x}"
axioms_1: fixes \<eta> :: "'b \<Rightarrow> 'c"   and M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and M' :: "'c set"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit' :: "'c" assumes "monoid_homomorphism \<eta> M composition unit M' composition' unit'" shows "map \<eta> M M'"
disjunctiveD: fixes x :: "'b \<Rightarrow> 'c"   and y :: "'b"   and z :: "'b" assumes "x \<in> Apply.disjunctive" shows "x (y \<squnion> z) = x y \<squnion> x z"
fail_bind: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and fail :: "'e"   and f :: "'d \<Rightarrow> 'e" assumes "monad_fail return bind fail" shows "bind fail f = fail"
A_RCOSETS_def: fixes G :: "('c, 'd) ring_scheme"   and H :: "'c set" shows "a_rcosets\<^bsub>G\<^esub> H = rcosets\<^bsub>add_monoid G\<^esub> H"
assms_6: shows "!p \<cdot> l \<cdot> p = (0::'a)"
vors_4: shows "vs2 = between (vertices f) ram2 ram1"
alg_dvd_uminus_right_iff: fixes x :: "'a"   and y :: "'a" shows "(x alg_dvd - y) = (x alg_dvd y)"
cong_exp_iff_simps_1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and n :: "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 n) (numeral.numeral one plus num.One) = zero) = True"
in_kD_after_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and nip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" shows "kD (update rt nip (dsn, dsk, flag, hops, nhip)) = insert nip (kD rt)"
proj_tuple_in_join_def: fixes pos :: "bool"   and bs :: "bool list"   and as :: "'b option list"   and t :: "'b option list set" shows "proj_tuple_in_join pos bs as t = join_cond pos t (proj_tuple bs as)"
execn_dynCall: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and s :: "'e"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>call init (p s) return c,Normal s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>dynCall init p return c,Normal s\<rangle> =n\<Rightarrow> t"
coeffs_def: fixes p :: "'a poly" shows "coeffs p = (if p = 0 then [] else map (poly.coeff p) [0..<Suc (degree p)])"
chine_assoc_props_1: shows "\<guillemotleft>doms.chine_assoc : Chn ((local.dom \<mu> \<star> local.dom \<nu>) \<star> local.dom \<pi>) \<rightarrow>\<^sub>C Chn (local.dom \<mu> \<star> local.dom \<nu> \<star> local.dom \<pi>)\<guillemotright>"
measurable_spmf_measure1: fixes M :: "'c spmf"   and N :: "'d measure" shows "measure_spmf M \<rightarrow>\<^sub>M N = UNIV \<rightarrow> space N"
inf_prefix_takeR: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<sqinter> ys = ys \<down> inf_prefix_length xs ys"
tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_trms_subset': fixes F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "\<Union> (trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s ` set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D)) \<subseteq> pair ` set F \<union> pair ` set D"
boundedI: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf"   and c :: "'b::semilattice_inf" assumes "(a::'b::semilattice_inf) \<le> (b::'b::semilattice_inf)"   and "(a::'b::semilattice_inf) \<le> (c::'b::semilattice_inf)" shows "(a::'b::semilattice_inf) \<le> (b::'b::semilattice_inf) \<sqinter> (c::'b::semilattice_inf)"
lex_trans: fixes x :: "nat list"   and y :: "nat list"   and z :: "nat list" assumes "x <\<^sub>l\<^sub>e\<^sub>x y"   and "y <\<^sub>l\<^sub>e\<^sub>x z" shows "x <\<^sub>l\<^sub>e\<^sub>x z"
