hcju/isabelleps huggingface dataset:
induct: fixes P :: "'b formula \<Rightarrow> bool"   and a0 :: "'b formula" assumes "\<And>F G. P (F \<^bold>\<rightarrow> G)"   and "\<And>F G. \<lbrakk>P F; P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<and> G)"   and "\<And>F G. \<lbrakk>P F; P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G)"   and "\<And>v. P (Atom v)"   and "P \<bottom>"   and "\<And>v. P (\<^bold>\<not> v)" shows "P a0"
delete_delete_commute: fixes i1 :: "'d set"   and e1 :: "'c"   and i2 :: "'d set"   and e2 :: "'c" shows "\<langle>Delete i1 e1\<rangle> \<rhd> \<langle>Delete i2 e2\<rangle> = \<langle>Delete i2 e2\<rangle> \<rhd> \<langle>Delete i1 e1\<rangle>"
weak_ranking_index_unique': fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'c set list"   and i :: "nat"   and x :: "'c" assumes "finite_total_preorder_on carrier le"   and "is_weak_ranking xs"   and "i < length xs"   and "x \<in> xs ! i" shows "i = find_index ((\<in>) x) xs"
assms: shows "(\<lambda>j. cos (\<theta> j)) \<longlonglongrightarrow> 1"
fmrestrict_set_add_distrib: fixes A :: "'d set"   and m :: "('d, 'e) fmap"   and n :: "('d, 'e) fmap" shows "fmrestrict_set A (m ++\<^sub>f n) = fmrestrict_set A m ++\<^sub>f fmrestrict_set A n"
type_wf_def: shows "ShadowRootClass.type_wf \<equiv> l_type_wf_def\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t.a_type_wf"
l2_inv4_trans: shows "{l2_inv4} TS.trans l2 {> l2_inv4}"
S1: shows "\<exists>n\<in>A. nat_to_ce_set n = {}"
y1_def: shows "y1 \<equiv> root 3 u - e / (3 * root 3 u)"
Proc_CFG_Call: fixes p :: "char list"   and es :: "expr list"   and rets :: "char list list" shows "Call p es rets \<turnstile> Label 0 -CEdge (p, es, rets)\<rightarrow>\<^sub>p Label 1"
invt_def: fixes \<sigma> :: "'tp"   and a :: "univ" shows "invt \<sigma> a \<equiv> if unprot \<sigma> \<or> protFw \<sigma> then a else SOME b. eintT \<sigma> b \<and> eintF (Tag \<sigma>) [b] = a"
l_append_child_wf_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and append_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool" shows "l_append_child_wf_axioms type_wf known_ptrs append_child heap_is_wellformed \<equiv> (\<forall>h ptr child h'. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> append_child ptr child \<rightarrow>\<^sub>h h' \<longrightarrow> type_wf h') \<and> (\<forall>h ptr child h'. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> append_child ptr child \<rightarrow>\<^sub>h h' \<longrightarrow> known_ptrs h') \<and> (\<forall>h ptr child h'. type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> heap_is_wellformed h \<longrightarrow> h \<turnstile> append_child ptr child \<rightarrow>\<^sub>h h' \<longrightarrow> heap_is_wellformed h')"
assms_1: shows "\<R>\<^sub>\<circ> xs \<subseteq>\<^sub>\<circ> X"
fmpt_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "fmpt M T \<equiv> mpt M T \<and> finite_measure M"
inf_sup_aci_1: fixes inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'f"   and y :: "'f" assumes "class.lattice inf less_eq less sup" shows "inf x y = inf y x"
hyps: fixes v :: "event_data list"   and v' :: "event_data list"   and e :: "String.literal \<times> event_data list" assumes "\<forall>x\<in>fv \<phi>. v ! x = v' ! x" shows "matches v \<phi> e = matches v' \<phi> e"
degree_eq_iff: fixes p :: "'a mpoly"   and x :: "nat"   and k :: "nat" shows "(MPoly_Type.degree p x = k) = (if k = 0 then \<forall>m\<in>monomials p. lookup m x = 0 else (\<exists>m\<in>monomials p. lookup m x = k) \<and> (\<forall>m\<in>monomials p. lookup m x \<le> k))"
