hcju/isabelleps huggingface dataset:
intros_1: fixes y :: "nat" shows "less_nat 0 (Suc y)"
weakening: fixes \<Gamma>1 :: "(name \<times> ty) list"   and t :: "lam"   and T :: "ty"   and \<Gamma>2 :: "(name \<times> ty) list" assumes "\<Gamma>1 \<turnstile> t : T"   and "valid \<Gamma>2"   and "\<Gamma>1 \<subseteq> \<Gamma>2" shows "\<Gamma>2 \<turnstile> t : T"
fps_mult_commute_lemma: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat" shows "(\<Sum>i = 0..n. f i (n - i)) = (\<Sum>i = 0..n. f (n - i) i)"
sum_f_non_negative: fixes X :: "(nat \<times> nat) set" shows "(0::'a) \<le> sum f X"
add_eq_conv_ex: fixes a :: "'a"   and M :: "'a multiset"   and b :: "'a"   and N :: "'a multiset" shows "(add_mset a M = add_mset b N) = (M = N \<and> a = b \<or> (\<exists>K. M = add_mset b K \<and> N = add_mset a K))"
vcpower_two_ftimesE: fixes x :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> A ^\<^sub>\<times> 2\<^sub>\<nat>" obtains "x \<in>\<^sub>\<circ> A \<times>\<^sub>\<bullet> A"
cont_Ch_R: fixes ch :: "'choice"   and c1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and n :: "nat" shows "local.cont (Ch ch c1 c2) s (Suc n) = c2"
arr_dom: fixes f :: "'c" assumes "C.arr f" shows "C.arr (C.dom f)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "inj_monoid_add_hom hom"   and "bijective hom" shows "monoid_add_isom hom"
preliminary1: fixes H :: "'a set"   and H1 :: "'a set"   and K :: "'a set"   and K1 :: "'a set" assumes "subgroup H G"   and "H1 \<lhd> G\<lparr>carrier := H\<rparr>"   and "subgroup K G"   and "K1 \<lhd> G\<lparr>carrier := K\<rparr>" shows "H \<inter> K \<inter> (H1 <#> H \<inter> K1) = H1 \<inter> K <#> H \<inter> K1"
lfinite_lmap: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist" shows "lfinite (lmap f xs) = lfinite xs"
assms: shows "get_S w1 = 0 \<and> w2 AND 128 = 0 \<and> w3 AND 128 = 0 \<and> w4 AND 128 = 0"
pred_of_succ_def: fixes s :: "'g \<Rightarrow> 'h set" shows "pred_of_succ s \<equiv> \<lambda>a b. b \<in> s a"
quantify_bdd_is_node: fixes N :: "bool list bdd list \<times> bool list"   and bdd :: "bool list bdd"   and v :: "nat" assumes "bdd_all (nfa_is_node N) bdd" shows "bdd_all (nfa_is_node N) (quantify_bdd v bdd)"
interpret_op_def: fixes oper :: "('d, 'c) operation"   and state :: "'c \<Rightarrow> 'd set" shows "\<langle>oper\<rangle> state \<equiv> let before = state (op_elem oper); after = case oper of Add i e \<Rightarrow> before \<union> {i} | Rem is e \<Rightarrow> before - is in Some (state(op_elem oper := after))"
llength_lmap: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist" shows "llength (lmap f xs) = llength xs"
antisym: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S"   and "b \<in> S"   and "le a b"   and "le b a" shows "a = b"
hsize_in_args: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" assumes "s \<in> set (args t)" shows "hsize s < hsize t"
frequently_eventually_frequently: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and Q :: "'a \<Rightarrow> bool" assumes "frequently P F"   and "eventually Q F" shows "\<exists>\<^sub>F x in F. P x \<and> Q x"
dist_c: shows "(\<Sum>z\<in>outcomes. pmf q z * (c * u z)) = c * (\<Sum>z\<in>outcomes. pmf q z * u z)"
