hcju/isabelleps huggingface dataset:
simps_2: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and x :: "('a, 'd, 'b) formula"   and n :: "nat" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "indexed_weak_nominal_ts.tau_steps \<tau> x (Suc n) = Act \<tau> (indexed_weak_nominal_ts.tau_steps \<tau> x n)"
lset_ltake_Suc: fixes xs :: "'c llist"   and x :: "'c"   and n :: "nat"   and A :: "'c set" assumes "\<not> lnull xs"   and "xs $ 0 = x"   and "lset (ltake (enat n) (ltl xs)) \<subseteq> A" shows "lset (ltake (enat (Suc n)) xs) \<subseteq> insert x A"
Incremental_State_Ops_Simplex_axioms: shows "Incremental_State_Ops_Simplex assert_bound init check"
lfilter_eq_LConsD: fixes P :: "'a \<Rightarrow> bool"   and ys :: "'a llist"   and x :: "'a"   and xs :: "'a llist" assumes "lfilter P ys = LCons x xs" shows "\<exists>us vs. ys = lappend us (LCons x vs) \<and> lfinite us \<and> (\<forall>u\<in>lset us. \<not> P u) \<and> P x \<and> xs = lfilter P vs"
fundamental_subgr: fixes A :: "'a set" assumes "finite A"   and "A \<subseteq> carrier G" obtains gs :: "'a list" where "set gs \<subseteq> generate G A" and "cycle gs" and "is_idirprod (generate G A) (\<lambda>g. generate G {g}) (set gs)" and "successively (dvd) (map ord gs)" and "card (set gs) \<le> card A"
rel_Grp: fixes A1 :: "'c set"   and f1 :: "'c \<Rightarrow> 'f"   and A2 :: "'d set"   and f2 :: "'d \<Rightarrow> 'g" shows "rel_gpv (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. results'_gpv x \<subseteq> A1 \<and> outs'_gpv x \<subseteq> A2} (map_gpv f1 f2)"
length_update_until: fixes args :: "Monitor.args"   and rel1 :: "event_data option list set"   and rel2 :: "event_data option list set"   and nt :: "nat"   and aux :: "(nat \<times> event_data option list set \<times> event_data option list set) list" shows "length (update_until args rel1 rel2 nt aux) = Suc (length aux)"
M_funcset_rcosets_H: shows "(\<lambda>x\<in>M. H #> (SOME g. g \<in> carrier G \<and> M1 #> g = x)) \<in> M \<rightarrow> rcosets H"
fin_J: shows "finite {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}"
A: shows "concat (take (Suc n) l) = concat (take (Suc x) l)"
Quotient_list: shows "Quotient (=) list_R1.list_R1_to_list list_R1.list_to_list_R1 list_R1.cr_list"
accepts_def: fixes A :: "'d \<times> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<times> ('d \<Rightarrow> bool)" shows "accepts A = (\<lambda>w. fin A (delta A w (start A)))"
bdd_above_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.preorder less_eq less" shows "preorder.bdd_above less_eq A = (\<exists>M. \<forall>x\<in>A. less_eq x M)"
nth_tl_length: fixes sl :: "'b list" assumes "tl sl \<noteq> []" shows "tl sl ! (length (tl sl) - Suc 0) = sl ! (length sl - Suc 0)"
of_nat_eq_of_nat_power_cancel_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and b :: "nat"   and w :: "nat"   and x :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(power.power one times (semiring_1.of_nat one plus zero b) w = semiring_1.of_nat one plus zero x) = (b ^ w = x)"
psubset_imp_subset: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subset> B" shows "A \<subseteq> B"
xor_zero_right: fixes x :: "'a" shows "x \<oplus> \<zero> = x"
chosen_right_adjoints_axioms: shows "chosen_right_adjoints (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"
FSwImorph_def: fixes h :: "'c \<Rightarrow> 'd"   and hA :: "'e \<Rightarrow> 'f"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'c, 'e, 'h) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'd, 'f, 'i) model_scheme" shows "FSwImorph h hA MOD MOD' \<equiv> ipresIGWlsAll h hA MOD MOD' \<and> ipresIGCons h hA MOD MOD' \<and> ipresIGFreshAll h hA MOD MOD' \<and> ipresIGSwapAll h hA MOD MOD'"
