hcju/isabelleps huggingface dataset:
F2set3_Lev2: fixes xf :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xd :: "'e"   and xb :: "'e"   and x :: "'e" assumes "xf \<in> Lev2 s1 s2 n xd"   and "rv2 s1 s2 xf xd = Inr xb"   and "x \<in> F2set3 (s2 xb)" shows "xf @ [Inr (tobd_F23 s2 xb x)] \<in> Lev2 s1 s2 (Suc n) xd"
Id_onI: fixes a :: "'g"   and A :: "'g set" assumes "a \<in> A" shows "(a, a) \<in> Id_on A"
support_on_nonneg_sum_subset': fixes X :: "'c set"   and f :: "'f \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'c \<Rightarrow> 'f set" shows "support_on X (\<lambda>x. \<Sum>i\<in>S x. f i x) \<subseteq> (\<Union>x\<in>X. \<Union>i\<in>S x. support_on X (f i))"
not_MOST: fixes P :: "'c \<Rightarrow> bool" shows "(\<not> (\<forall>\<^sub>\<infinity>x. P x)) = (\<exists>\<^sub>\<infinity>x. \<not> P x)"
rel_is_equivalence: shows "equivalence (Sigma I \<FF>1) {(x, y). stk1.rel x y}"
infer_simp_5: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and A :: "'a trm"   and B :: "'a trm" shows "infer \<Gamma> (SimplyTyped.Pair A B) = (case (infer \<Gamma> A, infer \<Gamma> B) of (None, b) \<Rightarrow> None | (Some \<tau>, None) \<Rightarrow> None | (Some \<tau>, Some \<sigma>) \<Rightarrow> Some (TPair \<tau> \<sigma>))"
lift_lift: fixes i :: "nat"   and k :: "nat"   and t :: "Core.term" assumes "i < k + 1" shows "lift (lift t i) (Suc k) = lift (lift t k) i"
C: shows "n \<notin> bad_shakey \<inter> bad_prikey \<inter> (bad_id_password \<union> bad_id_shak)"
eval\<^sub>c_def: fixes F :: "char list \<Rightarrow> 'b list \<Rightarrow> 'b"   and G :: "char list \<Rightarrow> 'b list \<Rightarrow> bool"   and C :: "fterm literal set" shows "eval\<^sub>c F G C = (\<forall>E. \<exists>l\<in>C. eval\<^sub>l E F G l)"
root_int_floor_pos_lower: fixes p :: "nat"   and x :: "int" assumes "p \<noteq> 0"   and "0 \<le> x" shows "root_int_floor_pos p x ^ p \<le> x"
subfrmlsr_finite: fixes \<phi> :: "'b ltlr" shows "finite (subfrmlsr \<phi>)"
MRList1: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<in> dom (Cp a)" shows "applied_rule_rev Cp x (b @ [a]) = \<lfloor>a\<rfloor>"
g: shows "(\<lambda>x. g x powr 1) \<in> powr_closure g"
CatDom: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme" assumes "PreFunctor F" shows "Category (CatDom F)"
pappus_iff_pascal: shows "is_pappus = pascal_prop"
unit_vec_def: fixes n :: "nat"   and i :: "nat" shows "unit_vec n i = Matrix.vec n (\<lambda>j. if j = i then 1::'a else (0::'a))"
abcd_dcba_only_8: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c d]]" shows "\<not> [[b c a d]]"
Join_assoc: fixes F :: "'b program"   and G :: "'b program"   and H :: "'b program" shows "F \<squnion> G \<squnion> H = F \<squnion> (G \<squnion> H)"
leq: shows "P \<turnstile> C \<preceq>\<^sup>* C'"
list_rel_simp_1: fixes l' :: "'f list"   and R :: "('e \<times> 'f) set" shows "(([], l') \<in> \<langle>R\<rangle>list_rel) = (l' = [])"
suc_least: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "a \<in> Field r"   and "\<And>b. b \<in> B \<Longrightarrow> a \<noteq> b \<and> (b, a) \<in> r" shows "(local.suc B, a) \<in> r"
inf_apply: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "inf f g x = inf (f x) (g x)"
br_chain: fixes \<beta> :: "'c \<Rightarrow> 'e"   and J :: "'c \<Rightarrow> bool"   and \<alpha> :: "'e \<Rightarrow> 'd"   and I :: "'e \<Rightarrow> bool" shows "br \<beta> J O br \<alpha> I = br (\<alpha> \<circ> \<beta>) (\<lambda>s. J s \<and> I (\<beta> s))"
