hcju/isabelleps huggingface dataset:
np_specD2: fixes np :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "np_spec np" shows "set hs \<times> (set gs \<union> set bs) \<subseteq> snd ` set (np gs bs hs data)"
insert_spec_inc_length: fixes xs :: "'c list"   and n :: "nat"   and oid :: "'c" assumes "distinct xs"   and "n < length xs" shows "length (insert_spec xs (oid, Some (xs ! n))) = Suc (length xs)"
distr_distr: fixes g :: "'a \<Rightarrow> 'b"   and N :: "'a measure"   and L :: "'b measure"   and f :: "'d \<Rightarrow> 'a"   and M :: "'d measure" assumes "g \<in> N \<rightarrow>\<^sub>M L"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "distr (distr M N f) L g = distr M L (g \<circ> f)"
bigo_alt_def: fixes f :: "'c \<Rightarrow> 'd" shows "O(f) = {h. \<exists>c>0::'d. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>}"
gauss_jordan_inverse_other_direction: fixes A :: "'c mat"   and n :: "nat"   and b :: "'d"   and B :: "'c mat"   and nc :: "nat" assumes "A \<in> Units (ring_mat TYPE('c) n b)"   and "B \<in> carrier_mat n nc" shows "fst (gauss_jordan A B) = 1\<^sub>m n"
normal_exponent_bounds_int_2: fixes x :: "('a, 'b) IEEE.float" assumes "is_normal x" shows "normal_exponent x \<le> 2 ^ (LENGTH('a) - 1) - int LENGTH('b) - 1"
assms: shows "A \<rightarrow>\<beta>\<^sup>* A'"
deg_zero: shows "deg R \<zero>\<^bsub>UP R\<^esub> = 0"
aless_diff_poss: fixes x :: "ant"   and y :: "ant" assumes "x < y" shows "0 < y - x"
ccSup_empty: 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" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
zero_less_divide_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide a b) = (less zero a \<and> less zero b \<or> less a zero \<and> less b zero)"
DOWN_def: fixes t :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "DOWN t \<equiv> the_elem (concrete_category.Dom t)"
eucl_rel_poly: fixes x :: "'a poly"   and y :: "'a poly" shows "eucl_rel_poly x y (pdivmod x y)"
coplanar_ABCA'B'C'P: fixes A :: "Points"   and A' :: "Points"   and B :: "Points"   and C :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points" assumes "rk {A, A'} = 2"   and "rk {A, B, C, A', B', C'} = 3"   and "rk {A, A', P} = 2" shows "rk {A, B, C, A', B', C', P} = 3"
leadsTo_induct_pre_lemma: fixes F :: "'c program"   and za :: "'c set"   and zb :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "F \<in> za \<longmapsto> zb"   and "P zb"   and "\<And>A B. \<lbrakk>F \<in> A ensures B; P B\<rbrakk> \<Longrightarrow> P A"   and "\<And>S. \<forall>A\<in>S. P A \<Longrightarrow> P (\<Union> S)" shows "P za"
assms: shows "f piecewise_differentiable_on (\<lambda>x. m *\<^sub>R x + c) ` S"
complex_cvec_eq_mix: fixes z1 :: "complex"   and z2 :: "complex"   and w1 :: "complex"   and w2 :: "complex" assumes "(z1, z2) \<noteq> vec_zero"   and "(w1, w2) \<noteq> vec_zero" shows "((z1, z2) \<approx>\<^sub>v (w1, w2)) = (z1 * w2 = z2 * w1)"
dvd_times_left_cancel_iff: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<noteq> (0::'c)" shows "(a * b dvd a * c) = (b dvd c)"
valid_return_list_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and cs :: "'a list"   and n :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.valid_return_list targetnode kind valid_edge get_proc cs n \<equiv> \<forall>cs' c cs''. cs = cs' @ c # cs'' \<longrightarrow> valid_edge c \<and> (\<exists>Q p f. kind c = Q\<hookleftarrow>\<^bsub>p\<^esub>f \<and> p = get_proc (case cs' of [] \<Rightarrow> n | a # list \<Rightarrow> last (CFG.targetnodes targetnode cs')))"