cA: shows "char_matrix A lam \<in> carrier_mat n n"
circle_top_bot_edges_neq': shows "circle_top_edge \<noteq> circle_bot_edge"
e_def: shows "e \<equiv> fst (normalize_quot (a, d))"
elims_2: fixes x :: "nat"   and xa :: "vname gexp" assumes "tests_input_equality x xa" obtains i :: "nat" and i' :: "nat" and uu_ :: "value" where "x = i" and "xa = Eq (V (I i')) (L uu_)" and "i = i'"
simps_6: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'b stream" shows "smap f (x1 ## x2) = f x1 ## smap f x2"
case_2: fixes f1 :: "'k"   and f2 :: "'l \<Rightarrow> 'k"   and x2 :: "'l" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
that: fixes n :: "nat" assumes "arity_sym\<^sub>h f = of_nat n" shows "thesis"
simps_6: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and \<phi>\<^sub>1 :: "'a formula"   and \<phi>\<^sub>2 :: "'a formula" shows "((w, I) \<Turnstile> FAnd \<phi>\<^sub>1 \<phi>\<^sub>2) = ((w, I) \<Turnstile> \<phi>\<^sub>1 \<and> (w, I) \<Turnstile> \<phi>\<^sub>2)"
ordIso_transitive: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and r'' :: "('e \<times> 'e) set" assumes "r =o r'"   and "r' =o r''" shows "r =o r''"
rep_eq: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'c \<Rightarrow>\<^sub>0 'd" assumes "inj f" shows "lookup (map_key f x) = lookup x \<circ> f"
psi_bounds_ln_fact_2: fixes n :: "nat" shows "psi n - psi (n div 2) \<le> ln (fact n) - 2 * ln (fact (n div 2))"
nth_least_Max: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "nth_least A (card A - 1) = Max A"
SATAxiomProofTree: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "SATAxiom (sequent gamma)" shows "proofTree (tree subs gamma)"
Inr_Field_csum: fixes a :: "'a"   and s :: "('a \<times> 'a) set"   and r :: "('b \<times> 'b) set" assumes "a \<in> Field s" shows "Inr a \<in> Field (BNF_Cardinal_Arithmetic.csum r s)"
zero_less_nat_eq: fixes z :: "int" shows "z \<in># nat = (0 < z)"
simps_1: shows "0\<^sub>\<nat> = []\<^sub>\<circ>"
enumerate_mono: fixes m :: "nat"   and n :: "nat"   and S :: "'a set" assumes "m < n"   and "infinite S" shows "wellorder_class.enumerate S m < wellorder_class.enumerate S n"
gcsort_def: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and xs :: "'c list" shows "gcsort index key p xs \<equiv> gcsort_out (gcsort_aux index key p (gcsort_in xs))"
exec_step_ind_Return_Static: fixes D :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "jvm_method"   and P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and h :: "heap"   and stk\<^sub>0 :: "val list"   and loc\<^sub>0 :: "val list"   and pc :: "nat"   and ics :: "init_call_status"   and stk' :: "val list"   and loc' :: "val list"   and C' :: "char list"   and m' :: "char list"   and pc' :: "nat"   and ics' :: "init_call_status"   and frs' :: "frame list"   and sh :: "sheap" assumes "(D, Static, Ts, T, m) = method P C\<^sub>0 M\<^sub>0"   and "M\<^sub>0 \<noteq> clinit" shows "exec_step_ind (StepI Return) P h stk\<^sub>0 loc\<^sub>0 C\<^sub>0 M\<^sub>0 pc ics ((stk', loc', C', m', pc', ics') # frs') sh (None, h, (hd stk\<^sub>0 # drop (length Ts) stk', loc', C', m', Suc pc', ics') # frs', sh)"
cases: fixes x :: "atomUni" obtains p :: "real \<times> real \<times> real" where "x = EqUni pa__"   | p :: "real \<times> real \<times> real" where "x = LessUni pb__"   | p :: "real \<times> real \<times> real" where "x = LeqUni pc__"   | p :: "real \<times> real \<times> real" where "x = NeqUni p"
