hcju/isabelleps huggingface dataset:
take_bit_not_eq_mask_diff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "take_bit n (not a) = minus (mask n) (take_bit n a)"
M: shows "\<forall>s\<in>set (snd (Ana m)). timpl_closure_set M TI \<turnstile>\<^sub>c s"
cases: fixes x :: "('a, 'e, 'f, 'g) SLin_state_scheme \<times> 'h" obtains s1 :: "('a, 'e, 'f, 'g) SLin_state_scheme" and s2 :: "'h" where "x = (s1, s2)"
document_ptr_kinds_eq3: shows "document_ptr_kinds h = document_ptr_kinds h'"
subsetD: fixes A :: "'c set"   and B :: "'c set"   and c :: "'c" assumes "A \<subseteq> B"   and "s2p A c" shows "s2p B c"
card_mono: fixes B :: "'g set"   and A :: "'g set" assumes "finite B"   and "A \<subseteq> B" shows "card A \<le> card B"
hd_prog_app_in_first: fixes p :: "'a"   and p' :: "'a"   and "is" :: "instr list"   and xs :: "'a memref list"   and q :: "'a"   and ys :: "'a memref list" assumes "Prog\<^sub>s\<^sub>b p p' is \<in> set xs" shows "hd_prog q (xs @ ys) = hd_prog q xs"
shift1_invD_2: fixes d :: "nat"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" assumes "shift1_inv d (qs, B)" shows "shift2_inv d qs"
image_map_of_conv_Image: fixes xys :: "('a \<times> 'b) list"   and A :: "'a set" assumes "distinct (map fst xys)" shows "map_of xys ` A = Some ` set xys `` A \<union> (if A \<subseteq> fst ` set xys then {} else {None})"
preserves_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>F'G'.\<epsilon> f : F'G'.FG.map a \<rightarrow>\<^sub>B B.map b\<guillemotright>"
assms: shows "insert_ops (pre @ [(oid, Some ref)] @ suf)"
supt_imp_supteq: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "s \<rhd> t" shows "s \<unrhd> t"
passive_subset_insert: fixes Cl :: "'f \<times> 'l"   and N :: "('f \<times> 'l) set" shows "passive_subset (insert Cl N) = (if snd Cl \<noteq> active then {Cl} else {}) \<union> passive_subset N"
cont_def: fixes c :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and i :: "nat" shows "local.cont c s i = fst (snd (wt_cont_eff c s i))"
simps_5: fixes t :: "'a local.term" shows "\<lbrace>\<^bold>\<l>\<^bold>[t\<^bold>]\<rbrace> = \<ll> \<lbrace>t\<rbrace>"
ll2f_None_le: fixes t :: "'b llist"   and j :: "nat"   and i :: "nat" assumes "t !! j = None"   and "j \<le> i" shows "t !! i = None"
finite_dom_map_of: fixes l :: "('b \<times> 'c) list" shows "finite (dom (table_of l))"
imaginary_eq_real_iff: fixes y :: "complex"   and x :: "complex" assumes "y \<in> \<real>"   and "x \<in> \<real>" shows "(\<i> * y = x) = (x = 0 \<and> y = 0)"
antimonoI: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f y \<le> f x" shows "order.antimono less_eq f"
lift_cm_inv_cm_all: fixes P :: "('loc, 't) Propagate.configuration \<Rightarrow> bool"   and \<Delta> :: "('loc \<times> 't) zmultiset"   and c0 :: "('loc, 't) Propagate.configuration" assumes "\<And>c0 c1 loc t n. \<lbrakk>P c0; next_change_multiplicity' c0 c1 loc t n\<rbrakk> \<Longrightarrow> P c1"   and "\<forall>(loc, t)\<in>#\<^sub>z\<Delta>. \<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc) \<and> t' \<le> t"   and "P c0" shows "P (cm_all c0 \<Delta>)"
trms\<^sub>s\<^sub>s\<^sub>t_unlabel_dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_eq: fixes A :: "('c strand_label \<times> ('a, 'b) stateful_strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
measurable_on_def: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "f measurable_on S \<equiv> \<exists>N g. negligible N \<and> (\<forall>n. continuous_on UNIV (g n)) \<and> (\<forall>x. x \<notin> N \<longrightarrow> (\<lambda>n. g n x) \<longlonglongrightarrow> (if x \<in> S then f x else (0::'d)))"