LNil_eq_lappend_iff: fixes xs :: "'c llist"   and ys :: "'c llist" shows "(LNil = lappend xs ys) = (xs = LNil \<and> ys = LNil)"
def_scheme: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and a2 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2"   and a3 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>3" shows "f self a1 a2 a3 \<equiv> \<lambda>\<tau>. SOME res. let res = \<lambda>_. res in if \<tau> \<Turnstile> \<delta> self \<and> \<tau> \<Turnstile> \<upsilon> a1 \<and> \<tau> \<Turnstile> \<upsilon> a2 \<and> \<tau> \<Turnstile> \<upsilon> a3 then \<tau> \<Turnstile> PRE self a1 a2 a3 \<and> \<tau> \<Turnstile> POST self a1 a2 a3 res else \<tau> \<Turnstile> res \<triangleq> invalid"
openin_imp_subset: fixes U :: "'b topology"   and S :: "'b set"   and T :: "'b set" assumes "openin (subtopology U S) T" shows "T \<subseteq> S"
reflect_rtranclp: fixes S :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "\<And>x y. S (f x) (f y) \<Longrightarrow> R x y"   and "bij_on_trancl S f"   and "S\<^sup>*\<^sup>* (f x) (f y)" shows "R\<^sup>*\<^sup>* x y"
inf_sup_aci_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "inf (inf x y) z = inf x (inf y z)"
Abs_NF: fixes t :: "dB"   and ts :: "dB list" assumes "NF (Abs t \<degree>\<degree> ts)" shows "ts = []"
acyclic_inv: fixes t :: "'a"   and v :: "'a"   and e :: "'a" assumes "pd_kleene_allegory_class.acyclic t"   and "vector v"   and "e \<le> v * - v\<^sup>T"   and "t \<le> v * v\<^sup>T" shows "pd_kleene_allegory_class.acyclic (t \<squnion> e)"
K: shows "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
right_hom_is_subcategory: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "subcategory (\<cdot>) (right \<mu>)"
SN_rel_on_mono': fixes R :: "('a \<times> 'a) set"   and R' :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and S' :: "('a \<times> 'a) set"   and T :: "'a set" assumes "R \<subseteq> R'"   and "S \<subseteq> R' \<union> S'"   and "SN_rel_on R' S' T" shows "SN_rel_on R S T"
\<E>_def: fixes \<tau> :: "'c \<Rightarrow> 's" shows "\<E> \<tau> = S.Fun (\<tau> a) (\<phi> (a, a) a)"
equiv_not_sym: fixes eqv :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.equiv eqv"   and "\<not> eqv x y" shows "\<not> eqv y x"
fresher_def: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and r' :: "nat \<times> k \<times> f \<times> nat \<times> nat" shows "r \<sqsubseteq> r' \<equiv> nsqn\<^sub>r r < nsqn\<^sub>r r' \<or> nsqn\<^sub>r r = nsqn\<^sub>r r' \<and> \<pi>\<^sub>5 r' \<le> \<pi>\<^sub>5 r"
right_pure_induction: fixes x :: "'a" shows "(x \<rightarrow> x)\<^sup>\<star> \<le> x \<rightarrow> x"
subcocycle_v: fixes n :: "nat"   and m :: "nat"   and x :: "'a" shows "v (n + m) x \<le> v n x + v m ((T ^^ n) x)"
iMOD_def: fixes r :: "nat"   and m :: "nat" shows "[ r, mod m ] \<equiv> {x. x mod m = r mod m \<and> r \<le> x}"
bddh_bddinsert: fixes x :: "nat"   and b :: "'a bdd"   and w :: "bool list"   and y :: "'a" assumes "bddh x b"   and "x \<le> length w" shows "bddh (length w) (bddinsert b w y)"
Int_space_eq2: fixes x :: "'c set"   and M :: "'c measure" assumes "x \<in> sets M" shows "x \<inter> space M = x"
sum_list_nonneg_eq_zero_iff: fixes xs :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> (0::'b) \<le> x" shows "(sum_list xs = (0::'b)) = (set xs \<subseteq> {0::'b})"
len1_h: shows "length qs = 1 \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<longrightarrow> satisfies_properties_R p qs (get_subsets_R (calculate_data_R p qs)) (get_signs_R (calculate_data_R p qs)) (get_matrix_R (calculate_data_R p qs))"