yz: shows "(y, z) \<in> {(N, M). \<exists>a M0 K. M = add_mset a M0 \<and> N = M0 + K \<and> (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"
diff_fun_add: fixes a :: "'a \<Rightarrow> 'c"   and b :: "'a \<Rightarrow> 'c" assumes "diff_fun k charts a"   and "diff_fun k charts b" shows "diff_fun k charts (a + b)"
prob_non_zero: fixes M :: "'a measure"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "prob_space_with_indicators M I A" shows "\<exists>i\<in>I. 0 < Sigma_Algebra.measure M (A i)"
alw_ev_stl: fixes P :: "'d stream \<Rightarrow> bool"   and \<omega> :: "'d stream" shows "alw (ev P) (stl \<omega>) = alw (ev P) \<omega>"
eventually_top: fixes P :: "'b \<Rightarrow> bool" shows "eventually P top = (\<forall>x. P x)"
scc_of_tail_inter: shows "tail G a \<in> G.scc_of (tail G a) \<inter> H.scc_of (tail G a)"
slide_surj: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "surj_to (slide i) {l. l \<le> j - i} (nset i j)"
rep_eq: fixes x :: "'b trace" shows "\<tau> x = (\<lambda>i. snd (Rep_trace x !! i))"
simps_2: fixes n :: "nat"   and us :: "'a vec list"   and w :: "'a vec"   and ws :: "'a vec list" shows "gram_schmidt.sub2_wit n us (w # ws) = (case gram_schmidt.adjuster_wit n [] w us of (wit, aw) \<Rightarrow> let u = aw + w in case gram_schmidt.sub2_wit n (u # us) ws of (wits, vvs) \<Rightarrow> (wit # wits, u # vvs))"
three\<^sub>2_def: shows "\<^bold>2 = fst (snd threes)"
set_inter_mem: fixes x :: "'a"   and l1 :: "'a list"   and l2 :: "'a list" assumes "x mem l1"   and "x mem l2" shows "set l1 \<inter> set l2 \<noteq> {}"
reduced_GB_unique: fixes G :: "('a \<Rightarrow>\<^sub>0 'd) set"   and F :: "('a \<Rightarrow>\<^sub>0 'd) set" assumes "finite G"   and "punit.is_reduced_GB G"   and "local.punit.pmdl G = local.punit.pmdl F" shows "punit.reduced_GB F = G"
Sup_dual_def_var: fixes X :: "('c \<times> 'd) set" shows "dual_prod (Sup_prod X) = Inf_prod (dual_prod ` X)"
continuous_on_Re: fixes s :: "'b set"   and g :: "'b \<Rightarrow> quat" assumes "continuous_on s g" shows "continuous_on s (\<lambda>x. quat.Re (g x))"
ereal_left_distrib: fixes a :: "ereal"   and b :: "ereal"   and r :: "ereal" assumes "0 \<le> a"   and "0 \<le> b" shows "(a + b) * r = a * r + b * r"
reaches1: fixes a :: "'a"   and b :: "'a" assumes "a \<rightarrow>\<^sup>+ b" shows "reaches1 a b"
map_snd_pair_pmf: fixes A :: "'d pmf"   and B :: "'c pmf" shows "map_pmf snd (pair_pmf A B) = B"
hurwitz_zeta_real_bound_aux: fixes a :: "real"   and x :: "real" assumes "0 < a"   and "1 < x" shows "(\<Sum>i. (a + real (Suc i)) powr - x) \<le> a powr (1 - x) / (x - 1)"
less_V_def: fixes x :: "V"   and y :: "V" shows "x < y \<equiv> x \<le> y \<and> x \<noteq> y"
d_type: shows "\<forall>P. (\<forall>x. x = ad (ad x) \<longrightarrow> P x) = (\<forall>x. P (ad (ad x)))"
winningAllocationUniqueness: fixes N :: "'c set"   and G :: "'d list"   and bids :: "'c \<times> 'd set \<Rightarrow> 'e"   and random :: "integer" assumes "N \<noteq> {}"   and "distinct G"   and "set G \<noteq> {}"   and "finite N" shows "winningAllocationRel N (set G) (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel N (set G) ((\<in>) x) bids) (allAllocationsAlg N G)) random) <| (N \<times> finestpart (set G))) N (set G)))) bids = {randomEl (takeAll (\<lambda>x. winningAllocationRel N (set G) ((\<in>) x) bids) (allAllocationsAlg N G)) random}"
