hcju/isabelleps huggingface dataset:
limit_is_suffix: fixes x :: "nat \<Rightarrow> 'b" assumes "finite (range x)" shows "\<exists>k. limit x = range (suffix k x)"
oVeblen_oSuc: fixes x :: "ordinal" shows "oVeblen F (oSuc x) = oDeriv (oVeblen F x)"
add_commute: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "dbm_add a b = dbm_add b a"
entail_union: fixes Bot :: "'b set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and N :: "'b set"   and N1 :: "'b set"   and N2 :: "'b set" assumes "consequence_relation Bot entails" shows "(entails N N1 \<and> entails N N2) = entails N (N1 \<union> N2)"
ex_in_conv: fixes A :: "'e set" shows "(\<exists>x. x \<in> A) = (A \<noteq> {})"
simps: fixes order :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b list"   and a2 :: "'b list" shows "lexp order a1 a2 = ((\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> order x y \<and> length xs = length ys) \<or> (\<exists>xs ys x. a1 = x # xs \<and> a2 = x # ys \<and> lexp order xs ys))"
STL4E: fixes F :: "'a \<Rightarrow> bool"   and sigma :: "behavior"   and G :: "'a \<Rightarrow> bool" assumes "sigma \<Turnstile> \<box>F"   and "\<turnstile> F \<longrightarrow> G" shows "sigma \<Turnstile> \<box>G"
complex_poly_decompose_multiset: fixes p :: "complex poly" shows "smult (lead_coeff p) (\<Prod>x\<in>#poly_roots_mset p. [:- x, 1:]) = p"
induct: fixes P :: "num \<Rightarrow> num \<Rightarrow> bool"   and a0 :: "num"   and a1 :: "num" assumes "P num.One num.One"   and "\<And>n. P num.One (num.Bit0 n)"   and "\<And>n. P num.One (num.Bit1 n)"   and "\<And>m. P (num.Bit0 m) num.One"   and "\<And>m n. P m n \<Longrightarrow> P (num.Bit0 m) (num.Bit0 n)"   and "\<And>m n. P m n \<Longrightarrow> P (num.Bit0 m) (num.Bit1 n)"   and "\<And>m. P (num.Bit1 m) num.One"   and "\<And>m n. P m n \<Longrightarrow> P (num.Bit1 m) (num.Bit0 n)"   and "\<And>m n. P m n \<Longrightarrow> P (num.Bit1 m) (num.Bit1 n)" shows "P a0 a1"
f: shows "f = minimal_antichain (\<Union>loc'. set_zmset (a loc'))"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D I\<^sub>D.map I\<^sub>D.cmp E.GF.map E.GF.cmp \<eta>\<^sub>0 \<eta>\<^sub>1"
asubg_subset1: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set"   and x :: "'c" assumes "aGroup A"   and "A +> H"   and "x \<in> H" shows "x \<in> carrier A"
osp__ntsp: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B C OSP P Q" shows "\<not> A B CTSP P Q"
inject_1: fixes x11 :: "bool"   and x12 :: "nat"   and x13 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and x14 :: "nat"   and x15 :: "nat set"   and x16 :: "nat set"   and x17 :: "nat set"   and x18 :: "nat set"   and y11 :: "bool"   and y12 :: "nat"   and y13 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and y14 :: "nat"   and y15 :: "nat set"   and y16 :: "nat set"   and y17 :: "nat set"   and y18 :: "nat set" shows "(Write\<^sub>s\<^sub>b x11 x12 x13 x14 x15 x16 x17 x18 = Write\<^sub>s\<^sub>b y11 y12 y13 y14 y15 y16 y17 y18) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13 \<and> x14 = y14 \<and> x15 = y15 \<and> x16 = y16 \<and> x17 = y17 \<and> x18 = y18)"
pathin_compose: fixes X :: "'b topology"   and g :: "real \<Rightarrow> 'b"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "pathin X g"   and "continuous_map X Y f" shows "pathin Y (f \<circ> g)"
numeral_le_one_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (numeral.numeral one plus n) one = (n \<le> num.One)"
fsafeG: shows "fsafe (foldr (&&) \<Gamma> TT)"
