hcju/isabelleps huggingface dataset:
assms_6: shows "b \<in> hyp2 \<union> S"
5: shows "infs (degen cs) (v ||| l ## sscan (count cs) v l)"
vfsequence_induct: fixes xs :: "V"   and P :: "V \<Rightarrow> bool" assumes "vfsequence xs"   and "P []\<^sub>\<circ>"   and "\<And>xs x. \<lbrakk>vfsequence xs; P xs\<rbrakk> \<Longrightarrow> P (xs #\<^sub>\<circ> x)" shows "P xs"
scaleR_add_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "real"   and b :: "real"   and x :: "'b" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR (a + b) x = plus (scaleR a x) (scaleR b x)"
Measurement_System_eq_intro: fixes x :: "('a, 'b, 'c, 'd) Measurement_System_scheme"   and y :: "('a, 'b, 'c, 'd) Measurement_System_scheme" assumes "mag x = mag y"   and "Quantity.dim x = Quantity.dim y"   and "Measurement_System.more x = Measurement_System.more y" shows "x = y"
IH: fixes k :: "nat"   and I :: "'a \<Rightarrow> 'b option" assumes "k < length (trace_parallel_plan_sas_plus I \<pi>) - 1" shows "are_all_operators_applicable_in (trace_parallel_plan_sas_plus I \<pi> ! k) (\<pi> ! k) \<and> are_all_operator_effects_consistent (\<pi> ! k)"
supp_empty: fixes f :: "'a \<Rightarrow> real"   and x :: "'a" assumes "supp f = {}" shows "f x = 0"
msg_sender_def: fixes m :: "msg" shows "msg_sender m \<equiv> case m of Rreq x xa xb xc xd xe xf ipc \<Rightarrow> ipc | Rrep x xa xb xc ipc \<Rightarrow> ipc | Rerr x ipc \<Rightarrow> ipc | Pkt x xa ipc \<Rightarrow> ipc"
L1_indicator: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets M" shows "eNorm (\<LL> 1 M) (indicat_real A) = emeasure M A"
prime_def: fixes p :: "'a" shows "prime p = (prime_elem p \<and> normalize p = p)"
mod7_lm: fixes n :: "nat" shows "n mod 7 = 0 \<or> n mod 7 = 1 \<or> n mod 7 = 2 \<or> n mod 7 = 3 \<or> n mod 7 = 4 \<or> n mod 7 = 5 \<or> n mod 7 = 6"
get_attribute_def: shows "get_attribute \<equiv> l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_attribute"
incoming'_def: fixes k :: "nat set" shows "cf.incoming' k \<equiv> {(u, v) |u v. u \<notin> k \<and> v \<in> k \<and> (u, v) \<in> cf.E}"
nth_drop_0: fixes ss :: "'c list" assumes "0 < length ss" shows "ss ! 0 # drop (Suc 0) ss = ss"
WT_gpv_inline: fixes \<I> :: "('b, 'c) \<I>"   and \<I>' :: "('d, 'e) \<I>"   and callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a, 'd, 'e) gpv"   and gpv :: "('f, 'b, 'c) gpv"   and s :: "'a" assumes "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> UNIV"   and "\<And>x s. x \<in> outs_\<I> \<I> \<Longrightarrow> \<I>' \<turnstile>g callee s x \<surd>"   and "\<I> \<turnstile>g gpv \<surd>" shows "\<I>' \<turnstile>g inline callee gpv s \<surd>"
finite_cone_iff_cone_list: fixes Vs :: "'a vec set"   and Vsl :: "'a vec list" assumes "Vs \<subseteq> carrier_vec m"   and "Vs = set Vsl" shows "m.finite_cone Vs = m.cone_list Vsl"
utp_order_carrier: fixes H :: "'a hrel \<Rightarrow> 'a hrel" shows "carrier (utp_order H) = \<lbrakk>H\<rbrakk>\<^sub>H"
False: shows "\<not> (\<forall>n>0. (deriv ^^ n) f \<xi> = 0)"
a_coset_add_zero: fixes G :: "('e, 'f) ring_scheme"   and M :: "'e set" assumes "abelian_group G"   and "M \<subseteq> carrier G" shows "M +>\<^bsub>G\<^esub> \<zero>\<^bsub>G\<^esub> = M"