rotate3: fixes A3 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A3 (insert A1 (insert A2 H)) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 H)) \<turnstile> B"
hyps_1: fixes x' :: "item"   and k' :: "nat"   and u' :: "nat" assumes "x' \<in> \<J> k' u'"   and "indexlt k' u' k u"   and "k' < item_end x'" shows "produced_by_scan_step x' k' u'"
t_def: shows "t \<equiv> \<lambda>cfg'. THE s'. s' \<in> set_pmf (rept s (action cfg)) \<and> s' ~ x"
prod_eq_iff: fixes s :: "'a \<times> 'b"   and t :: "'a \<times> 'b" shows "(s = t) = (fst s = fst t \<and> snd s = snd t)"
dghm_proj: shows "\<pi>\<^sub>D\<^sub>G I \<AA> i : dg_prod I \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA> i"
power_tendsto_0_iff: fixes n :: "nat"   and f :: "'c \<Rightarrow> real"   and F :: "'c filter" assumes "0 < n" shows "((\<lambda>x. f x ^ n) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> 0) F"
index_vec: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "i < n" shows "vec n f $ i = f i"
legacy_Complex_simps_15: fixes r :: "real" shows "\<i> * cor r = Complex 0 r"
Red: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" assumes "t = Branch R l k v r" shows "thesis"
vNM_utility_implies_rationality: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set" assumes "finite outcomes"   and "outcomes \<noteq> {}"   and "\<exists>u. ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)" shows "rational_preference (\<P> outcomes) \<R>"
strip_guards_guards: fixes F :: "'f set"   and gs :: "('f \<times> 'd set) list"   and c :: "('d, 'e, 'f) com" shows "strip_guards F (guards gs c) = guards (filter (\<lambda>(f, g). f \<notin> F) gs) (strip_guards F c)"
intro: fixes A :: "'b set" assumes "csubspace A"   and "closed A" shows "closed_csubspace A"
z_le_i: fixes x :: "int" shows "ant x \<le> \<infinity>"
elims_2: fixes x :: "'a list" assumes "strict_sorted x" obtains "xa__ = []"   | x :: "'a" and ys :: "'a list" where "xa__ = x # ys" and "Ball (set ys) ((<) x) \<and> strict_sorted ys"
msg_fresh_def: fixes \<sigma> :: "nat \<Rightarrow> state"   and m :: "msg" shows "msg_fresh \<sigma> m \<equiv> case m of Rreq hopsc x xa xb oipc osnc sipc xc \<Rightarrow> 1 \<le> osnc \<and> (sipc \<noteq> oipc \<longrightarrow> oipc \<in> kD (rt (\<sigma> sipc)) \<and> osnc \<le> nsqn (rt (\<sigma> sipc)) oipc \<and> (nsqn (rt (\<sigma> sipc)) oipc = osnc \<longrightarrow> the (dhops (rt (\<sigma> sipc)) oipc) \<le> hopsc \<or> the (flag (rt (\<sigma> sipc)) oipc) = Aodv_Basic.inv)) | Rrep hopsc dipc dsnc x sipc \<Rightarrow> 1 \<le> dsnc \<and> (sipc \<noteq> dipc \<longrightarrow> dipc \<in> kD (rt (\<sigma> sipc)) \<and> dsnc \<le> nsqn (rt (\<sigma> sipc)) dipc \<and> (nsqn (rt (\<sigma> sipc)) dipc = dsnc \<longrightarrow> the (dhops (rt (\<sigma> sipc)) dipc) \<le> hopsc \<or> the (flag (rt (\<sigma> sipc)) dipc) = Aodv_Basic.inv)) | Rerr destsc sipc \<Rightarrow> \<forall>ripc\<in>dom destsc. ripc \<in> kD (rt (\<sigma> sipc)) \<and> the (destsc ripc) - 1 \<le> nsqn (rt (\<sigma> sipc)) ripc | _ \<Rightarrow> True"
