hcju/isabelleps huggingface dataset:
SUP_upper2: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and i :: "'b"   and A :: "'b set"   and u :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq u (f i)" shows "less_eq u (Sup (f ` A))"
nfv_regex_def: fixes fv :: "'b \<Rightarrow> nat set"   and r :: "'b regex" shows "nfv_regex fv r = Max (insert 0 (Suc ` fv_regex fv r))"
widen2: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and join :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Top :: "'a"   and widen :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and narrow :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "class.WN le join Top widen narrow" shows "le y (widen x y)"
star_plus_one: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<star> = x\<^sup>\<star>"
wf_semiJoin_2: fixes J :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and QQ :: "(nat set \<times> 'a option list set) set"   and st :: "nat set"   and t :: "'a option list" assumes "1 \<le> card J"   and "wf_query n J Q Qn"   and "non_empty_query Q"   and "covering J Q"   and "\<forall>X\<in>Q. 1 \<le> card (fst X \<inter> J)"   and "QQ = (\<lambda>tab. semiJoin tab (st, t)) ` Q" shows "non_empty_query QQ"
icount_def: fixes M :: "nat" shows "icount M \<equiv> ASSERT (2 < M) \<bind> (\<lambda>_. let c = op_array_replicate (M + 1) 0; c = c[0 := 1, 1 := 1, 2 := 1, 3 := 2] in ASSERT (\<forall>i<4. c ! i = lcount i) \<bind> (\<lambda>_. nfoldli [4..<M + 1] (\<lambda>_. True) (\<lambda>n c. sum_prog (\<lambda>i. n - i - 1 < length c) 3 n (\<lambda>i. c ! (n - i - 1)) \<bind> (\<lambda>sum. ASSERT (n - 1 < length c \<and> n < length c) \<bind> (\<lambda>_. RETURN (c[n := c ! (n - 1) + 1 + sum])))) c \<bind> (\<lambda>c. ASSERT (\<forall>i\<le>M. c ! i = lcount i) \<bind> (\<lambda>_. ASSERT (M < length c) \<bind> (\<lambda>_. RETURN (c ! M))))))"
IREF: fixes it :: "'a set"   and \<sigma> :: "'b" assumes "it \<subseteq> S"   and "I it \<sigma>" shows "I' it \<sigma>"
map_gpv'_comp: fixes f :: "'m \<Rightarrow> 'j"   and g :: "'n \<Rightarrow> 'k"   and h :: "'l \<Rightarrow> 'o"   and f' :: "'p \<Rightarrow> 'm"   and g' :: "'q \<Rightarrow> 'n"   and h' :: "'o \<Rightarrow> 'r"   and gpv :: "('p, 'q, 'r) gpv" shows "map_gpv' f g h (map_gpv' f' g' h' gpv) = map_gpv' (f \<circ> f') (g \<circ> g') (h' \<circ> h) gpv"
dom_char: fixes f :: "'a" shows "local.dom f = (if arr f then C.dom f else C.null)"
gcd_neg1: fixes a :: "'a"   and b :: "'a" shows "gcd (- a) b = gcd a b"
words_infI_construct': fixes w :: "'transition stream"   and p :: "'state" assumes "\<And>k. \<exists>v. v \<le>\<^sub>F\<^sub>I w \<and> k < length v \<and> path v p" shows "run w p"
inorder_del: fixes t :: "('c \<times> 'd) tree234"   and x :: "'c" assumes "bal t"   and "sorted1 (inorder t)" shows "inorder (tree\<^sub>d (Tree234_Map.del x t)) = AList_Upd_Del.del_list x (inorder t)"
map_IInf_def: fixes F :: "'c upset \<Rightarrow> 'd upset" shows "map_IInf F = IInf \<circ> F \<circ> us"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) D.map GF.map \<eta>.map"
induct: fixes P :: "'b wordinterval \<Rightarrow> 'b wordinterval \<Rightarrow> bool"   and a0 :: "'b wordinterval"   and a1 :: "'b wordinterval" assumes "\<And>s e s' e'. P (WordInterval s e) (WordInterval s' e')"   and "\<And>r1 r2 t. \<lbrakk>P r1 t; P r2 t\<rbrakk> \<Longrightarrow> P (RangeUnion r1 r2) t"   and "\<And>v va r1 r2. \<lbrakk>P (WordInterval v va) r1; P (WordInterval v va) r2\<rbrakk> \<Longrightarrow> P (WordInterval v va) (RangeUnion r1 r2)" shows "P a0 a1"