not_outside_1: fixes x :: "'a"   and y :: "'a" assumes "le x y" shows "x \<in> carrier"
kD_Some: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> kD rt" shows "\<exists>dsn dsk flag hops nhip pre. rt dip = Some (dsn, dsk, flag, hops, nhip, pre)"
parent_opt: shows "h \<turnstile> get_parent node \<rightarrow>\<^sub>r parent_opt"
simps_7: fixes x3 :: "'a" shows "Diverges \<noteq> Goes_wrong x3"
simps: fixes t :: "'b tree"   and w :: "nat \<times> nat" shows "reusables t w = (if r t < fst w then [] else if fst w = l t then [t] else let v = lchild t; u = rchild t in if l u \<le> fst w then reusables u w else u # reusables v w)"
close_def: fixes C :: "('c, 'd) form set set" shows "close C = {S. \<exists>S'\<in>C. S \<subseteq> S'}"
monoid_homomorphism_axioms: shows "monoid_homomorphism induced Partition ([\<cdot>]) (Class \<one>) M' (\<cdot>') \<one>'"
less: fixes p :: "'a poly" assumes "degree p < degree pa__" shows "Q p"
setmult_lcos_assoc: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "K \<subseteq> carrier G"   and "x \<in> carrier G" shows "x <# H <#> K = x <# (H <#> K)"
ListReds2: fixes es :: "expr list"   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 es' :: "expr list"   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 v :: "val" assumes "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es',s'\<rangle>" shows "P,E \<turnstile> \<langle>Val v # es,s\<rangle> [\<rightarrow>]* \<langle>Val v # es',s'\<rangle>"
build_rule_index_f_def: fixes \<delta> :: "('b, 'a) ta_rule dlist" shows "build_rule_index_f \<delta> \<equiv> build_rule_index rhsl \<delta>"
Lim_null: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and net :: "'a filter" shows "(f \<longlongrightarrow> l) net = ((\<lambda>x. f x - l) \<longlongrightarrow> (0::'b)) net"
balR_def: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "balR l k v r = (let rl = rsub l in if cl r = R then Node l R k v (Node (lsub r) B (key r) (val r) (rsub r)) else if l = Leaf then Node l R k v r else if cl l = B then balance (Node (lsub l) R (key l) (val l) (rsub l)) k v r else if rl = Leaf then Node l R k v r else if cl rl = B then Node (balance (paint R (lsub l)) (key l) (val l) (lsub rl)) R (key rl) (val rl) (Node (rsub rl) B k v r) else Node l R k v r)"
prv_ldsj_eqv: fixes \<phi>1s :: "'fmla list"   and \<phi>2s :: "'fmla list" assumes "set \<phi>1s \<subseteq> fmla"   and "set \<phi>2s = set \<phi>1s" shows "prv (eqv (ldsj \<phi>1s) (ldsj \<phi>2s))"
iffb_def: fixes \<phi> :: "'d set \<Rightarrow> 'c set" shows "fb\<^sup>-\<^sub>\<F> \<phi> = (\<lambda>x. \<Inter> {X. s2p (\<phi> X) x})"
inv_solve_right: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "c \<in> carrier Zp" shows "(a = b \<oplus> \<ominus> c) = (b = a \<oplus> c)"
ceiling_add_of_int: fixes x :: "'a"   and z :: "int" shows "\<lceil>x + of_int z\<rceil> = \<lceil>x\<rceil> + z"
fid_on_iff: fixes a :: "V"   and b :: "V"   and A :: "V" shows "([a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> fid_on A) = (a = b \<and> a \<in>\<^sub>\<circ> A)"
match_final_forward: fixes s1 :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state"   and s2 :: "('fenv_ubx, 'henv, ('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame) state" assumes "s1 \<sim> s2"   and "Sinca.final s1" shows "Subx.final s2"
strict_trans2: fixes b :: "'f"   and a :: "'f"   and c :: "'f" assumes "b < a"   and "c \<le> b" shows "c < a"
assms: shows "finite {(x, y). P x \<and> Q y}"
is_int_rat_def: fixes x :: "rat" shows "is_int_rat x \<equiv> snd (quotient_of x) = 1"
