hcju/isabelleps huggingface dataset:
nonzero_divide_eq_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero" shows "(divide b c = a) = (b = times a c)"
TESL_interp_stepwise_implies_not_coind_unfold: fixes master :: "clock"   and slave :: "clock"   and n :: "nat" shows "\<lbrakk> master implies not slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = (\<lbrakk> master \<not>\<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<union> \<lbrakk> master \<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk> slave \<not>\<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m) \<inter> \<lbrakk> master implies not slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^esup>"
ident: fixes i :: "nat"   and j :: "nat"   and a :: "'a" assumes "i < length as"   and "j < length as"   and "a \<in> as ! i"   and "a \<in> as ! j" shows "(fs ! i) a = (fs ! j) a"
convergent_def: fixes "open" :: "'b set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b" assumes "class.topological_space open" shows "topological_space.convergent open X = (\<exists>L. filterlim X (topological_space.nhds open L) sequentially)"
abc_1: shows "a \<noteq> b \<and> \<not> p a < p b"
trivial_subalg_cond_expect_AE: fixes N :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "subalgebra M N"   and "sets N = {{}, space M}"   and "integrable M f" shows "AEeq M (real_cond_exp M N f) (\<lambda>x. expectation f)"
star_n_zero_num: shows "0 = star_n (\<lambda>n. 0::'a)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "'b option" shows "P (case option of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = \<lfloor>x2\<rfloor> \<longrightarrow> P (f2 x2)))"
finprod_empty: fixes G :: "('c, 'd) monoid_scheme"   and f :: "'e \<Rightarrow> 'c" assumes "Group.comm_monoid G" shows "finprod G f {} = \<one>\<^bsub>G\<^esub>"
AxiomsSet7orig_def: fixes source :: "i \<Rightarrow> i"   and target :: "i \<Rightarrow> i"   and compositionF :: "i \<Rightarrow> i \<Rightarrow> i" shows "AxiomsSet7orig source target compositionF \<equiv> (((\<forall>x y. (target x \<cong> source y \<^bold>\<leftarrow> E (compositionF x y)) \<^bold>\<and> (E (compositionF x y) \<^bold>\<leftarrow> target x \<cong> source y)) \<and> (\<forall>x. target (source x) \<cong> source x)) \<and> (\<forall>x. source (target x) \<cong> source x) \<and> (\<forall>x. compositionF (source x) x \<cong> x)) \<and> ((\<forall>x. compositionF x (target x) \<cong> x) \<and> (\<forall>x y. source (compositionF x y) \<cong> source (compositionF x (source y)))) \<and> (\<forall>x y. target (compositionF x y) \<cong> target (compositionF (target x) y)) \<and> (\<forall>x y z. compositionF x (compositionF y z) \<cong> compositionF (compositionF x y) z)"
permute_finite: fixes p :: "perm"   and X :: "'b set" shows "finite (p \<bullet> X) = finite X"
bounded_imp_convergent_subsequence: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'c" assumes "class.heine_borel dist uniformity open"   and "metric_space.bounded dist (range f)" shows "\<exists>l r. strict_mono r \<and> filterlim (f \<circ> r) (topological_space.nhds open l) sequentially"
term_eq_def: fixes t :: "'a \<Rightarrow> 'b"   and w :: "'b" shows "term_eq t w = {s. t s = w}"
periodN_def: fixes w :: "'b list"   and n :: "nat" shows "periodN w n = w \<le>p take n w\<^sup>\<omega>"
m_inv_closed: fixes H :: "'b set"   and G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "subgroup H G"   and "x \<in> H" shows "inv\<^bsub>G\<^esub> x \<in> H"
