hcju/isabelleps huggingface dataset:
opnet_net_ips_net_tree_ips: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and ns :: "'d net_state"   and np :: "nat \<Rightarrow> ((nat \<Rightarrow> 'c) \<times> 'd, 'e seq_action) automaton"   and p :: "net_tree"   and S :: "(nat \<Rightarrow> 'c) \<Rightarrow> (nat \<Rightarrow> 'c) \<Rightarrow> 'e node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'c) \<Rightarrow> (nat \<Rightarrow> 'c) \<Rightarrow> bool" assumes "(\<sigma>, ns) \<in> oreachable (opnet np p) S U" shows "net_ips ns = net_tree_ips p"
x0: assumes "b $\<^sub>e 0 = a $\<^sub>e 0" shows "x $\<^sub>e 0 = (a $\<^sub>e 0 * b $\<^sub>e Suc 0 - b $\<^sub>e 0 * a $\<^sub>e Suc 0 + (b $\<^sub>e 0 - a $\<^sub>e 0) * x $\<^sub>e Suc 0) / (b $\<^sub>e Suc 0 - a $\<^sub>e Suc 0)"
case_cong: fixes nat :: "nat"   and nat' :: "nat"   and f1 :: "'a"   and g1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a"   and g2 :: "nat \<Rightarrow> 'a" assumes "nat = nat'"   and "nat' = 0 \<Longrightarrow> f1 = g1"   and "\<And>x2. nat' = Suc x2 \<Longrightarrow> f2 x2 = g2 x2" shows "(case nat of 0 \<Rightarrow> f1 | Suc x2 \<Rightarrow> f2 x2) = (case nat' of 0 \<Rightarrow> g1 | Suc x2 \<Rightarrow> g2 x2)"
induct: fixes x1 :: "'node SDG_node set"   and x2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and x3 :: "'node list"   and x4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and x5 :: "'edge list"   and x6 :: "'node list"   and x7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and P :: "'node SDG_node set \<Rightarrow> ('edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind) \<Rightarrow> 'node list \<Rightarrow> (('var \<Rightarrow> 'val option) \<times> 'ret) list \<Rightarrow> 'edge list \<Rightarrow> 'node list \<Rightarrow> (('var \<Rightarrow> 'val option) \<times> 'ret) list \<Rightarrow> bool" assumes "x1,x2 \<turnstile> (x3,x4) =x5\<Rightarrow>* (x6,x7)"   and "\<And>ms s S f. length ms = length s \<Longrightarrow> P S f ms s [] ms s"   and "\<And>S f ms s as ms' s' as' ms'' s''. \<lbrakk>S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s'); S,f \<turnstile> (ms',s') =as'\<Rightarrow>* (ms'',s''); P S f ms' s' as' ms'' s''\<rbrakk> \<Longrightarrow> P S f ms s (last as # as') ms'' s''" shows "P x1 x2 x3 x4 x5 x6 x7"
ResF: fixes P :: "pi"   and \<alpha> :: "Late_Semantics.freeRes"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto>\<^sub>l \<alpha> \<prec> P'"   and "y \<sharp> \<alpha>" shows "<\<nu>y>P \<longmapsto>\<^sub>l \<alpha> \<prec> <\<nu>y>P'"
unique_expression7_1: fixes H :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and t :: "nat \<Rightarrow> 'b" assumes "free_generator R M H"   and "f \<in> {j. j \<le> n} \<rightarrow> H"   and "inj_on f {j. j \<le> n}"   and "s \<in> {j. j \<le> n} \<rightarrow> carrier R"   and "g \<in> {j. j \<le> m} \<rightarrow> H"   and "inj_on g {j. j \<le> m}"   and "t \<in> {j. j \<le> m} \<rightarrow> carrier R"   and "l_comb R M n s f = l_comb R M m t g"   and "\<forall>j\<in>{j. j \<le> n}. s j \<noteq> \<zero>\<^bsub>R\<^esub>"   and "\<forall>k\<in>{j. j \<le> m}. t k \<noteq> \<zero>\<^bsub>R\<^esub>" shows "n = m"
orbit_cyclic_eq3: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and y :: "'b" assumes "cyclic_on f S"   and "y \<in> S" shows "orbit f y = S"
image_subsetI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "order.greater_eq B (f ` A)"
bij_betw_inv_into_RIGHT_LEFT: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and A' :: "'c set"   and B' :: "'c set"   and B :: "'b set" assumes "bij_betw f A A'"   and "B' \<subseteq> A'"   and "inv_into A f ` B' = B" shows "f ` B = B'"