abs_summable_on_Times_swap: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and A :: "'c set"   and B :: "'d set" shows "(f abs_summable_on A \<times> B) = ((\<lambda>(x, y). f (y, x)) abs_summable_on B \<times> A)"
d_delta_arctan_upper_44_pos: fixes x :: "real" shows "diff_delta_arctan_upper_44 x < 0"
is_ground_cls_def: fixes C :: "('a, 'b) Term.term literal multiset" shows "is_ground_cls C = (\<forall>L. L \<in># C \<longrightarrow> is_ground_lit L)"
fps_shift_add: fixes n :: "nat"   and f :: "'a fps"   and g :: "'a fps" shows "fps_shift n (f + g) = fps_shift n f + fps_shift n g"
poly_ge_trans: fixes p1 :: "('c monom \<times> 'd) list"   and p2 :: "('c monom \<times> 'd) list"   and p3 :: "('c monom \<times> 'd) list" assumes "p1 \<ge>p p2"   and "p2 \<ge>p p3" shows "p1 \<ge>p p3"
cproper_interval_finite_2_def: shows "cproper_interval = proper_interval"
get_impl_zip_simp: shows "get_impl (zip (TopoS_Composition_Theory_impl.get_IFS (get_impl M)) (TopoS_Composition_Theory.get_IFS (get_spec M))) = TopoS_Composition_Theory_impl.get_IFS (get_impl M)"
closed_except_self: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and t :: "'a" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.closed_except frees t (frees t)"
pr_iso_test1_corr: fixes Fs\<^sub>1 :: "'a list list"   and Fs\<^sub>2 :: "'b list list" assumes "\<forall>F\<in>set Fs\<^sub>1. distinct F"   and "\<forall>F\<in>set Fs\<^sub>2. distinct F"   and "[] \<notin> set Fs\<^sub>2"   and "distinct Fs\<^sub>1"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>1)"   and "distinct Fs\<^sub>2"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>2)" shows "pr_iso_test1 Map.empty Fs\<^sub>1 Fs\<^sub>2 = (\<exists>\<phi>. is_pr_iso \<phi> Fs\<^sub>1 Fs\<^sub>2)"
cball_def: fixes x :: "'b"   and e :: "real" shows "cball x e = {y. dist x y \<le> e}"
INF_greatest: fixes A :: "'b set"   and u :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> A \<Longrightarrow> u \<le> f i" shows "u \<le> \<Sqinter>(f ` A)"
simps: fixes a :: "('test, 'atom, 'choice) cmd" shows "siso a = (\<exists>c. a = c \<and> (\<forall>x xa. xa < brn c \<longrightarrow> siso (local.cont c x xa)) \<and> (\<forall>x xa xb. xb < brn c \<and> x \<approx> xa \<longrightarrow> eff c x xb \<approx> eff c xa xb \<and> wt c x xb = wt c xa xb \<and> local.cont c x xb = local.cont c xa xb))"
"case": fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and x1 :: "'f"   and x2 :: "'g" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
rev_append: fixes xs :: "'k list"   and ys :: "'k list" shows "rev (xs @ ys) = rev ys @ rev xs"
divisor_set: fixes m :: "nat" shows "divisor_set m = {n. n divisor m}"
ev_cong: fixes P :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream"   and Q1 :: "'b stream \<Rightarrow> bool"   and Q2 :: "'b stream \<Rightarrow> bool" assumes "alw P \<omega>"   and "\<And>\<omega>. P \<omega> \<Longrightarrow> Q1 \<omega> = Q2 \<omega>" shows "ev Q1 \<omega> = ev Q2 \<omega>"
starfun_n: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and X :: "nat \<Rightarrow> 'b" shows "(*fn* f) (star_n X) = star_n (\<lambda>n. f n (X n))"
assms_2: shows "normalized_n_primitive (disc2, sel2) P m"
mk_linear_orders_Linear_order: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and r :: "'a list"   and C :: "'a set" assumes "f_range_on A f"   and "r \<in> mk_linear_orders f C A" shows "Linear_order (linord_of_list r)"
has_integral_spike: fixes S :: "'b set"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"   and "(f has_integral y) T" shows "(g has_integral y) T"