divide_non_zero: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "x \<noteq> 0" shows "(x * y = z) = (y = z / x)"
atLeast_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c" shows "ord.atLeast less_eq l \<equiv> {x. less_eq l x}"
assms: shows "a \<in>\<^sub>\<circ> xs #\<^sub>\<circ> x"
assms_3: shows "j < length (global (s_inst \<S> ! i))"
sorted_wrt_concat_map_map: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'b" assumes "sorted_wrt Q xs"   and "sorted_wrt Q ys"   and "\<And>a x y. Q x y \<Longrightarrow> P (f x a) (f y a)"   and "\<And>x y u v. \<lbrakk>x \<in> set xs; y \<in> set xs; Q u v\<rbrakk> \<Longrightarrow> P (f x u) (f y v)" shows "sorted_wrt P (concat (map (\<lambda>y. map (\<lambda>x. f x y) xs) ys))"
ord_term_strict_canc_left: fixes t :: "'a"   and v :: "'t"   and s :: "'a" assumes "t \<oplus> v \<prec>\<^sub>t s \<oplus> v" shows "t \<prec> s"
inres_simps_3: fixes x :: "'e" shows "inres (RETURN x) = (=) x"
EndPhase2_HInv3_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase2 s s' p"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
Inf_atLeast: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (ord.atLeast less_eq x) = x"
bot: fixes s :: "val option" shows "\<exists>rs. s = \<bottom> \<star> rs"
add_ac_2: fixes a :: "'f"   and b :: "'f" shows "a + b = b + a"
lead_coeff_minus: fixes p :: "'a poly" shows "lead_coeff (- p) = - lead_coeff p"
subst_all_empty: fixes \<theta> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "{} \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta> = {}"
rGamma_Weierstrass_complex: fixes z :: "complex" shows "rGamma_series_Weierstrass z \<longlonglongrightarrow> rGamma z"
flow_def: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open T f X" shows "ll_on_open.flow T f X t0 x0 = (\<lambda>t. if t \<in> ll_on_open.existence_ivl T f X t0 x0 then ll_on_open.csol T f X t0 x0 t t else (0::'b))"
single_MP_oalist: fixes k :: "'b"   and v :: "'c" shows "monomial v k = distr\<^sub>0 nat_term_order_of_le [(k, v)]"
no_smaller_prerec_violation: shows "\<forall>j\<in>{Max {i. postrecording_event t i \<and> 0 < card {j \<in> {i + 1..<length t}. prerecording_event t j}} + 1..<length t}. j < Min {j \<in> {Max {i. postrecording_event t i \<and> 0 < card {j \<in> {i + 1..<length t}. prerecording_event t j}} + 1..<length t}. prerecording_event t j} \<longrightarrow> \<not> prerecording_event t j"
map_def: fixes f :: "'a" shows "\<ll>.map f = (if S.arr f then \<l>[S.cod f] \<cdot>\<^sub>S L f else S.null)"
assms_1: shows "preprocess_part_2 as t = (t', tv)"
hyps_11: shows "x = (SOME x'. x' \<in> U - V0)"
Anno_code_def: fixes P :: "'b J_prog" shows "Anno_code P = Anno (is_lub_sup P) P"
IH: assumes "a \<in> {0..<m}" shows "gen_lucas_lehmer_sequence' m a n = gen_lucas_lehmer_sequence a n mod m"
meas: shows "set_borel_measurable borel (einterval A B \<union> einterval B C) f"
mustT_If_L: fixes tst :: "'test"   and s :: "'state"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "tval tst s"   and "mustT (if tst then c1 else c2) s" shows "mustT c1 s"
path_conforms_with_strategy: shows "lset P \<subseteq> VV p**"
split_minNoneN0: fixes t :: "'c bro"   and h :: "nat" assumes "t \<in> B h"   and "split_min t = None" shows "t = N0"
simple_ownership_distinct_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> 'e) list" shows "simple_ownership_distinct ts \<equiv> \<forall>i j p\<^sub>i is\<^sub>i \<O>\<^sub>i \<R>\<^sub>i \<D>\<^sub>i \<theta>\<^sub>i sb\<^sub>i p\<^sub>j is\<^sub>j \<O>\<^sub>j \<R>\<^sub>j \<D>\<^sub>j \<theta>\<^sub>j sb\<^sub>j. i < length ts \<longrightarrow> j < length ts \<longrightarrow> i \<noteq> j \<longrightarrow> ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i) \<longrightarrow> ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j) \<longrightarrow> \<O>\<^sub>i \<inter> \<O>\<^sub>j = {}"
