hcju/isabelleps huggingface dataset:
step: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and u :: "'e \<Rightarrow> 'f"   and l' :: "'d"   and u' :: "'e \<Rightarrow> 'f"   and l'' :: "'d"   and u'' :: "'e \<Rightarrow> 'f" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"   and "A \<turnstile> \<langle>l', u'\<rangle> \<rightarrow>* \<langle>l'', u''\<rangle>" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l'', u''\<rangle>"
extract_SomeE: fixes P :: "'k \<Rightarrow> bool"   and xs :: "'k list"   and ys :: "'k list"   and y :: "'k"   and zs :: "'k list" assumes "List.extract P xs = Some (ys, y, zs)" shows "xs = ys @ y # zs \<and> P y \<and> \<not> (\<exists>y\<in>set ys. P y)"
aux_range: fixes k :: "nat"   and n :: "nat" assumes "k < 2 ^ (Suc n + 1)"   and "2 \<le> k" shows "k = 2 \<or> k = 3 \<or> (\<exists>l\<ge>2. l \<le> 2 ^ (n + 1) - 1 \<and> (k = 2 * l \<or> k = 2 * l + 1))"
connected_continuous_image: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on s f"   and "connected s" shows "connected (f ` s)"
quasi_sturm_seq_Cons: fixes p :: "real poly"   and ps :: "real poly list" assumes "quasi_sturm_seq (p # ps)"   and "ps \<noteq> []" shows "quasi_sturm_seq ps"
mat_kernel_mult_subset: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat n nr" shows "mat_kernel A \<subseteq> mat_kernel (B * A)"
crsp_step_dec_b: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and n :: "nat"   and la :: "cell list"   and ra :: "cell list"   and e :: "nat" assumes "ly = layout_of ap"   and "crsp ly (as, lm) (s, l, r) ires"   and "inv_locate_a (as, lm) (n, la, ra) ires"   and "abc_fetch as ap = Some (Dec n e)" shows "\<exists>stp>0. crsp ly (abc_step_l (as, lm) (Some (Dec n e))) (steps (start_of ly as + 2 * n, la, ra) (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp) ires"
cprank_max1_IH: shows "cprank_max1 (tensors_from_net (insert_weights s (shallow_model' Z M N) (\<lambda>i. w (i + count_weights s (shallow_model' Z M 0)))) $ y)"
psubst_fml: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'a \<Rightarrow> ('sf, 'sc, 'sz) formula"   and \<alpha> :: "('sf, 'sc + 'a, 'sz) hp"   and \<phi> :: "('sf, 'sc + 'a, 'sz) formula" assumes "is_interp I" shows "(PPadmit \<sigma> \<alpha> \<longrightarrow> hpsafe \<alpha> \<longrightarrow> (\<forall>i. fsafe (\<sigma> i)) \<longrightarrow> (\<forall>\<nu> \<omega>. ((\<nu>, \<omega>) \<in> prog_sem I (PPsubst \<alpha> \<sigma>)) = ((\<nu>, \<omega>) \<in> prog_sem (PFadjoint I \<sigma>) \<alpha>))) \<and> (PFadmit \<sigma> \<phi> \<longrightarrow> fsafe \<phi> \<longrightarrow> (\<forall>i. fsafe (\<sigma> i)) \<longrightarrow> (\<forall>\<nu>. (\<nu> \<in> fml_sem I (PFsubst \<phi> \<sigma>)) = (\<nu> \<in> fml_sem (PFadjoint I \<sigma>) \<phi>)))"
sel_6: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "'b option"   and x24 :: "'b tree"   and x25 :: "'b tree" shows "val (Node x21 x22 x23 x24 x25) = x23"
ListMem_iff: fixes x :: "'c"   and xs :: "'c list" shows "ListMem x xs = (x \<in> set xs)"
Suc: shows "Stirling (Suc (Suc n)) (Suc (Suc 0)) = 2 ^ Suc n - 1"
T_leaves: fixes as :: "'a list" shows "T_leaves as = length as + 1"
mthd_simp: fixes C :: "'b"   and m :: "'a" shows "mthd (C, m) = m"