eval_wlp_Mu: fixes T :: "(bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "wlp (\<mu>t. T t) = gfp_trans (\<lambda>t. wlp (T (Embed t)))"
functor_implies_equals_cod: assumes "functor (\<cdot>\<^sub>J) (\<cdot>) A.map" shows "A.G.map = A.map"
valid_proj_neq_succ_restricted_neq_succ: fixes x' :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and vs :: "'a set"   and s :: "('a, 'b) fmap" assumes "x' \<in> prob_proj PROB vs"   and "state_succ s x' \<noteq> s" shows "fmrestrict_set vs (state_succ s x') \<noteq> fmrestrict_set vs s"
size_del_max: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and t :: "('b \<times> nat) tree" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "t \<noteq> \<langle>\<rangle>" shows "size t = Suc (size (snd (WBT.del_max \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 t)))"
hcomp_ide_isomorphic: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "g \<cong>\<^sub>D h"   and "src\<^sub>D f = trg\<^sub>D g" shows "f \<star>\<^sub>D g \<cong>\<^sub>D f \<star>\<^sub>D h"
R: shows "{..<brnL cl (length cl)} = (\<Union>n<length cl. {brnL cl n..<brnL cl n + brn (cl ! n)})"
rtr_nomac_e2: fixes rt :: "'b routing_rule list"   and fw :: "'b simple_rule list"   and mlf :: "'b word \<Rightarrow> 48 word option"   and ifl :: "interface list"   and pi :: "'b Simple_Packet.simple_packet_ext"   and po :: "'b Simple_Packet.simple_packet_ext" assumes "simple_linux_router rt fw mlf ifl pi = Some po" shows "\<exists>po'. simple_linux_router_nol12 rt fw pi = Some po'"
inject_2: fixes x2 :: "('g \<Rightarrow> 'h option) \<Rightarrow> bool"   and y2 :: "('g \<Rightarrow> 'h option) \<Rightarrow> bool" shows "((x2)\<^sub>\<surd> = (y2)\<^sub>\<surd>) = (x2 = y2)"
simps_12: fixes x6 :: "ty" shows "x6\<lfloor>\<rceil> \<noteq> Void"
prems: shows "avl \<langle>l, (a, h), r\<rangle>"
emeasure_pair_measure_alt: fixes X :: "('d \<times> 'e) set"   and N :: "'d measure"   and Xa :: "'e set"   and Aa :: "'e set set" assumes "X \<in> sets (N \<Otimes>\<^sub>M sigma Xa Aa)" shows "emeasure (N \<Otimes>\<^sub>M sigma Xa Aa) X = \<integral>\<^sup>+ x. emeasure (sigma Xa Aa) (Pair x -` X) \<partial>N"
le_cases3: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.linorder less_eq less" obtains "less_eq x y" and "less_eq y z"   | "less_eq y x" and "less_eq x z"   | "less_eq x z" and "less_eq z y"   | "less_eq z y" and "less_eq y x"   | "less_eq y z" and "less_eq z x"   | "less_eq z x" and "less_eq x y"
prems_1: shows "mod_type_class.to_nat i < Suc k"
\<psi>\<^sub>1_to_\<psi>\<^sub>2: shows "U\<^sub>f * \<psi>\<^sub>1 = Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[real (1 - f 0) / 2 - real (f 0) / 2, real (f 0) / 2 - real (1 - f 0) / 2, real (1 - f 1) / 2 - real (f 1) / 2, real (f 1) / 2 - real (1 - f 1) / 2]])"
minimalize_offending_overapprox_spec_impl: fixes G :: "'c list_graph"   and sinvar_spec :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and sinvar_impl :: "'c list_graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and nP :: "'c \<Rightarrow> 'd"   and fs :: "('c \<times> 'c) list"   and keeps :: "('c \<times> 'c) list" assumes "wf_list_graph G"   and "\<And>G nP. wf_list_graph G \<Longrightarrow> sinvar_spec (list_graph_to_graph G) nP = sinvar_impl G nP" shows "TopoS_Interface_impl.minimalize_offending_overapprox (\<lambda>G. sinvar_impl G nP) fs keeps G = TopoS_withOffendingFlows.minimalize_offending_overapprox (\<lambda>G. sinvar_spec G nP) fs keeps (list_graph_to_graph G)"