sing_pow_len: fixes r :: "'b"   and l :: "nat" shows "\<^bold>|[r] \<^sup>@ l\<^bold>| = l"
skl1_refines_a0i_r: shows "refines (R0sk1iar \<inter> reach a0i \<times> (skl1_inv4 \<inter> skl1_inv5)) med0sk1iar a0i skl1"
alphas_abs_eqvt_2: fixes bs :: "atom set"   and x :: "'a"   and cs :: "atom set"   and y :: "'a"   and p :: "perm" assumes "(bs, x) \<approx>abs_res (cs, y)" shows "(p \<bullet> bs, p \<bullet> x) \<approx>abs_res (p \<bullet> cs, p \<bullet> y)"
continuous_IVT_local_extremum: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> real" assumes "continuous_on (closed_segment a b) f"   and "a \<noteq> b"   and "f a = f b" obtains z :: "'a" where "z \<in> open_segment a b" and "(\<forall>w\<in>closed_segment a b. f w \<le> f z) \<or> (\<forall>w\<in>closed_segment a b. f z \<le> f w)"
zero_le_power: fixes a :: "'b"   and n :: "nat" assumes "(0::'b) \<le> a" shows "(0::'b) \<le> a ^ n"
Var_def: fixes N :: "nat" shows "Var N = Abs_Exp (exprel `` {VAR N})"
interleave_rev_cons1: fixes l :: "'b list"   and a :: "'b list"   and b :: "'b list"   and x :: "'b" assumes "l \<in> a \<otimes> b" shows "l @ [x] \<in> a @ [x] \<otimes> b"
A: shows "A \<in> carrier_mat (n + m) (n + m)"
intros_2: fixes s' :: "('a, 'b, 'c) state"   and do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and s :: "('a, 'b, 'c) state"   and sl :: "('a, 'b, 'c) state list" assumes "s' \<in> GM_sec_model.\<delta> do s"   and "GM_sec_model.wffp do (s' # sl)" shows "GM_sec_model.wffp do (s # s' # sl)"
fmadd_restrict_right_dom: fixes n :: "('a, 'b) fmap"   and m :: "('a, 'b) fmap" shows "fmrestrict_fset (fmdom n) (m ++\<^sub>f n) = n"
IH: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "vc C Q" shows "\<turnstile>\<^sub>t {pre C Q} strip C {Q}"
simps_295: fixes f1 :: "String.literal \<Rightarrow> Formula.trm list \<Rightarrow> 'a"   and f2 :: "String.literal \<Rightarrow> Formula.formula \<Rightarrow> Formula.formula \<Rightarrow> 'a"   and f3 :: "Formula.trm \<Rightarrow> Formula.trm \<Rightarrow> 'a"   and f4 :: "Formula.trm \<Rightarrow> Formula.trm \<Rightarrow> 'a"   and f5 :: "Formula.trm \<Rightarrow> Formula.trm \<Rightarrow> 'a"   and f6 :: "Formula.formula \<Rightarrow> 'a"   and f7 :: "Formula.formula \<Rightarrow> Formula.formula \<Rightarrow> 'a"   and f8 :: "Formula.formula \<Rightarrow> Formula.formula \<Rightarrow> 'a"   and f9 :: "Formula.formula list \<Rightarrow> 'a"   and f10 :: "Formula.formula \<Rightarrow> 'a"   and f11 :: "nat \<Rightarrow> Formula.agg_type \<times> event_data \<Rightarrow> nat \<Rightarrow> Formula.trm \<Rightarrow> Formula.formula \<Rightarrow> 'a"   and f12 :: "\<I> \<Rightarrow> Formula.formula \<Rightarrow> 'a"   and f13 :: "\<I> \<Rightarrow> Formula.formula \<Rightarrow> 'a"   and f14 :: "Formula.formula \<Rightarrow> \<I> \<Rightarrow> Formula.formula \<Rightarrow> 'a"   and f15 :: "Formula.formula \<Rightarrow> \<I> \<Rightarrow> Formula.formula \<Rightarrow> 'a"   and f16 :: "\<I> \<Rightarrow> Formula.formula Regex.regex \<Rightarrow> 'a"   and f17 :: "\<I> \<Rightarrow> Formula.formula Regex.regex \<Rightarrow> 'a"   and x6 :: "Formula.formula" shows "(case formula.Neg x6 of formula.Pred x xa \<Rightarrow> f1 x xa | formula.Let x xa xb \<Rightarrow> f2 x xa xb | formula.Eq x xa \<Rightarrow> f3 x xa | formula.Less x xa \<Rightarrow> f4 x xa | formula.LessEq x xa \<Rightarrow> f5 x xa | formula.Neg x \<Rightarrow> f6 x | formula.Or x xa \<Rightarrow> f7 x xa | formula.And x xa \<Rightarrow> f8 x xa | formula.Ands x \<Rightarrow> f9 x | formula.Exists x \<Rightarrow> f10 x | formula.Agg x xa xb xc xd \<Rightarrow> f11 x xa xb xc xd | formula.Prev x xa \<Rightarrow> f12 x xa | formula.Next x xa \<Rightarrow> f13 x xa | formula.Since x xa xb \<Rightarrow> f14 x xa xb | formula.Until x xa xb \<Rightarrow> f15 x xa xb | formula.MatchF x xa \<Rightarrow> f16 x xa | formula.MatchP x xa \<Rightarrow> f17 x xa) = f6 x6"