iprev_singleton_cut_less_not_empty_iff: fixes t0 :: "nat"   and I :: "nat set" shows "({iprev t0 I} \<down>< t0 \<noteq> {}) = (I \<down>< t0 \<noteq> {} \<and> t0 \<in> I)"
curry_simp: fixes f1 :: "'a1"   and F :: "'a1 \<times> 'a2 \<Rightarrow> 'b"   and G :: "'a1 \<times> 'a2 \<Rightarrow> 'b"   and \<tau> :: "'a1 \<times> 'a2 \<Rightarrow> 'b" assumes "A1.arr f1" shows "local.curry F G \<tau> f1 = A2_B.MkArr (\<lambda>f2. F (A1.dom f1, f2)) (\<lambda>f2. G (A1.cod f1, f2)) (\<lambda>f2. \<tau> (f1, f2))"
rtlE_complete: fixes L :: "'a tree set"   and A :: "'a \<Rightarrow> nat option" assumes "L \<in> regular_languages A" obtains TA :: "(nat, 'a) tree_automaton_rec" where "L = ta_lang TA" and "complete_tree_automaton TA A"
exp_conj_idem: fixes P :: "'c \<Rightarrow> bool" shows "\<guillemotleft> P \<guillemotright> && \<guillemotleft> P \<guillemotright> = \<guillemotleft> P \<guillemotright>"
equal_mod_timpls_inv_1: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and g :: "('f, 'g, 'e) prot_fun"   and S :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" assumes "equal_mod_timpls TI (Fun f T) (Fun g S)" shows "length T = length S"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_on_left_moving_1 ires rs (l, r) = (wcode_on_left_moving_1_B ires rs (l, r) \<or> wcode_on_left_moving_1_O ires rs (l, r))"
Hoare_def: fixes p :: "'e"   and S :: "'f \<Rightarrow> 'e"   and q :: "'f" shows "\<Turnstile> p{| S |}q = (p \<sqsubseteq> S q)"
simps_5: fixes l :: "nat"   and u :: "nat" shows "asubst\<^sub>2 l u (Eq 0 0) = TrueF"
induct: fixes x1 :: "int"   and x2 :: "(int \<times> Instr) list"   and x3 :: "Expr"   and x4 :: "(int \<times> Instr) list \<times> int"   and P :: "int \<Rightarrow> (int \<times> Instr) list \<Rightarrow> Expr \<Rightarrow> (int \<times> Instr) list \<times> int \<Rightarrow> bool" assumes "(x1, x2, x3, x4) \<in> compileExpr"   and "\<And>l code p code1 l1 OUT. \<lbrakk>(l, code, p, code1, l1) \<in> compilePrim; OUT = (code1[l1\<mapsto>vreturn], l1 + 1)\<rbrakk> \<Longrightarrow> P l code (PrimE p) OUT"   and "\<And>l code p code1 l1 code2 l2 x e OUT. \<lbrakk>(l, code, p, code1, l1) \<in> compilePrim; (code2, l2) = (code1[l1\<mapsto>store x], l1 + 1); (l2, code2, e, OUT) \<in> compileExpr; P l2 code2 e OUT\<rbrakk> \<Longrightarrow> P l code (LetE x p e) OUT"   and "\<And>l code e2 codeElse XXX e1 codeThen YYY OUT x. \<lbrakk>(l + 2, code, e2, codeElse, XXX) \<in> compileExpr; P (l + 2) code e2 (codeElse, XXX); (XXX, codeElse, e1, codeThen, YYY) \<in> compileExpr; P XXX codeElse e1 (codeThen, YYY); OUT = (codeThen[l\<mapsto>load x][l + 1\<mapsto>iftrue XXX], YYY)\<rbrakk> \<Longrightarrow> P l code (CondE x e1 e2) OUT"   and "\<And>l code e2 codeCons lNil e1 codeNil lRes OUT x FALSE h t. \<lbrakk>(l + 9, code, e2, codeCons, lNil) \<in> compileExpr; P (l + 9) code e2 (codeCons, lNil); (lNil, codeCons, e1, codeNil, lRes) \<in> compileExpr; P lNil codeCons e1 (codeNil, lRes); OUT = (codeNil[l\<mapsto>load x][l + 1\<mapsto>unop (\<lambda>v. if v = RVal Nullref then TRUE else FALSE)][l + 2\<mapsto>iftrue lNil][l + 3\<mapsto>load x][l + 4\<mapsto>getfield LIST HD][l + 5\<mapsto>store h][l + 6\<mapsto>load x][l + 7\<mapsto>getfield LIST TL][l + 8\<mapsto>store t], lRes)\<rbrakk> \<Longrightarrow> P l code (MatchE x e1 h t e2) OUT" shows "P x1 x2 x3 x4"
game_sem_loop_back: fixes I :: "interp"   and \<alpha> :: "game" shows "(\<lambda>X. lfp (\<lambda>Z. X \<union> game_sem I \<alpha> Z)) = game_sem I \<alpha>**"