smult_diff: fixes a :: "'b"   and A :: "('b, 'c) sq_matrix"   and B :: "('b, 'c) sq_matrix" shows "a *\<^sub>S (A - B) = a *\<^sub>S A - a *\<^sub>S B"
finite_greaterThanLessThan: fixes l :: "nat"   and u :: "nat" shows "finite {l<..<u}"
exec_meth_append_xt: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ins' :: "'addr instr list"   and xt' :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list" assumes "exec_meth ci P ins xt t h s ta h' s'" shows "exec_meth ci P (ins @ ins') (xt @ xt') t h s ta h' s'"
r_zero: fixes x :: "complex vec" assumes "x \<in> carrier_vec d" shows "x + 0\<^sub>v d = x"
greatest_prev_image: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat" assumes "dilating f sub r"   and "\<nexists>n\<^sub>0. f n\<^sub>0 = n" shows "\<exists>n\<^sub>p. f n\<^sub>p < n \<and> (\<forall>k. f n\<^sub>p < k \<and> k \<le> n \<longrightarrow> (\<nexists>k\<^sub>0. f k\<^sub>0 = k))"
arctan_upper_43_def: shows "arctan_upper_43 \<equiv> \<lambda>x. (256 * x ^ 6 + 5943 * x ^ 4 + 19250 * x\<^sup>2 + 15015) * x / (35 * (35 * x ^ 6 + 315 * x ^ 4 + 693 * x\<^sup>2 + 429))"
Rep_contour_inverse: fixes x :: "contour" shows "Abs_contour (Rep_contour x) = x"
writes_subset: fixes S :: "('x, 'y, 'z) prog set"   and f :: "('x, 'y, 'aa) prog"   and h :: "'x"   and h' :: "'x"   and S' :: "('x, 'y, 'z) prog set" assumes "writes S f h h'"   and "S \<subseteq> S'" shows "writes S' f h h'"
2: shows "row_length (M1 \<otimes> M2 \<otimes> M3) = row_length M1 * row_length M2 * row_length M3"
hyps_9: shows "A\<^sub>P \<sharp>* \<Psi>"
pt_eqvt_fun1a: fixes f :: "'a \<Rightarrow> 'c" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "supp f = {}" shows "\<forall>pi. pi \<bullet> f = f"
preserves_cod: fixes f :: "'t" assumes "S'.arr f" shows "S.cod (\<Psi> f) = \<Psi> (S'.cod f)"
T_Sub_5: fixes S' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> U <: S"   and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'"   and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'" shows "P"
induct: fixes P :: "('a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'd) list \<Rightarrow> ('a \<times> 'e) list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "('a \<times> 'd) list"   and a2 :: "('a \<times> 'e) list" assumes "\<And>f. P f [] []"   and "\<And>f ky vy ys. P f [] ys \<Longrightarrow> P f [] ((ky, vy) # ys)"   and "\<And>f kx vx xs. P f xs [] \<Longrightarrow> P f ((kx, vx) # xs) []"   and "\<And>f kx vx xs ky vy ys. \<lbrakk>comp kx ky = Eq \<Longrightarrow> P f xs ys; comp kx ky = Lt \<Longrightarrow> P f xs ((ky, vy) # ys); comp kx ky = Gt \<Longrightarrow> P f ((kx, vx) # xs) ys\<rbrakk> \<Longrightarrow> P f ((kx, vx) # xs) ((ky, vy) # ys)" shows "P a0 a1 a2"
distinct_4: fixes name' :: "name"   and lam' :: "lam"   and name :: "name" shows "Lam [name'].lam' \<noteq> Var name"
awalk_last_in_verts: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v" shows "v \<in> verts G"
nonstrict_eq: fixes i :: "nat"   and j :: "nat" assumes "non_strict_dominate i j" shows "\<not> strict_dominate i j"
dependent_zero: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set" assumes "module scale"   and "(0::'c) \<in> A" shows "module.dependent scale A"