bin_code_alpha: fixes us :: "'a list list"   and vs :: "'a list list" assumes "us \<in> lists {u\<^sub>0, u\<^sub>1}"   and "vs \<in> lists {u\<^sub>0, u\<^sub>1}"   and "hd us \<noteq> hd vs" shows "concat us \<cdot> symcode.\<alpha> \<and>\<^sub>p concat vs \<cdot> symcode.\<alpha> = symcode.\<alpha>"
from_nat_to_nat: fixes j :: "nat"   and k :: "'b" assumes "from_nat j = k"   and "j < CARD('b)" shows "to_nat k = j"
icd: shows "l icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
axioms_4: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "digraph \<alpha> \<BB>"
mult_right_cancel: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.semiring_no_zero_divisors_cancel plus zero times"   and "c \<noteq> zero" shows "(times a c = times b c) = (a = b)"
fps_of_ratfps_altdef: fixes x :: "'a ratfps" shows "fps_of_ratfps x = (case quot_of_ratfps x of (a, b) \<Rightarrow> fps_of_poly a / fps_of_poly b)"
liset_finite: fixes w :: "'b llist"   and A :: "'b set" assumes "lfinite w" shows "finite (liset A w)"
well_order_on_Field: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "well_order_on A r" shows "A = Field r"
FrameStatImpEqvt: fixes p :: "(name \<times> name) list"   and F :: "'b frame"   and G :: "'b frame" shows "p \<bullet> (F \<hookrightarrow>\<^sub>F G) = p \<bullet> F \<hookrightarrow>\<^sub>F p \<bullet> G"
assms: fixes x :: "'b" assumes "x \<in> A" shows "f x dvd g x"
the_leftI: fixes j :: "nat" assumes "local.has_left j" shows "local.the_left j < j \<and> xs ! local.the_left j < xs ! j"
gauss2complex_mult: fixes x :: "gauss_int"   and y :: "gauss_int" shows "gauss2complex (x * y) = gauss2complex x * gauss2complex y"
finite_rangeC: fixes A :: "('a \<Rightarrow> 'b) set" assumes "\<forall>f\<in>A. inj f"   and "\<forall>f\<in>A. \<forall>g\<in>A. f \<noteq> g \<longrightarrow> (\<forall>x y. f x \<noteq> g y)" shows "finite (rangeC A) = (finite A \<and> (A \<noteq> {} \<longrightarrow> finite UNIV))"
rep_eq: fixes x :: "('a, 'c, 'b) formula" shows "S_transform_formula x = S_transform_Tree\<^sub>\<alpha> (Rep_formula x)"
vec_to_iarray_column: fixes i :: "'d"   and A :: "(('b, 'd) vec, 'c) vec" shows "vec_to_iarray (column i A) = column_iarray (mod_type_class.to_nat i) (matrix_to_iarray A)"
eventually_INF_base: fixes B :: "'c set"   and F :: "'c \<Rightarrow> 'd filter"   and P :: "'d \<Rightarrow> bool" assumes "B \<noteq> {}"   and "\<And>a b. \<lbrakk>a \<in> B; b \<in> B\<rbrakk> \<Longrightarrow> \<exists>x\<in>B. F x \<le> inf (F a) (F b)" shows "eventually P (Inf (F ` B)) = (\<exists>b\<in>B. eventually P (F b))"
lift_lift2: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and inp' :: "'a \<Rightarrow> 'd option"   and inp :: "'a \<Rightarrow> 'e option" shows "lift f (lift2 g inp' inp) = lift2 (\<lambda>v' v. f (g v' v)) inp' inp"
psubsetE: fixes A :: "'f set"   and B :: "'f set" assumes "A \<subset> B" obtains "A \<subseteq> B" and "\<not> B \<subseteq> A"
cross3_simps_11: fixes a :: "'a"   and b :: "'a" shows "a * b = b * a"