op_strict_1: fixes pat :: "[:'r:]" shows "op\<cdot>pat\<cdot>\<bottom> = \<bottom>"
hol1f: shows "(\<lambda>z. 1 - (f z)\<^sup>2) holomorphic_on S"
directly_attracted_bounded_by_V: fixes p :: "Player"   and W :: "'a set" shows "directly_attracted p W \<subseteq> V"
find_index_not_in_set: fixes x :: "'a"   and xs :: "'a list" shows "(x \<notin> set xs) = (find_index xs x = length xs)"
f_approx_pos: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and integrable :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> bool"   and integral :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and C :: "real"   and x :: "real" assumes "akra_bazzi_real as bs hs k x\<^sub>0 x\<^sub>1 hb e p integrable integral f g C"   and "x\<^sub>0 \<le> x" shows "0 < akra_bazzi_real.f_approx x\<^sub>0 p integral g x"
rep_eq: fixes x :: "integer" shows "int_of_integer (log2_integer x) = log2 (int_of_integer x)"
assms_5: shows "\<not> (d \<or> False) \<or> c"
tauLaw4SimLeft: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<And>\<Psi> P. (\<Psi>, P, P) \<in> Rel"   and "\<And>\<Psi> P Q. \<Psi> \<rhd> P \<sim> Q \<Longrightarrow> (\<Psi>, P, Q) \<in> Rel" shows "\<Psi> \<rhd> (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q \<leadsto><Rel> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q"
map_of_eq_None_iff: fixes xys :: "('d \<times> 'c) list"   and x :: "'d" shows "(map_of xys x = None) = (x \<notin> fst ` set xys)"
R29_ac_1: shows "pmf (sds R29) a = 1 / 2"
le_IFrel_Comp: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x1 :: "'d IF1"   and y1 :: "'f IF1"   and x2 :: "'d IF2"   and y2 :: "'f IF2" shows "((IF1rel R OO IF1rel S) x1 y1 \<longrightarrow> IF1rel (R OO S) x1 y1) \<and> ((IF2rel R OO IF2rel S) x2 y2 \<longrightarrow> IF2rel (R OO S) x2 y2)"
simps_3: fixes x2 :: "'e" shows "Some x2 \<noteq> None"
Gauss_Jordan_in_ij_preserves_previous_elements: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and a :: "'c"   and j :: "'b"   and i :: "'c" assumes "reduced_row_echelon_form_upt_k A k"   and "\<not> is_zero_row_upt_k a k A"   and "\<exists>m. A $ m $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> (GREATEST m. \<not> is_zero_row_upt_k m k A) + (1::'c) \<le> m"   and "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'c) \<noteq> (0::'c)"   and "mod_type_class.to_nat j < k" shows "Gauss_Jordan_in_ij A ((GREATEST m. \<not> is_zero_row_upt_k m k A) + (1::'c)) (mod_type_class.from_nat k) $ i $ j = A $ i $ j"
colossless_restrict_gpvI: fixes gpv :: "('b, 'out, 'in) gpv" assumes "colossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>" shows "colossless_gpv \<I> (local.restrict_gpv gpv)"
has_dom_arev: shows "has_dom (edge_rev M) (arcs (del_vert u))"
S_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "S x \<le> S y"
removeProcs_length': fixes ps :: "'b pState_scheme list" shows "length (fst (snd (snd (removeProcs ps)))) \<le> length ps"
matrix_mul_assoc: fixes A :: "(('c, 'f) vec, 'e) vec"   and B :: "(('c, 'g) vec, 'f) vec"   and C :: "(('c, 'd) vec, 'g) vec" shows "A ** (B ** C) = A ** B ** C"
split_snil: fixes pat :: "[:'r:]" assumes "pat \<noteq> \<bottom>" shows "split\<cdot>pat\<cdot>[::] = ([::], pat)"
u_g_is_u_g: shows "usesOf \<circ> (\<lambda>g. Mapping (\<lambda>n. if step.u_g g n = {} then None else Some (step.u_g g n))) = step.u_g"
lookup_single: fixes k :: "'b"   and v :: "'a"   and k' :: "'b" shows "lookup (Poly_Mapping.single k v) k' = (v when k = k')"
regular_complement_top: fixes x :: "'a" assumes "regular x" shows "x \<squnion> - x = top"