exec_instr_Store: fixes n :: "nat"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_instr (Store n) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh = (None, h, (tl stk, loc[n := hd stk], C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
\<omega>_limit_set_in_compact_existence: fixes K :: "'a set"   and x :: "'a"   and y :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K"   and "y \<in> \<omega>_limit_set x" shows "existence_ivl0 y = UNIV"
f4: fixes r :: "real" shows "r + (1 - 1) = r"
leg0_commutes: shows "cod.leg0 \<cdot> chine = dom.leg0"
x_2: shows "\<forall>y\<in>s. (norm \<circ> f) x \<le> (norm \<circ> f) y"
C2: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "(\<Psi>', R, S) \<in> Rel"   and "xvec \<sharp>* \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>*xvec\<rparr>R, \<lparr>\<nu>*xvec\<rparr>S) \<in> Rel"
of_natural_def: fixes R :: "('a, 'b) ring_scheme"   and n :: "nat" shows "\<guillemotleft>n\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub> = ((\<oplus>\<^bsub>R\<^esub>) \<one>\<^bsub>R\<^esub> ^^ n) \<zero>\<^bsub>R\<^esub>"
H: fixes x :: "'b \<Rightarrow>\<^sub>0 nat"   and x' :: "'b" assumes "0 < lookup x x'" shows "(\<lambda>k. (if x' = k then Suc 0 else 0) + (if k = x' \<and> 0 < lookup x k then lookup x k - 1 else lookup x k)) = lookup x"
exhaust: fixes y :: "opinl" obtains "y = OpAddNum"   | "y = OpMulNum"
id2: shows "snd ` set (to_ns (map (Pair ()) csl)) = set (concat (map constraint_to_qdelta_constraint csl))"
prefixes_fin: fixes x :: "'c list" shows "let prefixes = {t. \<exists>t2. x = t @ t2} in finite prefixes \<and> card prefixes = length x + 1"
vector_smult_lneg: fixes c :: "'c"   and x :: "('c, 'd) vec" shows "- c *s x = - (c *s x)"
ample_nonempty: fixes q :: "'state"   and A :: "'action set" assumes "ample_set q A"   and "A \<subset> {a. en a q}" shows "A \<noteq> {}"
w2: fixes r :: "complex"   and b :: "'a"   and y :: "'b" shows "h (r *\<^sub>C b) y = r *\<^sub>C h b y"
sgn2_zero_eq_arg_zero: fixes x :: "nat" shows "(sgn2 x = 0) = (0 < x)"
oneof_result: fixes xs :: "char list list"   and s :: "char list"   and y :: "char list"   and r :: "char list" assumes "oneof xs s = Inr (y, r)" shows "\<exists>w. s = y @ w @ r \<and> y \<in> set xs"
1: shows "\<forall>x\<in>MPLS_charles PROB. x \<le> 2 ^ card (prob_dom PROB) - 1"
assms_1: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
hr_comp_def: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> assn"   and R2 :: "('d \<times> 'f) set"   and a :: "'f"   and c :: "'e" shows "hr_comp R1 R2 a c \<equiv> \<exists>\<^sub>Ab. R1 b c * \<up> ((b, a) \<in> R2)"
invariance_of_domain_Euclidean_space: fixes n :: "nat"   and U :: "(nat \<Rightarrow> real) set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "openin (Euclidean_space n) U"   and "continuous_map (subtopology (Euclidean_space n) U) (Euclidean_space n) f"   and "inj_on f U" shows "openin (Euclidean_space n) (f ` U)"
prob_eq_0_AE: fixes P :: "'a \<Rightarrow> bool" assumes "AE x in M. \<not> P x" shows "prob {x \<in> space M. P x} = 0"
degree_higher_pderiv: fixes n :: "nat"   and p :: "'a poly" shows "Polynomial.degree ((pderiv ^^ n) p) = Polynomial.degree p - n"