dim_sumlist: fixes xs :: "'a vec list"   and n :: "nat" assumes "\<forall>x\<in>set xs. dim_vec x = n" shows "dim_vec (abelian_monoid.sumlist (module_vec TYPE('a) n) xs) = n"
eval_zero: fixes l :: "dag"   and v :: "ref"   and r :: "dag"   and var :: "ref \<Rightarrow> nat"   and x :: "bdt" assumes "bdt (Node l v r) var = Some x"   and "var (root (Node l v r)) = 0" shows "x = Zero"
fixp_unfold: fixes f :: "'b \<Rightarrow> 'b" assumes "monotone ordb ordb f" shows "b.fixp f = f (b.fixp f)"
cospoly_upper_nonneg: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x" shows "cos x \<le> cospoly (Suc (4 * n)) x"
induct: fixes P :: "('b \<times> bal) tree \<Rightarrow> bool"   and a0 :: "('b \<times> bal) tree" assumes "\<And>l a ba r. (r \<noteq> \<langle>\<rangle> \<Longrightarrow> P r) \<Longrightarrow> P \<langle>l, (a, ba), r\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
plen_take_prefix: fixes i :: "nat"   and \<sigma> :: "'b trace" shows "plen (take_prefix i \<sigma>) = i"
fbrelation_fcomp: shows "fbrelation (r\<^sub>1 \<circ>\<^sub>\<bullet> r\<^sub>2)"
A6_Acc_L1: shows "Acc level1 sA6 = {sA71, sA72, sA81, sA82, sA91, sA92, sA93}"
agent_abs_spr_repRels: fixes t :: "('a, 'es, 'as) BEState Trace"   and x :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState"   and a :: "'a"   and y :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState" assumes "t \<in> SPR.jkbpC"   and "x \<in> agent_abs a t"   and "y \<in> agent_abs a t" shows "(x, y) \<in> local.spr_repRels a"
nAct_active: fixes i :: "nat"   and t :: "cnf llist"   and c :: "'id" assumes "enat i < llength t"   and "\<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>" shows "\<langle>c #\<^bsub>enat (Suc i)\<^esub>t\<rangle> = eSuc \<langle>c #\<^bsub>enat i\<^esub>t\<rangle>"
point_ordinate_add: fixes A :: "point"   and B :: "point" shows "ordinate (A + B) = ordinate A + ordinate B"
substA_o: fixes \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm"   and at :: "('c, 'd) atm" shows "substA (subst \<pi>1 \<circ> \<pi>2) at = substA \<pi>1 (substA \<pi>2 at)"
add_le_same_cancel1: fixes b :: "'b multiset"   and a :: "'b multiset" shows "(b + a \<subseteq># b) = (a \<subseteq># {#})"
closed_subspace: fixes s :: "'b set" assumes "subspace s" shows "closed s"
join_least: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> z"   and "y \<sqsubseteq> z" shows "x \<squnion> y \<sqsubseteq> z"
sum_nonneg_eq_0_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq zero (f x)" shows "(comm_monoid_add.sum plus zero f A = zero) = (\<forall>x\<in>A. f x = zero)"
simps_1: fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and p :: "real mpoly" shows "elimVar var L F (Eq p) = (let (a, b, c) = get_coeffs var p in Or (And (And (fm.Atom (Eq a)) (fm.Atom (Neq b))) (list_conj (map (\<lambda>a. fm.Atom (linear_substitution var (- c) b a)) L @ map (linear_substitution_fm var (- c) b) F))) (And (fm.Atom (Neq a)) (And (fm.Atom (Leq (- b\<^sup>2 + 4 * a * c))) (Or (list_conj (map (quadratic_sub var (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a)) L @ map (quadratic_sub_fm var (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a)) F)) (list_conj (map (quadratic_sub var (- b) (- 1) (b\<^sup>2 - 4 * a * c) (2 * a)) L @ map (quadratic_sub_fm var (- b) (- 1) (b\<^sup>2 - 4 * a * c) (2 * a)) F))))))"
strict_trans1: fixes b :: "'f"   and a :: "'f"   and c :: "'f" assumes "b \<le> a"   and "c < b" shows "c < a"