Integer_arith_strict_1: fixes x :: "Integer" shows "\<bottom> + x = \<bottom>"
initialized2_def: fixes A :: "int \<times> day \<Rightarrow> int"   and i :: "int" shows "initialized2 A i = (\<forall>j\<in>{0..<i}. initialized3 A j 7)"
ab_semigroup_add_ow_axioms_def: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "ab_semigroup_add_ow_axioms S pls \<equiv> \<forall>a b. a \<in> S \<longrightarrow> b \<in> S \<longrightarrow> pls a b = pls b a"
crel_vs_executeD: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d Heap"   and heap :: "heap" assumes "crel_vs R a b"   and "P heap"   and "P heap"   and "state_dp_consistency.cmem heap" obtains x :: "'d" and heap' :: "heap" where "execute b heap = Some (x, heap')" and "P heap'" and "P heap'" and "state_dp_consistency.cmem heap'" and "R a x"
abel_semigroup_axioms: shows "abel_semigroup (\<oplus>)"
assms_5: shows "\<forall>w'. w \<rightarrow>\<^sub>e\<^sub>m\<^sub>b w' \<longrightarrow> \<not> subst \<rho> w' \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"
simps_30: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x51 :: "nat"   and x52 :: "nat"   and x53 :: "nat" shows "(case Pkt x51 x52 x53 of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f5 x51 x52 x53"
flowD_capacity: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" assumes "flow \<Delta> f" shows "f e \<le> capacity \<Delta> e"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_backto_standard_pos_2 ires rs (l, r) = (wcode_backto_standard_pos_2_O ires rs (l, r) \<or> wcode_backto_standard_pos_2_B ires rs (l, r))"
Abs'_0: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.Abs' Abs (int CARD('b)) = zero"
zero_index: fixes i :: "'b" shows "0 $h i = (0::'a)"
Liminf_def: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'b" shows "Liminf F f = (SUP P\<in>{P. eventually P F}. Inf (f ` {x. P x}))"
OAlist_tc_sorted_domain_def: fixes xs :: "('a, 'b) oalist_tc" shows "OAlist_tc_sorted_domain xs = map fst (list_of_oalist_tc xs)"
step_nonneg': fixes i :: "nat"   and x :: "real" assumes "i < k"   and "x\<^sub>1 \<le> x" shows "0 \<le> bs ! i + (hs ! i) x / x"
val_imp_close_world: fixes M :: "object atom formula set"   and \<phi> :: "object atom formula" assumes "valuation M \<Turnstile> \<phi>" shows "M \<^sup>c\<TTurnstile>\<^sub>= \<phi>"
assms: shows "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 c, $Grow_memory] : ts _> ts'"
homh: shows "k \<in> hom (free_Abelian_group T) (free_Abelian_group S)"
neg_0_less_iff_less: fixes a :: "'a" shows "((0::'a) < - a) = (a < (0::'a))"
iFROM_Min: fixes n :: "nat" shows "iMin [n\<dots>] = n"
alphaBoundOutput: fixes x' :: "name"   and P :: "pi"   and a :: "name"   and x :: "name" assumes "x' \<sharp> P" shows "a<\<nu>x> \<prec>\<^sub>e P = a<\<nu>x'> \<prec>\<^sub>e [(x, x')] \<bullet> P"
2: shows "subst (subst (rawpsubst \<phi> (zip (map Var vs) xs)) (Var u) x) t u = subst (subst (rawpsubst \<phi> (zip (map Var vs) xs)) (Var v) x) t v"
nc_zero: fixes x :: "'a" shows "(x \<sqinter> nc) \<cdot> (0::'a) = (0::'a)"
assms_3: shows "q * f\<^sup>T \<le> q"
Sup_pres_Sup: fixes F :: "('b \<Rightarrow> 'c) set" assumes "\<forall>f\<in>F. Sup_pres f" shows "Sup_pres (\<Squnion> F)"
fmdom'_alt_def: fixes m :: "('c, 'd) fmap" shows "fmdom' m = fset (fmdom m)"
preserves_reflects_arr: fixes f :: "'a" shows "arr \<t>[f] = arr f"
isCont_tendsto_compose: fixes l :: "'d"   and g :: "'d \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'd"   and F :: "'f filter" assumes "isCont g l"   and "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