simps_2: fixes x :: "ref"   and h :: "ref \<Rightarrow> ref"   and y :: "ref"   and p :: "ref"   and ps :: "ref list" shows "Path x h y (p # ps) = (x = p \<and> x \<noteq> Null \<and> Path (h x) h y ps)"
optimize_matches_a_def: fixes f :: "action \<Rightarrow> 'b match_expr \<Rightarrow> 'b match_expr"   and rs :: "'b rule list" shows "optimize_matches_a f rs = map (\<lambda>r. Rule (f (get_action r) (get_match r)) (get_action r)) rs"
lfp_ordinal_induct: fixes f :: "'c \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool" assumes "mono f"   and "\<And>S. \<lbrakk>P S; S \<le> lfp f\<rbrakk> \<Longrightarrow> P (f S)"   and "\<And>M. \<forall>S\<in>M. P S \<Longrightarrow> P (Sup M)" shows "P (lfp f)"
Int_Ico: fixes a :: "'a"   and b :: "'a" shows "{a..} \<inter> {b..} = {max a b..}"
that: shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'a) \<le> len m i i xs"
predicate2D: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "P \<le> Q"   and "P x y" shows "Q x y"
inv: fixes x :: "'a" assumes "x \<in> space N" shows "f' (f x) = x"
integrable_moments: fixes k :: "nat" assumes "k \<le> n" shows "integrable M (\<lambda>x. x ^ k)"
validc_fresh: fixes a :: "coname"   and \<Delta> :: "(coname \<times> ty) list" assumes "a \<sharp> \<Delta>" shows "\<nexists>B. (a, B) \<in> set \<Delta>"
Phi_map: fixes w :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_direct_product w x"   and "is_direct_product y z" shows "is_map (\<Phi> w x y z)"
foldl_conv_foldr: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and a :: "'c"   and xs :: "'d list" shows "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
list_fast_def: fixes t :: "'b tree" shows "list_fast t = list_fast_rec [t]"
rel_reflp: fixes R1 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "reflp R1"   and "reflp R2" shows "reflp (rel_sum R1 R2)"
ipassmt_sanity_nowildcards_def: fixes ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option" shows "ipassmt_sanity_nowildcards ipassmt \<equiv> \<forall>iface\<in>dom ipassmt. \<not> iface_is_wildcard iface"
triv_phis'_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and "next" :: "'node \<times> 'val"   and triv_phis :: "('node \<times> 'val) set"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping" shows "uninst_code.triv_phis' p g next triv_phis nodes_of_phis = Set.remove next triv_phis \<union> Set.filter (\<lambda>n. uninst_code.ssa.trivial_code (snd n) (the (Mapping.lookup (p g) n))) (case Mapping.lookup nodes_of_phis (snd next) of None \<Rightarrow> {} | Some x \<Rightarrow> Set.remove next x)"
WHILE_def: shows "WHILE \<equiv> WHILE\<^bsup>\<lambda>_. True\<^esup>"
intersecting_sets_spec: fixes X :: "'b set"   and Y :: "'b set" shows "intersecting_sets X Y \<le> split_intersecting X Y"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a arrow_of_spans_data \<Rightarrow> 'a arrow_of_spans_data" assumes "natural_isomorphism (\<bullet>) (\<bullet>) R local.map \<phi>" shows "equivalence_functor (\<bullet>) (\<bullet>) R"
id_func_preserves_objects: shows "id_func AA \<^bsub>\<o>\<^esub> \<in> Ob \<rightarrow> Ob"
gcd_greatest: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c dvd a"   and "c dvd b" shows "c dvd gcd a b"
