hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "'c \<Rightarrow> 'c list"   and x2 :: "('c, 'd) Term.term"   and x3 :: "('c, 'd) Term.term"   and P :: "('c \<Rightarrow> 'c list) \<Rightarrow> ('c, 'd) Term.term \<Rightarrow> ('c, 'd) Term.term \<Rightarrow> bool" assumes "term_variants_pred x1 x2 x3"   and "\<And>P x. P P (Var x) (Var x)"   and "\<And>T S P g f. \<lbrakk>length T = length S; \<And>i. i < length T \<Longrightarrow> term_variants_pred P (T ! i) (S ! i); \<And>i. i < length T \<Longrightarrow> P P (T ! i) (S ! i); g \<in> set (P f)\<rbrakk> \<Longrightarrow> P P (Fun f T) (Fun g S)"   and "\<And>T S P f. \<lbrakk>length T = length S; \<And>i. i < length T \<Longrightarrow> term_variants_pred P (T ! i) (S ! i); \<And>i. i < length T \<Longrightarrow> P P (T ! i) (S ! i)\<rbrakk> \<Longrightarrow> P P (Fun f T) (Fun f S)" shows "P x1 x2 x3"
upt_merge: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j \<and> j \<le> k" shows "[i..<j] @ [j..<k] = [i..<k]"
sublist_altdef': fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<le>f ys = (\<exists>ys'. ys' \<le>s ys \<and> xs \<le>p ys')"
hassoc_map_assn_hassoc_lookup: fixes m :: "'b \<Rightarrow> 'a option"   and p :: "('b \<times> 'a) list"   and k :: "'b" shows "<hassoc_map_assn m p> hassoc_lookup k p <\<lambda>r. hassoc_map_assn m p * \<up> (r = m k)>\<^sub>t"
not_iRelease_iUntil_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(\<not> (P t'. t' \<R> t I. Q t)) = (\<not> P t'. t' \<U> t I. \<not> Q t)"
left_neutral: fixes a :: "'c list" shows "[] \<cdot> a = a"
direct_decomp_map_Int_zero: fixes A :: "'c set"   and f :: "'d \<Rightarrow> 'c set"   and ss :: "'d list"   and s1 :: "'d"   and s2 :: "'d" assumes "direct_decomp A (map f ss)"   and "s1 \<in> set ss"   and "s2 \<in> set ss"   and "s1 \<noteq> s2"   and "\<And>s. s \<in> set ss \<Longrightarrow> (0::'c) \<in> f s" shows "f s1 \<inter> f s2 = {0::'c}"
rep_eq: fixes x :: "ivl" shows "\<gamma>_ivl x = \<gamma>_rep (rep_ivl x)"
tuple_since'_def: shows "tuple_since' \<equiv> Mapping.filter (\<lambda>as _. as \<in> all_tuples) tuple_since"
receiveT: fixes \<xi> :: "'e"   and l :: "'h"   and u\<^sub>m\<^sub>s\<^sub>g :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and msg :: "'f"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" shows "((\<xi>, {l}receive(u\<^sub>m\<^sub>s\<^sub>g) . p), receive msg, u\<^sub>m\<^sub>s\<^sub>g msg \<xi>, p) \<in> seqp_sos \<Gamma>"
lessThan_strict_subset_iff: fixes m :: "'a"   and n :: "'a" shows "({..<m} \<subset> {..<n}) = (m < n)"
mat_addrow_def: fixes ad :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mul :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "mat_addrow_gen ad mul a k l A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then ad (mul a (A $$ (l, j))) (A $$ (i, j)) else A $$ (i, j))"
F1_f: shows "Finca_get F1 f = Some fd1"
is_NPreflow: shows "NPreflow c s t (flow_of_cf cf)"
exec_move_raise_xcp_pcD: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and E :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and a :: "'addr" assumes "exec_move ci P t E h (stk, loc, pc, None) ta h' (stk', loc', pc', \<lfloor>a\<rfloor>)" shows "pc' = pc"
not_rt_fresher_single: fixes ip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "\<not> rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2" shows "\<not> the (rt1 ip) \<sqsubseteq> the (rt2 ip)"
perm_closed: fixes as :: "'a list"   and bs :: "'a list" assumes "as <~~> bs"   and "set as \<subseteq> carrier R" shows "set bs \<subseteq> carrier R"