not_inf_on_unit_circle': shows "\<not> on_circline unit_circle \<infinity>\<^sub>h"
map_add_restrict_comp_right_dom: fixes m :: "'c \<Rightarrow> 'd option"   and S :: "'c set" shows "m |` S ++ m |` (dom m - S) = m"
KbaseP_def: fixes K :: "('c, 'd) Ring_scheme"   and P :: "nat \<Rightarrow> ('c \<Rightarrow> ant) set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" shows "KbaseP K P n f = ((\<forall>j\<le>n. f j \<in> carrier K) \<and> (\<forall>j\<le>n. \<forall>l\<le>n. (\<nu>\<^bsub>K P j\<^esub>) (f l) = \<delta>\<^bsub>j l\<^esub>))"
strand_sem_mono_ik_1: fixes M :: "('fun, 'var) Term.term set"   and M' :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "M \<subseteq> M'"   and "\<lbrakk>M; S\<rbrakk>\<^sub>c \<theta>" shows "\<lbrakk>M'; S\<rbrakk>\<^sub>c \<theta>"
map_iterator_linord_foldli_conv: fixes iti :: "('d \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'a \<Rightarrow> 'e option" shows "map_iterator_linord iti m = (\<exists>l0. distinct (map fst l0) \<and> m = map_of l0 \<and> sorted (map fst l0) \<and> iti = foldli l0)"
neqE: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "linorder less_eq less"   and "x \<noteq> y" obtains "less x y"   | "less y x"
kozen_induct_r: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<cdot> x + z \<le> y" shows "z \<cdot> x\<^sup>\<star> \<le> y"
generator_def: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme"   and H :: "'g set" shows "generator R M H \<equiv> H \<subseteq> carrier M \<and> linear_span R M (carrier R) H = carrier M"
mono_Inf: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "('b \<Rightarrow> 'c) set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> {f. order.mono less_eq f}" shows "order.mono less_eq (\<Sqinter>A)"
cardinalityOneTheElem: fixes X :: "'a set"   and Y :: "'a set" assumes "card X = 1"   and "X \<subseteq> Y" shows "the_elem X \<in> Y"
exists_trace_for_any_i_j: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and i :: "nat"   and j :: "nat" assumes "\<exists>c'. trace c t c'"   and "i \<le> j" shows "trace (s c t i) (take (j - i) (drop i t)) (s c t j)"
\<mu>: shows "\<guillemotleft>\<mu> : k \<star> v \<Rightarrow> h \<star> u\<guillemotright> \<and> local.iso \<mu>"
sscan_scons: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b stream"   and a :: "'a" shows "sscan f (x ## xs) a = f x a ## sscan f xs (f x a)"
get_mode_pure: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap" assumes "l_get_mode type_wf get_mode get_mode_locs" shows "pure (get_mode shadow_root_ptr) h"
subforest_def: fixes F :: "nat uprod set" shows "subforest F \<equiv> forest F \<and> F \<subseteq> E"
induct: fixes P :: "MIR.num \<Rightarrow> bool"   and a0 :: "MIR.num" assumes "P (Bound 0)"   and "\<And>a b. \<lbrakk>P a; \<And>x. x = rsplit0 a \<Longrightarrow> P b\<rbrakk> \<Longrightarrow> P (Add a b)"   and "\<And>a b. P (Add a (Neg b)) \<Longrightarrow> P (Sub a b)"   and "\<And>a. P a \<Longrightarrow> P (Neg a)"   and "\<And>a. P a \<Longrightarrow> P (Floor a)"   and "\<And>c a. P a \<Longrightarrow> P (CN 0 c a)"   and "\<And>v c a. P a \<Longrightarrow> P (CN (Suc v) c a)"   and "\<And>c t s. P (Add (Mul c (Floor t)) s) \<Longrightarrow> P (CF c t s)"   and "\<And>c a. P a \<Longrightarrow> P (Mul c a)"   and "\<And>v. P (C v)"   and "\<And>va. P (Bound (Suc va))" shows "P a0"