spec_ops_exists_base: fixes ops :: "('c \<times> 'd) set"   and deps :: "'d \<Rightarrow> 'c set" assumes "finite ops"   and "\<And>oid op1 op2. \<lbrakk>(oid, op1) \<in> ops; (oid, op2) \<in> ops\<rbrakk> \<Longrightarrow> op1 = op2"   and "\<And>oid oper ref. \<lbrakk>(oid, oper) \<in> ops; ref \<in> deps oper\<rbrakk> \<Longrightarrow> ref < oid" shows "\<exists>op_list. set op_list = ops \<and> spec_ops op_list deps"
b_e_check_single_weaken_type: fixes \<C> :: "t_context"   and e :: "b_e"   and tn :: "t list"   and tm :: "t list"   and ts :: "t list" assumes "check_single \<C> e (Type tn) = Type tm" shows "check_single \<C> e (Type (ts @ tn)) = Type (ts @ tm)"
flip_altsublist_adjacent_lists: fixes ss :: "'w list"   and ts :: "'w list" assumes "ss \<in> lists S"   and "flip_altsublist_adjacent ss ts" shows "ts \<in> lists S"
mult_minus_distrib_mat: fixes A :: "'c mat"   and nr :: "nat"   and n :: "nat"   and B :: "'c mat"   and nc :: "nat"   and C :: "'c mat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc"   and "C \<in> carrier_mat n nc" shows "A * (B - C) = A * B - A * C"
dsr5: fixes x :: "'a"   and y :: "'a" shows "d (x + y) = d x + d y"
assms_2: shows "the (rt dip) \<sqsubseteq> r"
perfect_imp_proper_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "perfect_map X Y f" shows "proper_map X Y f"
rep_eq: shows "pdevs_apply zero_pdevs = (\<lambda>_. 0::'a)"
relcompE: fixes xz :: "'a \<times> 'b"   and r :: "('a \<times> 'c) set"   and s :: "('c \<times> 'b) set" assumes "xz \<in> r ; s" obtains x :: "'a" and y :: "'c" and z :: "'b" where "xz = (x, z)" and "(x, y) \<in> r" and "(y, z) \<in> s"
ipurge_tr_rev_t_invariance_1: fixes Y :: "('c, 'd) ipurge_rec"   and u :: "'d"   and D :: "'c \<Rightarrow> 'd"   and I :: "('d \<times> 'd) set"   and X :: "('c, 'd) ipurge_rec"   and ws :: "'c list" assumes "Y \<in> ipurge_tr_rev_t_set (X\<lparr>Pol := I, Map := D, Dom := u\<rparr>)"   and "ipurge_tr_rev_t_inv_1 I D u ws (X\<lparr>Pol := I, Map := D, Dom := u\<rparr>)" shows "ipurge_tr_rev_t_inv_1 I D u ws Y"
step'_def: shows "step' = Step asem (\<lambda>b S. S)"
add_zero_left: fixes z :: "complex_homo" shows "0\<^sub>h +\<^sub>h z = z"
pair_declclass_def: fixes p :: "'a \<times> 'b" shows "declclass p = declclass (fst p)"
essential_mono: fixes G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set"   and x :: "'a"   and S' :: "'a set" assumes "essential G B S x"   and "S' \<subseteq> S" shows "essential G B S' x"
ty_cmd_imp_ty_la: fixes \<Gamma> :: "(String.literal \<Rightarrow> unit option) \<times> (String.literal \<Rightarrow> unit option)"   and c :: "cmd"   and a :: "action"   and c' :: "cmd" assumes "ty_cmd (\<Gamma>, False) c"   and "cfg' c a c'" shows "ty_la \<Gamma> a"
tred: shows "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'"
normalize_ex_bound': fixes s :: "String.literal set"   and c :: "String.literal" assumes "finite s"   and "c \<in> s"   and "c \<notin> normalize_sort cs s"   and "sort_ex cs s" shows "\<exists>c'\<in>normalize_sort cs s. class_les cs c' c"
domA_append: fixes a :: "('b \<times> 'c) list"   and b :: "('b \<times> 'c) list" shows "domA (a @ b) = domA a \<union> domA b"
R'_prop: fixes k :: "'k"   and v :: "'v"   and u :: "'e"   and k' :: "'k"   and v' :: "'v"   and u' :: "'e" assumes "m k = Some v"   and "u \<in> S_b v"   and "m k' = Some v'"   and "u' \<in> S_b v'"   and "if k = k' then R_b v u u' else R_a (k, v) (k', v')" shows "R_ab (k, u) (k', u')"
