hcju/isabelleps huggingface dataset:
funpow_Suc_right: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b" shows "f^(Suc n) = f^n \<circ> f"
Ball_set_replicate: fixes n :: "nat"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>set (a \<up> n). P x) = (P a \<or> n = 0)"
monoid_add_hom_axioms: shows "monoid_add_hom (inv hom)"
connected_component_of_maximal: fixes X :: "'a topology"   and S :: "'a set"   and x :: "'a" assumes "connectedin X S"   and "x \<in> S" shows "S \<subseteq> connected_component_of_set X x"
min_key_val_raw_in: fixes xs :: "('a \<times> 'b) list \<times> 'c"   and rep_key_order :: "'c \<Rightarrow> 'a key_order"   and ko :: "'c" assumes "fst xs \<noteq> []" shows "oalist_raw.min_key_val_raw rep_key_order ko xs \<in> set (fst xs)"
setgeD: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x <=* S"   and "y \<in> S" shows "x \<le> y"
insert_compr: fixes a :: "'b"   and B :: "'b set" shows "a \<triangleright> B = {x. x = a \<or> x \<in> B}"
bisim_invI: assumes "\<And>s1 ta1 s1' s2 t. \<lbrakk>t \<turnstile> s1 \<approx> s2; t \<turnstile> s1 -1-ta1\<rightarrow> s1'\<rbrakk> \<Longrightarrow> \<exists>s2'. t \<turnstile> s1' \<approx> s2'"   and "\<And>s2 ta2 s2' s1 t. \<lbrakk>t \<turnstile> s1 \<approx> s2; t \<turnstile> s2 -2-ta2\<rightarrow> s2'\<rbrakk> \<Longrightarrow> \<exists>s1'. t \<turnstile> s1' \<approx> s2'" shows "bisim_inv"
aff_dim_negative_iff: fixes S :: "'c set" shows "(aff_dim S < 0) = (S = {})"
simps_2: fixes U :: "complex mat"   and P :: "complex mat" shows "ps_P.wlp (Utrans U) P = adjoint U * P * U"
I_or: fixes aneg :: "'a \<Rightarrow> 'a fm"   and anormal :: "'a \<Rightarrow> bool"   and I\<^sub>a :: "'a \<Rightarrow> 'b list \<Rightarrow> bool"   and depends\<^sub>0 :: "'a \<Rightarrow> bool"   and decr :: "'a \<Rightarrow> 'a"   and \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm"   and xs :: "'b list" assumes "ATOM aneg anormal I\<^sub>a depends\<^sub>0 decr" shows "interpret I\<^sub>a (or \<phi>\<^sub>1 \<phi>\<^sub>2) xs = interpret I\<^sub>a (Or \<phi>\<^sub>1 \<phi>\<^sub>2) xs"
indRelLSTPO_impl_SRel_and_TRel_weakly_reflect_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_reflects_barbs (indRelLSTPO SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_reflects_barbs TRel TWB"
Goal: fixes \<Psi>' :: "'b"   and zvec :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<simeq> \<Psi>'"   and "\<Psi>' \<rhd> R \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> R'"   and "\<Psi>' \<otimes> \<Psi>\<^sub>R \<turnstile> M \<leftrightarrow> K"   and "\<langle>A\<^sub>Q, \<Psi>' \<otimes> \<Psi>\<^sub>R\<rangle> \<hookrightarrow>\<^sub>F \<langle>A\<^sub>P, (\<Psi> \<otimes> \<Psi>\<^sub>P) \<otimes> \<Psi>\<^sub>R\<rangle>"   and "A\<^sub>R \<sharp>* zvec"   and "A\<^sub>P \<sharp>* zvec"   and "zvec \<sharp>* R"   and "zvec \<sharp>* P"   and "A\<^sub>R \<sharp>* \<Psi>'" shows "\<exists>K'. \<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> R \<longmapsto> ROut K' (\<lparr>\<nu>*xvec\<rparr>N \<prec>' R') \<and> \<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>R \<turnstile> M \<leftrightarrow> K' \<and> zvec \<sharp>* K' \<and> A\<^sub>R \<sharp>* K'"
paths_singles': fixes xs :: "'b list"   and S :: "'b set" shows "(xs \<in> paths (singles S)) = (\<forall>x\<in>set xs \<inter> S. one_call_in_path x xs)"
invertible_right_inverse: fixes A :: "(('a, 'b) vec, 'b) vec" shows "invertible A = (\<exists>B. A ** B = mat (1::'a))"
p: shows "p \<in> \<PP> \<and> charslength p = length Doc \<and> terminals p \<in> \<L>"
inverts_components: fixes a :: "'a" assumes "A.ide a" shows "B.inverse_arrows (\<tau> a) (local.map a)"