alg_closed_imp_factorization: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "\<exists>A. size A = degree p \<and> p = smult (lead_coeff p) (\<Prod>x\<in>#A. [:- x, 1::'b:])"
prod3_intro: fixes A :: "'a \<times> 'b \<times> 'c"   and a :: "'a"   and b :: "'b"   and c :: "'c" assumes "fst A = a"   and "fst (snd A) = b"   and "snd (snd A) = c" shows "A = (a, b, c)"
simps_2: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and ta :: "('a, 'd, 'e) new_thread_action"   and tas :: "('a, 'd, 'e) new_thread_action list" shows "thread_oks ts (ta # tas) = (thread_ok ts ta \<and> thread_oks (redT_updT' ts ta) tas)"
Comp: fixes A :: "triple set"   and P :: "assn"   and c1 :: "stmt"   and Q :: "assn"   and c2 :: "stmt"   and R :: "assn" assumes "A \<turnstile> {P} c1 {Q}"   and "A \<turnstile> {Q} c2 {R}" shows "A \<turnstile> {P} c1;; c2 {R}"
ccpo_Sup_singleton: fixes b :: "'a"   and x :: "'a" shows "flat_lub b {x} = x"
\<W>_montone: fixes P :: "(symbol \<times> character list) list set"   and Q :: "(symbol \<times> character list) list set"   and k :: "nat" assumes "P \<subseteq> Q" shows "\<W> P k \<subseteq> \<W> Q k"
sprod_map_map: fixes f1 :: "'f \<rightarrow> 'e"   and g1 :: "'h \<rightarrow> 'g"   and f2 :: "'i \<rightarrow> 'f"   and g2 :: "'j \<rightarrow> 'h"   and p :: "'i \<otimes> 'j" assumes "f1\<cdot>\<bottom> = \<bottom>"   and "g1\<cdot>\<bottom> = \<bottom>" shows "sprod_map\<cdot>f1\<cdot>g1\<cdot>(sprod_map\<cdot>f2\<cdot>g2\<cdot>p) = sprod_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
lincomb_def: fixes a :: "'e \<Rightarrow> 'a"   and A :: "'e set" shows "W.module.lincomb a A = (\<Oplus>\<^bsub>W\<^esub>v\<in>A. a v \<odot>\<^bsub>W\<^esub> v)"
ldrop_fin_iffT: fixes t :: "'b llist"   and i :: "nat" shows "(t \<up> i \<in> UNIV\<^sup>\<star>) = (t \<in> UNIV\<^sup>\<star>)"
w\<theta>\<nu>: shows "ide w \<and> \<guillemotleft>\<theta> : f' \<star> w \<Rightarrow> f\<guillemotright> \<and> \<guillemotleft>\<nu> : g \<Rightarrow> g' \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> \<rho> = ((r \<star> \<theta>) \<cdot> \<a>[r, f', w] \<cdot> (\<rho>' \<star> w)) \<cdot> \<nu>"
composition_closed: fixes a :: "('c set \<times> 'd) set"   and b :: "('c set \<times> 'd) set" assumes "a \<in> stfx.carrier_direct_lim"   and "b \<in> stfx.carrier_direct_lim" shows "stfx.add_rel a b \<in> stfx.carrier_direct_lim"
le_fininf_not_member': fixes a :: "'action"   and u :: "'action list"   and v :: "'action stream" assumes "[a] \<preceq>\<^sub>F\<^sub>I u @- v"   and "a \<notin> set u" shows "[a] \<preceq>\<^sub>F\<^sub>I v"
successively_if_sorted_wrt: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" assumes "sorted_wrt P xs" shows "successively P xs"
hom_one: shows "h \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>G\<^esub>"
map_cong: fixes x :: "'b formula"   and ya :: "'b formula"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = ya"   and "\<And>z. z \<in> set_formula ya \<Longrightarrow> f z = g z" shows "map_formula f x = map_formula g ya"
flip_at_base_simps_3: fixes a :: "'c"   and c :: "'c"   and b :: "'c" assumes "a \<noteq> c"   and "b \<noteq> c" shows "(a \<leftrightarrow> b) \<bullet> c = c"
inj_on_inverseI: fixes A :: "'c set"   and g :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. s2p A x \<Longrightarrow> g (f x) = x" shows "inj_on f A"
of_int_integer_of: fixes k :: "integer" shows "of_int (int_of_integer k) = k"
ide_dom: fixes f :: "'t" assumes "S'.arr f" shows "S'.ide (S'.dom f)"
finite_distinct_list: fixes A :: "'c set" assumes "finite A" shows "\<exists>xs. set xs = A \<and> cycle xs"