\<R>\<^sub>2_elim: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and c\<^sub>1 :: "('b, 'a, 'c) Stmt"   and mds :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'd"   and c\<^sub>2 :: "('b, 'a, 'c) Stmt"   and mem\<^sub>2 :: "'b \<Rightarrow> 'd"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma> :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> sifum_types.\<R>\<^sub>2 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>" obtains \<Gamma>\<^sub>1 :: "'b \<Rightarrow> Sec option" and \<Gamma>\<^sub>2 :: "'b \<Rightarrow> Sec option" where "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> sifum_security.mm_equiv dma (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B)" and "\<forall>x\<in>dom \<Gamma>. \<Gamma> x = Some High" and "sifum_types.has_type aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>\<^sub>1 c\<^sub>1 \<Gamma>" and "sifum_types.has_type aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>\<^sub>2 c\<^sub>2 \<Gamma>" and "sifum_types.mds_consistent dma mds \<Gamma>\<^sub>1" and "sifum_types.mds_consistent dma mds \<Gamma>\<^sub>2"
submonoid_nonempty: fixes G :: "('a, 'b) monoid_scheme" shows "\<not> submonoid {} G"
finite2: fixes A :: "'a set"   and f :: "'v \<Rightarrow> 'a" shows "(\<Sum>\<^sup>+ y. g (x, y) * indicator A (f y)) \<noteq> \<top>"
v11_eq_iff: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s"   and "b \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s" shows "(s\<lparr>a\<rparr> = s\<lparr>b\<rparr>) = (a = b)"
A1: shows "Esigma 69615 = Esigma (3\<^sup>2 * 7 * 13 * 5 * 17)"
coprime_prod_int2Tr: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set" shows "(\<forall>k\<le>Suc n. ideal R (J k)) \<and> (\<forall>i\<le>Suc n. \<forall>j\<le>Suc n. i \<noteq> j \<longrightarrow> coprime_ideals R (J i) (J j)) \<longrightarrow> \<Inter> {I. \<exists>k\<le>Suc n. I = J k} = i\<Pi>\<^bsub>R,Suc n\<^esub> J"
IH: fixes x :: "real" assumes "0 < x" shows "Polygamma_approx (Suc (Suc j)) m x = stirling_sum' (Suc j) m x"
elims_2: fixes x :: "'b routing_rule list" assumes "has_default_route x" obtains r :: "'b routing_rule" and rs :: "'b routing_rule list" where "x = r # rs" and "routing_prefix r = 0 \<or> has_default_route rs"
f': fixes tx :: "real \<times> 'a" assumes "tx \<in> T \<times> X" shows "((\<lambda>(t, x). f t x) has_derivative f' tx) (at tx within T \<times> X)"
derived_in_carrier: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" assumes "Group.group G"   and "H \<subseteq> carrier G" shows "derived G H \<subseteq> carrier G"
padic_int_simps_5: fixes p :: "int" shows "carrier (padic_int p) = padic_set p"
tpart2_su: fixes wtFsym :: "'b \<Rightarrow> bool"   and wtPsym :: "'c \<Rightarrow> bool"   and V :: "Preliminaries.var set" assumes "Signature TYPE('a) wtFsym wtPsym"   and "Signature.tinfinite TYPE('a) V" shows "Signature.tpart2 TYPE('a) V \<subseteq> V"
prems_1: shows "map dim_vec inputs = input_sizes (Pool m1 m2)"
perfect_sec_P1_def: fixes m1 :: "nat"   and m2 :: "nat" shows "local.sim_non_det_def.perfect_sec_P1 m1 m2 \<equiv> R1 m1 m2 = funct m1 m2 \<bind> (\<lambda>(s1, s2). local.sim_non_det_def.Ideal1 m1 m2 s1)"
It: fixes A :: "'t" assumes "A \<in> Ty\<^bsub>iS\<^esub>" shows "Ty\<lbrakk>A\<rbrakk> \<in> obj\<^bsub>iC\<^esub>"