elims: fixes x :: "'d DBMEntry"   and xa :: "'d DBMEntry"   and y :: "'d DBMEntry" assumes "dbm_add x xa = y" obtains uu_ :: "'d DBMEntry" where "x = \<infinity>" and "xa = uu_" and "y = \<infinity>"   | v :: "'d" where "x = Le va__" and "xa = \<infinity>" and "y = \<infinity>"   | v :: "'d" where "x = Lt v" and "xa = \<infinity>" and "y = \<infinity>"   | a :: "'d" and b :: "'d" where "x = Le a__" and "xa = Le b__" and "y = Le (a__ + b__)"   | a :: "'d" and b :: "'d" where "x = Le aa__" and "xa = Lt ba__" and "y = Lt (aa__ + ba__)"   | a :: "'d" and b :: "'d" where "x = Lt ab__" and "xa = Le bb__" and "y = Lt (ab__ + bb__)"   | a :: "'d" and b :: "'d" where "x = Lt a" and "xa = Lt b" and "y = Lt (a + b)"
trans: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "xs <~~> ys"   and "ys <~~> zs" shows "xs <~~> zs"
get_mode_locs_impl: shows "get_mode_locs = get_mode_locs"
annir_def: fixes x :: "'a" shows "annir x = (\<forall>y. D y x \<and> y \<oplus> x = x)"
simps_14: fixes f11 :: "'b"   and f12 :: "X \<Rightarrow> 'b \<Rightarrow> 'b"   and f13 :: "Y \<Rightarrow> 'a \<Rightarrow> 'b"   and f21 :: "X \<Rightarrow> 'b \<Rightarrow> 'a"   and f22 :: "Y \<Rightarrow> 'a \<Rightarrow> 'a"   and f23 :: "'a" shows "rec_Y f11 f12 f13 f21 f22 f23 F = f23"
hnorm_add_less: fixes x :: "'a star"   and r :: "real star"   and y :: "'a star"   and s :: "real star" assumes "hnorm x < r"   and "hnorm y < s" shows "hnorm (x + y) < r + s"
stutter_invariant_next_free: fixes \<phi> :: "'b pltl" assumes "stutter_invariant \<phi>" obtains \<psi> :: "'b pltl" where "next_free \<psi>" and "atoms_pltl \<psi> \<subseteq> atoms_pltl \<phi>" and "\<forall>\<sigma>. \<sigma> \<Turnstile>\<^sub>p \<psi> = \<sigma> \<Turnstile>\<^sub>p \<phi>"
assms_3: fixes h :: "'t \<Rightarrow>\<^sub>0 'b" assumes "h \<in> H" shows "u \<preceq>\<^sub>t lt h"
has_integral_add_cbox: fixes f :: "'a \<Rightarrow> 'b"   and k :: "'b"   and a :: "'a"   and b :: "'a"   and g :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "(f has_integral k) (cbox a b)"   and "(g has_integral l) (cbox a b)" shows "((\<lambda>x. f x + g x) has_integral k + l) (cbox a b)"
list_induct2: fixes xs :: "'b list"   and ys :: "'c list"   and P :: "'b list \<Rightarrow> 'c list \<Rightarrow> bool" assumes "length xs = length ys"   and "P [] []"   and "\<And>x xs y ys. \<lbrakk>length xs = length ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
supp_map_transform_step: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and t :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>" assumes "\<And>x e a. (x, e) \<in> set \<Gamma> \<Longrightarrow> supp (t a e) \<subseteq> supp e" shows "supp (map_transform t ae \<Gamma>) \<subseteq> supp \<Gamma>"
delete_shadow_root_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and ptr :: "(_) document_ptr"   and getter :: "('c, 'a, 'b, 'j option) RDocument_scheme \<Rightarrow> 'm" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
from_hma\<^sub>m_diff: fixes a :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and b :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>m a - Mod_Type_Connect.from_hma\<^sub>m b = Mod_Type_Connect.from_hma\<^sub>m (a - b)"