iteratei_abs_simple_rule_insert_P: assumes "set_iterator_abs_genord \<alpha> invar it S0 (\<lambda>_ _. True)"   and "\<And>xc. \<lbrakk>invar xc; \<alpha> xc \<in> S0\<rbrakk> \<Longrightarrow> f xc = f' (\<alpha> xc)"   and "I {} \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S0 - S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S0 - S - {x}. True; \<forall>y\<in>S. True\<rbrakk> \<Longrightarrow> I (insert x S) (f' x \<sigma>)"   and "\<And>\<sigma>. I S0 \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> S0; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S0 - S. \<forall>y\<in>S. True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
\<Z>_sgrp_basis_def: fixes \<alpha> :: "V"   and \<SS> :: "V" shows "\<Z>_sgrp_basis \<alpha> \<SS> \<equiv> \<Z>_vfsequence \<alpha> \<SS> \<and> binop (\<SS>\<lparr>\<A>\<rparr>) (\<SS>\<lparr>mbinop\<rparr>) \<and> \<Z>_sgrp_basis_axioms \<SS>"
g_def: fixes x :: "'a" shows "g x = (SOME y. y \<in> f x)"
setdist1D: fixes a :: "'a"   and b :: "'b" assumes "setdist {a} (- U) *\<^sub>R b = One" shows "setdist {a} (- U) \<noteq> 0"
a_get_disconnected_nodes_locs_def: fixes document_ptr :: "(_) document_ptr" shows "local.a_get_disconnected_nodes_locs document_ptr = {preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes)}"
A: shows "A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier R \<and> lincomb a A = \<zero>\<^bsub>M\<^esub> \<and> v \<in> A \<and> a v \<noteq> \<zero>"
loc_reach_mem_diff_eq: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and mds :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and mem :: "'a \<Rightarrow> 'b"   and c :: "'c" assumes "sifum_security_init dma \<C>_vars eval"   and "\<forall>x. var_asm_not_written mds x \<longrightarrow> mem' x = mem x \<and> dma mem' x = dma mem x" shows "sifum_security_init.loc_reach dma eval ((c, mds), mem) = sifum_security_init.loc_reach dma eval ((c, mds), mem')"
lemma_Infinitesimal_hyperpow: fixes x :: "real star"   and N :: "nat star" assumes "x \<in> Infinitesimal"   and "0 < N" shows "\<bar>x pow N\<bar> \<le> \<bar>x\<bar>"
simps: fixes a1 :: "'b nmultiset"   and a2 :: "'b nmultiset" shows "((a1, a2) \<in> sub_nmset) = (\<exists>X M. a1 = X \<and> a2 = MSet M \<and> X \<in># M)"
typing_elims_1: fixes e :: "nat \<Rightarrow> LambdaType.type"   and i :: "nat"   and T :: "LambdaType.type" assumes "e \<turnstile> dB.Var i : T" obtains "e i = T"
replicate_empty: fixes n :: "nat"   and x :: "'a" shows "(x \<up> n = []) = (n = 0)"
simps_1: fixes r :: "rat" shows "floor_2 (Rational r) = \<lfloor>r\<rfloor>"
f_ind: fixes anything :: "nat list" shows "{\<lambda>nl. nl = xs @ r # 0 \<up> (fft - far) @ anything} fap {\<lambda>nl. nl = xs @ r # 0 # 0 \<up> (fft - Suc far) @ anything}"
lexLessState1_def: fixes Vbl :: "nat set" shows "lexLessState1 Vbl = {(state1, state2). getSATFlag state1 = UNDEF \<and> getSATFlag state2 = UNDEF \<and> (getM state1, getM state2) \<in> lexLessRestricted Vbl}"
in_measure_of: fixes M :: "'c set set"   and \<Omega> :: "'c set"   and A :: "'c set"   and \<mu> :: "'c set \<Rightarrow> ennreal" assumes "M \<subseteq> Pow \<Omega>"   and "A \<in> M" shows "A \<in> sets (measure_of \<Omega> M \<mu>)"
mahler_measure_l2norm: fixes f :: "int poly" shows "mahler_measure f \<le> sqrt (real_of_int \<parallel>f\<parallel>\<^sup>2)"
infinite: fixes A :: "'g set" assumes "infinite A" shows "sorted_list_of_set A = []"