D_T: fixes s :: "'a event list"   and P :: "'a process" assumes "s \<in> D P" shows "s \<in> T P"
coprime_normalize: fixes a :: "'a"   and b :: "'a" assumes "comm_monoid_mult_class.coprime a b" shows "comm_monoid_mult_class.coprime (normalize a) b"
nxt: fixes q :: "hf"   and x :: "'c" assumes "q \<in> dfa.states \<lparr>dfa.states = {0}, init = 0, final = {}, nxt = \<lambda>q x. q\<rparr>" shows "dfa.nxt \<lparr>dfa.states = {0}, init = 0, final = {}, nxt = \<lambda>q x. q\<rparr> q x \<in> dfa.states \<lparr>dfa.states = {0}, init = 0, final = {}, nxt = \<lambda>q x. q\<rparr>"
bisimulation_into_delay_flip: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool" assumes "bisimulation_into_delay trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2" shows "bisimulation_into_delay trsys2 trsys1 (flip bisim) (flip tlsim) \<tau>move2 \<tau>move1"
bind_strict_mono_strong1: fixes m :: "('d, 'e) state"   and f :: "'e \<Rightarrow> ('d, 'f) state" assumes "mono_state m"   and "\<And>x s s'. run_state m s = (x, s') \<Longrightarrow> strict_mono_state (f x)" shows "strict_mono_state (m \<bind> f)"
option2list_def: fixes n :: "'b option" shows "option2list n \<equiv> case n of None \<Rightarrow> [] | Some s \<Rightarrow> [s]"
Ueq: shows "inter_scheme l U = concat [c, a, d, b] @ interact as bs"
ffb_ffd_demorgan: fixes f :: "'d \<Rightarrow> 'c set" shows "\<partial> \<circ> fb\<^sub>\<F> f = fd\<^sub>\<F> f \<circ> \<partial>"
valid_B_first: fixes xs :: "color list"   and i :: "nat" assumes "valid_point xs i"   and "i < length xs" shows "valid_point (B # xs) (i + 1)"
construct_lhs_matches_solve_for_lhs_R: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "(nat list \<times> nat list) list"   and signs :: "rat list list" assumes "satisfy_equation_R p qs subsets signs"   and "invertible_mat (M_mat_R signs subsets)" shows "w_vec_R p qs signs = solve_for_lhs_R p qs subsets (M_mat_R signs subsets)"
1: shows "before x xs \<inter> before x ys \<inter> (after x xs \<inter> before x ys) = {}"
Inr': fixes s2 :: "'s2" shows "Inr s2 \<in> terminates_on (append_has_next (has_next1, next1) (has_next2, next2), append_next (has_next1, next1) (has_next2, next2))"
g_def: fixes y :: "'v" shows "g y = (if y \<in> \<^bold>E `` X then weight \<Gamma> y else 0)"
matrix_to_iarray_fst_Gauss_Jordan_det_P: fixes A :: "(('b, 'c) vec, 'd) vec" shows "fst (Gauss_Jordan_det_P A) = fst (Gauss_Jordan_det_P_iarrays (matrix_to_iarray A))"
Suc_0_mod_eq_Suc_0_iff: fixes n :: "nat" shows "(Suc 0 mod n = Suc 0) = (n \<noteq> Suc 0)"
shunt_bijective: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "bijective z" shows "(x \<le> y * z) = (x * z\<^sup>T \<le> y)"
measurable_merge: fixes I :: "'a set"   and J :: "'a set"   and M :: "'a \<Rightarrow> 'b measure" shows "merge I J \<in> Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M \<rightarrow>\<^sub>M Pi\<^sub>M (I \<union> J) M"
fresh_star_permute_iff: fixes p :: "perm"   and a :: "atom set"   and x :: "'e" shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) = a \<sharp>* x"
conj_zero_left: fixes x :: "'a" shows "\<one> \<^bold>\<squnion> x = \<one>"