seqll_def: fixes i :: "'h"   and P :: "('i \<times> 'j) \<times> 'k \<times> 'i \<times> 'j \<Rightarrow> bool" shows "seqll i P \<equiv> \<lambda>((\<sigma>, p), a, \<sigma>', p'). P ((\<sigma> i, p), a, \<sigma>' i, p')"
g_def: shows "g = (\<lambda>t. (Re b - Re a) * t + (Re a - Re z))"
Compl_Un: fixes A :: "'b set"   and B :: "'b set" shows "- (A \<union> B) = - A \<inter> - B"
czero_le_sgn_iff: fixes x :: "complex" shows "(0 \<le> sgn x) = (0 \<le> x)"
hmem_eclose_le: fixes y :: "hf"   and x :: "hf" assumes "y \<^bold>\<in> x" shows "eclose y \<le> eclose x"
strict_boundedE: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "max b c < a" obtains "b < a" and "c < a"
Eb_1: shows "Eb \<subseteq> Pow (space B)"
R_a_def: fixes i :: "I" shows "R_a i = (case i of loop \<Rightarrow> R2_a | _ \<Rightarrow> R1_a)"
co_out2c: shows "connected (outside (path_image c2 \<union> path_image c))"
convergent_eq_Cauchy: fixes S :: "nat \<Rightarrow> 'b" shows "(\<exists>l. S \<longlonglongrightarrow> l) = Cauchy S"
assms_3: shows "S \<union> {p \<^bold>\<and> q} \<in> C"
A: shows "\<forall>\<^sub>F n in sequentially. rGamma_series (1::'a) n = of_nat (Suc n) / of_nat n"
Tree\<^sub>\<alpha>_abs_rep: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>" shows "Formula.abs_Tree\<^sub>\<alpha> (Formula.rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
mbisim_def: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "s1 \<approx>m s2 \<equiv> finite (dom (thr s1)) \<and> locks s1 = locks s2 \<and> wset s1 = wset s2 \<and> wset_thread_ok (wset s1) (thr s1) \<and> interrupts s1 = interrupts s2 \<and> (\<forall>t. tbisim (wset s2 t = None) t (thr s1 t) (shr s1) (thr s2 t) (shr s2))"
open_surjective_linear_image: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "open A"   and "linear f"   and "surj f" shows "open (f ` A)"
IH: shows "\<exists>j. q (sdrop j (stl \<omega>)) \<and> (\<forall>k<j. p (sdrop k (stl \<omega>)))"
inf_idem: fixes inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and x :: "'a::type" assumes "class.semilattice_inf (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)" shows "(inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (x::'a::type) x = x"
assms: shows "b \<in>\<^sub>A implied_frontier_alt c loc2"
accepting_pair\<^sub>G\<^sub>R_simp: fixes \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and F :: "('d \<times> 'e \<times> 'd) set"   and \<I> :: "('d \<times> 'e \<times> 'd) set set"   and w :: "nat \<Rightarrow> 'e" shows "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (F, \<I>) w \<equiv> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> F = {} \<and> (\<forall>I\<in>\<I>. limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> I \<noteq> {})"
orbit_stab_fun_is_bij: fixes x :: "'c" assumes "x \<in> E" shows "bij_betw (\<lambda>R. \<phi> (inv (SOME h. h \<in> R)) x) (rcosets stabilizer G \<phi> x) (orbit G \<phi> x)"
rel_eq_onp: fixes P1 :: "'d \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> bool" shows "rel_gpv (eq_onp P1) (eq_onp P2) = eq_onp (pred_gpv P1 P2)"
H_seq: fixes p :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and r :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" assumes "rel_aka.H p x r"   and "rel_aka.H r y q" shows "rel_aka.H p (x ; y) q"