delete_point_strong_block_subset: fixes p :: "'a" shows "str_del_point_blocks p \<subseteq># \<B>"
is_class_xcpt: fixes C :: "String.literal"   and wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "C \<in> sys_xcpts"   and "wf_prog wf_md P" shows "is_class P C"
authK_authentic: fixes A :: "agent"   and authK :: "nat"   and Peer :: "msg"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Key authK, Peer, Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> kerbIV" shows "Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Peer, Ta, authTicket\<rbrace>) \<in> set evs"
RecvMarker: shows "ev = RecvMarker i' (occurs_on ev) r"
app_unapp: fixes u :: "'a"   and u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a" assumes "unapp u = Some (u\<^sub>1, u\<^sub>2)" shows "u = app u\<^sub>1 u\<^sub>2"
fls_nth_vanishes_belowE: fixes f :: "'a fls" obtains N :: "int" where "\<forall>n<N. fls_nth f n = (0::'a)"
code: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "iterates f x = x ## iterates f (f x)"
IH_1: fixes t' :: "'a" assumes "no_abs t'"   and "convert_term t\<^sub>1 = convert_term t'" shows "t\<^sub>1 = t'"
my_f_def: shows "my_f = FDisj (FAll (FConj (NAtom 0 [0]) (NAtom 1 [0]))) (FDisj (FEx (PAtom 1 [0])) (FEx (PAtom 0 [0])))"
vcomp_atE: fixes s :: "V"   and r :: "V"   and a :: "V"   and c :: "V" assumes "(s \<circ>\<^sub>\<circ> r)\<lparr>a\<rparr> = c"   and "vsv r"   and "vsv s"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s" obtains b :: "V" where "r\<lparr>a\<rparr> = b" and "s\<lparr>b\<rparr> = c"
Abs_bset_inject: fixes x :: "'c set"   and y :: "'c set" assumes "x \<in> {A. |A| <o natLeq +c |UNIV|}"   and "y \<in> {A. |A| <o natLeq +c |UNIV|}" shows "(Abs_bset x = Abs_bset y) = (x = y)"
cod_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VV.seq g f" shows "B.VV.cod (B.VV.comp g f) = B.VV.cod g"
intro: fixes \<U> :: "'c set"   and \<A> :: "'c set multiset"   and \<V> :: "'c set"   and \<B> :: "'c set multiset" assumes "sub_set_system \<U> \<A> \<V> \<B>"   and "incidence_system \<U> \<A>" shows "sub_incidence_system \<U> \<A> \<V> \<B>"
sep_add_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x ## y"   and "y ## z"   and "x ## z" shows "x + y + z = x + (y + z)"
BC: fixes r :: "real" shows "B r \<subseteq> C r"
finite_union_of_inc: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "P S" shows "(finite union_of P) S"
wp1WhileFalse': fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and l :: "char list \<Rightarrow> nat" assumes "\<not> bval b s" shows "wp\<^sub>1 (WHILE b DO c) Q l s = Q l s"
path_subpath: fixes g :: "real \<Rightarrow> 'a"   and u :: "real"   and v :: "real" assumes "path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}" shows "path (subpath u v g)"
pj_Hom: fixes R :: "('d, 'e) Ring_scheme"   and I :: "'d set" assumes "Ring R"   and "ideal R I" shows "pj R I \<in> rHom R (R /\<^sub>r I)"
invert_side_of_triangle_1: fixes h :: "'a"   and f :: "'a"   and g :: "'a" assumes "B.arr h"   and "f \<cdot>\<^sub>B g = h"   and "B.iso f" shows "B.seq (B.inv f) h \<and> g = B.inv f \<cdot>\<^sub>B h"
case_1: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and x1 :: "'b" shows "(case Inl x1 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f1 x1"
full_rank_injective: fixes A :: "((real, 'a) vec, 'b) vec" shows "(rank A = CARD('a)) = inj ((*v) A)"
disjoint: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> P"   and "B \<in> P"   and "A \<noteq> B" shows "A \<inter> B = {}"
simps_2: fixes h :: "'c \<Rightarrow> 'c"   and x :: "'c"   and a :: "'c"   and as :: "'c list"   and y :: "'c" shows "Path h x (a # as) y = (x \<noteq> Null \<and> x = a \<and> Path h (h a) as y)"