cases: fixes a1 :: "char list \<Rightarrow> com option"   and a2 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and a3 :: "(com \<times> (char list \<Rightarrow> int \<Rightarrow> int)) option" assumes "small_steps a1 a2 a3" obtains \<pi> :: "char list \<Rightarrow> com option" and cs :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)" where "a1 = \<pi>'__" and "a2 = cs__" and "a3 = Some cs__"   | \<pi> :: "char list \<Rightarrow> com option" and cs :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)" where "a1 = \<pi>''__" and "a2 = csa__" and "a3 = None" and "small_step \<pi>''__ csa__ = None"   | \<pi> :: "char list \<Rightarrow> com option" and cs :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)" and cs1 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)" and cs2 :: "(com \<times> (char list \<Rightarrow> int \<Rightarrow> int)) option" where "a1 = \<pi>" and "a2 = cs" and "a3 = cs2" and "small_step \<pi> cs = Some cs1" and "small_steps \<pi> cs1 cs2"
bilinear_conv_bounded_bilinear: fixes h :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" shows "bilinear h = bounded_bilinear h"
inverse_inverse_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse (inverse a) = a"
prefix_split_trace: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and t' :: "('a, 'b, 'c) event list" assumes "\<exists>c'. trace c t c'"   and "prefix t' t" shows "\<exists>c''. trace c t' c''"
B: shows "pre_well_ordered_set B (\<sqsubseteq>)"
map_le_produces_same_raw_match': fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and lesubs :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match' T U subs = Some subs'"   and "dom subs \<subseteq> tvsT T"   and "lesubs \<subseteq>\<^sub>m subs" shows "raw_match' T U lesubs = Some subs'"
ide_dom: fixes f :: "'j" assumes "J.arr f" shows "J.ide (J.dom f)"
inv: shows "TS_inv' ([a, b], h) a [a, b]"
fun_upd_eq_iff: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'b"   and g :: "'a \<Rightarrow> 'b"   and b' :: "'b"   and d :: "'b" shows "(f(a := b) = g(a := b')) = (b = b' \<and> f(a := d) = g(a := d))"
ih: assumes "set_mset N \<subseteq> A"   and "set_mset M \<subseteq> A" shows "\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = N - X + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> R k a))"
ltln_to_ltlc_semantics: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>c ltln_to_ltlc \<phi> = w \<Turnstile>\<^sub>n \<phi>"
gfp_fusion_var: fixes F :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c"   and x :: "'b" assumes "mono F"   and "mono g"   and "\<forall>h. F h x = g (h x)" shows "gfp F x = gfp g"
lunit_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bicategory V H \<a> \<i> src trg" shows "bicategory.lunit V H \<a> \<i> src trg a \<equiv> prebicategory_with_homs_and_units.lunit V H \<a> (bicategory.\<i>' V H \<i> src) trg a"
LeftDerives1_implies_Derives1: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "LeftDerives1 a i r b" shows "Derives1 a i r b"
simps_6: fixes f :: "form" shows "fv (FEx f) = preSuc (fv f)"
prems_2: shows "read_only_unowned \<S> (t # ts)"
directed: fixes i :: "'a"   and j :: "'a" assumes "i \<in> I"   and "j \<in> I" shows "\<exists>k\<in>I. f k + g k \<le> f i + g j"
zero_component: shows "\<exists>x. snd (rep_nat_term x) = 0"