results_gpv_mono: fixes \<I>' :: "('a, 'b) \<I>"   and \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" assumes "\<I>' \<le> \<I>"   and "\<I>' \<turnstile>g gpv \<surd>" shows "results_gpv \<I> gpv \<subseteq> results_gpv \<I>' gpv"
n2_type1: fixes t1 :: "'c bro"   and h :: "nat"   and t2 :: "'c bro"   and a :: "'c" assumes "t1 \<in> Um h"   and "t2 \<in> B h" shows "n2 t1 a t2 \<in> T (Suc h)"
SPLIT2: shows "((rh, ch), e, v # r', c') \<in> trss fg"
inf_compl_bot_left1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf (uminus x) (inf x y) = bot"
supp_is_subset: fixes S :: "atom set"   and x :: "'a" assumes "S supports x"   and "finite S" shows "supp x \<subseteq> S"
simps_1: fixes x :: "'c"   and y :: "'c"   and xs :: "'c list" shows "ALG_P [] x y xs = 0"
sorted_hd_le: fixes xs :: "'a list"   and x :: "'a" assumes "sorted xs"   and "x \<in> list.set xs" shows "hd xs \<le> x"
dist_add_cancel2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "dist (plus b a) (plus c a) = dist b c"
clo: shows "closedin (top_of_set (S \<times> T)) U"
compact_space_def: fixes X :: "'d topology" shows "compact_space X \<equiv> compactin X (topspace X)"
enclen_def: fixes z :: "nat" shows "enclen z = BMax_rec (\<lambda>x. inside z (x - 1)) z"
Bs_def: shows "Bs = map (\<lambda>i. unit_vec (hd (dims A)) i \<otimes> subtensor A i) [0..<hd (dims A)]"
envalid_def: fixes n :: "nat"   and t :: "etriple" shows "\<Turnstile>n:\<^sub>e t \<equiv> let (P, e, Q) = t in \<forall>s v t. s -e\<succ>v-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q v t"
phiDefNodes_aux_cases: fixes g :: "'g"   and v :: "'var"   and unvisited :: "'node list"   and n :: "'node" obtains "phiDefNodes_aux g v unvisited n = {}" and "n \<notin> set unvisited \<or> v \<in> defs g n"   | "phiDefNodes_aux g v unvisited n = fold (\<union>) (map (phiDefNodes_aux g v (removeAll n unvisited)) (predecessors g n)) (if length (predecessors g n) = 1 then {} else {n})" and "n \<in> set unvisited" and "v \<notin> defs g n"
one_stalk_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and x :: "'c"   and V :: "'c set" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x" shows "stalk.one_stalk \<FF> \<rho> one_str I V \<equiv> direct_lim.class_of \<FF> \<rho> I V (one_str V)"
derivation_one_eq_zero: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" assumes "X \<in> tangent_space p"   and "p \<in> carrier" shows "X (restrict0 carrier (\<lambda>x. 1)) = 0"
oreachable_pair_induct: fixes \<sigma> :: "'e"   and p :: "'f"   and A :: "('e \<times> 'f, 'g) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "\<And>\<sigma> p. (\<sigma>, p) \<in> init A \<Longrightarrow> P \<sigma> p"   and "\<And>\<sigma> p \<sigma>'. \<lbrakk>(\<sigma>, p) \<in> oreachable A S U; P \<sigma> p; U \<sigma> \<sigma>'\<rbrakk> \<Longrightarrow> P \<sigma>' p"   and "\<And>\<sigma> p \<sigma>' p' a. \<lbrakk>(\<sigma>, p) \<in> oreachable A S U; P \<sigma> p; ((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A; S \<sigma> \<sigma>' a\<rbrakk> \<Longrightarrow> P \<sigma>' p'" shows "P \<sigma> p"
set_times_plus_distribs_1: fixes a :: "'c"   and b :: "'c"   and C :: "'c set" shows "a *o (b +o C) = a * b +o a *o C"
Pd_singletons_for_ds_def: fixes X :: "'a set"   and ds :: "'b set" shows "Pd_singletons_for_ds X ds \<equiv> override_on Pd (\<lambda>d. Y.dX X d \<times> Y.dX X d) ds"