inverse_fract: fixes a :: "'a"   and b :: "'a" shows "inverse (Fraction_Field.Fract a b) = Fraction_Field.Fract b a"
C: shows "(c, ps) \<Rightarrow>\<^sub>A m \<Down> ps' \<and> n = e + m"
intros_9: fixes \<iota> :: "nat"   and V :: "('exp, 'id) MWLsCom list"   and m :: "'id \<Rightarrow> 'val" shows "SRabbr (spawn\<^bsub>\<iota>\<^esub> V) m V None m"
distinct_set_replace: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" assumes "distinct xs" shows "set (replace x ys xs) = (if x \<in> set xs then set xs - {x} \<union> set ys else set xs)"
sub_add_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "sub_add t = (\<forall>P Q s. sound P \<and> sound Q \<longrightarrow> t P s + t Q s \<le> t (\<lambda>s'. P s' + Q s') s)"
induction_axioms_imp_Pf: fixes \<alpha> :: "fm"   and e :: "(name, hf) finfun" assumes "\<alpha> \<in> induction_axioms" shows "Pf \<lbrakk>\<guillemotleft>\<alpha>\<guillemotright>\<rbrakk>e"
simps_4: fixes r :: "'b mrexp2"   and s :: "'b mrexp2"   and uu :: "bool"   and uv :: "bool" shows "strip2 (Plus2 r s uu uv) = Plus (strip2 r) (strip2 s)"
smallstepV_ps_append: fixes \<pi> :: "Syntax.term list"   and eV :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and eV' :: "Syntax.term"   and X :: "Syntax.term list" shows "\<lless>\<pi>, eV\<ggreater> V\<rightarrow> \<lless>\<pi>', eV'\<ggreater> = \<lless>\<pi> @ X, eV\<ggreater> V\<rightarrow> \<lless>\<pi>' @ X, eV'\<ggreater>"
has_bochner_integral_simple_bochner_integrable: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_bochner_integrable M f" shows "has_bochner_integral M f (simple_bochner_integral M f)"
set_iterator_genord_foldli_conv: fixes iti :: "('b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'c set"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "set_iterator_genord iti S R = (\<exists>l0. distinct l0 \<and> S = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0)"
castmsg_simps_7: fixes P :: "'d \<Rightarrow> bool" shows "castmsg P \<tau>\<^sub>n = True"
step: fixes 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" assumes "\<tau>Exec_movest ci P t es h (stk, loc, pc, xcp) (stk', loc', pc', xcp')"   and "\<tau>exec_moves ci P t es h (stk', loc', pc', xcp') (stk'', loc'', pc'', xcp'')"   and "Q stk' loc' pc' xcp'" shows "Q stk'' loc'' pc'' xcp''"
total_onI: fixes A :: "'g set"   and r :: "('g \<times> 'g) set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r" shows "total_on A r"
eventually_True: fixes F :: "'a filter" shows "\<forall>\<^sub>F x in F. True"
IH: fixes ts :: "t list"   and ts' :: "t list" assumes "const_list xs"   and "\<S>\<bullet>\<C> \<turnstile> xs : ts _> ts'" shows "\<exists>tvs. ts' = ts @ tvs \<and> length xs = length tvs \<and> \<S>\<bullet>\<C>' \<turnstile> xs : [] _> tvs"
pi_half_neq_zero: shows "pi / 2 \<noteq> 0"
varnum_le_iff: fixes X :: "'c set"   and t :: "'c \<Rightarrow>\<^sub>0 'd"   and n :: "nat" shows "(varnum X t \<le> n) = (keys t \<subseteq> X \<union> {x. elem_index x < n})"
Pi_mem: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'e \<Rightarrow> 'f set"   and x :: "'e" assumes "f \<in> Pi A B"   and "x \<in> A" shows "f x \<in> B x"
spec\<^sub>3_def: fixes p :: "prog" shows "spec\<^sub>3 p \<equiv> para p = [''x'', ''y''] \<and> (\<exists>e\<^sub>1 e\<^sub>2. body p = Add e\<^sub>1 e\<^sub>2 \<and> wfe \<Gamma>\<^sub>x\<^sub>y e\<^sub>1 \<and> wfe \<Gamma>\<^sub>x\<^sub>y e\<^sub>2 \<and> (\<forall>x y. eval (\<E>\<^sub>x\<^sub>y x y) e\<^sub>1 \<oplus> eval (\<E>\<^sub>x\<^sub>y x y) e\<^sub>2 = Some (f x y)) \<and> coste e\<^sub>1 + coste e\<^sub>2 \<le> 2)"