num_params_approx_tm: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and env :: "taylor_model list"   and t :: "taylor_model" assumes "approx_tm prec ord I a f env = Some t"   and "\<And>tm. tm \<in> set env \<Longrightarrow> num_params (tm_poly tm) \<le> length I" shows "num_params (tm_poly t) \<le> length I"
Qring_fix1: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c"   and I :: "'c set" assumes "Ring R"   and "a \<in> carrier R"   and "ideal R I"   and "a \<in> I" shows "a \<uplus>\<^bsub>R\<^esub> I = I"
L: shows "Chn ((\<mu> \<star> \<lparr>Chn = \<nu>\<pi>\<rho>.chine_assoc, Dom = Dom ((\<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<nu> \<star> \<pi> \<star> \<rho>)\<rparr>) \<bullet> \<lparr>Chn = \<mu>_\<nu>\<pi>_\<rho>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<mu> \<star> (\<nu> \<star> \<pi>) \<star> \<rho>)\<rparr> \<bullet> (\<lparr>Chn = \<mu>\<nu>\<pi>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu>) \<star> \<pi>), Cod = Cod (\<mu> \<star> \<nu> \<star> \<pi>)\<rparr> \<star> \<rho>)) = chine_hcomp \<mu> \<lparr>Chn = \<nu>\<pi>\<rho>.chine_assoc, Dom = Dom ((\<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<nu> \<star> \<pi> \<star> \<rho>)\<rparr> \<cdot> \<mu>_\<nu>\<pi>_\<rho>.chine_assoc \<cdot> chine_hcomp \<lparr>Chn = \<mu>\<nu>\<pi>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu>) \<star> \<pi>), Cod = Cod (\<mu> \<star> \<nu> \<star> \<pi>)\<rparr> \<rho>"
cmp_simps_1: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.arr (\<Phi> (f, g))"
lfinite_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" shows "lfinite (lappend xs ys) = (lfinite xs \<and> lfinite ys)"
l_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" shows "l_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs to_tree_order known_ptrs get_parent get_parent_locs heap_is_wellformed parent_child_rel get_disconnected_nodes \<equiv> (l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs to_tree_order \<and> l_get_parent type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs) \<and> l_get_parent_wf type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_parent \<and> l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"
star_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "top \<le> top"
floor_uminus_of_int: fixes z :: "int" shows "\<lfloor>- of_int z\<rfloor> = - z"
mset_heap_insert: fixes x :: "'b"   and t :: "'b tree list" shows "mset_heap (insert x t) = {#x#} + mset_heap t"
simps: fixes I :: "'b set"   and v :: "nat \<Rightarrow> rat"   and cs :: "('b \<times> constraint) set" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s cs = v \<Turnstile>\<^sub>c\<^sub>s (snd ` (cs \<inter> I \<times> UNIV))"
FAssThrow2: fixes v :: "val"   and F :: "char list"   and D :: "char list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>Val v\<bullet>F{D} := throw e,s\<rangle> \<rightarrow> \<langle>throw e,s\<rangle>"
ennreal_suminf_neq_top: fixes f :: "nat \<Rightarrow> real" assumes "summable f"   and "\<And>i. 0 \<le> f i" shows "(\<Sum>i. ennreal (f i)) \<noteq> top"
upper_bound_right: fixes y :: "'a"   and x :: "'a" shows "- y \<le> - x \<squnion> - y"