greaterThanLessThan_subseteq_atLeastAtMost_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.dense_linorder less_eq less" shows "(ord.greaterThanLessThan less a b \<subseteq> ord.atLeastAtMost less_eq c d) = (less a b \<longrightarrow> less_eq c a \<and> less_eq b d)"
mset_del_min: fixes h :: "'b tree" shows "mset_tree (del_min h) = mset_tree h - {#get_min h#}"
intro: fixes sep_size :: "nat"   and paths :: "'c list set" assumes "sep_size \<noteq> 0"   and "card paths = sep_size" shows "ProofStepInduct_axioms paths sep_size"
fds_commutes: fixes f :: "'b fds"   and g :: "'b fds" assumes "\<And>m n. \<lbrakk>0 < m; 0 < n\<rbrakk> \<Longrightarrow> fds_nth f m * fds_nth g n = fds_nth g n * fds_nth f m" shows "f * g = g * f"
set_sel_5: fixes a :: "('a, 'b) tm"   and xi :: "'b" assumes "is_App a"   and "xi \<in> vars (fun a)" shows "xi \<in> vars a"
2: shows "(0::'a) \<in> rel_interior ((+) (- a) ` S)"
bal_sub_height: fixes ls :: "('b btree \<times> 'b) list"   and a :: "'b btree \<times> 'b"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "bal (Node (ls @ a # rs) t)" shows "case rs of [] \<Rightarrow> True | (sub, sep) # x \<Rightarrow> height sub = height t"
emeasure_eq_measure: fixes na :: "nat"   and A :: "(nat \<Rightarrow>\<^sub>F 'a) set" shows "emeasure (P' na) A = ennreal (P'.prob na A)"
finite_vars_clause: fixes x :: "('b, 'a) Term.term literal multiset" shows "finite (vars_clause x)"
epiE: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "epi f"   and "seq g f"   and "seq g' f"   and "g \<cdot> f = g' \<cdot> f" shows "g = g'"
Union_ran_override2: fixes S :: "'d"   and G :: "'d \<Rightarrow> 'e set option"   and SA :: "'e" assumes "S \<in> dom G" shows "\<Union> (ran (G(S \<mapsto> insert SA (the (G S))))) = insert SA (\<Union> (ran G))"
simps_2: fixes v :: "'a"   and va :: "'a list" shows "p_fusion (v # va) [] = []"
Ln_minus_ii: shows "Ln (- \<i>) = - (\<i> * complex_of_real pi / 2)"
simps_3: fixes x :: "('b, 'c) Combinators"   and v :: "('b, 'c) Combinators"   and va :: "('b, 'c) Combinators list" shows "list2FWpolicy (x # v # va) = x \<oplus> list2FWpolicy (v # va)"
ord_iso_inv1: fixes f :: "'b \<Rightarrow> 'c" assumes "ord_iso f" shows "the_inv f \<circ> f = id"
pth_4_1: fixes x :: "'c" shows "0 *\<^sub>R x \<equiv> 0::'c"
Inf_inf_closed: fixes X :: "'b set" assumes "Inf_closed_set X" shows "inf_closed_set X"
cone_axioms: shows "D.cone a' (D.cones_map f \<chi>)"
get_root_node_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok Shadow_DOM.get_root_node ptr" shows "ptr |\<in>| object_ptr_kinds h"
foo'_ax2: fixes x :: "bool"   and y :: "'a"   and z :: "bool" assumes "foo' x y y" shows "x \<and> z"
priorities_finite: shows "finite (\<omega>\<^bsub>G'\<^esub> ` V\<^bsub>G'\<^esub>)"
set_drop: shows "set (drop (Suc i) as') \<subseteq> carrier G"
nxtAct_active: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat" assumes "active c (t i)" shows "dynamic_component.nxtAct active c t i = i"
map_of_zip_in_range: fixes A :: "'b list"   and B :: "'c list"   and b :: "'c" assumes "distinct A"   and "length B = length A"   and "b \<in> set B" shows "\<exists>a\<in>set A. the (map_of (zip A B) a) = b"
fSum_zero: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<in>nset n (n + m). f j = \<zero>" shows "\<Sigma>\<^sub>f A f n (n + m) = \<zero>"
to_Arity_inverse: fixes y :: "nat" assumes "y \<in> UNIV" shows "Rep_Arity (to_Arity y) = y"
snd'_def: shows "snd' \<equiv> op_sem Snd"
continuous_cos: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "continuous F f" shows "continuous F (\<lambda>x. cos (f x))"