append_simpl2: fixes xs :: "'b list"   and i :: "nat"   and ys :: "'b list" shows "length xs \<le> i \<longrightarrow> (xs @ ys) ! i = ys ! (i - length xs)"
lucas_lehmer_hom_def: fixes m :: "nat" shows "lucas_lehmer_hom m = (\<lambda>(x, y). (nat (x mod int m), nat (y mod int m)))"
integral_std_normal_distribution_moment_odd: fixes k :: "nat" assumes "odd k" shows "LINT x|std_normal_distribution. x ^ k = 0"
map_add_SomeD: fixes m :: "'c \<Rightarrow> 'b option"   and n :: "'c \<Rightarrow> 'b option"   and k :: "'c"   and x :: "'b" assumes "(m ++ n) k = \<lfloor>x\<rfloor>" shows "n k = \<lfloor>x\<rfloor> \<or> n k = None \<and> m k = \<lfloor>x\<rfloor>"
indep_sets_collect_sigma: fixes E :: "'c \<Rightarrow> 'a set set"   and I :: "'d \<Rightarrow> 'c set"   and J :: "'d set" assumes "indep_sets E (\<Union> (I ` J))"   and "\<And>i j. \<lbrakk>j \<in> J; i \<in> I j\<rbrakk> \<Longrightarrow> Int_stable (E i)"   and "disjoint_family_on I J" shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union> (E ` I j))) J"
elims: fixes x :: "'fsym trm"   and y :: "('fsym, 'tp) efsym trm" assumes "tNN x = y" obtains x :: "Preliminaries.var" where "xa__ = Var x" and "y = (if unprot (TE.tpOfV x) \<or> protFw (TE.tpOfV x) then Var x else Fn (Tag (TE.tpOfV x)) [Var x])"   | f :: "'fsym" and Tl :: "'fsym trm list" where "xa__ = Fn f Tl" and "y = (if unprot (resOf f) \<or> protFw (resOf f) then Fn (Oldf f) (map tNN Tl) else Fn (Tag (resOf f)) [Fn (Oldf f) (map tNN Tl)])"
zero_map_sq_matrix: fixes f :: "'d \<Rightarrow> 'c" assumes "f (0::'d) = (0::'c)" shows "map_sq_matrix f 0 = 0"
sturm_def: fixes p :: "real poly" shows "sturm p = sturm_aux p (pderiv p)"
\<R>\<^sub>3_sym: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" shows "sym (\<R>\<^sub>3 \<Gamma> \<S> P)"
assms_1: shows "f \<in> A \<rightarrow>\<^sub>E B"
wp_loop_step_mono_trans: fixes P :: "'a \<Rightarrow> real"   and body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "sound P"   and "healthy (wp body)" shows "mono_trans (\<lambda>Q s. \<guillemotleft> G \<guillemotright> s * wp body Q s + \<guillemotleft> \<N> G \<guillemotright> s * P s)"
measurable_limit: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure"   and c :: "'d" assumes "\<And>n. f n \<in> borel_measurable M" shows "Measurable.pred M (\<lambda>x. (\<lambda>n. f n x) \<longlonglongrightarrow> c)"
eventually_gt_ereal_at_top: fixes c :: "ereal" assumes "c \<noteq> \<infinity>" shows "\<forall>\<^sub>F x in at_top. c < ereal x"
IN_plus_current: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current (quotient_web \<Gamma> f) g" shows "d_IN (plus_current f g) x = (if x \<in> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) then d_IN f x else d_IN g x)"
qqpp_tendsto: shows "((\<lambda>x. qq x / pp x) \<longlongrightarrow> lead_coeff q / lead_coeff p) at_top"
as1: fixes x :: "'a" shows "ad x \<cdot> x = (zero_class.zero::'a)"
cases: fixes a :: "('s \<times> ('c \<Rightarrow> real)) pmf"   and st :: "'s \<times> ('c \<Rightarrow> real)" assumes "a \<in> K st" obtains l :: "'s" and u :: "'c \<Rightarrow> real" and t :: "real" where "a = return_pmf (l__, u__ \<oplus> t)" and "st \<in> S" and "st = (l__, u__)" and "0 \<le> t" and "u__ \<oplus> t \<turnstile> PTA.inv_of A l__"   | l :: "'s" and u :: "'c \<Rightarrow> real" and g :: "('c, real) cconstraint" and \<mu> :: "('c set \<times> 's) pmf" where "a = map_pmf (\<lambda>(X, l). (l, [(SOME r. set r = X)\<rightarrow>0]u)) \<mu>" and "st \<in> S" and "st = (l, u)" and "(l, g, \<mu>) \<in> PTA.trans_of A" and "u \<turnstile> g"   | "a = return_pmf st"