cases: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "'t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action"   and a3 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "redT a1 a2 a3" obtains t :: "'t" and x :: "'x" and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x' :: "'x" and m' :: "'m" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" where "a1 = s__" and "a2 = (tb__, ta)" and "a3 = s'__" and "tb__ \<turnstile> \<langle>xa__, shr s__\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>" and "thr s__ tb__ = \<lfloor>(xa__, no_wait_locks)\<rfloor>" and "actions_ok s__ tb__ ta" and "redT_upd s__ tb__ ta x' m' s'__"   | s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and t :: "'t" and x :: "'x" and n :: "'l" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and ln :: "'l \<Rightarrow>f nat" where "a1 = s" and "a2 = (t, K$ [], [], [], [], [], convert_RA ln)" and "a3 = s'" and "thr s t = \<lfloor>(x, ln)\<rfloor>" and "\<not> waiting (wset s t)" and "may_acquire_all (locks s) t ln" and "0 < ln $ n" and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)"
f_def: shows "f = (\<lambda>n t. except_pp t (- {x. elem_index x < n}))"
random_treap_fold: fixes t :: "('a \<times> real) tree" assumes "t \<in> space (tree_sigma (count_space A \<Otimes>\<^sub>M lborel))" shows "rinss xs t A = distr (Pi\<^sub>M (set xs) (\<lambda>x. \<U> 0 1)) (tree_sigma (count_space A \<Otimes>\<^sub>M lborel)) (\<lambda>p. foldl (\<lambda>t x. ins x (p x) t) t xs)"
one_closed: shows "\<one>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> \<in> carrier (G\<lparr>carrier := S\<rparr>)"
induct: fixes P :: "rat poly \<Rightarrow> bool"   and a0 :: "rat poly" assumes "\<And>p. (\<And>x x2. \<lbrakk>x = degree p; x \<noteq> 0; x \<noteq> 1; x \<noteq> 2; rational_root_test p = Some x2\<rbrakk> \<Longrightarrow> P (p div [:- x2, 1:])) \<Longrightarrow> P p" shows "P a0"
ba_iso_left_invertible: fixes pf :: "('c, 'd) lifted_pair regular" shows "ba_iso_inv (ba_iso pf) = pf"
locally_constant_imp_constant: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "connected S"   and "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x\<in>T. f x = f a)" shows "f constant_on S"
inf_chain_offset: fixes f :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "local.inf_chain f" shows "local.inf_chain (\<lambda>j. f (j + i))"
LFOc_refine: fixes li :: "'c list"   and l :: "'d set"   and A :: "('c \<times> 'd) set"   and R :: "('f \<times> 'e) set"   and ci :: "'f \<Rightarrow> bool"   and c :: "'e \<Rightarrow> bool"   and fi :: "'c \<Rightarrow> 'f \<Rightarrow> 'f nres"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e nres"   and s0i :: "'f"   and s0 :: "'e" assumes "(li, l) \<in> \<langle>A\<rangle>list_set_rel"   and "\<And>s si. (si, s) \<in> R \<Longrightarrow> ci si = c s"   and "\<And>x xi s si. \<lbrakk>(xi, x) \<in> A; (si, s) \<in> R\<rbrakk> \<Longrightarrow> fi xi si \<le> \<Down> R (f x s)"   and "(s0i, s0) \<in> R" shows "nfoldli li ci fi s0i \<le> \<Down> R (FOREACH\<^sub>C l c f s0)"
atMost_iff: fixes i :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a" shows "(i \<in> ord.atMost less_eq k) = less_eq i k"
inf_bot_iff_le: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y = \<bottom>) = (x \<le> - y)"
hull_inc: fixes x :: "'c"   and S :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "x \<in> S" shows "x \<in> P hull S"
nl4: fixes l' :: "real"   and r' :: "real"   and ll :: "real"   and r :: "real" assumes "1 \<le> l'"   and "1 \<le> r'"   and "1 \<le> ll"   and "1 \<le> r" shows "4 * (l' + r')\<^sup>2 * (l' + ll) * (r' + r) \<le> (l' + ll + r') * (l' + ll + r' + r) ^ 3"