non_strict_into_strict: fixes NS :: "('b \<times> 'b) set"   and S :: "('b \<times> 'c) set"   and s :: "'b"   and t :: "'c" assumes "NS O S \<subseteq> S"   and "(s, t) \<in> NS\<^sup>* O S" shows "(s, t) \<in> S"
dg_GRPH_components_2: fixes \<alpha> :: "V" shows "dg_GRPH \<alpha>\<lparr>Arr\<rparr> = all_dghms \<alpha>"
set_replicate: fixes n :: "nat"   and x :: "'c" assumes "n \<noteq> 0" shows "set (replicate n x) = {x}"
bounds_1: fixes n :: "nat" shows "real_of_float (lb_exp_horner prec n (((\<lambda>i. i + 1) ^^ 0) 1) (fact 0) x) \<le> (\<Sum>j = 0..<n. 1 / real (fact (0 + j)) * real_of_float x ^ j)"
proj2_Col_permute_2: fixes a :: "proj2"   and b :: "proj2"   and c :: "proj2" assumes "proj2_Col a b c" shows "proj2_Col b a c"
Class_cong: fixes a :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a" assumes "additive.Class a = additive.Class a'"   and "additive.Class b = additive.Class b'"   and "a \<in> R"   and "a' \<in> R"   and "b \<in> R"   and "b' \<in> R" shows "additive.Class (a \<cdot> b) = additive.Class (a' \<cdot> b')"
fall2_intro: fixes \<pi>l :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list"   and F :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream \<Rightarrow> (('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream \<Rightarrow> (('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list \<Rightarrow> bool" assumes "\<And>\<pi> \<pi>'. \<lbrakk>wfp UNIV \<pi>; wfp UNIV \<pi>'; \<pi>l \<noteq> [] \<Longrightarrow> stateOf \<pi> = stateOf (last \<pi>l); \<pi>l = [] \<Longrightarrow> stateOf \<pi> = s0; stateOf \<pi>' = stateOf \<pi>\<rbrakk> \<Longrightarrow> F \<pi> \<pi>' \<pi>l" shows "fall2 F \<pi>l"
Image_Id_on: fixes A :: "'b set"   and B :: "'b set" shows "Id_on A `` B = A \<inter> B"
mapl_FGco_def: fixes l1 :: "'p \<Rightarrow> 'ac" shows "mapl_FGco l1 = map_FGco l1 id id id id id id id id id id"
eqvt_1: fixes x1 :: "binding list"   and x2 :: "trm"   and x3 :: "ty"   and pi :: "(vrs \<times> vrs) list" assumes "x1 \<turnstile> x2 : x3" shows "pi \<bullet> x1 \<turnstile> pi \<bullet> x2 : pi \<bullet> x3"
prognosis_Var2: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and e :: "exp"   and x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and S :: "stack_elem list" assumes "CardinalityPrognosisVar prognosis"   and "isVal e"   and "x \<notin> domA \<Gamma>" shows "prognosis ae as 0 ((x, e) # \<Gamma>, e, S) \<sqsubseteq> prognosis ae as 0 (\<Gamma>, e, Upd x # S)"
pred_map: fixes Q :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b bdd" shows "bdd_all Q (bdd_map f x) = bdd_all (Q \<circ> f) x"
Wbis_matchT_M: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>w d" shows "matchT_M c d"
forest_def: fixes E :: "'b uprod set" shows "forest E \<equiv> \<nexists>u p. decycle E u p"
new_pre_hitting: fixes x' :: "'a" assumes "x' \<in> set new_pre" shows "\<not> hitting_paths x'"
poly_exp_max_constant_bound: fixes b :: "'b"   and c :: "'b"   and deg :: "nat" assumes "(0::'b) < b"   and "b < (1::'b)" shows "\<exists>p. \<forall>x. c * b ^ x * max (1::'b) (of_nat x ^ deg) \<le> p"
leadsTo_cancel_Diff2: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto> A' \<union> B"   and "F \<in> B - A' \<longmapsto> B'" shows "F \<in> A \<longmapsto> A' \<union> B'"
P2: shows "M0 Midpoint P' P \<and> Col A B M0"
assms_1: shows "bij_betw \<phi> S (Collect arr)"
null_set_Int1: fixes B :: "'b set"   and M :: "'b measure"   and A :: "'b set" assumes "B \<in> null_sets M"   and "A \<in> sets M" shows "A \<inter> B \<in> null_sets M"