sympartp_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" shows "sympartp r x y \<equiv> r x y \<and> r y x"
F: shows "\<nexists>f. t = Fun f [] \<or> t' = Fun f []"
point_dist: fixes xA :: "real"   and yA :: "real"   and xB :: "real"   and yB :: "real" shows "dist (Point xA yA) (Point xB yB) = sqrt ((xA - xB)\<^sup>2 + (yA - yB)\<^sup>2)"
Arcsin_range_lemma: fixes z :: "complex" assumes "\<bar>Re z\<bar> < 1" shows "0 < Re (\<i> * z + csqrt (1 - z\<^sup>2))"
simps_5: fixes f1 :: "'a"   and f2 :: "'a" shows "rec_T1 f1 f2 T1.A = f1"
sorted_nths: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and I :: "nat set" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (nths xs I)"
Quadratic_Reciprocity_Jacobi': fixes p :: "int"   and q :: "int" assumes "coprime p q"   and "2 < p"   and "2 < q"   and "odd p"   and "odd q" shows "Jacobi q p = (if p mod 4 = 3 \<and> q mod 4 = 3 then - 1 else 1) * Jacobi p q"
f_bij: shows "bij_betw f {..<length ss2} {..<length ss1}"
induct: fixes x1 :: "'node SDG_node"   and x2 :: "'node SDG_node list"   and x3 :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> 'node SDG_node list \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "x1 ics-x2\<rightarrow>\<^sub>d* x3"   and "\<And>n. valid_SDG_node n \<Longrightarrow> P n [] n"   and "\<And>n ns n'' n'. \<lbrakk>n ics-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' s\<longrightarrow>\<^bsub>cd\<^esub> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'"   and "\<And>n ns n'' V n'. \<lbrakk>n ics-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' s-V\<rightarrow>\<^sub>d\<^sub>d n'; n'' \<noteq> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'"   and "\<And>n ns n'' p n'. \<lbrakk>n ics-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'"   and "\<And>n ns n'' p n'. \<lbrakk>n ics-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' s-p\<rightarrow>\<^bsub>call\<^esub> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'"   and "\<And>n ns n'' p V n'. \<lbrakk>n ics-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' s-p:V\<rightarrow>\<^bsub>in\<^esub> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'" shows "P x1 x2 x3"
assms_1: shows "w \<in> path_image (part_circlepath z r s t)"
fixp_above_unfold: fixes a :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "a \<in> Field leq" shows "fixp_above a = link (fixp_above a)"
d: shows "length (select l i j) = length (selecthelp k (j - i))"
rcosetsI: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #> x \<in> rcosets H"
poly_deg_one: shows "poly_deg 1 = (0::'b)"
length_seqs: fixes xs :: "bool list"   and n :: "nat" assumes "xs \<in> seqs n" shows "length xs = n"
induct: fixes x1 :: "dB"   and x2 :: "dB"   and P :: "dB \<Rightarrow> dB \<Rightarrow> bool" assumes "x1 \<rightarrow>\<^sub>\<beta> x2"   and "\<And>s t. P (Abs s \<degree> t) (s[t/0])"   and "\<And>s t u. \<lbrakk>s \<rightarrow>\<^sub>\<beta> t; P s t\<rbrakk> \<Longrightarrow> P (s \<degree> u) (t \<degree> u)"   and "\<And>s t u. \<lbrakk>s \<rightarrow>\<^sub>\<beta> t; P s t\<rbrakk> \<Longrightarrow> P (u \<degree> s) (u \<degree> t)"   and "\<And>s t. \<lbrakk>s \<rightarrow>\<^sub>\<beta> t; P s t\<rbrakk> \<Longrightarrow> P (Abs s) (Abs t)" shows "P x1 x2"
oInv_ex: fixes a :: "ordinal" assumes "F 0 \<le> a" shows "\<exists>q. F q \<le> a \<and> a < F (oSuc q)"
induces_hom_adjunction: shows "hom_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) replete_setcat.comp \<phi>C \<phi>D F G \<Phi> \<Psi>"
le_cases: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less_eq y x"