par_subst_upd_left_in: fixes A :: "'e \<Longrightarrow> 'f"   and B :: "'g \<Longrightarrow> 'f"   and x :: "'h \<Longrightarrow> 'f"   and \<sigma> :: "'f \<Rightarrow> 'f"   and v :: "('h, 'f) uexpr"   and \<rho> :: "'f \<Rightarrow> 'f" assumes "vwb_lens A"   and "A \<bowtie> B"   and "x \<subseteq>\<^sub>L A" shows "\<sigma>(x \<mapsto>\<^sub>s v) [A|B]\<^sub>s \<rho> = (\<sigma> [A|B]\<^sub>s \<rho>)(x \<mapsto>\<^sub>s v)"
that: fixes C :: "'a set"   and T :: "'a set" assumes "fsigma C"   and "T \<in> null_sets lebesgue"   and "C \<union> T = S"   and "disjnt C T" shows "thesis"
const_map_notin_set: fixes x :: "'a"   and A :: "'a set"   and v :: "'b" assumes "x \<notin> A" shows "const_map v A x = None"
bnd: fixes a :: "'a"   and i :: "'i" assumes "a \<in> A"   and "i \<in> set_pmf M" shows "norm (f i a) \<le> B"
par_beta_cases_1: fixes x :: "char list"   and t :: "sterm" assumes "Fvar x => t" obtains "t = Fvar x"
minWait_le_pos: fixes rs :: "'rule stream"   and r :: "'rule"   and s :: "'state" assumes "fair rs"   and "r \<in> R"   and "enabled r s" shows "minWait rs s \<le> pos rs r"
list_update_id: fixes xs :: "'d list"   and i :: "nat" shows "xs[i := xs ! i] = xs"
swap_swapped: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" shows "(X, X #[x \<and> y]_zs) \<in> swapped"
aff_dim_eq_dim_subtract: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> affine hull S" shows "aff_dim S = int (dim ((\<lambda>x. x - a) ` S))"
exec_callUndefined: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and init :: "'d \<Rightarrow> 'd"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com"   and s :: "'d" assumes "\<Gamma> p = None" shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> \<Rightarrow> Stuck"
spanE_rep: fixes x :: "'c"   and B :: "'c set" assumes "x \<in> ideal B" obtains r :: "'c \<Rightarrow>\<^sub>0 'c" where "keys r \<subseteq> B" and "x = ideal.rep r"
abs_nat_pp_pp_inverse: fixes t :: "(nat, nat) pp" shows "rep_nat_pp (abs_nat_pp t) = t"
diagseq_seqseq: fixes k :: "nat" shows "nat.diagseq \<circ> (+) k = nat.seqseq k \<circ> (\<lambda>x. nat.fold_reduce k x (k + x))"
active_subset_insert: fixes Cl :: "'f \<times> 'l"   and N :: "('f \<times> 'l) set" shows "active_subset (insert Cl N) = (if snd Cl = active then {Cl} else {}) \<union> active_subset N"
mismatch_assn_def: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and R2 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b" shows "mismatch_assn R1 R2 x y \<equiv> R1 x y \<or>\<^sub>A R2 x y"
differentiable_ident: fixes F :: "'a filter" shows "(\<lambda>x. x) differentiable F"
ConsCons3: fixes e1 :: "'a \<times> 'a"   and e2 :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list" assumes "\<not> (e1 = (fst e, w) \<and> e2 = (w, snd e))"   and "\<not> (e1 = (snd e, w) \<and> e2 = (w, fst e))"   and "P e w (e2 # es)" shows "P e w (e1 # e2 # es)"
rk2_slp_impl_def: fixes ode_ops :: "'c \<times> ('d \<times> 'e \<times> 'f \<times> 'a \<times> 'g) \<times> 'h \<times> 'i" shows "rk2_slp_impl ode_ops = (case ode_ops of (xa, (xb, xc, xd, x, xe), xf, xg) \<Rightarrow> x)"
SDF_def: shows "SDF \<equiv> Sup (sum (\<lambda>k. norm (integral k f)) ` {d. d division_of \<Union> d})"
disc_filtr_def: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure" shows "disc_filtr M F = ((\<forall>n. subalgebra M (F n)) \<and> (\<forall>n m. n \<le> m \<longrightarrow> subalgebra (F m) (F n)))"
insertl_replicate: fixes m :: "nat"   and n :: "nat"   and x :: "'a" assumes "m \<le> n" shows "insertl m x (replicate n x) = x # replicate n x"
pr1_dup: fixes a :: "'a" assumes "ide a" shows "\<l>[a] \<cdot> (\<t>[a] \<otimes> a) \<cdot> \<d>[a] = a"