next_def: fixes g :: "('b, 'a) generator" shows "list.next g = snd (generator g)"
iT_Plus_neg_Max: fixes I :: "nat set"   and k :: "nat" assumes "finite I"   and "I \<oplus>- k \<noteq> {}" shows "Max (I \<oplus>- k) = Max I - k"
rdv'': fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z \<sqinter> x \<cdot> \<top>) \<cdot> y = z \<cdot> y \<sqinter> x \<cdot> \<top>"
spec_impl: fixes G :: "'v list_graph"   and nP :: "'v \<Rightarrow> 'a" assumes "wf_list_graph G" shows "sinvar_spec (list_graph_to_graph G) nP = sinvar_impl G nP"
coset_join3: fixes x :: "Vertex \<Rightarrow> Vertex"   and H :: "(Vertex \<Rightarrow> Vertex) set" assumes "x \<in> carrier tetrahedral_group"   and "subgroup H tetrahedral_group"   and "x \<in> H" shows "x <#\<^bsub>tetrahedral_group\<^esub> H = H"
RN_1: fixes M :: "'b measure"   and N :: "'b measure" shows "RN_deriv M N \<in> borel_measurable M"
n2_def: shows "n2 \<equiv> alpha2 \<or> beta2 \<or> gamma2"
emp_not_in_Cb: fixes b :: "'a list" shows "\<epsilon> \<notin> no_head_gen \<C> b"
phis_same_var: fixes g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list"   and v' :: "'val" assumes "phis g (n, v) = Some vs"   and "v' \<in> set vs" shows "var g v' = var g v"
word_sub_mono: fixes a :: "'b word"   and c :: "'b word"   and d :: "'b word"   and b :: "'b word" assumes "a \<le> c"   and "d \<le> b"   and "a - b \<le> a"   and "c - d \<le> c" shows "a - b \<le> c - d"
red_proc_pres: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and C :: "char list"   and sfs :: "sfields" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "not_init C e"   and "sh C = \<lfloor>(sfs, Processing)\<rfloor>" shows "not_init C e' \<and> (\<exists>sfs'. sh' C = \<lfloor>(sfs', Processing)\<rfloor>)"
cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) node_ptr" shows "cast ptr = object_ptr.Ext (Inl ptr)"
induct: fixes P :: "nat list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "nat list"   and a1 :: "'a list"   and a2 :: "'a list" assumes "\<And>uu_ xs. P [] uu_ xs"   and "\<And>v va xs. P (v # va) [] xs"   and "\<And>i is u us xs. P is us (xs[i := u]) \<Longrightarrow> P (i # is) (u # us) xs" shows "P a0 a1 a2"
conversepI: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e" assumes "r a b" shows "r\<inverse>\<inverse> b a"
redex_size: fixes c :: "('d, 'e, 'f) com"   and r :: "('d, 'e, 'f) com" assumes "redex c = r" shows "size r \<le> size c"
compl_inf_bot: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf (uminus x) x = bot"
gallery_betw_map: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "folding.gallery (C # Cs @ [D])" shows "folding.gallery (f ` C # f \<Turnstile> Cs @ [f ` D])"
respection_of_binary_predicates_and_closures_3: fixes Rel :: "('a \<times> 'a) set"   and Pred :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_respects_binary_pred Rel Pred" shows "rel_respects_binary_pred (Rel\<^sup>+) Pred"
circ_loop_fixpoint: fixes y :: "'a"   and z :: "'a" shows "y * (y\<^sup>\<circ> * z) \<squnion> z = y\<^sup>\<circ> * z"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> woolam"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> woolam; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evs1 A B. \<lbrakk>evs1 \<in> woolam; P evs1\<rbrakk> \<Longrightarrow> P (Says A B (Agent A) # evs1)"   and "\<And>evs2 A' B A NB. \<lbrakk>evs2 \<in> woolam; P evs2; Says A' B (Agent A) \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B A (Nonce