fsubset_fsingletonD: fixes A :: "'a fset"   and x :: "'a" assumes "A |\<subseteq>| {|x|}" shows "A = {||} \<or> A = {|x|}"
max_ge_y: fixes y :: "'a"   and x :: "'a" shows "y \<le> max x y"
conv_radius_leI': fixes f :: "nat \<Rightarrow> 'a"   and z :: "'a" assumes "\<not> summable (\<lambda>n. f n * z ^ n)" shows "conv_radius f \<le> ereal (norm z)"
abs_eq: shows "E! = make\<Pi>\<^sub>1 (\<lambda>u s w. case u of \<omega>\<upsilon> x \<Rightarrow> ConcreteInWorld x w | \<sigma>\<upsilon> \<sigma> \<Rightarrow> False)"
plus_base: fixes A :: "'b tensor"   and B :: "'b tensor" assumes "dims A = dims B" shows "A + B = plus_base A B"
composite_functor_def: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and C :: "'ai \<Rightarrow> 'ai \<Rightarrow> 'ai"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ah \<Rightarrow> 'ai" shows "composite_functor A B C F G \<equiv> functor A B F \<and> functor B C G"
Minr_is_Suc_Suc: fixes Rr :: "nat list \<Rightarrow> bool"   and xs :: "nat list"   and w :: "nat" assumes "Minr Rr xs w = Suc w"   and "\<not> Rr (xs @ [Suc w])" shows "Minr Rr xs (Suc w) = Suc (Suc w)"
prm_apply_unequal: fixes x :: "'b"   and y :: "'b"   and \<pi> :: "'b prm" assumes "x \<noteq> y" shows "\<pi> $ x \<noteq> \<pi> $ y"
wfTransitionRelation: fixes decisionVars :: "nat set"   and state0 :: "State"   and F0 :: "Literal list list" assumes "finite decisionVars"   and "isInitialState state0 F0" shows "wf {(stateB, stateA). (state0, stateA) \<in> transitionRelation F0 decisionVars \<and> transition stateA stateB F0 decisionVars}"
aezfun_setI: fixes f :: "'a \<Rightarrow> 'b" assumes "finite (supp f)" shows "f \<in> aezfun_set"
s_Tr_2: fixes \<pi> :: "Syntax.term list" assumes "\<pi>\<^sub>2 = \<pi>\<^sub>1 @ \<pi>"   and "\<lless>\<pi>\<^sub>1, eP\<^sub>1\<ggreater> P\<rightarrow>i \<lless>\<pi>\<^sub>1 @ \<pi>, eP\<^sub>2\<ggreater>"   and "{$$} \<turnstile> e, eP\<^sub>1, eV : \<tau>"   and "{$$} \<turnstile> e, eP\<^sub>1, eV : \<tau>" shows "\<exists>e' eV'. {$$} \<turnstile> e', eP\<^sub>2, eV' : \<tau> \<and> \<lless>[], e\<ggreater> I\<rightarrow>i \<lless>[], e'\<ggreater> \<and> \<lless>\<pi>, eV\<ggreater> V\<rightarrow>i \<lless>[], eV'\<ggreater>"
prems_2: shows "\<O> \<inter> read_only \<S> = {}"
SeqConstP_Eats: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and s2 :: "tm"   and k1 :: "tm"   and k2 :: "tm"   and t1 :: "tm"   and t2 :: "tm" assumes "atom s \<sharp> (k, s1, s2, k1, k2, t1, t2)"   and "atom k \<sharp> (s1, s2, k1, k2, t1, t2)" shows "{SeqConstP s1 k1 t1, SeqConstP s2 k2 t2} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqConstP (Var s) (Var k) (Q_Eats t1 t2)))"
elims_2: fixes x :: "('b, 'c) Combinators"   and xa :: "('b, 'c) Combinators" assumes "member x xa" obtains a :: "('b, 'c) Combinators" and xb :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators" where "x = a__" and "xa = xb \<oplus> xs" and "member a__ xb \<or> member a__ xs"   | a :: "('b, 'c) Combinators" where "x = aa__" and "xa = DenyAll" and "aa__ = DenyAll"   | a :: "('b, 'c) Combinators" and v :: "'b" and va :: "'b" where "x = ab__" and "xa = DenyAllFromTo v__ va__" and "ab__ = DenyAllFromTo v__ va__"   | a :: "('b, 'c) Combinators" and v :: "'b" and va :: "'b" and vb :: "'c" where "x = a" and "xa = AllowPortFromTo v va vb" and "a = AllowPortFromTo v va vb"