edom_def: fixes m :: "'d \<Rightarrow> 'e" shows "edom m = {x. m x \<noteq> \<bottom>}"
th3_7_1: fixes b :: "(real, 'a) vec"   and c :: "(real, 'a) vec"   and a :: "(real, 'a) vec"   and d :: "(real, 'a) vec" assumes "b \<noteq> c"   and "B\<^sub>\<real> a b c"   and "B\<^sub>\<real> b c d" shows "B\<^sub>\<real> a c d"
LFP_lemma3: fixes f :: "'a \<Rightarrow> 'a" assumes "Mono f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "LFP f \<sqsubseteq> f (LFP f)"
THE_default1_equality: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and d :: "'a" assumes "\<exists>!x. P x"   and "P a" shows "THE_default d P = a"
perzeta_one_half_1: shows "perzeta (1 / 2) 1 = - Ln 2"
intro: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and test_num' :: "int \<Rightarrow> 'a \<Rightarrow> bool"   and filter_plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"   and filter_less' :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"   and \<gamma> :: "'a \<Rightarrow> int set" assumes "Abs_Int1_mono num' plus' test_num' filter_plus' filter_less' \<gamma>" shows "Abs_Int2 num' plus' test_num' filter_plus' filter_less' \<gamma>"
isl_map_sum: fixes f :: "'j \<Rightarrow> 'h"   and g :: "'k \<Rightarrow> 'i"   and s :: "'j + 'k" shows "isl (map_sum f g s) = isl s"
beh_obs_upd: fixes x :: "'a \<Rightarrow> 'b"   and S :: "('a, 'b, 'c) spec_scheme" shows "beh (S\<lparr>obs := x\<rparr>) = beh S"
CA_image_eq: fixes x :: "nat"   and A :: "nat set" shows "(CA x \<in> CA ` A) = (x \<in> A)"
isDiagonal: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and a :: "'d"   and b :: "'c" assumes "isDiagonal A"   and "to_nat a \<noteq> to_nat b" shows "A $h a $h b = (0::'b)"
INF_SUP_set: 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 g :: "'d \<Rightarrow> 'c"   and A :: "'d set set" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "Inf ((\<lambda>B. Sup (g ` B)) ` A) = Sup ((\<lambda>B. Inf (g ` B)) ` {f ` A |f. \<forall>C\<in>A. f C \<in> C})"
that: fixes x0 :: "'a" assumes "x0 \<in> s"   and "\<And>x. x \<in> {x0..} \<inter> s \<Longrightarrow> dist (f x) l < e" shows "thesis"
distinct_280: fixes x151 :: "floatarith"   and x152 :: "nat"   and x131 :: "floatarith"   and x132 :: "floatarith" shows "x151 ^\<^sub>e x152 \<noteq> Powr x131 x132"
distinct_3: fixes x1 :: "'b" shows "Le x1 \<noteq> \<infinity>"
cast_shadow_root_ptr_not_node_ptr_1: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and node_ptr :: "(_) node_ptr" shows "cast shadow_root_ptr \<noteq> cast node_ptr"
simps_5: fixes \<phi> :: "atomUni fmUni"   and \<psi> :: "atomUni fmUni"   and x :: "real" shows "evalUni (OrUni \<phi> \<psi>) x = (evalUni \<phi> x \<or> evalUni \<psi> x)"
partitioningIps_nonempty: fixes ts :: "'a wordinterval list"   and ss :: "'a wordinterval list" assumes "ts \<noteq> []" shows "partitioningIps ss ts \<noteq> []"
red: shows "\<tau>reds1r P t h (es', xs) (map Val vs @ Throw a # es'', loc)"
mcont_lprefix_case_aux: fixes lub :: "'d set \<Rightarrow> 'd"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'c llist \<Rightarrow> 'd"   and bot :: "'d" assumes "\<And>x. mcont lSup (\<sqsubseteq>) lub ord (\<lambda>xs. f x xs (LCons x xs))"   and "class.ccpo lub ord (mk_less ord)"   and "\<And>x. ord bot x" shows "mcont lSup (\<sqsubseteq>) lub ord (\<lambda>xs. case xs of LNil \<Rightarrow> bot | LCons x xs' \<Rightarrow> f x xs' xs)"
continuous_discrete_range_constant_eq: fixes S :: "'a set" shows "connected S = (\<forall>f. continuous_on S f \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)) \<longrightarrow> f constant_on S)"