scaleC_right: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and r :: "complex"   and b :: "'b" shows "cblinfun_apply a (r *\<^sub>C b) = r *\<^sub>C cblinfun_apply a b"
app_invimageE: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A" obtains b :: "V" where "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" and "b \<in>\<^sub>\<circ> A"
a: shows "limsup (u \<circ> r) \<noteq> \<infinity>"
finsert_commute: fixes x :: "'b"   and y :: "'b"   and A :: "'b fset" shows "finsert x (finsert y A) = finsert y (finsert x A)"
infinite_chain_below: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set"   and x :: "'b" assumes "Complete_Partial_Order.chain ord C"   and "infinite C"   and "x \<in> C"   and "finite {y \<in> C. ord x y}" shows "infinite {y \<in> C. ord y x}"
map\<^sub>0_simps_1: fixes a :: "'a" assumes "C.obj a" shows "D.obj (map\<^sub>0 a)"
map_poly_hom_mult_monom: fixes p :: "'a poly"   and a :: "'a"   and n :: "nat" shows "map_poly hom (p * monom a n) = map_poly hom p * monom (hom a) n"
keys_rep_nat_pp_pp: fixes t :: "('a, 'b) pp" shows "keys_pp (rep_nat_pp t) = rep_nat ` keys_pp t"
double_dual_iso_eq_iff: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "finite_comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "(finite_comm_group.double_dual_iso G x = finite_comm_group.double_dual_iso G y) = (x = y)"
False: shows "\<not> integrable (lebesgue_on {a..b}) f"
Semantics_optimize_primitive_univ_common_matcher: fixes \<gamma> :: "'a common_primitive \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and m :: "'a common_primitive match_expr"   and p :: "('a, 'b) tagged_packet_scheme" assumes "matcher_agree_on_exact_matches \<gamma> common_matcher" shows "Semantics.matches \<gamma> (optimize_primitive_univ m) p = Semantics.matches \<gamma> m p"
algebraic_int_nth_root_real_iff: fixes n :: "nat"   and x :: "real" shows "algebraic_int (root n x) = (n = 0 \<or> algebraic_int x)"
surj_image_vimage_eq: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'a set" assumes "surj f" shows "f ` f -` A = A"
matroid_ax_2_alt: fixes X :: "Points set"   and x :: "Points" shows "rk X \<le> rk (X \<union> {x}) \<and> rk (X \<union> {x}) \<le> rk X + 1"
elim_p_sign_r_pos: shows "sign_r_pos (pderiv p * q * p) x = sign_r_pos q x"
f: shows "ring_isomorphism f A addA multA zeroA oneA B addB multB zeroB oneB"
intros: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> actions E"   and "is_write_action (action_obs E a)" shows "a \<in> write_actions E"
sg_tassoc: fixes H :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "G \<guillemotright> H "   and "x \<in> H"   and "y \<in> H"   and "z \<in> H" shows "x \<cdot> y \<cdot> z = x \<cdot> (y \<cdot> z)"
insert_union: fixes A :: "'d set"   and a :: "'d"   and B :: "'d set" shows "A \<union> insert a B = insert a A \<union> B"
F_Mprefix: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" shows "F (Mprefix A P) = {(tr, ref). tr = [] \<and> ref \<inter> ev ` A = {}} \<union> {(tr, ref). tr \<noteq> [] \<and> hd tr \<in> ev ` A \<and> (\<exists>a. ev a = hd tr \<and> (tl tr, ref) \<in> F (P a))}"
f_Exec_N_Suc2: fixes n :: "nat"   and xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "n < length xs" shows "f_Exec_Comp_N trans_fun (Suc n) xs c = trans_fun (xs ! n) (f_Exec_Comp_N trans_fun n xs c)"