elims_2: fixes x :: "cell list"   and xa :: "nat"   and xb :: "cell list \<times> cell list" assumes "wcode_erase1 x xa xb" obtains ires :: "cell list" and rs :: "nat" and l :: "cell list" and r :: "cell list" where "x = ires" and "xa = rs" and "xb = (l, r)" and "\<exists>ln rn. l = Oc # ires \<and> tl r = Bk \<up> ln @ Bk # Bk # Oc \<up> Suc rs @ Bk \<up> rn"
carrier_polynomial: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "polynomial K p" shows "polynomial (carrier R) p"
simps_13: fixes user_reg' :: "('b word \<Rightarrow> word5 \<Rightarrow> word32) \<Rightarrow> 'b word \<Rightarrow> word5 \<Rightarrow> word32"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "user_reg_update user_reg' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg' user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
polyn_expr_restrict: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "pol_coeff S (Suc n, f)" shows "polyn_expr R X n (Suc n, f) = polyn_expr R X n (n, f)"
SN_rel_ext_map: fixes P' :: "('a \<times> 'a) set"   and Pw' :: "('a \<times> 'a) set"   and R' :: "('a \<times> 'a) set"   and Rw' :: "('a \<times> 'a) set"   and M' :: "'a \<Rightarrow> bool"   and M :: "'a \<Rightarrow> bool"   and P :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'a"   and I :: "'a \<Rightarrow> bool"   and Pw :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and Rw :: "('a \<times> 'a) set" assumes "SN_rel_ext P' Pw' R' Rw' M'"   and "\<And>s t. \<lbrakk>M s; M t; (s, t) \<in> P\<rbrakk> \<Longrightarrow> (f s, f t) \<in> ((P' \<union> Pw' \<union> R' \<union> Rw') \<inter> {(s, t). M' t})\<^sup>* O (P' \<inter> {(s, t). M' t}) O ((P' \<union> Pw' \<union> R' \<union> Rw') \<inter> {(s, t). M' t})\<^sup>* \<and> I t"   and "\<And>s t. \<lbrakk>M s; M t; (s, t) \<in> Pw\<rbrakk> \<Longrightarrow> (f s, f t) \<in> ((P' \<union> Pw' \<union> R' \<union> Rw') \<inter> {(s, t). M' t})\<^sup>* O ((P' \<union> Pw') \<inter> {(s, t). M' t}) O ((P' \<union> Pw' \<union> R' \<union> Rw') \<inter> {(s, t). M' t})\<^sup>* \<and> I t"   and "\<And>s t. \<lbrakk>I s; M s; M t; (s, t) \<in> R\<rbrakk> \<Longrightarrow> (f s, f t) \<in> ((P' \<union> Pw' \<union> R' \<union> Rw') \<inter> {(s, t). M' t})\<^sup>* O ((P' \<union> R') \<inter> {(s, t). M' t}) O ((P' \<union> Pw' \<union> R' \<union> Rw') \<inter> {(s, t). M' t})\<^sup>* \<and> I t"   and "\<And>s t. \<lbrakk>I s; M s; M t; (s, t) \<in> Rw\<rbrakk> \<Longrightarrow> (f s, f t) \<in> ((P' \<union> Pw' \<union> R' \<union> Rw') \<inter> {(s, t). M' t})\<^sup>* \<and> I t" shows "SN_rel_ext P Pw R Rw M"
unit_simps_5: fixes a :: "'a" assumes "C.obj a" shows "C.cod \<i>\<^sub>C[a] = a"
that: fixes g :: "complex \<Rightarrow> complex" assumes "\<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> g field_differentiable at x" shows "thesis"
strongly_connected_def: fixes G :: "('c, 'd) pre_digraph" shows "strongly_connected G \<equiv> verts G \<noteq> {} \<and> (\<forall>u\<in>verts G. \<forall>v\<in>verts G. u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v)"
symrun_interp_left_idem: fixes \<Gamma>\<^sub>1 :: "'a constr list"   and \<Gamma>\<^sub>2 :: "'a constr list" shows "\<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
1: shows "\<guillemotleft>f \<cdot> mkPoint (local.dom f) t : unity \<rightarrow> cod f\<guillemotright>"