homotopic_loops_imp_subset: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "homotopic_loops s p q" shows "path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
ri: fixes q :: "int poly"   and ri :: "root_info" assumes "(q, ri) \<in> set polys" shows "root_info_cond ri q"
conjl2_iff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<lhd> y \<le> z) = (y \<le> - (- z \<rhd> x))"
set_sorted_list_of_set: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
b': shows "b' = e_append b (Partial_Recursive.list_encode (replicate (y - e_length b) 0 @ [1 - the (\<phi> i y)]))"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Add of Eq \<Rightarrow> f1 | Add \<Rightarrow> f2) = f2"
connected_component_of_subset_topspace: fixes X :: "'c topology"   and x :: "'c" shows "connected_component_of_set X x \<subseteq> topspace X"
is_path_rev: fixes v' :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v :: "'v" shows "is_path v' (rev_path ps) v = is_path v ps v'"
toList_empty: fixes t :: "('a, 'b) FingerTreeStruc" shows "(FingerTreeStruc.toList t = []) = (t = Empty)"
root_position: fixes t :: "'b trm" shows "[] \<in> pos_of t"
infdist_def: fixes x :: "'c"   and A :: "'c set" shows "infdist x A = (if A = {} then 0 else Inf (dist x ` A))"
check_poly_weak_anti_mono: fixes p :: "('c monom \<times> 'd) list"   and v :: "'c" assumes "check_poly_weak_anti_mono p v" shows "poly_weak_anti_mono p v"
"3_4": fixes t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (insert\<langle>t,s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>) \<union> fv (s \<cdot> \<theta>)"
connected_component_eq_self: fixes S :: "'c set"   and x :: "'c" assumes "connected S"   and "x \<in> S" shows "connected_component_set S x = S"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat list \<times> nat"   and y :: "(nat list \<times> nat) list" assumes "incs x xa xb = y" obtains a :: "nat" and x :: "nat" and xs :: "nat list" and s :: "nat" where "xa__ = a" and "xa = x" and "xb = (xs, s)" and "y = (let t = s + a * x in if C (x # xs) t then (x # xs, t) # incs a (Suc x) (xs, s) else [])"
ntrs_c_cases: fixes s :: "'a list"   and c :: "'a list multiset"   and ww :: "('b, 'c) label list list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s, c), ww, s', c') \<in> trcl (ntrs fg)" obtains csp :: "'a list multiset" where "c' = csp + c" and "\<And>s. s \<in># csp \<Longrightarrow> \<exists>p u v. s = [entry fg p] \<and> (u, Spawn p, v) \<in> edges fg \<and> initialproc fg p"
enn2real_less: fixes e :: "ennreal"   and r :: "real" assumes "enn2real e < r"   and "e \<noteq> \<top>" shows "e < ennreal r"
weight_spmf_of_set: fixes A :: "'a set" shows "weight_spmf (spmf_of_set A) = (if finite' A then 1 else 0)"
Ineq_def: shows "Ineq \<equiv> \<lambda>\<delta>. list_ex (\<lambda>f. fst f \<cdot> \<delta> \<circ>\<^sub>s \<I> \<noteq> snd f \<cdot> \<delta> \<circ>\<^sub>s \<I>)"
ae1: shows "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Px (fst x) = 0 \<longrightarrow> Pxyz x = 0"
rt_fresh_as_inc_invalidate: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option" assumes "dip \<in> kD rt"   and "\<forall>rip\<in>dom dests. rip \<in> vD rt \<and> the (dests rip) = inc (sqn rt rip)" shows "rt \<approx>\<^bsub>dip\<^esub> invalidate rt dests"
term_ok_code: fixes \<Theta> :: "exetheory"   and t :: "Core.term" assumes "exe_wf_theory' \<Theta>" shows "term_ok (translate_theory \<Theta>) t = exeterm_ok \<Theta> t"