interp_closed: fixes \<phi> :: "'aprop dfmla"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<pi>l1 :: "('state \<times> 'aprop set) stream list"   and env :: "pvar \<Rightarrow> nat"   and env1 :: "pvar \<Rightarrow> nat" assumes "wff \<phi>"   and "FV \<phi> = {}"   and "pointerOf \<pi>l = pointerOf \<pi>l1" shows "sem \<phi> env \<pi>l = sem \<phi> env1 \<pi>l1"
continuous_right_inverse_imp_quotient_map: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'b"   and U :: "'c set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "continuous_on T g"   and "g ` T \<subseteq> S"   and "\<And>y. y \<in> T \<Longrightarrow> f (g y) = y"   and "U \<subseteq> T" shows "openin (top_of_set S) (S \<inter> f -` U) = openin (top_of_set T) U"
strip2_shift2: fixes r :: "'b mrexp2"   and m :: "bool"   and c :: "'b" assumes "ok2 r" shows "strip2 (shift2 m r c) = shift m (strip2 r) c"
map_eq_append_conv: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list"   and ys :: "'e list"   and zs :: "'e list" shows "(map f xs = ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"
wlsPar_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and P :: "('b, 'c, 'd, 'a, 'f, 'e) paramS" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.wlsPar Delta P \<equiv> (\<forall>s. \<forall>X\<in>termsOfS P s. FixSyn.wls Delta s X) \<and> (\<forall>xs s. \<forall>A\<in>absOfS P (xs, s). FixSyn.wlsAbs Delta (xs, s) A) \<and> (\<forall>rho\<in>envsOfS P. FixSyn.wlsEnv Delta rho)"
conforms_gext: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option"   and s' :: "st" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "s\<le>|s'"   and "\<forall>r. ! obj:globs s' r: G,s'\<turnstile>obj\<Colon>\<preceq>\<surd>r"   and "locals s' = locals s" shows "(x, s')\<Colon>\<preceq>(G, L)"
a1: shows "\<lbrace>\<iota> p\<rbrace> x \<lbrace>\<iota> q\<rbrace>"
strict_mono_less: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'f" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "(f x < f y) = less x y"
lfind_def: fixes P :: "'b \<Rightarrow> bool"   and lst :: "'b llist" shows "lfind P lst = (LEAST i. P (lst $ i))"
hom_def: fixes G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme" shows "hom G H = {h \<in> carrier G \<rightarrow> carrier H. \<forall>x\<in>carrier G. \<forall>y\<in>carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y}"
"11_4": shows "b =\<^sub>F b\<^sub>1 @ b\<^sub>2"
E4: shows "(\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init}. \<Sum>i | i < length qs \<and> qs ! i = y. ALG x qs i (swaps (snd (Strat ! i)) (steps' init qs Strat i), ())) = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x \<noteq> y}. \<Sum>i | i < length qs \<and> qs ! i = y. ALG x qs i (swaps (snd (Strat ! i)) (steps' init qs Strat i), ()))"
Rep_vector_inject: fixes x :: "'b vector"   and y :: "'b vector" shows "(Rep_vector x = Rep_vector y) = (x = y)"
downset_set_prop_var: fixes X :: "('b \<times> 'c) set" shows "prod.downset_set X = \<Union> (prod.downset ` X)"
measurable_measure_of: fixes N :: "'b set set"   and \<Omega> :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and M :: "'c measure"   and \<mu> :: "'b set \<Rightarrow> ennreal" assumes "N \<subseteq> Pow \<Omega>"   and "f \<in> space M \<rightarrow> \<Omega>"   and "\<And>y. y \<in> N \<Longrightarrow> f -` y \<inter> space M \<in> sets M" shows "f \<in> M \<rightarrow>\<^sub>M measure_of \<Omega> N \<mu>"
cases: fixes a :: "'b set" assumes "finite a" obtains "aa__ = {}"   | A :: "'b set" and a :: "'b" where "aa__ = insert a A" and "finite A"
sum4sq_int_nat_eq: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" shows "int (sum4sq_nat a b c d) = sum4sq_int (int a, int b, int c, int d)"