restrict_root_Node: fixes t :: "dag"   and repc :: "ref \<Rightarrow> ref"   and nb :: "nat"   and levellista :: "ref list list"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and var :: "ref \<Rightarrow> nat"   and pret :: "dag" assumes "t \<in> Dags (repc ` Nodes (Suc nb) levellista) (repc \<propto> low) (repc \<propto> high)"   and "t \<notin> Dags (repc ` Nodes nb levellista) (repc \<propto> low) (repc \<propto> high)"   and "ordered t var"   and "\<forall>no\<in>Nodes (Suc nb) levellista. var (repc no) \<le> var no \<and> repc (repc no) = repc no"   and "wf_ll pret levellista var"   and "nb < length levellista"   and "repc ` Nodes (Suc nb) levellista \<subseteq> Nodes (Suc nb) levellista" shows "\<exists>q. Dag (repc q) (repc \<propto> low) (repc \<propto> high) t \<and> q \<in> set (levellista ! nb)"
suffix_def: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and i :: "nat" shows "\<sigma> |\<^sub>s i \<equiv> \<lambda>j. \<sigma> (j + i)"
arg_max_nat_le: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and f :: "'a \<Rightarrow> nat"   and b :: "nat" assumes "P x"   and "\<forall>y. P y \<longrightarrow> f y < b" shows "f x \<le> f (arg_max f P)"
fail_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set" shows "mojmir_def.fail \<Sigma> \<delta> q\<^sub>0 w F = {x. mojmir_def.token_fails \<Sigma> \<delta> q\<^sub>0 w F x}"
assms_2: fixes n' :: "nat" assumes "n \<le> n'" shows "eval c t t' n' \<gamma>"
simps_12: fixes a :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Exp a) vs = exp (interpret_floatarith a vs)"
ListReds1: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e # es,s\<rangle> [\<rightarrow>]* \<langle>e' # es,s'\<rangle>"
pmc_recovery: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<subseteq> (K \<div>\<^bsub>\<gamma>\<^esub> \<phi>) \<oplus> \<phi>"
nexts_set: fixes y :: "node"   and g :: "(node \<times> node) list"   and x :: "node" shows "(y \<in> set (nexts g x)) = ((x, y) \<in> set g)"
points_arc: fixes x :: "'a"   and y :: "'a" assumes "point x"   and "point y" shows "arc (x * y\<^sup>T)"
assms_8: shows "y \<in># arr_mset mod_arr (m - 1) r\<^sub>0"
assms_1: shows "arr1 = arr2 on - {l..<r}"
isl_map_sum: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and s :: "'e + 'f" shows "isl (map_sum f g s) = isl s"
MOST_coeff_eq_0: fixes p :: "'b poly" shows "\<forall>\<^sub>\<infinity>n. poly.coeff p n = (0::'b)"
that: fixes x0 :: "'a" assumes "\<And>x. x0 \<le> x \<Longrightarrow> dist (f x) l < e" shows "thesis"
feven_def: shows "PROP feven \<equiv> (\<And>n kvs t k v kvs'. \<lbrakk>0 < n; n \<le> length kvs; P n kvs; rbtreeify_f n kvs = (t, (k, v) # kvs'); n \<le> Suc (length kvs'); Q n kvs'\<rbrakk> \<Longrightarrow> P (2 * n) kvs)"
remove_child_in_disconnected_nodes: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap"   and owner_document :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'"   and "h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \<rightarrow>\<^sub>r owner_document"   and "h' \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes" shows "child \<in> set disc_nodes"
normalize_right_idem: fixes a :: "'a"   and b :: "'a" shows "lcm a (normalize b) = lcm a b"
that: shows "set_pmf p \<inter> A \<noteq> {}"
pj_zero: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set"   and x :: "'e" assumes "Ring R"   and "ideal R I"   and "x \<in> carrier R" shows "(pj R I x = \<zero>\<^bsub>R /\<^sub>r I\<^esub>) = (x \<in> I)"