smallo_real_nat_transfer: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "f \<in> o(g)" shows "(\<lambda>x. f (real x)) \<in> o(\<lambda>x. g (real x))"
qGood_iff_good_asTerm: fixes qX :: "('e, 'f, 'varSort, 'var, 'g) qTerm" shows "good (asTerm qX) = qGood qX"
trace_preserves_simple_ownership_distinct: fixes c :: "nat \<Rightarrow> ('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat"   and k :: "nat"   and l :: "nat" assumes "simple_ownership_distinct (fst (c n))"   and "trace c n k"   and "\<forall>x<k. safe_delayed (c (n + x))"   and "l \<le> k" shows "simple_ownership_distinct (fst (c (n + l)))"
ide_is_iso: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "category C"   and "partial_magma.ide C a" shows "category.iso C a"
pCons_2: shows "pm_of_poly x (p * q) = pm_of_poly x p * pm_of_poly x q"
sf_norm_help: fixes K :: "'b set"   and j :: "'b"   and t :: "'c"   and C :: "'b \<Rightarrow> 'c set"   and z :: "'b \<Rightarrow> real" assumes "finite K"   and "j \<in> K"   and "t \<in> C j"   and "\<forall>i\<in>K - {j}. t \<notin> C i" shows "(\<Sum>i\<in>K. z i * \<chi> (C i) t) = z j"
restrict_bind_spmf: fixes p :: "'d spmf"   and g :: "'d \<Rightarrow> 'c spmf"   and A :: "'c set" shows "(p \<bind> g) \<upharpoonleft> A = p \<bind> (\<lambda>x. g x \<upharpoonleft> A)"
open_segment_eq_empty: fixes a :: "'b"   and b :: "'b" shows "(open_segment a b = {}) = (a = b)"
order_iff: fixes b :: "'c"   and a :: "'c" shows "(b \<le> a) = (a = a + b)"
termdiffs_strong: fixes c :: "nat \<Rightarrow> 'a"   and K :: "'a"   and x :: "'a" assumes "summable (\<lambda>n. c n * K ^ n)"   and "norm x < norm K" shows "((\<lambda>x. \<Sum>n. c n * x ^ n) has_field_derivative (\<Sum>n. diffs c n * x ^ n)) (at x)"
SUBMAP_FUNION_DRESTRICT_i: fixes v :: "'d"   and vsa :: "'d set"   and vsb :: "'d set"   and vs :: "'d set"   and f :: "('d, 'e) fmap" assumes "v \<in> vsa" shows "fmlookup (fmrestrict_set ((vsa \<union> vsb) \<inter> vs) f) v = fmlookup (fmrestrict_set (vsa \<inter> vs) f) v"
fdomainD: fixes a :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> r" shows "\<exists>b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
shortestPath_is_path: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "cf.isShortestPath u p v" shows "cf.isPath u p v"
tiny_dghm_ObjMap_in_Vset: shows "\<FF>\<lparr>ObjMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
Aexp_heap_below_Aheap: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity"   and x :: "Vars.var"   and a' :: "Arity"   and e' :: "exp" assumes "(Aheap \<Gamma> e\<cdot>a) x = up\<cdot>a'"   and "map_of \<Gamma> x = Some e'" shows "\<A>\<^bsub>a'\<^esub> e' \<sqsubseteq> Aheap \<Gamma> e\<cdot>a \<squnion> \<A>\<^bsub>a\<^esub> (Terms.Let \<Gamma> e)"
fMin_eqI: fixes A :: "'a fset"   and x :: "'a" assumes "\<And>y. y |\<in>| A \<Longrightarrow> x \<le> y"   and "x |\<in>| A" shows "fMin A = x"
finite_theFT: fixes cl :: "('g, 'h, 'i) cmd list" shows "finite (theFT cl)"
arity_sym\<^sub>h_lt_\<omega>_if_\<delta>\<^sub>h_gt_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and f :: "'b" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "0 < of_nat \<delta>" shows "hmset_of_enat (arity_sym f) < \<omega>"