NB) # evs2)"   and "\<And>evs3 A B B' NB. \<lbrakk>evs3 \<in> woolam; P evs3; Says A B (Agent A) \<in> set evs3; Says B' A (Nonce NB) \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says A B (Crypt (shrK A) (Nonce NB)) # evs3)"   and "\<And>evs4 A' B X A'' A. \<lbrakk>evs4 \<in> woolam; P evs4; Says A' B X \<in> set evs4; Says A'' B (Agent A) \<in> set evs4\<rbrakk> \<Longrightarrow> P (Says B Server \<lbrace>Agent A, Agent B, X\<rbrace> # evs4)"   and "\<And>evs5 B' A B NB. \<lbrakk>evs5 \<in> woolam; P evs5; Says B' Server \<lbrace>Agent A, Agent B, Crypt (shrK A) (Nonce NB)\<rbrace> \<in> set evs5\<rbrakk> \<Longrightarrow> P (Says Server B (Crypt (shrK B) \<lbrace>Agent A, Nonce NB\<rbrace>) # evs5)" shows "P x"
fix_index': fixes r :: "'b"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry"   and i :: "nat"   and j :: "nat"   and n :: "nat"   and v :: "'c \<Rightarrow> nat" assumes "Le r \<le> M i j"   and "Le (- r) \<le> M j i"   and "cycle_free M n"   and "canonical M n"   and "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "(\<forall>u. u \<turnstile>\<^bsub>v,n\<^esub> \<lambda>i' j'. if i' = i \<and> j' = j then Le r else if i' = j \<and> j' = i then Le (- r) else M i' j' \<longrightarrow> u \<turnstile>\<^bsub>v,n\<^esub> M) \<and> cycle_free (\<lambda>i' j'. if i' = i \<and> j' = j then Le r else if i' = j \<and> j' = i then Le (- r) else M i' j') n"
cs_length_g_one: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "IFC prob"   and "length (IFC_def.cs prob \<pi> i) \<noteq> 1" obtains k :: "nat" where "IFC_def.cs prob \<pi> i = IFC_def.cs prob \<pi> k @ [\<pi> i]" and "IFC_def.is_icdi prob i \<pi> k"
add_uminus_conv_diff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "plus a (uminus b) = minus a b"
1: shows "{i \<in> s. enat i < llength w} \<inter> liset A w = s \<inter> liset A w"
term_variants_pred_inv_3: fixes P :: "'a \<Rightarrow> 'a list"   and f :: "'a"   and T :: "('a, 'b) Term.term list"   and h :: "'a"   and S :: "('a, 'b) Term.term list" assumes "term_variants_pred P (Fun f T) (Fun h S)"   and "f \<noteq> h" shows "h \<in> set (P f)"
infinite: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'a measure" assumes "infinite A" shows "sup_measure.F g A = bot"
meas_M_eq: fixes \<rho> :: "nat \<Rightarrow> val" assumes "\<rho> \<in> space (dens_ctxt_measure \<Y> \<rho>)" shows "(\<rightarrow>\<^sub>M) (M \<rho>) = (\<rightarrow>\<^sub>M) (stock_measure t)"
InvMsgInGlob_def: fixes c :: "('b, 'loc \<times> 't, 'c) Exchange.configuration_scheme" shows "cri.InvMsgInGlob c = (\<forall>p q t. c_msg c p q \<noteq> [] \<longrightarrow> t \<in>#\<^sub>z hd (c_msg c p q) \<longrightarrow> (\<exists>t'. t'\<le>\<^sub>pt \<and> 0 < zcount (c_glob c q) t'))"
simps_3: fixes T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty"   and Y :: "tyvrs"   and T :: "ty" shows "(T\<^sub>1 \<rightarrow> T\<^sub>2)[Y \<mapsto> T]\<^sub>\<tau> = T\<^sub>1[Y \<mapsto> T]\<^sub>\<tau> \<rightarrow> T\<^sub>2[Y \<mapsto> T]\<^sub>\<tau>"
sscan_sdrop: fixes k :: "nat"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b stream"   and a :: "'a" shows "sdrop k (sscan f xs a) = sscan f (sdrop k xs) (fold f (stake k xs) a)"
F_axioms_5: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme"   and F :: "('d, 'e, 'g, 'h) functor" assumes "Functor F : AA \<longrightarrow> BB" shows "\<forall>A\<in>Ob\<^bsub>AA\<^esub>. F \<^bsub>\<a>\<^esub> (Id\<^bsub>AA\<^esub> A) = Id\<^bsub>BB\<^esub> F \<^bsub>\<o>\<^esub> A"
map_nth: fixes xs :: "'c list" shows "map ((!) xs) [0..<length xs] = xs"