simps: fixes a :: "('Q set, 'L) ta_rule" shows "\<delta>ssp a = (\<exists>f ss s. a = s \<rightarrow> f ss \<and> A f = Some (length ss) \<and> listsp (\<lambda>s. s \<subseteq> Q) ss \<and> s = \<delta>ss_lhs f ss)"
neg_eqvt: fixes pi :: "('a \<times> 'a) list"   and A :: "bool" shows "pi \<bullet> (\<not> A) = (\<not> pi \<bullet> A)"
progress_regex_le: fixes j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and r :: "Formula.formula Regex.regex" assumes "pred_mapping (\<lambda>x. x \<le> j) P" shows "progress_regex \<sigma> P r j \<le> j"
fresh_star_Pair: fixes as :: "atom set"   and x :: "'b"   and y :: "'c" shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)"
harm_at_top: shows "harm \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"
rel_mono: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R1a :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R3 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R3a :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a"   and "R3 \<le> R3a" shows "rel_pre_coG\<^sub>M R1 R2 R3 \<le> rel_pre_coG\<^sub>M R1a R2a R3a"
resChainSupp: fixes xvec :: "name list"   and P :: "('a, 'b, 'c) psi" shows "supp (\<lparr>\<nu>*xvec\<rparr>P) = supp P - set xvec"
assms_2: fixes x :: "'a" assumes "x \<in> carrier G" shows "u \<otimes> x = x"
auxiliary_ineq: fixes x :: "nat" assumes "2 \<le> x" shows "x + 1 < 2 * x"
rPar1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P M N P'"   and "distinct A\<^sub>Q"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* M"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) M N (P' \<parallel> Q)"
Vagree_sym: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real"   and V :: "variable set" shows "Vagree \<nu> \<nu>' V = Vagree \<nu>' \<nu> V"
induct: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and x1 :: "'a language"   and x2 :: "'a language"   and P :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool" assumes "regular_cong R x1 x2"   and "\<And>x y. x = y \<Longrightarrow> P x y"   and "\<And>x y. \<lbrakk>regular_cong R x y; P x y\<rbrakk> \<Longrightarrow> P y x"   and "\<And>x y z. \<lbrakk>regular_cong R x y; P x y; regular_cong R y z; P y z\<rbrakk> \<Longrightarrow> P x z"   and "\<And>x y. R x y \<Longrightarrow> P x y"   and "\<And>x y x' y'. \<lbrakk>regular_cong R x y; P x y; regular_cong R x' y'; P x' y'\<rbrakk> \<Longrightarrow> P (Plus x x') (Plus y y')"   and "\<And>x y x' y'. \<lbrakk>regular_cong R x y; P x y; regular_cong R x' y'; P x' y'\<rbrakk> \<Longrightarrow> P (Times x x') (Times y y')"   and "\<And>x y. \<lbrakk>regular_cong R x y; P x y\<rbrakk> \<Longrightarrow> P (Star x) (Star y)"   and "\<And>x y x' y'. \<lbrakk>regular_cong R x y; P x y; regular_cong R x' y'; P x' y'\<rbrakk> \<Longrightarrow> P (Inter x x') (Inter y y')"   and "\<And>x y. \<lbrakk>regular_cong R x y; P x y\<rbrakk> \<Longrightarrow> P (Coinductive_Language.Not x) (Coinductive_Language.Not y)"   and "\<And>x y x' y'. \<lbrakk>regular_cong R x y; P x y; regular_cong R x' y'; P x' y'\<rbrakk> \<Longrightarrow> P (Shuffle x x') (Shuffle y y')" shows "P x1 x2"
rQuot_compl: fixes a :: "'b"   and A :: "'b list set" shows "rQuot a (- A) = - rQuot a A"
set_permutations_step: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "set (permutations xs) = (\<Union>x\<in>set xs. (#) x ` set (permutations (remove1 x xs)))"