intros_31: fixes vs :: "e list"   and n :: "nat"   and es :: "e list" assumes "const_list vs" shows "\<lparr>[Label n es vs]\<rparr> \<leadsto> \<lparr>vs\<rparr>"
pA: shows "\<Theta>,\<Gamma> \<turnstile> A"
weak_coinduct: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>\<^sup>^<X> Q \<and> (Q, P) \<in> X" shows "P \<approx> Q"
n_bits_of_eq_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'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 n :: "nat"   and a :: "'a"   and b :: "'a" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "(unique_euclidean_semiring_with_bit_shifts.n_bits_of divide plus times one n a = unique_euclidean_semiring_with_bit_shifts.n_bits_of divide plus times one n b) = (take_bit n a = take_bit n b)"
simps_3: fixes uu :: "poscheckvariant"   and t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>t\<^sub>p \<langle>uu: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t = fv t \<union> fv t'"
collapse_5: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "is_Delete stateful_strand_step" shows "delete\<langle>the_elem_term stateful_strand_step,the_set_term stateful_strand_step\<rangle> = stateful_strand_step"
DiaP: fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and i :: "'c"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>\<diamond> p) at a in' set ((ps, a) # branch)"   and "i \<notin> A \<union> branch_nominals ((ps, a) # branch)"   and "\<nexists>a. p = Nom a"   and "\<not> witnessed p a ((ps, a) # branch)"   and "A, Suc n \<turnstile> ((\<^bold>@ i p) # (\<^bold>\<diamond> Nom i) # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
simps_10: fixes e :: "expr"   and Copt :: "char list option"   and M :: "char list"   and es :: "expr list"   and A :: "char list set option" shows "\<D> (Call e Copt M es) A = (\<D> e A \<and> \<D>s es (A \<squnion> \<A> e))"
sel_3: fixes x21 :: "'a"   and x22 :: "('a, 'b) tllist" shows "thd (TCons x21 x22) = x21"
no: fixes t :: "real"   and x :: "'a" assumes "t \<in> {0..1}"   and "x \<notin> S" shows "norm (h t - g t) < norm (g t - x)"
subst_Var_Suc: fixes t :: "dB"   and i :: "nat" shows "t[Var i/i] = t[Var i/i + 1]"
caseCases: fixes \<Psi> :: "'b"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and Rs :: "('a, 'b, 'c) residual" assumes "\<Psi> \<rhd> Cases Cs \<longmapsto> Rs" obtains \<phi> :: "'c" and P :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> P \<longmapsto> Rs" and "(\<phi>, P) mem Cs" and "\<Psi> \<turnstile> \<phi>" and "guarded P"
wr_com_seq: fixes c1 :: "command"   and c2 :: "command" shows "wr_com (c1 ;; c2) = wr_com c1 \<union> wr_com c2"
simps_23: shows "cospan_shape.comp cospan_shape.BT cospan_shape.AA = cospan_shape.Null"
INF_Lim: fixes X :: "nat \<Rightarrow> 'b"   and l :: "'b" assumes "decseq X"   and "X \<longlonglongrightarrow> l" shows "Inf (range X) = l"
assms_5: shows "Y\<^sub>1 \<in> Sols f U S t\<^sub>0 s"
pt_set_bij1a: fixes x :: "'a"   and pi :: "('b \<times> 'b) list"   and X :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "(x \<in> pi \<bullet> X) = (rev pi \<bullet> x \<in> X)"
find_shortest_augmenting_spec_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity" shows "find_shortest_augmenting_spec f \<equiv> assert (NFlow c s t f) \<bind> (\<lambda>_. select p. Graph.isShortestPath (cf_of f) s p t)"
homeomorphism_locally: fixes S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b"   and P :: "'b set \<Rightarrow> bool"   and Q :: "'c set \<Rightarrow> bool" assumes "homeomorphism S T f g"   and "\<And>S T. homeomorphism S T f g \<Longrightarrow> P S = Q T" shows "locally P S = locally Q T"