that: fixes as :: "'edge list" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'"   and "n' \<notin> set (sourcenodes as)" shows "thesis"
has_unique_favorites_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "has_unique_favorites R = (\<forall>i. favorites R i = {} \<or> is_singleton (favorites R i))"
NatTransCompAssoc: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and g :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and h :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "f \<approx>>\<bullet> g"   and "g \<approx>>\<bullet> h" shows "f \<bullet> g \<bullet> h = f \<bullet> (g \<bullet> h)"
Field_card_order: fixes r :: "('c \<times> 'c) set" assumes "card_order r" shows "Field r = UNIV"
elims_1: fixes x :: "('b \<times> color) tree"   and y :: "bool" assumes "invh x = y" obtains "x = \<langle>\<rangle>" and "y"   | l :: "('b \<times> color) tree" and xa :: "'b" and c :: "color" and r :: "('b \<times> color) tree" where "x = \<langle>l, (xa, c), r\<rangle>" and "y = (bheight l = bheight r \<and> invh l \<and> invh r)"
src_dom: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "src\<^sub>D (D.dom \<mu>) = src\<^sub>D \<mu>"
P_subterm: shows "\<forall>f T t. M \<turnstile> Fun f T \<longrightarrow> P (Fun f T) \<longrightarrow> t \<in> set T \<longrightarrow> P t"
intros_4: shows "c_snd \<in> PrimRec1'"
hd_replicate: fixes n :: "nat"   and x :: "'c" assumes "n \<noteq> 0" shows "lead_coeff (replicate n x) = x"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr ((G \<circ> FG.\<epsilon>) f) = A.arr f"
carrier_qring1: fixes I :: "'a set" assumes "ideal R I" shows "carrier (R /\<^sub>r I) = set_ar_cos R I"
product_in_second_factor: fixes x :: "i"   and y :: "i" assumes "O x y" shows "P (x \<otimes> y) y"
assms: shows "\<C> \<turnstile> [C v1, C v2] : t1s _> t2s"
simps_3: fixes t :: "('c, 'd) BinomialTree"   and t' :: "('c, 'd) BinomialTree"   and bq :: "('c, 'd) BinomialTree list" shows "rank_invar (t # t' # bq) = (rank t < rank t' \<and> rank_invar (t' # bq))"
irreducible_imp_prime_elem: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.irreducible times one zero x" shows "comm_semiring_1.prime_elem times one zero x"
nat_mult_distrib_neg: fixes z :: "int"   and z' :: "int" assumes "z \<le> 0" shows "nat (z * z') = nat (- z) * nat (- z')"
D_def: shows "D = (\<lambda>i. {real i..real (Suc i)}) ` {m..<n}"
assms_2: shows "tfr\<^sub>s\<^sub>s\<^sub>t A"
simps_7: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case Lh of Lh \<Rightarrow> f1 | Bal \<Rightarrow> f2 | Rh \<Rightarrow> f3) = f1"
sum_list_map_vec_to_iarray: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> ('b, 'c) vec" assumes "xs \<noteq> []" shows "sum_list (map (vec_to_iarray \<circ> f) xs) = vec_to_iarray (sum_list (map f xs))"
append_perm_imp_perm: fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "zs @ xs <~~> zs @ ys" shows "xs <~~> ys"
nat_to_set_srj: fixes D :: "nat set" assumes "finite D" shows "nat_to_set (set_to_nat D) = D"
substInp_def: fixes xs :: "'varSort"   and X :: "('f, 'g, 'varSort, 'var, 'h) qTerm set"   and x :: "'var" shows "substInp xs X x \<equiv> psubstInp (idEnv [x \<leftarrow> X]_xs)"
1: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" shows "(hom_induced p X {} X T id \<circ> hom_induced p (subtopology X S) {} X {} id) x = hom_induced p (subtopology X S) {} X T id x"