cobounded2: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "signed.min a b \<le>s b"
x_2: shows "(x, \<tau>') \<notin> fv (mk_eq (Abs \<tau>' s) (Abs \<tau>' t))"
print_msg_def: fixes uu :: "char list" shows "print_msg uu = ()"
ordinal_linear: fixes x :: "ordinal"   and y :: "ordinal" shows "x \<le> y \<or> y \<le> x"
assms_4: fixes x :: "real" shows "(F has_real_derivative f x) (at x)"
simps_2: fixes x :: "'b"   and xs :: "'b llist"   and ys :: "'b llist" shows "(x :@ xs) :++ ys = x :@ xs :++ ys"
lset_lappend_lfinite: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lfinite xs" shows "lset (lappend xs ys) = lset xs \<union> lset ys"
simps: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and x :: "nat"   and n :: "nat" shows "semi_mojmir_def.senior \<delta> q\<^sub>0 w x n = the (semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n) n)"
g: fixes s :: "complex" assumes "b < Re s" shows "(g has_laplace F) s"
bound_invD_3: fixes outside :: "bool"   and upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_bound_invariant outside upw i fs" shows "g_bound fs"
closedin_t1_singleton: fixes X :: "'c topology"   and a :: "'c" assumes "t1_space X"   and "a \<in> topspace X" shows "closedin X {a}"
comp_assoc: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and h :: "'ag"   and g :: "'ag"   and f :: "'ag" assumes "category C" shows "C (C h g) f = C h (C g f)"
pw_ESPEC_3: fixes \<Phi> :: "'c \<Rightarrow> bool"   and \<Psi> :: "'d \<Rightarrow> bool"   and x :: "'d" shows "inres (ESPEC \<Phi> \<Psi>) (Inr x) = \<Psi> x"
cpoly_of_eq_0_iff: fixes pR :: "real poly"   and pI :: "real poly" shows "(cpoly_of pR pI = 0) = (pR = 0 \<and> pI = 0)"
BuildSeqI: fixes s :: "hf"   and k :: "hf"   and y :: "hf"   and B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool" assumes "LstSeq s k y"   and "\<And>l. l \<^bold>\<in> succ k \<Longrightarrow> Builds B C s l" shows "BuildSeq B C s k y"
prodag_iOp: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "mop (a\<Pi>\<^bsub>I\<^esub> A) = prod_mOp I A"
inverse_p32_def: fixes p :: "uint32"   and x :: "uint32" shows "inverse_p32 p x = (if x = 0 then 0 else power_p32 p x (p - 2))"
partition_main_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b"   and lo :: "nat"   and hi :: "nat"   and xs0 :: "'a list" shows "partition_main R h lo hi xs0 = ASSERT (hi < length xs0) \<bind> (\<lambda>_. RETURN (h (xs0 ! hi)) \<bind> (\<lambda>pivot. WHILE\<^sub>T\<^bsup>partition_main_inv R h lo hi xs0\<^esup> (\<lambda>(i, j, xs). j < hi) (\<lambda>(i, j, xs). ASSERT (i < length xs \<and> j < length xs) \<bind> (\<lambda>_. if R (h (xs ! j)) pivot then RETURN (i + 1, j + 1, swap xs i j) else RETURN (i, j + 1, xs))) (lo, lo, xs0) \<bind> (\<lambda>(i, j, xs). ASSERT (i < length xs \<and> j = hi \<and> lo \<le> i \<and> hi < length xs \<and> mset xs = mset xs0) \<bind> (\<lambda>_. RETURN (swap xs i hi, i)))))"
eval_fps_expansion': fixes f :: "complex \<Rightarrow> complex"   and z0 :: "complex"   and r :: "ereal"   and z :: "complex" assumes "f holomorphic_on eball z0 r"   and "ereal (cmod z) < r" shows "eval_fps (fps_expansion f z0) z = f (z0 + z)"
plus_Nat_def: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and RepNat :: "'a \<Rightarrow> nat"   and AbsNat :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.nat minus plus zero RepNat AbsNat" shows "plus a b = AbsNat (RepNat a + RepNat b)"