4: shows "map ((!) (as @ bs) \<circ> (+) (length as)) (snd ys) = map ((!) bs) (snd ys)"
has_fields_rec: fixes P :: "'b prog"   and C :: "char list"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'b mdecl list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and FDTs' :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "class P C = \<lfloor>(D, fs, ms)\<rfloor>"   and "C \<noteq> Object"   and "P \<turnstile> D has_fields FDTs"   and "FDTs' = map (\<lambda>(F, b, T). ((F, C), b, T)) fs @ FDTs" shows "P \<turnstile> C has_fields FDTs'"
cases: fixes x :: "'b list list" obtains "x = []"   | xss :: "'b list list" where "x = [] # xss__"   | xa :: "'b" and xs :: "'b list" and xss :: "'b list list" where "x = (xa # xs) # xss"
simps_5: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c" shows "rel_dual R (\<partial> x) (\<partial> y) = R x y"
less_irrefl: fixes x :: "('a, 'b) Term.term" shows "\<not> x \<sqsubset> x"
taylor_id: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "p \<in> carrier P" shows "p = Cring_Poly.compose R (T\<^bsub>a\<^esub> p) (X_poly_minus R a)"
sscan_snth: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b stream"   and a :: "'a"   and i :: "nat" shows "sscan f xs a !! i = fold f (stake (Suc i) xs) a"
E_LetV: fixes v :: "trm"   and p :: "pat"   and ts :: "trm list"   and t :: "trm" assumes "v \<in> value"   and "\<turnstile> p \<rhd> v \<Rightarrow> ts" shows "(LET p = v IN t) \<longmapsto> t[0 \<mapsto>\<^sub>s ts]"
preserves_comp: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "V.seq g f" shows "trg (g \<bullet> f) = trg g \<bullet> trg f"
modelAssignsD: fixes f :: "vbl \<Rightarrow> object"   and M :: "model" assumes "f \<in> modelAssigns M" shows "range f \<subseteq> objects M"
alpha': fixes a :: "name"   and x :: "'a"   and b :: "name"   and y :: "'a" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> [(b, a)] \<bullet> x = y \<and> b \<sharp> x)"
crypts_crypt_2: fixes K :: "key"   and X :: "msg" shows "Crypt K ` crypts {X} \<subseteq> crypts {Crypt K X}"
ran_def: fixes m :: "'d \<Rightarrow> 'c option" shows "ran m = {b. \<exists>a. m a = Some b}"
cases: fixes a1 :: "Player"   and a2 :: "'a llist"   and a3 :: "'a \<Rightarrow> 'a" assumes "path_conforms_with_strategy a1 a2 a3" obtains p :: "Player" and \<sigma> :: "'a \<Rightarrow> 'a" where "a1 = pa__" and "a2 = LNil" and "a3 = \<sigma>''__"   | p :: "Player" and v :: "'a" and \<sigma> :: "'a \<Rightarrow> 'a" where "a1 = pb__" and "a2 = LCons v__ LNil" and "a3 = \<sigma>'''__"   | v :: "'a" and p :: "Player" and w :: "'a" and \<sigma> :: "'a \<Rightarrow> 'a" and Ps :: "'a llist" where "a1 = pc__" and "a2 = LCons va__ (LCons w Ps__)" and "a3 = \<sigma>''''__" and "va__ \<in> VV pc__" and "w = \<sigma>''''__ va__" and "path_conforms_with_strategy pc__ (LCons w Ps__) \<sigma>''''__"   | v :: "'a" and p :: "Player" and Ps :: "'a llist" and \<sigma> :: "'a \<Rightarrow> 'a" where "a1 = p" and "a2 = LCons v Ps" and "a3 = \<sigma>" and "v \<notin> VV p" and "path_conforms_with_strategy p Ps \<sigma>"
is_nextElem: shows "is_nextElem (rotate m ls) x y"
S5: shows "c_snd (Suc (Suc 3)) = 0"
language_eq_nextl: shows "language = eq_nextl `` \<Union> (left_lang ` final M)"
prime_prime_factor_sqrt: fixes n :: "nat" shows "prime n = (n \<noteq> 0 \<and> n \<noteq> 1 \<and> (\<nexists>p. prime p \<and> p dvd n \<and> p\<^sup>2 \<le> n))"