zip_Cons1: fixes x :: "'d"   and xs :: "'d list"   and ys :: "'e list" shows "x # xs || ys = (case ys of [] \<Rightarrow> [] | y # ys \<Rightarrow> (x, y) # (xs || ys))"
homotopic_with_eq: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b" assumes "homotopic_with P X Y f g"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f' x = f x"   and "\<And>x. x \<in> topspace X \<Longrightarrow> g' x = g x"   and "\<And>h k. (\<And>x. x \<in> topspace X \<Longrightarrow> h x = k x) \<Longrightarrow> P h = P k" shows "homotopic_with P X Y f' g'"
fps_to_fls_def: fixes f :: "'a fps" shows "fps_to_fls f \<equiv> Abs_fls (\<lambda>k. if k < 0 then 0::'a else f $ nat k)"
J_1: fixes i :: "nat" shows "J i \<subseteq> I"
assms_2: shows "Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P"
Dev_preserves_fresh: fixes M :: "lam"   and N :: "lam"   and x :: "name" assumes " M \<longrightarrow>\<^sub>d N"   and "x \<sharp> M" shows "x \<sharp> N"
IH: assumes "xs prefix of i" shows "apply_operations xs \<noteq> None"
num_params_tm_mul_le: fixes t1 :: "taylor_model"   and X :: "nat"   and t2 :: "taylor_model"   and prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list" assumes "num_params (tm_poly t1) \<le> X"   and "num_params (tm_poly t2) \<le> X" shows "num_params (tm_poly (tm_mul prec ord I a t1 t2)) \<le> X"
dim_col_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "dim_col (mat nr nc f) = nc"
coeff_int_def: fixes p :: "'c poly"   and i :: "int" shows "coeff_int p i = (if i < 0 then 0::'c else coeff p (nat i))"
finite_neg_img: shows "finite (\<phi> -` (s' \<inter> {0..1}))"
filter_mset_sum_list: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b multiset list" shows "filter_mset P (sum_list xs) = sum_list (map (filter_mset P) xs)"
symmetry_preserves_midpoint: fixes Z :: "'p"   and A :: "'p"   and D :: "'p"   and B :: "'p"   and E :: "'p"   and C :: "'p"   and F :: "'p" assumes "Z Midpoint A D"   and "Z Midpoint B E"   and "Z Midpoint C F"   and "B Midpoint A C" shows "E Midpoint D F"
fresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm set" shows "fresh xs x X \<equiv> qFresh xs x (pick X)"
subterm_subst_neq: fixes t :: "('a, 'b) Term.term"   and t' :: "('a, 'b) Term.term"   and s :: "'b \<Rightarrow> ('a, 'c) Term.term" assumes "t \<sqsubset> t'" shows "t \<cdot> s \<noteq> t' \<cdot> s"
inject: fixes x :: "'a \<times> 'a rose_tree list"   and ya :: "'a \<times> 'a rose_tree list" shows "(Tree x = Tree ya) = (x = ya)"
pi_ahm: shows "proper_it' ahm_iteratei ahm_iteratei"
vcomp_assoc: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<rho> :: "'a \<Rightarrow> 'b"   and H :: "'a \<Rightarrow> 'b"   and \<sigma> :: "'a \<Rightarrow> 'b"   and K :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_transformation A B F G \<rho>"   and "natural_transformation A B G H \<sigma>"   and "natural_transformation A B H K \<tau>" shows "vertical_composite.map A B (vertical_composite.map A B \<rho> \<sigma>) \<tau> = vertical_composite.map A B \<rho> (vertical_composite.map A B \<sigma> \<tau>)"
lower: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "Turn_into_distance f" shows "f x y \<le> 2 * turn_into_distance f x y"
add_diff_eq_ereal: fixes x :: "ereal"   and y :: "ereal"   and z :: "ereal" shows "x + (y - z) = x + y - z"
cross_ratio_0inf: fixes y :: "complex"   and x :: "complex" assumes "y \<noteq> 0" shows "cross_ratio (of_complex x) 0\<^sub>h (of_complex y) \<infinity>\<^sub>h = of_complex (x / y)"