leg1_uniquely_isomorphic_2: shows "\<exists>!\<nu>. \<guillemotleft>\<nu> : local.dom \<sigma> \<Rightarrow> s\<^sub>1 \<star> src s\<^sub>0\<guillemotright>"
induct: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>nat. P (Atom nat)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 \<rightarrow> ty2)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 \<otimes> ty2)" shows "P ty"
stutter_selection_stutter_sampler: fixes w :: "'a llist"   and s :: "nat set" assumes "linfinite w"   and "stutter_selection s w" shows "stutter_sampler (nth_least_ext s) ((?!) w)"
fib_work_final_fib_work_eq: shows "fib_work_final = fib_work"
validFrom_validTrans: fixes s :: "'state"   and a :: "'act"   and ou :: "'out"   and s' :: "'state"   and tr :: "('state, 'act, 'out) trans list" assumes "validTrans (Trans s a ou s')"   and "validFrom s' tr" shows "validFrom s (Trans s a ou s' # tr)"
vreal_def: shows "\<real>\<^sub>\<circ> = ZFC_in_HOL.set (range vreal_of_real)"
long_division_def: fixes p :: "'a list"   and q :: "'a list" shows "long_division p q = (THE t. long_divides p q t)"
card_multisets: fixes A :: "'b set"   and k :: "nat" assumes "finite A" shows "card {M. set_mset M \<subseteq> A \<and> size M = k} = card A + k - 1 choose k"
comp_null_2: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "local.comp f null = null"
path_append: fixes A :: "('c, 'd) nba"   and r :: "('c \<times> 'd) list"   and p :: "'d"   and s :: "('c \<times> 'd) list" assumes "path A r p"   and "path A s (target r p)" shows "path A (r @ s) p"
nth_eq_iff_index_eq: fixes xs :: "'g list"   and i :: "nat"   and j :: "nat" assumes "distinct xs"   and "i < length xs"   and "j < length xs" shows "(xs ! i = xs ! j) = (i = j)"
hd_w_in_verts: shows "hd w \<in> set (g_V G)"
cong_solve_nat: fixes a :: "nat"   and n :: "nat" shows "\<exists>x. [a * x = gcd a n] (mod n)"
span_eq: fixes S :: "('c \<Rightarrow>\<^sub>0 'd) set"   and T :: "('c \<Rightarrow>\<^sub>0 'd) set" shows "(phull S = phull T) = (S \<subseteq> phull T \<and> T \<subseteq> phull S)"
assms_3: shows "g \<in> space\<^sub>N (\<LL> \<infinity> M)"
unG: shows "(\<Union>x. {xa \<in> G. real_cond_exp M Invariants f xa / (real x + 1) < d}) = G"
Q'_def: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'a option"   and u :: "'v" shows "Q' Q \<pi> u \<equiv> (Qinter Q \<pi> u)(u := \<infinity>)"
Inf_lower: fixes x :: "'c \<times> 'd"   and A :: "('c \<times> 'd) set" assumes "s2p A x" shows "less_eq_prod (Inf_prod A) x"
abs_set_union: fixes A :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and B :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and a :: "nat \<times> bool \<Rightarrow> 'c set" shows "(A \<union> B) \<cdot>\<^sub>\<alpha>\<^sub>s\<^sub>e\<^sub>t a = A \<cdot>\<^sub>\<alpha>\<^sub>s\<^sub>e\<^sub>t a \<union> B \<cdot>\<^sub>\<alpha>\<^sub>s\<^sub>e\<^sub>t a"
bigger_prime: shows "\<exists>p. prime p \<and> n < p"
is_def: shows "is = weave (Collect even) (digit_encode (nths (dims Aw) (Collect even)) i) (digit_encode (nths (dims Aw) {n. odd n}) j)"
transpose_mat_eq: fixes A :: "'a mat"   and B :: "'a mat" shows "(A\<^sup>t = B\<^sup>t) = (A = B)"
invert_side_of_triangle_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "'b"   and f :: "'b"   and g :: "'b" assumes "category C"   and "partial_magma.arr C h"   and "C f g = h"   and "category.iso C g" shows "partial_magma.arr C (C h (category.inv C g)) \<and> f = C h (category.inv C g)"
test_num': fixes n :: "int"   and a :: "'av" shows "test_num' n a = (n \<in> \<gamma> a)"