asIMOD_igFresh: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and ys :: "'varSort"   and y :: "'var"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b" shows "igFresh (asIMOD SEM) ys y X = (\<forall>val val'. sWlsVal SEM val \<and> sWlsVal SEM val' \<and> eqBut val val' ys y \<longrightarrow> X val = X val')"
E\<^sub>i\<^sub>i\<^sub>iImpl: fixes x :: "i"   and y :: "i" shows "dom x \<cong> cod y \<^bold>\<and> E (cod y) \<^bold>\<leftarrow> E (x \<cdot> y)"
pos_caps_pos_records: fixes c :: "('b, 'loc \<times> 't) Exchange.configuration"   and p :: "'b"   and x :: "'loc \<times> 't" assumes "cri.InvCapsNonneg c"   and "0 < zcount (c_caps c p) x" shows "0 < zcount (cri.records c) x"
spec\<^sub>1_def: fixes p :: "prog" shows "spec\<^sub>1 p \<equiv> vars p = vars\<^sub>0 \<and> wfs \<Gamma>\<^sub>0 (body p) \<and> low_proc_non0\<^sub>1 p \<and> low_proc_0\<^sub>1 p \<and> high_proc\<^sub>1 p"
int_eq: fixes X :: "'a \<Rightarrow> 'b"   and Y :: "'a \<Rightarrow> 'b" assumes "\<turnstile> X = Y" shows "X = Y"
reachable_awalk: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric G)\<^esub> v) = (\<exists>p. S.awalk u p v)"
IS_PURE_def: fixes P :: "('a \<times> 'b) set \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn" shows "IS_PURE P R \<equiv> is_pure R \<and> P (the_pure R)"
Elim_2: fixes F :: "'b"   and S :: "'b set" assumes "F \<in> S" shows "{F} \<subseteq> S"
inj_power_mod: fixes n :: "nat"   and a :: "nat" assumes "coprime n a" shows "inj_on (\<lambda>k. a ^ k mod n) {..<ord n a}"
comp_univalent_below_complement: fixes x :: "'a"   and y :: "'a" assumes "univalent x" shows "x * - y \<le> - (x * y)"
flowD_outside: fixes g :: "'v \<times> 'v \<Rightarrow> ennreal"   and e :: "'v \<times> 'v" assumes "flow \<Delta>''' g"   and "e \<notin> \<^bold>E\<^bsub>\<Delta>'''\<^esub>" shows "g e = 0"
all_same_arities_def: fixes F1 :: "'d \<Rightarrow> 'e fundef option"   and F2 :: "'d \<Rightarrow> 'f fundef option" shows "all_same_arities F1 F2 \<equiv> \<forall>f. rel_option (\<lambda>fd gd. arity fd = arity gd) (F1 f) (F2 f)"
igFreshIGAbs2_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igFreshIGAbs2 MOD \<equiv> \<forall>ys y xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igFresh MOD ys y X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD xs x X)"
le_iff_add: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (\<exists>c. b = a + c)"
pos_cw_len: fixes i :: "'b" shows "0 < 1 / b ^ cw_len i"
face\<M>: fixes S :: "'a set"   and F :: "'a set" assumes "S \<in> \<M>"   and "F face_of S" shows "F \<in> \<M>"
add_neg_neg: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subset># {#}"   and "b \<subset># {#}" shows "a + b \<subset># {#}"
inres_simps_2: fixes X :: "'b set" shows "inres (RES X) = (\<lambda>x. x \<in> X)"
find_seg_impl_def: fixes SBIP :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list"   and j :: "nat" shows "GS.find_seg_impl SBIP j \<equiv> find_max_nat (length (GS.B SBIP)) (\<lambda>i. GS.B SBIP ! i \<le> j)"
setbydecision_all_def: fixes iface :: "iface"   and rs :: "'b common_primitive rule list"   and dec :: "final_decision" shows "setbydecision_all TYPE('c) iface rs dec = {ip. \<forall>p. approximating_bigstep_fun (common_matcher, in_doubt_allow) (p\<lparr>p_iiface := iface_sel iface, p_src := ip\<rparr>) rs Undecided = Decision dec}"
Cons1_1: fixes a :: "(real \<times> real \<times> real) list"   and b :: "(real \<times> real \<times> real) list"   and c :: "(real \<times> real \<times> real) list"   and d :: "(real \<times> real \<times> real) list" assumes "separateAtoms L = (a, b, c, d)" shows "(\<forall>l\<in>set L. aEvalUni l x) = ((\<forall>a\<in>set a. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c = 0) \<and> (\<forall>a\<in>set b. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c < 0) \<and> (\<forall>a\<in>set c. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<le> 0) \<and> (\<forall>a\<in>set d. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<noteq> 0))"