g_ind: shows "\<forall>y<x. \<forall>anything. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything}"
card_sccs_da: shows "card H.sccs = card G.sccs"
assms_7: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and e0 :: "exp"   and D :: "nat"   and C :: "nat" assumes "CT;\<Gamma> \<turnstile> e0 : D"   and "P CT \<Gamma> e0 D"   and "CT \<turnstile> C <: D"   and "C \<noteq> D" shows "P CT \<Gamma> (Cast C e0) C"
par__coplanar: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Par C D" shows "Coplanar A B C D"
id: fixes r :: "sequent list \<times> sequent"   and R :: "(sequent list \<times> sequent) set"   and S :: "sequent" assumes "r \<in> R"   and "case r of (x, xa) \<Rightarrow> (x, xa) \<in> Ax" shows "extendRule S r \<in>  R*"
mu': shows "fs.gs.\<mu> k (k - 1) * (\<parallel>fs.gs.gso (k - 1)\<parallel>\<^sup>2 / \<parallel>fs'.gs.gso (k - 1)\<parallel>\<^sup>2) = fs'.gs.\<mu> k (k - 1)"
eta_cfun: fixes f :: "'e \<rightarrow> 'f" shows "(\<Lambda> x. f\<cdot>x) = f"
sin_cos_eq: fixes x :: "'a" shows "sin x = cos (of_real pi / (2::'a) - x)"
induct: fixes P :: "wall \<Rightarrow> wall \<Rightarrow> bool"   and a0 :: "wall"   and a1 :: "wall" assumes "\<And>x y. P (basic x) (basic y)"   and "\<And>x xs y. (codomain_block y \<noteq> 0 \<Longrightarrow> P xs (basic (make_vert_block (nat (codomain_block y))))) \<Longrightarrow> P (x * xs) (basic y)"   and "\<And>x y ys. (codomain_block x \<noteq> 0 \<Longrightarrow> P (basic (make_vert_block (nat (codomain_block x)))) ys) \<Longrightarrow> P (basic x) (y * ys)"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x * xs) (y * ys)" shows "P a0 a1"
var: fixes i :: "'b" shows "Collect ((\<in>) i) \<in> fba"
simps_1: fixes xs_ys :: "(int \<times> int) list" shows "interpolation_poly_int Newton xs_ys = newton_interpolation_poly_int xs_ys"
mask_eqI: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word" assumes "x AND mask n = y AND mask n"   and "x AND NOT (mask n) = y AND NOT (mask n)" shows "x = y"
cont2cont_case_prod: fixes f :: "'d \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'b \<times> 'c" assumes "\<And>a b. cont (\<lambda>x. f x a b)"   and "\<And>x b. cont (\<lambda>a. f x a b)"   and "\<And>x a. cont (f x a)"   and "cont g" shows "cont (\<lambda>x. case g x of (a, b) \<Rightarrow> f x a b)"
inheritedUNEq: fixes A :: "'b set"   and subs :: "'c \<Rightarrow> 'c set"   and P :: "(nat \<times> 'c) set \<Rightarrow> bool"   and B :: "'b \<Rightarrow> (nat \<times> 'c) set" assumes "finite A"   and "inherited subs P" shows "(\<forall>x\<in>A. P (B x)) = P (\<Union> (B ` A))"
iterSTR_termFSbImorph_errMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSb MOD" shows "termFSbImorph (iterSTR (errMOD MOD)) (iterAbsSTR (errMOD MOD)) (errMOD MOD)"
N: fixes k :: "nat" assumes "N \<le> k" shows "u k * indicat_real A k < e / 4"
P4_b: fixes x :: "'a" shows "x r\<rightarrow> (1::'a) = (1::'a)"
martingaleAE: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure"   and X :: "'d \<Rightarrow> 'c \<Rightarrow> real"   and t :: "'d"   and s :: "'d" assumes "martingale M F X"   and "t \<le> s" shows "AE w in M. real_cond_exp M (F t) (X s) w = X t w"
the_inv_into_into: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'b"   and B :: "'a set" assumes "inj_on f A"   and "x \<in> f ` A"   and "order.greater_eq B A" shows "the_inv_into A f x \<in> B"