cdens_ctxt_invar_insert_bool: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and b :: "expr"   and f :: "cexpr"   and \<delta> :: "cexpr"   and v :: "bool" assumes "dens_ctxt_\<alpha> ([], vs @ vs', \<Gamma>, CReal 1) \<turnstile>\<^sub>d b \<Rightarrow> (\<lambda>\<rho> x. ennreal (eval_cexpr f \<rho> x))"   and "is_density_expr ([], vs @ vs', \<Gamma>, CReal 1) BOOL f"   and "\<Gamma> \<turnstile> b : BOOL"   and "free_vars b \<subseteq> set vs \<union> set vs'"   and "cdens_ctxt_invar vs vs' \<Gamma> \<delta>" shows "cdens_ctxt_invar vs vs' \<Gamma> (\<delta> *\<^sub>c cexpr_subst_val f (BoolVal v))"
is_valid_operator_sas_plus_then_range_of_sas_plus_op_is_set_range_of_op: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c, 'd) sas_plus_operator"   and v :: "'c"   and a :: "'d" assumes "is_valid_operator_sas_plus \<Psi> op"   and "(v, a) \<in> set (sas_plus_operator.precondition_of op) \<or> (v, a) \<in> set (effect_of op)" shows "\<R>\<^sub>+ \<Psi> v = set (the (range_of \<Psi> v))"
takeWhile_eq_all_conv: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "(takeWhile P xs = xs) = (\<forall>x\<in>set xs. P x)"
gseq_iff: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "((f, g) \<in> gseq) = (f \<in> SEQ trees \<and> g \<in> SEQ trees \<and> (\<exists>i. size (g i) < size (f i) \<and> (\<forall>j<i. f j = g j)))"
update_PSR_icc_def: fixes n_val :: "word1"   and z_val :: "word1"   and v_val :: "word1"   and c_val :: "word1"   and psr_val :: "word32" shows "update_PSR_icc n_val z_val v_val c_val psr_val \<equiv> let n_val_32 = if n_val = 0 then 0 else 8388608; z_val_32 = if z_val = 0 then 0 else 4194304; v_val_32 = if v_val = 0 then 0 else 2097152; c_val_32 = if c_val = 0 then 0 else 1048576; tmp0 = psr_val AND 4279238655; tmp1 = tmp0 OR n_val_32; tmp2 = tmp1 OR z_val_32; tmp3 = tmp2 OR v_val_32; tmp4 = tmp3 OR c_val_32 in tmp4"
simps_2: fixes l :: "'b btree"   and a :: "'b"   and r :: "'b btree" shows "height_up\<^sub>i (Up\<^sub>i l a r) = max (height l) (height r)"
\<alpha>_\<beta>': shows "LeftDerives1 \<alpha> (i - length \<delta>) r \<beta>'"
assms_2: fixes i :: "nat" shows "graph_homomorphism (S i) C (g i)"
nlt: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "\<not> Tarski_neutral_dimensionless.Lt Bet Cong A B A B"
similar'_down: fixes n :: "nat"   and d :: "Value"   and e :: "CValue" assumes "d \<triangleleft>\<triangleright>\<^bsub>Suc n\<^esub> e" shows "\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>d \<triangleleft>\<triangleright>\<^bsub>n\<^esub> \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>e"
basis_null_space_iarrays_def: fixes A :: "'b iarray iarray" shows "basis_null_space_iarrays A = (let GJ = Gauss_Jordan_iarrays_PA (transpose_iarray A); rank_A = length (filter (\<lambda>x. \<not> is_zero_iarray x) (IArray.list_of (snd GJ))) in set (map (\<lambda>i. row_iarray i (fst GJ)) [rank_A..<ncols_iarray A]))"
filter_IFS_no_violations_def: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and Es :: "('b \<times> 'b) list" shows "TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations G M Es = TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations_accu G M [] Es"
simps_4: fixes uu :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and v :: "'b" shows "option_assn uu None (Some v) = false"
jumpF_poly_noroot_1: fixes p :: "real poly"   and a :: "real"   and q :: "real poly" assumes "poly p a \<noteq> 0" shows "jumpF_polyL q p a = 0"