xcpt_app_def: fixes i :: "instr"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and pc :: "nat"   and mxs :: "nat"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and \<tau> :: "ty list \<times> ty err list" shows "xcpt_app i P pc mxs xt \<tau> = (\<forall>(f, t, C, h, d)\<in>set (relevant_entries P i pc xt). is_class P C \<and> d \<le> length (fst \<tau>) \<and> d < mxs)"
ndp: shows "Reflected_Multivariate_Polynomial.degree p = n"
mem_eqvt: fixes p :: "perm"   and x :: "'a"   and A :: "'a set" shows "p \<bullet> (x \<in> A) = (p \<bullet> x \<in> p \<bullet> A)"
SCp_swap_applies_2: fixes F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "F, G, H, \<Gamma> \<Rightarrow> \<Delta>" shows "G, H, F, \<Gamma> \<Rightarrow> \<Delta>"
rp: fixes p :: "int poly"   and x :: "'a" shows "(rp p x = (0::'a)) = (ipoly p x = (0::'a))"
enforce_option_eq_Some_iff: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c option"   and y :: "'c" shows "(enforce_option P x = Some y) = (x = Some y \<and> P y)"
left_unique_alt_def: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "left_unique R = (R OO R\<inverse>\<inverse> \<le> (=))"
simps_7: fixes r :: "'a rexp"   and s :: "'a rexp" shows "pnPlus (Plus r s) One = pnPlus r (pnPlus s One)"
rtrancl_Un_subset: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" shows "R\<^sup>* \<union> S\<^sup>* \<subseteq> (R \<union> S)\<^sup>*"
med_a0iim1_ir_def: fixes o1 :: "m1_state" shows "med_a0iim1_ir o1 \<equiv> \<lparr>signals = ir_runs2sigs (runs o1), corrupted = {}\<rparr>"
da_elim_cases_37: fixes Env :: "env"   and B :: "lname set"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>Super\<rangle>\<guillemotright> A" obtains "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>" and "This \<in> B"
unfold_while: fixes b :: "bexp"   and c :: "com" shows "WHILE b DO c \<sim> IF b THEN c;; WHILE b DO c ELSE SKIP"
simply_connected_eq_holomorphic_sqrt: fixes S :: "complex set" assumes "open S" shows "simply_connected S = (connected S \<and> (\<forall>f. f holomorphic_on S \<and> (\<forall>z\<in>S. f z \<noteq> 0) \<longrightarrow> (\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S. f z = (g z)\<^sup>2))))"
relprime_mult: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "somegcd G a b \<sim> \<one>"   and "somegcd G a c \<sim> \<one>"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "somegcd G a (b \<otimes> c) \<sim> \<one>"
real_root_ge_1_iff: fixes n :: "nat"   and y :: "real" assumes "0 < n" shows "(1 \<le> root n y) = (1 \<le> y)"
strip_while_append: fixes P :: "'f \<Rightarrow> bool"   and x :: "'f"   and xs :: "'f list" assumes "\<not> P x" shows "strip_while P (xs @ [x]) = xs @ [x]"
prob0_deutsch_algo_balanced: assumes "is_balanced" shows "prob0 2 deutsch_algo 0 = 0"
intro: fixes U\<^sub>1 :: "'a set"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top\<^sub>1 :: "'a"   and U\<^sub>2 :: "'b set"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top\<^sub>2 :: "'b" assumes "order_top_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1 top\<^sub>1"   and "order_top_ow U\<^sub>2 le\<^sub>2 ls\<^sub>2 top\<^sub>2" shows "order_top_pair_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1 top\<^sub>1 U\<^sub>2 le\<^sub>2 ls\<^sub>2 top\<^sub>2"
sem_def: fixes c :: "'a com" shows "sem c \<equiv> \<lambda>s. {t. \<exists>Ts. (c, s) -P*\<rightarrow> (Parallel Ts, t) \<and> All_None Ts}"