add: fixes x :: "'a"   and y :: "'a"   and ns :: "('a \<times> 'a) set"   and X :: "'a multiset"   and Y :: "'a multiset" assumes "(x, y) \<in> ns"   and "(X, Y) \<in> multpw ns" shows "(add_mset x X, add_mset y Y) \<in> multpw ns"
Ord_minimal: fixes k :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "Ord k"   and "P k" shows "\<exists>n. Ord n \<and> P n \<and> (\<forall>m. Ord m \<and> P m \<longrightarrow> n \<le> m)"
t_design_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and index :: "int" shows "t_design point_set block_collection u_block_size grouping index \<equiv> incomplete_design point_set block_collection u_block_size \<and> t_wise_balance point_set block_collection grouping index \<and> t_design_axioms u_block_size grouping"
fold_congs_8: fixes r :: "('a, 'b) state_scheme"   and r' :: "('a, 'b) state_scheme"   and v' :: "('a \<times> 'a) set"   and f :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   and f' :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" assumes "r = r'"   and "cross_edges r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "cross_edges_update f r = cross_edges_update f' r'"
that: fixes K' :: "'a" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> R \<longmapsto> K'\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> R'"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>R \<turnstile> M \<leftrightarrow> K'"   and "A\<^sub>R \<sharp>* K'" shows "thesis"
oinvariant_initD: fixes A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and s :: "'a \<times> 'b" assumes "A \<Turnstile> (S, U \<rightarrow>) P"   and "s \<in> init A" shows "P s"
abs_eq_iff: fixes x :: "complex_homo_coords"   and y :: "complex_homo_coords" shows "(abs_complex_homo x = abs_complex_homo y) = (x \<approx> y)"
simps_9: fixes v :: "'b match_expr" shows "opt_MatchAny_match_expr_once (MatchAnd MatchAny (MatchNot v)) = opt_MatchAny_match_expr_once (MatchNot v)"
ss_422v: fixes x :: "'a"   and y :: "'a" assumes "is_p_fun x" shows "x ; - y = x ; 1 \<cdot> - (x ; y)"
co_path_simps_2: 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 = (fst e, w)"   and "e2 = (w, snd e)" shows "co_path e w (e1 # e2 # es) = e # co_path e w es"
length_pos_if_in_set: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "xs \<in># length"
node_rel_def: fixes Re :: "('b \<times> 'd) set"   and R :: "('a \<times> 'c) set" shows "\<langle>Re, R\<rangle>node_rel = {(\<lparr>name_impl = namei, incoming_impl = inci, new_impl = newi, old_impl = oldi, next_impl = nexti, \<dots> = morei\<rparr>, \<lparr>name = name, incoming = inc, new = new, old = old, next = next, \<dots> = more\<rparr>) |namei name inci inc newi new oldi old nexti next morei more. (namei, name) \<in> nat_rel \<and> (inci, inc) \<in> \<langle>nat_rel\<rangle>dflt_rs_rel \<and> (newi, new) \<in> \<langle>\<langle>R\<rangle>ltlr_rel\<rangle>lss.rel \<and> (oldi, old) \<in> \<langle>\<langle>R\<rangle>ltlr_rel\<rangle>lss.rel \<and> (nexti, next) \<in> \<langle>\<langle>R\<rangle>ltlr_rel\<rangle>lss.rel \<and> (morei, more) \<in> Re}"
tailrec_transform: fixes m :: "'d nres"   and a :: "'d \<Rightarrow> 'd nres"   and b :: "'d \<Rightarrow> 'c nres" shows "m \<bind> REC (\<lambda>D s. sup (a s \<bind> D) (b s)) = lfp (\<lambda>x. sup m (x \<bind> a)) \<bind> b"
gauge_def: fixes \<gamma> :: "'a \<Rightarrow> 'a set" shows "gauge \<gamma> = (\<forall>x. x \<in> \<gamma> x \<and> open (\<gamma> x))"