divide_divide_eq_right: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide a (divide b c) = divide (times a c) b"
right_commute: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ac_operator f" shows "f (f a b) c = f (f a c) b"
Expr2ZFDepth0: fixes e :: "(ZF, ZF) Expression" assumes "ZFDepth (Expr2ZF e) = 0" shows "ZFType (Expr2ZF e) = 0"
ncols: fixes A :: "'b matrix"   and n :: "nat"   and m :: "nat" assumes "ncols A \<le> n" shows "Rep_matrix A m n = (0::'b)"
sums_if': fixes g :: "nat \<Rightarrow> real"   and x :: "real" assumes "g sums x" shows "(\<lambda>n. if even n then 0 else g ((n - 1) div 2)) sums x"
nGt_le_conv: fixes xa :: "'o"   and x :: "'a"   and y :: "'a" shows "(key_compare (rep_key_order xa) x y \<noteq> Gt) = le xa x y"
intros: fixes \<sigma> :: "char list \<Rightarrow> nat option"   and p :: "prog"   and \<sigma>' :: "char list \<Rightarrow> nat option" assumes "\<sigma> \<in> states p"   and "body p \<rhd> \<sigma> \<leadsto> Some \<sigma>'" shows "\<lparr>initial = \<sigma>, final = \<sigma>'\<rparr> \<in> traces p"
Smith_normal_form_upt_k_diagonal_to_Smith_aux: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec" assumes "is_bezout_ext bezout"   and "k < min (nrows A) (ncols A)"   and "isDiagonal A" shows "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<k] bezout) k"
assms_3: fixes x :: "real" assumes "x \<in> S" shows "f' x = g' x"
exist_extension_mhom: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and H :: "'e set"   and f :: "'e \<Rightarrow> 'i" assumes "R module M"   and "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N" shows "\<exists>g\<in>mHom R M N. \<forall>x\<in>H. g x = f x"
endpoints_in_S_cross_ratio_correct: fixes a :: "proj2"   and b :: "proj2" assumes "a \<noteq> b"   and "a \<in> hyp2"   and "b \<in> hyp2" shows "cross_ratio_correct (endpoint_in_S a b) (endpoint_in_S b a) a b"
S4: shows "nat_to_sch (sch_to_nat t) = t"
sim_def: fixes M :: "('e, 'f, 'g) KripkeStructure"   and M' :: "('e, 'f, 'h) KripkeStructure"   and f :: "'g \<Rightarrow> 'h" shows "sim M M' f \<equiv> sim_range M M' f \<and> sim_val M M' f \<and> sim_f M M' f \<and> sim_r M M' f"
convex_closure_rel_interior: fixes S :: "'c set" assumes "convex S" shows "closure (rel_interior S) = closure S"
Try: fixes G :: "prog"   and s0 :: "st"   and c1 :: "stmt"   and n :: "nat"   and s1 :: "state"   and s2 :: "state"   and tn :: "qtname"   and vn :: "vname"   and c2 :: "stmt"   and s3 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2"   and "if G,s2\<turnstile>catch tn then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<midarrow>n\<rightarrow> s3 else s3 = s2" shows "G\<turnstile>Norm s0 \<midarrow>Try c1 Catch(tn vn) c2\<midarrow>n\<rightarrow> s3"
matrix_rank_le_cprank_max: fixes r :: "nat"   and A :: "'b tensor"   and I :: "nat set" assumes "cprank_max r A" shows "mrank (matricize I A) \<le> r"
RBT_lookup_empty: fixes t :: "('a, unit) rbt" shows "(ord.rbt_lookup cless t = Map.empty) = (t = rbt.Empty)"
predecessors_def: fixes g :: "'g"   and n :: "'node" shows "predecessors g n \<equiv> map getFrom (inEdges g n)"
stutinv_strictly_stronger: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b" assumes "STUTINV F" shows "NSTUTINV F"
h: shows "|~ _actrans A f \<longrightarrow> _actrans B g"
supports_def: fixes S :: "'a set"   and x :: "'b" shows "S supports x = (\<forall>a b. a \<notin> S \<and> b \<notin> S \<longrightarrow> [(a, b)] \<bullet> x = x)"