strong_cases: fixes a1 :: "(name \<times> ty) list"   and a2 :: "lam"   and a3 :: "ty"   and x :: "name" assumes "a1 \<turnstile> a2 : a3" obtains \<Gamma> :: "(name \<times> ty) list" and x :: "name" and T :: "ty" where "a1 = \<Gamma>__" and "a2 = Var x" and "a3 = T" and "valid \<Gamma>__" and "(x, T) \<in> set \<Gamma>__"   | \<Gamma> :: "(name \<times> ty) list" and t\<^sub>1 :: "lam" and T\<^sub>1 :: "ty" and T\<^sub>2 :: "ty" and t\<^sub>2 :: "lam" where "a1 = \<Gamma>'__" and "a2 = App t\<^sub>1 t\<^sub>2" and "a3 = T\<^sub>2__" and "\<Gamma>'__ \<turnstile> t\<^sub>1 : T\<^sub>1__ \<rightarrow> T\<^sub>2__" and "\<Gamma>'__ \<turnstile> t\<^sub>2 : T\<^sub>1__"   | \<Gamma> :: "(name \<times> ty) list" and T\<^sub>1 :: "ty" and t :: "lam" and T\<^sub>2 :: "ty" where "\<lbrakk>x__ \<sharp> a1; x__ \<sharp> a2; x__ \<sharp> a3\<rbrakk> \<Longrightarrow> a1 = \<Gamma> \<and> a2 = Lam [x__].t \<and> a3 = T\<^sub>1 \<rightarrow> T\<^sub>2 \<and> x__ \<sharp> \<Gamma> \<and> (x__, T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2"
simps_2: fixes varT :: "variable \<Rightarrow> 'a"   and objT :: "object \<Rightarrow> 'a"   and c :: "object" shows "ty_term varT objT (term.CONST c) = objT c"
select_result_I2: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog"   and x :: "'z" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "|h \<turnstile> f|\<^sub>r = x"
valuation_models_encoding_cnf_formula_equals_corollary: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>" shows "\<A> \<Turnstile> \<Phi> \<Pi> t = (\<forall>C\<in>cnf (\<Phi> \<Pi> t). \<exists>L\<in>C. lit_semantics \<A> L)"
infinite_iff_natLeq_ordLeq: fixes A :: "'b set" shows "finite A \<noteq> (natLeq \<le>o |A|)"
B_2: shows "S \<inter> T \<subseteq> span B"
sinks_aux_subset_ipurge_tr_aux: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and I' :: "('d \<times> 'd) set"   and D' :: "'c \<Rightarrow> 'd"   and U' :: "'d set"   and xs :: "'c list" shows "sinks_aux I D U (ipurge_tr_aux I' D' U' xs) \<subseteq> sinks_aux I D U xs"
prv_dsj_assoc_imp2: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (dsj (dsj \<phi> \<chi>) \<psi>) (dsj \<phi> (dsj \<chi> \<psi>)))"
continuous_at_Sup_antimono: fixes f :: "'b \<Rightarrow> 'd"   and S :: "'b set" assumes "antimono f"   and "continuous (at_left (Sup S)) f"   and "S \<noteq> {}"   and "bdd_above S" shows "f (Sup S) = Inf (f ` S)"
rep_eq: fixes x :: "'a fset" shows "sorted_list_of_fset x = sorted_list_of_set (fset x)"
add_comm': fixes x :: "'a"   and y :: "'a" assumes "D x y" shows "x \<oplus> y = y \<oplus> x"
1: shows "(S, T) \<in> \<langle>A\<rangle>set_rel"
ins_queue_invar: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and q :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.tree_invar t"   and "BinomialHeapStruc.queue_invar q" shows "BinomialHeapStruc.queue_invar (BinomialHeapStruc.ins t q)"
token_run_intial_state: fixes x :: "nat" shows "token_run x x = q\<^sub>0"
fresh_fmap_fresh_Some: fixes x :: "'a"   and F :: "('a, 'b) fmap"   and y :: "'a"   and a :: "'b" assumes "atom x \<sharp> F"   and "x \<noteq> y"   and "F $$ y = Some a" shows "atom x \<sharp> a"
subst_domain_def: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "subst_domain \<sigma> = {x. \<sigma> x \<noteq> Var x}"
size_Union_distinct_cart_prod_filter2: fixes V :: "'a multiset"   and B :: "'b multiset"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "nat" assumes "distinct_mset V"   and "\<And>b. b \<in># B \<Longrightarrow> size {#v \<in># V. P v b#} = c" shows "size (\<Sum>b\<in>#B. {#v \<in># V. P v b#} \<times># {#b#}) = size B * c"
b: shows "x \<cdot> (d z \<cdot> y\<^sup>\<star>) \<le> d z \<cdot> (y \<cdot> y\<^sup>\<star>)"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>)). \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = b"