emb_step_at_left_context: fixes p :: "dir list"   and dir :: "dir"   and t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "emb_step_at (dir.Left # p) dir (App t s) = App (emb_step_at p dir t) s"
assms_1: shows "finite {t. l1 t \<bullet> c = d \<and> 0 \<le> t \<and> t \<le> 1}"
bind_returns_result_E4: fixes h :: "'l"   and f :: "('l, 'm, 'o) prog"   and g :: "'o \<Rightarrow> ('l, 'm, 'n) prog"   and y :: "'n"   and x :: "'o" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"   and "h \<turnstile> f \<rightarrow>\<^sub>r x" obtains h' :: "'l" where "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>r y"
tm_induct_apps: fixes P :: "('c, 'd) tm \<Rightarrow> bool"   and s :: "('c, 'd) tm" assumes "\<And>\<zeta> ss. (\<And>s. s \<in> set ss \<Longrightarrow> P s) \<Longrightarrow> P (apps (Hd \<zeta>) ss)" shows "P s"
partList4_empty: fixes ts :: "'b set list"   and s :: "'b set" assumes "{} \<notin> set ts" shows "{} \<notin> set (partList4 s ts)"
not_iEx: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "(\<not> (\<diamond> t I. P t)) = (\<box> t I. \<not> P t)"
basis_proj_def: shows "local.basis_proj = (SOME B. countable B \<and> topological_basis B)"
Int_Un_eq_1: fixes S :: "'b set"   and T :: "'b set" shows "S \<inter> T \<union> S = S"
convex_halfspace_le: fixes a :: "'a"   and b :: "real" shows "convex {x. a \<bullet> x \<le> b}"
mod_add_left_eq: 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 a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (modulo a c) b) c = modulo (plus a b) c"
dropWhile_eq_self_iff: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(dropWhile P xs = xs) = (xs = [] \<or> \<not> P (hd xs))"
isPath_edgeset: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat"   and e :: "nat \<times> nat" assumes "Graph.isPath c u p v"   and "e \<in> set p" shows "e \<in> Graph.E c"
increasing_measurable_info: fixes f :: "'a \<Rightarrow> 'c"   and s :: "nat"   and N :: "'c measure"   and t :: "nat" assumes "f \<in> F s \<rightarrow>\<^sub>M N"   and "s \<le> t" shows "f \<in> F t \<rightarrow>\<^sub>M N"
intro: fixes x :: "'b \<Longrightarrow> 'a" assumes "\<And>\<sigma> v. get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v" shows "weak_lens x"
id2: shows "(\<Sum>i\<leftarrow>[Suc j..<Suc n]. local.c i * poly (local.N i) (local.x j)) = (0::'a)"
Fun_incl: fixes f :: "'s" assumes "S.incl f" shows "S.Fun f = (\<lambda>x\<in>S.set (S.dom f). x)"
simps_7: fixes v :: "'b val"   and vs :: "'b val list" shows "flat (Stars (v # vs)) = flat v @ flat (Stars vs)"
simps: fixes x :: "'b ext0"   and y :: "'b ext0" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
proj_plus: fixes k :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly k (p + q) = proj_poly k p + proj_poly k q"
card_eq_sum: fixes A :: "'d set" shows "card A = (\<Sum>x\<in>A. 1)"
notin_fset: fixes x :: "'e"   and S :: "'e fset" shows "(x |\<notin>| S) = (x \<notin> fset S)"
il_idempotent: fixes x :: "'a" shows "x \<squnion> x = x"
FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m_refines_FORK: fixes i :: "nat" shows "FORK i \<sqsubseteq> FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i 0"
\<Phi>_simp_2: fixes y :: "'d"   and x :: "'c" assumes "D.ide y"   and "C.ide x" shows "\<Phi> (y, x) = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x)) (\<phi>D (y, G x) \<circ> \<phi> y \<circ> \<psi>C (F y, x))"
prod_mset_subset_imp_dvd: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "A \<subseteq># B" shows "\<Prod>\<^sub># A dvd \<Prod>\<^sub># B"