rell_G_def: fixes L1 :: "'p \<Rightarrow> 'w \<Rightarrow> bool"   and L2 :: "'q \<Rightarrow> 'x \<Rightarrow> bool" shows "rell_G L1 L2 = rel_G L1 L2 (=) (=) (=) (=)"
rel_intros: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" assumes "R x y" shows "rel_id R (return_id x) (return_id y)"
chain_on_imp_subset: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set" assumes "chain_on P C A" shows "C \<subseteq> A"
exp_conj_idem: fixes P :: "'a \<Rightarrow> bool" shows "\<guillemotleft> P \<guillemotright> && \<guillemotleft> P \<guillemotright> = \<guillemotleft> P \<guillemotright>"
simps_3: fixes x :: "'ubx2" shows "norm_unboxed (OpUbx2 x) = box_ubx2 x"
less_poly_mapping_def: shows "(<) \<equiv> map_fun lookup (map_fun lookup id) less_fun"
inject_2: fixes x21 :: "'c"   and x22 :: "'d"   and x23 :: "('c, 'd) chain"   and y21 :: "'c"   and y22 :: "'d"   and y23 :: "('c, 'd) chain" shows "(\<lbrace> x21 \<rbrace> \<cdot> x22 \<cdot> x23 = \<lbrace> y21 \<rbrace> \<cdot> y22 \<cdot> y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
equa: shows "R = take_cols R (map snd (pivot_positions R)) * take_rows R [0..<length (pivot_positions R)]"
cdiv_in_iff': fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "((\<lambda>x. f x / c) \<in> \<omega>[F](g)) = (f \<in> \<omega>[F](g))"
maximal_def: fixes A :: "'b set"   and S :: "(('c, 'b) fm list \<times> 'b) set" shows "maximal A S \<equiv> consistent A S \<and> (\<forall>block. block \<notin> S \<longrightarrow> \<not> consistent A ({block} \<union> S))"
ran_map_addD: fixes x :: "'a"   and m :: "'b \<Rightarrow> 'a option"   and f :: "'b \<Rightarrow> 'a option" assumes "x \<in> ran (m ++ f)" shows "x \<in> ran m \<or> x \<in> ran f"
fold_insert: fixes A :: "'b set"   and x :: "'b"   and z :: "'a" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold (pls' \<circ> g) z (insert x A) = (pls' \<circ> g) x (Finite_Set.fold (pls' \<circ> g) z A)"
lessThan_Suc: fixes g :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "sum g {..<Suc n} = sum g {..<n} + g n"
worst_def: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>" shows "worst outcomes \<R> = {x \<in> \<P> outcomes. \<forall>y\<in>\<P> outcomes. y \<succeq>[\<R>] x}"
supp_1: fixes x1 :: "name" shows "supp (InputS x1) = supp x1"
complete_iff_size1: fixes t :: "'c tree" shows "complete t = (size1 t = 2 ^ height t)"
sum_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> less zero (f i)" shows "less zero (comm_monoid_add.sum plus zero f I)"
e_c_inc_self: fixes f :: "'a \<Rightarrow> 'c"   and I :: "'a set" assumes "f \<in> rHom R R'"   and "ideal R I" shows "I \<subseteq> i_contract f R R' (i_extension f R R' I)"
is_non_volatile_Read\<^sub>s\<^sub>b_simps_3: fixes p :: "'b"   and p' :: "'b"   and "is" :: "instr list" shows "is_non_volatile_Read\<^sub>s\<^sub>b (Prog\<^sub>s\<^sub>b p p' is) = False"
stoch_proc_filt_adapt: fixes X :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "\<And>n. X n \<in> M \<rightarrow>\<^sub>M N" shows "adapt_stoch_proc (stoch_proc_filt M X N) X N"
co_path_simps_3: fixes e1 :: "'b \<times> 'b"   and e :: "'b \<times> 'b"   and w :: "'b"   and e2 :: "'b \<times> 'b"   and es :: "('b \<times> 'b) list" assumes "e1 = (snd e, w)"   and "e2 = (w, fst e)" shows "co_path e w (e1 # e2 # es) = (snd e, fst e) # co_path e w es"
pdevs_val_sum: fixes e :: "nat \<Rightarrow> real"   and X :: "'b pdevs" shows "pdevs_val e X = (\<Sum>i<degree X. e i *\<^sub>R pdevs_apply X i)"