qSwapAll_simps_2: fixes zs :: "'c"   and x :: "'d"   and y :: "'d"   and delta :: "'e"   and inp :: "'a \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm option"   and binp :: "'b \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs option" shows "qOp delta inp binp #[[x \<and> y]]_zs = qOp delta (lift (qSwap zs x y) inp) (lift (qSwapAbs zs x y) binp)"
snd_min_key_val_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs"   and "fst xs \<noteq> []" shows "snd (min_key_val_raw ko xs) = lookup_raw xs (fst (min_key_val_raw ko xs))"
simps_2: fixes uv :: "'a" shows "less_eq_option (Some uv) None = False"
field_long_division_theorem: fixes K :: "'a set"   and p :: "'a list"   and b :: "'a list" assumes "subfield K R"   and "polynomial K p"   and "polynomial K b"   and "b \<noteq> []" shows "\<exists>q r. polynomial K q \<and> polynomial K r \<and> p = b \<otimes>\<^bsub>K [X]\<^esub> q \<oplus>\<^bsub>K [X]\<^esub> r \<and> (r = [] \<or> degree r < degree b)"
token_run_enter_final_states: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set"   and x :: "nat"   and n :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n \<in> F" shows "\<exists>m\<ge>x. semi_mojmir_def.token_run \<delta> q\<^sub>0 w x m \<notin> F - {q\<^sub>0} \<and> semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (Suc m) \<in> F"
eventually_finite_subsets_at_top_weakI: fixes A :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "\<And>X. \<lbrakk>finite X; X \<subseteq> A\<rbrakk> \<Longrightarrow> P X" shows "eventually P (finite_subsets_at_top A)"
set_intros_1: fixes y :: "'e"   and x :: "('e, 'f, ('e, 'f) T\<^sub>m, 'f T\<^sub>h) F\<^sub>m" assumes "y \<in> set1_F\<^sub>m x" shows "y \<in> set1_T\<^sub>m (T\<^sub>m x)"
y_prop: shows "r < y' \<and> (\<forall>x\<in>{r<..y'}. poly [:- c, - b, - a:] x < 0)"
invc_baliR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invc l"   and "invc2 r" shows "invc (baliR l a r)"
subst_compose_def: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'g) Term.term"   and \<tau> :: "'g \<Rightarrow> ('e, 'f) Term.term" shows "\<sigma> \<circ>\<^sub>s \<tau> = (\<lambda>x. \<sigma> x \<cdot> \<tau>)"
nin_nodes_reachable: fixes G :: "nat graph_rec"   and n :: "nat" assumes "cfg_doms G"   and "n \<notin> set (g_V G)" shows "\<not> cfg_doms.reachable G n"
graph_eq_intro: fixes G :: "'a graph"   and G' :: "'a graph" assumes "nodes G = nodes G'"   and "edges G = edges G'" shows "G = G'"
monotone_fun_ord_apply: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "monotone orda (fun_ord ordb) f = (\<forall>x. monotone orda ordb (\<lambda>y. f y x))"
add_divide_distrib: 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 a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
of_nat_monom: fixes n :: "nat" shows "of_nat n = monom (of_nat n) 0"
dg_Set_Dom_vrange: fixes \<alpha> :: "V" shows "\<R>\<^sub>\<circ> (dg_Set \<alpha>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> dg_Set \<alpha>\<lparr>Obj\<rparr>"
Chain_def: fixes D :: "('h, 'i) Order_scheme"   and C :: "'h set" shows "Chain D C = (C \<subseteq> carrier D \<and> Torder (Iod D C))"
Send_def: fixes return :: "nat \<Rightarrow> 'a Frame list"   and send :: "nat \<Rightarrow> 'a Frame list"   and get :: "nat \<Rightarrow> nat list"   and activation :: "nat \<Rightarrow> nat list" shows "Send return send get activation \<equiv> \<forall>t. if activation t = [] then get t = [] \<and> send t = [] else get t = activation t \<and> send t = return t"