rComm: fixes a :: "act"   and P' :: "ccs"   and P'' :: "ccs"   and \<C> :: "'a" assumes "P \<longmapsto>a \<prec> P'"   and "!P \<longmapsto>coAction a \<prec> P''"   and "\<And>\<C>. Prop \<C> !P (coAction a) P''"   and "a \<noteq> \<tau>" shows "Prop \<C> (P \<parallel> !P) (\<tau>) (P' \<parallel> P'')"
info_3_fun: fixes x :: "real_alg_3"   and y :: "real_alg_3" assumes "real_of_3 x = real_of_3 y" shows "info_3 x = info_3 y"
runit'_in_vhom: fixes f :: "'a"   and a :: "'a" assumes "B.ide f"   and "src\<^sub>B f = a" shows "\<guillemotleft>B.runit' f : f \<rightarrow>\<^sub>B f \<star>\<^sub>B a\<guillemotright>"
nextnat_empty_gzero: fixes s :: "nat \<Rightarrow> 'b" assumes "notemptyseq s" shows "0 < nextnat s"
One_not_eq_j0: shows "quiver.One \<noteq> parallel_pair.j0"
STAR_sin_Infinitesimal_divide: fixes x :: "'a star" assumes "x \<in> Infinitesimal"   and "x \<noteq> 0" shows "(*f* sin) x / x \<approx> 1"
dom_dom: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "VV.dom (VV.dom f) = VV.dom f"
n0: shows "\<forall>n\<ge>n\<^sub>0. s (z \<triangleright> n) \<down>= Suc i"
ipresOp_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'd, 'e, 'opSym, 'f, 'b, 'c, 'g) model_scheme" shows "ipresOp h hA MOD \<equiv> \<forall>delta inp binp. wlsInp delta inp \<and> wlsBinp delta binp \<longrightarrow> h (Op delta inp binp) = igOp MOD delta (lift h inp) (lift hA binp)"
weakSimWeakEqSim: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" assumes "P \<leadsto><Rel> Q" shows "P \<leadsto>\<^sup>^<Rel> Q"
eucl_truncate_down_zero: fixes p :: "nat" shows "eucl_truncate_down p (0::'a) = (0::'a)"
InvariantCEntailedAfterExplain: fixes conflictFlag :: "bool"   and F :: "Literal list list"   and C :: "Literal list"   and reason :: "Literal list"   and C' :: "Literal list"   and l :: "Literal" assumes "InvariantCEntailed conflictFlag F C"   and "formulaEntailsClause F reason"   and "C' = resolve C reason (opposite l)" shows "InvariantCEntailed conflictFlag F C'"
admissible_leI: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "mcont luba orda luba orda f"   and "mcont luba orda luba orda g" shows "ccpo.admissible luba orda (\<lambda>x. orda (f x) (g x))"
0: shows "rawpsubstT (suc r) (zip (map Var vs) (map snd txs)) = suc (rawpsubstT r (zip (map Var vs) (map snd txs)))"
eliminate_entries_gen_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and v :: "nat \<Rightarrow> 'b"   and A :: "'b mat"   and I :: "nat"   and J :: "nat" shows "eliminate_entries_gen minus times v A I J = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if i \<noteq> I then minus (A $$ (i, j)) (times (v i) (A $$ (I, j))) else A $$ (i, j))"
"case": fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b list \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'c"   and x3 :: "'b list" shows "(case x1 \<rightarrow> x2 x3 of x \<rightarrow> xa xb \<Rightarrow> f x xa xb) = f x1 x2 x3"
upper_dim_implies_one_or_two_sides: assumes "UpperDimAxiom" shows "\<forall>A B X Y. \<not> Col X A B \<and> \<not> Col Y A B \<longrightarrow> A B TS X Y \<or> A B OS X Y"
V_disjoint_iff: fixes x :: "V"   and y :: "V" shows "(x \<sqinter> y = 0) = (elts x \<inter> elts y = {})"
get_ancestors_parent_child_rel: fixes h :: "(_) heap"   and child :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> Shadow_DOM.get_ancestors child \<rightarrow>\<^sub>r ancestors"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "((ptr, child) \<in> (Shadow_DOM.parent_child_rel h)\<^sup>*) = (ptr \<in> set ancestors)"