le_INF_iff: fixes Inf :: "'a::type set \<Rightarrow> 'a::type"   and Sup :: "'a::type set \<Rightarrow> 'a::type"   and inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and sup :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and bot :: "'a::type"   and top :: "'a::type"   and u :: "'a::type"   and f :: "'b::type \<Rightarrow> 'a::type"   and A :: "'b::type set" assumes "class.complete_lattice (Inf::'a::type set \<Rightarrow> 'a::type) (Sup::'a::type set \<Rightarrow> 'a::type) (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (sup::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (bot::'a::type) (top::'a::type)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (u::'a::type) ((Inf::'a::type set \<Rightarrow> 'a::type) ((f::'b::type \<Rightarrow> 'a::type) ` (A::'b::type set))) = (\<forall>i::'b::type\<in>A. less_eq u (f i))"
\<alpha>_ide: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and h :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.ide g"   and "Span.ide h"   and "Span.src f = Span.trg g"   and "Span.src g = Span.trg h" shows "Span.assoc (fst (f, g, h)) (fst (snd (f, g, h))) (snd (snd (f, g, h))) = \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<odot>) Maps.PRJ\<^sub>0 Maps.PRJ\<^sub>1 f g h, Dom = Dom ((f \<circ> g) \<circ> h), Cod = Cod (f \<circ> g \<circ> h)\<rparr>"
r_minus: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier K"   and "y \<in> carrier K" shows "x \<otimes>\<^bsub>K\<^esub> \<ominus>\<^bsub>K\<^esub> y = \<ominus>\<^bsub>K\<^esub> (x \<otimes>\<^bsub>K\<^esub> y)"
is_processT2_TR: fixes P :: "'a process" shows "\<forall>s. s \<in> T P \<longrightarrow> front_tickFree s"
simps_15: fixes v :: "'a rexp" shows "local.le_rexp (Star v) One = False"
IsCompFun_HA: fixes HA :: "('a, 'b, 'c) hierauto" shows "IsCompFun (SAs HA) (CompFun HA)"
join_subpaths_middle: fixes p :: "real \<Rightarrow> 'b" shows "subpath 0 (1 / 2) p +++ subpath (1 / 2) 1 p = p"
return_app_return: fixes f :: "'d \<Rightarrow> ['c]"   and x :: "'d" shows "\<langle>f\<rangle> . \<langle>x\<rangle> = f x"
attach_callee_id_oracle: fixes "oracle" :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf" shows "attach_callee 1\<^sub>I oracle = \<dagger>oracle"
hom_power: fixes x :: "rat poly"   and n :: "nat" shows "map_poly of_rat (x ^ n) = map_poly of_rat x ^ n"
Done_bind: fixes a :: "'g option"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) Resumption.resumption" shows "Resumption.resumption.Done a \<bind> f = (case a of None \<Rightarrow> Resumption.resumption.Done None | Some a \<Rightarrow> f a)"
"case": fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x1 :: "'d"   and x2 :: "'e" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
completeness_game_def: fixes h :: "'pub_input"   and w :: "'witness"   and e :: "'challenge" shows "completeness_game h w e = init h w \<bind> (\<lambda>(r, a). response r w e \<bind> (\<lambda>z. return_spmf (check h a e z)))"
l1_inv4_init: shows "init l1 \<subseteq> l1_inv4"
StepI: fixes P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and Cs :: "char list list"   and i :: "instr" assumes "exec_step_input P C M pc (Called Cs) = StepI i" shows "instrs_of P C M ! pc = i \<and> Cs = []"
corec_llist_never_stop: fixes IS_LNIL :: "'d \<Rightarrow> bool"   and LHD :: "'d \<Rightarrow> 'c"   and MORE :: "'d \<Rightarrow> 'c llist"   and LTL :: "'d \<Rightarrow> 'd"   and x :: "'d" shows "corec_llist IS_LNIL LHD (\<lambda>_. False) MORE LTL x = unfold_llist IS_LNIL LHD LTL x"
vimage_vdiff: fixes r :: "V"   and A :: "V"   and B :: "V" shows "r `\<^sub>\<circ> A -\<^sub>\<circ> r `\<^sub>\<circ> B \<subseteq>\<^sub>\<circ> r `\<^sub>\<circ> (A -\<^sub>\<circ> B)"