in_homI: fixes f :: "'a1 \<times> 'a2"   and a :: "'a1 \<times> 'a2"   and b :: "'a1 \<times> 'a2" assumes "\<guillemotleft>fst f : fst a \<rightarrow>\<^sub>A\<^sub>1 fst b\<guillemotright>"   and "\<guillemotleft>snd f : snd a \<rightarrow>\<^sub>A\<^sub>2 snd b\<guillemotright>" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 b\<guillemotright>"
ih_8: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state')"
objE: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "horizontal_homs V src trg"   and "horizontal_homs.obj V src a" obtains "partial_magma.ide V a" and "src a = a" and "trg a = a"
wellformed_deltas: fixes s :: "'s1"   and w :: "'a1 list" assumes "wellformed1 s"   and "w \<in> lists (set alphabet1)" shows "wellformed1 (fold delta1 w s)"
a_l: fixes l :: "nat"   and i :: "nat" assumes "l \<le> i"   and "i < length fs" shows "a i l = d l \<cdot>\<^sub>v (fs ! i + M.sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<l]))"
affine_affine_hull: fixes s :: "'b set" shows "affine (affine hull s)"
arr_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "\<F>C.arr (\<F>C.mkarr t)"
top_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and a :: "'a" assumes "class.order_top less_eq less top"   and "less_eq top a" shows "a = top"
apex_def: shows "gf.apex \<equiv> C.dom gf.leg0"
zeroS_closed: fixes G :: "'a set" assumes "Group G" shows "0 \<subseteq> G"
fps_nth_induct: fixes y :: "'c list \<Rightarrow> 'd"   and P :: "('c list \<Rightarrow> 'd) \<Rightarrow> bool" assumes "y \<in> {f. True}"   and "\<And>x. P (($) x)" shows "P y"
enum_less: fixes a :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "a \<in> t"   and "i < n" shows "(t.enum i < a) = (t.enum (Suc i) \<le> a)"
\<ff>_Suc: fixes i :: "nat"   and w :: "'b list" assumes "Suc i \<le> length w" shows "\<ff> w (Suc i) \<le> \<ff> w i + 1"
chart_last_inv_def: shows "chart_last_inv \<equiv> (id ---> Proj) chart_last_inv_nonzero"
normalize_coeff: fixes p :: "'a list" shows "local.coeff p = local.coeff (local.normalize p)"
hyps_2: shows "monomial (lookup p v) v \<in> pmdl B"
semiring_simprules_11: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "semiring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = x"
locally_sound_mode_use_def2: fixes lc :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" shows "conc.locally_sound_mode_use lc \<equiv> \<forall>lc'\<in>conc.loc_reach lc. conc.respects_own_guarantees (fst lc')"
neither: shows "\<forall>T\<ge>LEAST t. \<forall>T. gap i T \<le> gap i t. change_on_neither (block i T) (gap i T)"
double_cadjoint: fixes U :: "'a \<Rightarrow> 'b" assumes "bounded_clinear U" shows "U\<^sup>\<dagger>\<^sup>\<dagger> = U"
G_is_sd: shows "G = subdivide G' (u, v) w"
assms_16: shows "A * x \<le> sparse_row_matrix b"
fup_CCexp_eq: fixes e :: "exp"   and a :: "Arity\<^sub>\<bottom>" shows "ccExp_bot_syn a e = snd (fup\<cdot>(cCCexp e)\<cdot>a)"
sel_4: fixes x21 :: "'a process"   and x22 :: "'a process" shows "ch2Of (Choice x21 x22) = x22"
simps_31: fixes r :: "'a rexp"   and r' :: "'a rexp"   and s :: "'a rexp"   and s' :: "'a rexp" shows "local.le_rexp (Times r r') (Times s s') = (if r = s then local.le_rexp r' s' else local.le_rexp r s)"
tfst: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and P :: "'a trm"   and \<tau> :: "PreSimplyTyped.type"   and \<sigma> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> P : TPair \<tau> \<sigma>" shows "\<Gamma> \<turnstile> Fst P : \<tau>"
sign_simps_33: fixes am :: "'d"   and bm :: "'d" shows "am * bm = bm * am"