set_bst_of_list: fixes xs :: "'a list" shows "set_tree (bst_of_list xs) = set xs"
tj_stack_reach_hd_stack: fixes v :: "'v" assumes "v \<in> set (tj_stack s)" shows "(v, hd (stack s)) \<in> E\<^sup>*"
f3: shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') = (snd y' \<otimes> fst x' \<oplus> snd x' \<otimes> fst y' |\<^bsub>rel\<^esub> snd x' \<otimes> snd y')"
assms: fixes m :: "'a"   and n :: "'a" assumes "m \<in> M"   and "n \<in> M" shows "P m n"
gt'_total: fixes y :: "'a"   and x :: "'a" shows "gt' y x \<or> gt' x y \<or> y = x"
Rep_filter_cases: fixes y :: "('c \<Rightarrow> bool) \<Rightarrow> bool" assumes "y \<in> {F. is_filter F}" obtains x :: "'c Filter.filter" where "y = Filter.filter.Rep_filter x"
elims_2: fixes x :: "ground_action"   and xa :: "(predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option" assumes "sound_opr x xa" obtains pre :: "object atom formula" and add :: "object atom formula list" and del :: "object atom formula list" and f :: "(predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option" where "x = Ground_Action pre (Effect add del)" and "xa = f" and "(\<forall>s. close_eq s \<Turnstile> pre \<longrightarrow> (\<exists>s'. f s = Some s' \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set del \<and> close_eq s \<Turnstile> atm \<longrightarrow> close_eq s' \<Turnstile> atm) \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set add \<and> close_eq s \<Turnstile> \<^bold>\<not> atm \<longrightarrow> close_eq s' \<Turnstile> \<^bold>\<not> atm) \<and> (\<forall>fmla. fmla \<in> set add \<longrightarrow> close_eq s' \<Turnstile> fmla) \<and> (\<forall>fmla. fmla \<in> set del \<and> fmla \<notin> set add \<longrightarrow> close_eq s' \<Turnstile> \<^bold>\<not> fmla))) \<and> Ball (set add) is_predAtom"
apply_operations_Snoc: fixes xs :: "('a \<times> ('a, 'b) operation) list"   and x :: "'a \<times> ('a, 'b) operation" shows "hb.apply_operations (xs @ [x]) = hb.apply_operations xs \<rhd> interp_msg x"
parallel_converter2_eq_\<I>_cong: fixes \<I>1 :: "('h, 'i) \<I>"   and \<I>1' :: "('j, 'k) \<I>"   and conv1 :: "('h, 'i, 'j, 'k) converter"   and conv1' :: "('h, 'i, 'j, 'k) converter"   and \<I>2 :: "('l, 'm) \<I>"   and \<I>2' :: "('n, 'o) \<I>"   and conv2 :: "('l, 'm, 'n, 'o) converter"   and conv2' :: "('l, 'm, 'n, 'o) converter" assumes "\<I>1,\<I>1' \<turnstile>\<^sub>C conv1 \<sim> conv1'"   and "\<I>2,\<I>2' \<turnstile>\<^sub>C conv2 \<sim> conv2'" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2,\<I>1' \<oplus>\<^sub>\<I> \<I>2' \<turnstile>\<^sub>C conv1 |\<^sub>= conv2 \<sim> conv1' |\<^sub>= conv2'"
project_LeadsTo_D: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set" assumes "F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> A \<longmapsto>w B" shows "extend h F \<squnion> G \<in> extend_set h A \<longmapsto>w extend_set h B"
sorted_domain_def: fixes ko :: "'c nat_term_order"   and xs :: "('c, 'd) oalist_ntm" shows "OAlist_sorted_domain_ntm ko xs = sorted_domain_ko_ntm ko (list_of_oalist_ntm xs)"
pair_no_loops: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs (list_digraph G)" shows "fst e \<noteq> snd e"
int_mod_le': fixes b :: "int"   and n :: "int" assumes "0 \<le> b - n" shows "b mod n \<le> b - n"