split_list_first_prop: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>y\<in>set ys. \<not> P y)"
assigned_nodes_ptr_in_heap: fixes h :: "(_) heap"   and slot :: "(_) element_ptr" assumes "h \<turnstile> ok assigned_nodes slot" shows "slot |\<in>| element_ptr_kinds h"
that: fixes k :: "nat"   and qk :: "'Q" assumes "i \<le> k"   and "k < j"   and "r k = (qk, n)"   and "n \<in> acc qk" shows "thesis"
llcp_commute: fixes xs :: "'c llist"   and ys :: "'c llist" shows "llcp xs ys = llcp ys xs"
iso_counit: fixes a :: "'d" assumes "D.obj a" shows "D.iso (\<eta>'.counit a)"
distinct_27: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> While x61 x62"
term_ok_eq_ax_4: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "is_std_sig (sig \<Theta>)" shows "term_ok \<Theta> eq_intr_ax"
adjacent_def: fixes w :: "nat \<times> nat"   and ts :: "'b tree list" shows "adjacent w ts = (Leaf \<notin> set ts \<and> list_all2 (\<lambda>t u. l t = Suc (r u)) (butlast ts) (tl ts) \<and> (ts = [] \<or> l (last ts) = fst w \<and> r (hd ts) = snd w))"
vertex_set_split: shows "\<Union> X = f ` \<Union> X \<union> g ` \<Union> X"
mhd_resTy_simp: fixes m :: "ref_ty \<times> mhead" shows "resTy (mhd m) = resTy m"
constants_ax: shows "0 < \<beta> \<and> 0 < \<mu> \<and> 0 < rmin \<and> rmin \<le> rmax \<and> 0 < \<rho> \<and> 0 < np \<and> maxfaults \<le> np"
weak_conv_imp_continuity_set_conv: fixes \<mu> :: "nat \<Rightarrow> real measure"   and M :: "real measure"   and A :: "real set" assumes "\<And>n. real_distribution (\<mu> n)"   and "real_distribution M"   and "weak_conv_m \<mu> M"   and "A \<in> sets borel"   and "emeasure M (frontier A) = 0" shows "(\<lambda>n. Sigma_Algebra.measure (\<mu> n) A) \<longlonglongrightarrow> Sigma_Algebra.measure M A"
fw_impl_correct: fixes n :: "nat" shows "(fw_impl n, fw_spec n) \<in> (mtx_curry_assn n)\<^sup>d \<rightarrow>\<^sub>a mtx_curry_assn n"
expand: fixes writerT :: "('b, 'c, 'a) writerT"   and writerT' :: "('b, 'c, 'a) writerT" assumes "run_writer writerT = run_writer writerT'" shows "writerT = writerT'"
lower_bound_right: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y \<le> - y"
tape_of_prod_def: shows "tape_of \<equiv> tape_of_nat_prod"
in_span_listI: fixes v :: "'a vec"   and c :: "nat \<Rightarrow> 'a"   and vs :: "'a vec list" assumes "True"   and "v = lincomb_list c vs" shows "v \<in> span_list vs"
idx_spec: fixes fP :: "'a list \<Rightarrow> bool"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "cnvSpec fP P" shows "fP xs = (\<forall>i<length xs - 1. P (xs ! i) (xs ! Suc i))"
check_in_heap_reads: fixes object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing)} (check_in_heap object_ptr) h h'"
that: fixes l :: "'b" assumes "lstep l (s i) (s (Suc i))" shows "thesis"
frechet_derivative_unique_within: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and S :: "'a set"   and f'' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within S)"   and "(f has_derivative f'') (at x within S)"   and "\<And>i e. \<lbrakk>i \<in> Basis; 0 < e\<rbrakk> \<Longrightarrow> \<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> S" shows "f' = f''"
P3: shows "A \<noteq> B \<and> C \<noteq> B"