simps_1: fixes uu :: "bool"   and uv :: "'a tree" shows "mrbst_inter_diff uu \<langle>\<rangle> uv = return_pmf \<langle>\<rangle>"
mult_of_int_commute: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "int"   and y :: "'c" assumes "class.ring_1 one times plus zero minus uminus" shows "times (ring_1.of_int one plus zero minus x) y = times y (ring_1.of_int one plus zero minus x)"
load_deterministic: fixes p :: "'prog"   and s :: "'state"   and s' :: "'state" assumes "load p s"   and "load p s'" shows "s = s'"
a: shows "idc \<Delta> y cmaps b to Some (x, M)"
fut_def: shows "fut = (if b then nFOR else nFAND) (snd (while_default fut_default (\<lambda>(\<phi>, \<Phi>). \<phi> \<notin> set \<Phi>) fut_step (\<psi>, [])))"
cong_pow: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo (power.power one times b n) (power.power one times c n) a"
\<nu>_in_hom_1: shows "\<guillemotleft>\<nu> : src f \<rightarrow> trg r\<guillemotright>"
set_val_typess_preserved: fixes w :: "((_) heap, exception, unit) prog"   and character_data_ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_val_locs character_data_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "ShadowRootClass.type_wf h = ShadowRootClass.type_wf h'"
bounded_clinear_minus: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_clinear f" shows "bounded_clinear (\<lambda>x. - f x)"
hyps: fixes t :: "('a, 'b) tm" shows "emb_step_at (p @ q) d2 (emb_step_at p d1 t) = emb_step_at p d1 (emb_step_at (p @ [d1] @ q) d2 t)"
simps_1: fixes n :: "nat"   and i :: "nat"   and k :: "nat" shows "liftn n (Var i) k = (if i < k then Var i else Var (i + n))"
f_Exec_Stream_Init_nth_Suc_eq_f_Exec_Stream_nth: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d"   and n :: "nat" shows "f_Exec_Comp_Stream_Init trans_fun xs c ! Suc n = f_Exec_Comp_Stream trans_fun xs c ! n"
simps_3: fixes ops :: "'b arith_ops_record"   and v :: "'b"   and va :: "'b list" shows "minus_poly_i ops [] (v # va) = uminus_poly_i ops (v # va)"
orbit_not_empty: fixes A :: "'c set" assumes "finite M"   and "A \<in> orbits" shows "0 < card A"
induct: fixes P :: "('b \<times> nat) tree \<Rightarrow> ('b \<times> nat) tree \<Rightarrow> bool"   and a0 :: "('b \<times> nat) tree"   and a1 :: "('b \<times> nat) tree" assumes "\<And>t. P \<langle>\<rangle> t"   and "\<And>v va vb. P \<langle>v, va, vb\<rangle> \<langle>\<rangle>"   and "\<And>l1 a1 n1 r1 l2 a2 n2 r2. \<lbrakk>a1 \<le> a2 \<Longrightarrow> P r1 \<langle>l2, (a2, n2), r2\<rangle>; \<not> a1 \<le> a2 \<Longrightarrow> P \<langle>l1, (a1, n1), r1\<rangle> r2\<rbrakk> \<Longrightarrow> P \<langle>l1, (a1, n1), r1\<rangle> \<langle>l2, (a2, n2), r2\<rangle>" shows "P a0 a1"
submodule_def: fixes R :: "('e, 'f) ring_scheme"   and N :: "'g set"   and M :: "('e, 'g, 'h) module_scheme" shows "LinearCombinations.submodule R N M \<equiv> (module R M \<and> N \<subseteq> carrier M) \<and> (\<forall>v w. v \<in> N \<longrightarrow> w \<in> N \<longrightarrow> v \<oplus>\<^bsub>M\<^esub> w \<in> N) \<and> \<zero>\<^bsub>M\<^esub> \<in> N \<and> (\<forall>c v. c \<in> carrier R \<longrightarrow> v \<in> N \<longrightarrow> c \<odot>\<^bsub>M\<^esub> v \<in> N)"
y3_prop: shows "r < y3 \<and> (\<forall>(d, e, f)\<in>set d. \<forall>x\<in>{r<..y3}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
cross_zero_left: fixes x :: "(real, 3) vec" shows "0 \<times> x = 0"