double_SNa: fixes a :: "trm"   and b :: "trm"   and P :: "trm \<Rightarrow> trm \<Rightarrow> bool" assumes "SNa a"   and "SNa b"   and "\<forall>x z. (\<forall>y. x \<longrightarrow>\<^sub>a y \<longrightarrow> P y z) \<and> (\<forall>u. z \<longrightarrow>\<^sub>a u \<longrightarrow> P x u) \<longrightarrow> P x z" shows "P a b"
sum_sqs_eq: fixes x :: "'c"   and y :: "'c" assumes "x * x + y * y = x * (y * (2::'c))" shows "y = x"
ball_correct: fixes m :: "('a, 'b) assoc_list"   and P :: "'a \<times> 'b \<Rightarrow> bool" assumes "lm.invar m" shows "lm.ball m P = (\<forall>u v. lm.\<alpha> m u = Some v \<longrightarrow> P (u, v))"
Plus_assoc: fixes r :: "'b language"   and s :: "'b language"   and t :: "'b language" shows "Plus (Plus r s) t = Plus r (Plus s t)"
queue_invar_def: fixes q :: "('e, 'f) BinomialTree list" shows "queue_invar q \<equiv> \<forall>t\<in>set q. tree_invar t"
range_binary_eq: fixes a :: "'b"   and b :: "'b" shows "range (binary a b) = {a, b}"
p2_def: shows "p2 = Prog (compile_env F) H main"
cr_cr_vfsequence_def: fixes a :: "V"   and b :: "V list list" shows "cr_cr_vfsequence a b = (a = vfsequence_of_vlist (map vfsequence_of_vlist b))"
R_def: shows "R = {(C, C'). \<exists>i W W'. W \<approx>\<^bsub>d\<^esub> W' \<and> i < length W \<and> C = [W ! i] \<and> C' = [W' ! i]}"
relation_rules_7: fixes P :: "'c \<times> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d"   and Q :: "'d \<times> 'e \<Rightarrow> bool"   and c :: "'e" assumes "P (a, b)"   and "Q (b, c)" shows "(P ;; Q) (a, c)"
eventually_at_le: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and S :: "'b set" shows "eventually P (at a within S) = (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a \<le> d \<longrightarrow> P x)"
bounded_closed_imp_seq_compact: fixes S :: "'a set" assumes "bounded S"   and "closed S" shows "seq_compact S"
sample_bits_fusion2: fixes v :: "nat"   and f :: "nat \<Rightarrow> 'a pmf" assumes "0 < v" shows "pmf_of_set UNIV \<bind> (\<lambda>b. pmf_of_set {..<v} \<bind> (\<lambda>c. f (c + v * (if b then 1 else 0)))) = pmf_of_set {..<2 * v} \<bind> f"
unwinding_theorem_SD: fixes \<V>' :: "'e V_rec"   and ur :: "('s \<times> 's) set" assumes "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>"   and "Unwinding.lrf SES \<V>' ur"   and "Unwinding.osc SES \<V>' ur" shows "SD \<V> Tr\<^bsub>(induceES SES)\<^esub>"
S_ArrowE_left: fixes \<Gamma> :: "binding list"   and S\<^sub>1 :: "ty"   and S\<^sub>2 :: "ty"   and T :: "ty" assumes "\<Gamma>\<turnstile>S\<^sub>1 \<rightarrow> S\<^sub>2<:T" obtains "T = Top" and "\<turnstile> \<Gamma> ok" and "S\<^sub>1 \<rightarrow> S\<^sub>2 closed_in \<Gamma>"   | T\<^sub>1 :: "ty" and T\<^sub>2 :: "ty" where "T = T\<^sub>1 \<rightarrow> T\<^sub>2" and "\<Gamma>\<turnstile>T\<^sub>1<:S\<^sub>1" and "\<Gamma>\<turnstile>S\<^sub>2<:T\<^sub>2"
assms: shows "\<forall>p'. prefix p p' \<longrightarrow> augment_list s1 p' = augment_list s2 p'"
cINF_union: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset"   and B :: "'c set" assumes "A \<noteq> {}"   and "subset_mset.bdd_below (f ` A)"   and "B \<noteq> {}"   and "subset_mset.bdd_below (f ` B)" shows "Inf (f ` (A \<union> B)) = Inf (f ` A) \<inter># Inf (f ` B)"
RPDs_aux2: fixes i :: "nat"   and r :: "mregex" shows "RPDi i r \<subseteq> lfp (\<lambda>S. insert r S \<union> (insert r S \<bind> RPD))"
lincomb_insert2: fixes S :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a"   and v :: "'a vec" assumes "finite S"   and "S \<subseteq> carrier_vec n"   and "a \<in> insert v S \<rightarrow> UNIV"   and "v \<notin> S"   and "v \<in> carrier_vec n" shows "lincomb a (insert v S) = a v \<cdot>\<^sub>v v + lincomb a S"
x2_in_type: shows "(s, right x) \<in> {r. fst r \<le> snd r}"