intersect_poincare_circles_x_axis: fixes z1 :: "complex"   and z2 :: "complex"   and r1 :: "real"   and r2 :: "real"   and x1 :: "complex_homo"   and x2 :: "complex_homo" assumes "is_real z1"   and "is_real z2"   and "0 < r1"   and "0 < r2"   and "- 1 < Re z1"   and "Re z1 < 1"   and "- 1 < Re z2"   and "Re z2 < 1"   and "z1 \<noteq> z2"   and "x1 \<in> poincare_circle (of_complex z1) r1 \<inter> poincare_circle (of_complex z2) r2"   and "x2 \<in> poincare_circle (of_complex z1) r1 \<inter> poincare_circle (of_complex z2) r2"   and "x1 \<noteq> x2" shows "x1 = conjugate x2"
obtain_p_1: shows "x = Inf (PLS (fst p) (snd p))"
lift_insertion: fixes i :: "nat"   and p :: "real mpoly" shows "\<forall>init. length init = i \<longrightarrow> (\<forall>I xs. insertion (nth_default 0 (init @ xs)) p = insertion (nth_default 0 (init @ I @ xs)) (liftPoly i (length I) p))"
not_Liminf_llist_imp_exists_index: fixes Xs :: "'b set llist"   and x :: "'b"   and i :: "nat" assumes "\<not> lnull Xs"   and "x \<notin> Liminf_llist Xs"   and "enat i < llength Xs" shows "\<exists>j\<ge>i. enat j < llength Xs \<and> x \<notin> lnth Xs j"
finite_f_in: fixes A :: "'v set" shows "\<integral>\<^sup>+y\<in>A. f (y, x)\<partial>count_space UNIV \<noteq> \<top>"
hom_of_snd: fixes G :: "('b, 'c) monoid_scheme"   and f :: "'d \<Rightarrow> 'e"   and H :: "('d, 'g) monoid_scheme"   and K :: "('e, 'f) monoid_scheme" assumes "Group.group G" shows "(f \<circ> snd \<in> hom (G \<times>\<times> H) K) = (f \<in> hom H K)"
coboundedI2: fixes b :: "'d multiset"   and c :: "'d multiset"   and a :: "'d multiset" assumes "b \<subseteq># c" shows "a \<inter># b \<subseteq># c"
enum_pos'_inverse: fixes xs :: "'c list"   and x :: "'c" assumes "List.member xs x" shows "xs ! enum_pos' xs x = x"
Number: fixes n :: "nat"   and H :: "msg set" shows "Number n \<in> synth H"
sub_num_simps_4: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one (num.Bit0 k) num.One = numeral.numeral one plus (Num.BitM k)"
assms: shows "FI m (SLN * up) qs SLN (q * uq) f"
preimage_identity_self: fixes A :: "'c set"   and B :: "'c set" shows "identity A \<^sup>\<inverse> A B = B \<inter> A"
swapInp_def2: fixes xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var"   and inp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option" shows "inp %[x1 \<and> x2]_xs = asInp (qSwapInp xs x1 x2 (pickInp inp))"
oprod_congL: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set"   and t :: "('f \<times> 'f) set" assumes "r =o s"   and "Well_order t" shows "r *o t =o s *o t"
le_infI1: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and x :: "'e"   and b :: "'e" assumes "class.semilattice_inf inf less_eq less"   and "less_eq a x" shows "less_eq (inf a b) x"
Hermite_of_iarrays_def: fixes A :: "'a iarray iarray"   and ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "Hermite_of_iarrays A ass res bezout = (let A' = echelon_form_of_iarrays A bezout in Hermite_of_upt_row_i_iarrays A' (nrows_iarray A) ass res)"
lunit_char_1: fixes f :: "'a" assumes "B.ide f" shows "\<guillemotleft>B.lunit f : R f \<rightarrow>\<^sub>B f\<guillemotright>"
rel_cases: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "('a, 'c) blindable\<^sub>m"   and b :: "('b, 'd) blindable\<^sub>m" assumes "rel_blindable\<^sub>m R1 R2 a b" obtains x :: "'a" and y :: "'b" where "a = Unblinded x" and "b = Unblinded y" and "R1 x y"   | xa :: "'c blindable\<^sub>h" and ya :: "'d blindable\<^sub>h" where "a = Blinded xa" and "b = Blinded ya" and "rel_blindable\<^sub>h R2 xa ya"
rel_upd1: fixes a :: "'b"   and b :: "'b"   and r :: "'b \<Rightarrow> 'b ref"   and q :: "'b"   and t :: "'b ref" assumes "(a, b) \<notin> rel (r(q := t))"   and "(a, b) \<in> rel r" shows "a = q"
ex_countable_subbasis: fixes "open" :: "'b set \<Rightarrow> bool" assumes "class.second_countable_topology open" shows "\<exists>B. countable B \<and> open = generate_topology B"