fetch_instruction_def: fixes s :: "'b sparc_state" shows "fetch_instruction s \<equiv> let pc_val = cpu_reg_val PC s; psr_val = cpu_reg_val PSR s; s_val = get_S psr_val; asi = if s_val = 0 then word_of_int 8 else word_of_int 9 in if uint (3 AND pc_val) = 0 then let (mem_result, n_s) = memory_read asi pc_val s in case mem_result of None \<Rightarrow> Inl [fetch_instruction_error] | Some v \<Rightarrow> Inr v else Inl [fetch_instruction_error]"
R'_prop: fixes k1 :: "'a"   and v1 :: "'b"   and k2 :: "'a"   and v2 :: "'b" assumes "m k1 = Some v1"   and "P (k1, v1)"   and "m k2 = Some v2"   and "P (k2, v2)"   and "R (k1, v1) (k2, v2)" shows "R' k1 k2"
that: fixes ax :: "'edge" assumes "sourcenode a = sourcenode ax"   and "valid_edge ax"   and "\<not> n postdominates targetnode ax" shows "thesis"
IH: fixes y :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n" assumes "y < n"   and "\<And>\<pi>' n'. \<lbrakk>is_path \<pi>'; \<pi>' 0 = x; \<pi>' n' = local.return; \<forall>i>0. \<pi>' i \<noteq> x\<rbrakk> \<Longrightarrow> thesis"   and "is_path \<pi>"   and "\<pi> 0 = x"   and "x \<noteq> local.return"   and "\<pi> y = local.return" shows "thesis"
mod_mod_cancel: 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 c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "dvd.dvd times c b" shows "modulo (modulo a b) c = modulo a c"
mult_period: fixes g :: "nat \<Rightarrow> complex"   and k :: "nat"   and q :: "nat" assumes "periodic_arithmetic g k" shows "periodic_arithmetic g (k * q)"
elims_2: fixes x :: "('var, 'val, 'ret, 'pname) edge_kind list"   and xa :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "preds x xa" obtains s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = []" and "xa = s__"   | et :: "('var, 'val, 'ret, 'pname) edge_kind" and ets :: "('var, 'val, 'ret, 'pname) edge_kind list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = et # ets" and "xa = s" and "pred et s \<and> preds ets (transfer et s)"
rank_mat_mul_right: fixes A :: "'b mat"   and n :: "nat"   and d :: "nat"   and B :: "'b mat"   and nc :: "nat" assumes "A \<in> carrier_mat n d"   and "B \<in> carrier_mat d nc" shows "vec_space.rank n (A * B) \<le> vec_space.rank n A"
is_SDG_path_slice2: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node"   and n'' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'"   and "n' \<in> SDG.sum_SDG_slice2 sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n''" shows "n \<in> SDG.sum_SDG_slice2 sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n''"
freshEnv_getEnv: fixes ys :: "'a"   and y :: "'b"   and rho :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qTerm set option" assumes "freshEnv ys y rho" shows "rho ys y = None"
funs_term_type_iff: fixes t :: "('fun, 'var) Term.term"   and f :: "'fun" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "0 < arity f" shows "(f \<in> funs_term (\<Gamma> t)) = (f \<in> funs_term t \<or> (\<exists>x\<in>fv t. f \<in> funs_term (\<Gamma> (Var x))))"
set_sort: fixes xs :: "('a, 'b) Combinators list"   and l :: "'a set list" shows "set (FWNormalisationCore.sort xs l) = set xs"
ball_one_point2: fixes A :: "'b set"   and a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>A. a = x \<longrightarrow> P x) = (a \<in> A \<longrightarrow> P a)"
assms_3: shows "\<forall>i<n. qs ! i \<in> set init"
Min_eq_iff: fixes A :: "'a set"   and m :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(Min A = m) = (m \<in> A \<and> (\<forall>a\<in>A. m \<le> a))"
snth_run: fixes r :: "'c stream"   and p :: "'a" assumes "\<And>k. r !! k \<in> en (target (stake k r) p)" shows "run r p"
tc_eq: fixes x :: "'a" assumes "x \<cdot> x = x" shows "x\<^sup>\<star> \<cdot> x = x"