if_reg_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and c\<^sub>A_then :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>A_else :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_then :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_else :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = Stmt.If (bexp.Eq x 0) c\<^sub>A_then c\<^sub>A_else ;; c\<^sub>A_tail"   and "c\<^sub>C = Stmt.If (bexp\<^sub>C.Eq reg\<^sub>C 0) c\<^sub>C_then c\<^sub>C_else ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "x \<in> mds\<^sub>A AsmNoWrite \<and> x \<notin> mds\<^sub>A AsmNoReadOrWrite"   and "ev\<^sub>B\<^sub>C mem\<^sub>C (bexp\<^sub>C.Eq reg\<^sub>C 0) = ev\<^sub>B mem\<^sub>A (bexp.Eq x 0)"   and "reg\<^sub>C \<notin> mds\<^sub>C GuarNoReadOrWrite"   and "reg\<^sub>C \<notin> mds\<^sub>C GuarNoWrite"   and "x \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "\<forall>x'. x \<in> \<C>_vars x' \<longrightarrow> x' \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "\<forall>mem\<^sub>A' mem\<^sub>C' mem\<^sub>A'' mem\<^sub>C''. mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> ev\<^sub>B\<^sub>C mem\<^sub>C'' (bexp\<^sub>C.Eq reg\<^sub>C 0) = ev\<^sub>B mem\<^sub>A'' (bexp.Eq x 0) \<and> ev\<^sub>B\<^sub>C mem\<^sub>C' (bexp\<^sub>C.Eq reg\<^sub>C 0) = ev\<^sub>B mem\<^sub>A' (bexp.Eq x 0) \<and> eval_abv\<^sub>A \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A \<langle>(if ev\<^sub>B mem\<^sub>A'' (bexp.Eq x 0) then c\<^sub>A_then else c\<^sub>A_else) ;; c\<^sub>A_tail, mds\<^sub>A, mem\<^sub>A'\<rangle>\<^sub>A \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>(if ev\<^sub>B\<^sub>C mem\<^sub>C'' (bexp\<^sub>C.Eq reg\<^sub>C 0) then c\<^sub>C_then else c\<^sub>C_else) ;; c\<^sub>C_tail, mds\<^sub>C, mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>(if ev\<^sub>B mem\<^sub>A'' (bexp.Eq x 0) then c\<^sub>A_then else c\<^sub>A_else) ;; c\<^sub>A_tail, mds\<^sub>A, mem\<^sub>A'\<rangle>\<^sub>A, \<langle>(if ev\<^sub>B\<^sub>C mem\<^sub>C'' (bexp\<^sub>C.Eq reg\<^sub>C 0) then c\<^sub>C_then else c\<^sub>C_else) ;; c\<^sub>C_tail, mds\<^sub>C, mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"   and "x \<noteq> buffer \<and> x \<noteq> high_var" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
simps_8: fixes arity :: "'c \<Rightarrow> nat"   and Ana :: "('c, 'd) Term.term \<Rightarrow> ('c, 'd) Term.term list \<times> ('c, 'd) Term.term list"   and public :: "'c \<Rightarrow> bool"   and M :: "('c, 'd) Term.term set"   and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set"   and X :: "'d list"   and F :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and F' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and S :: "('c, 'd) stateful_strand_step list" assumes "intruder_model arity Ana" shows "intruder_model.strand_sem_stateful arity public Ana M D (\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> # S) = (\<lambda>\<I>. intruder_model.negchecks_model \<I> D X F F' \<and> intruder_model.strand_sem_stateful arity public Ana M D S \<I>)"
changes_poly_neg_inf_def: fixes ps :: "'a poly list" shows "changes_poly_neg_inf ps = changes (map sgn_neg_inf ps)"
lagrange: fixes H :: "nat set" assumes "subgroup H G" shows "card (rcosets\<^bsub>G\<^esub> H) * card H = Coset.order G"
hyps_3: shows "\<langle>c'',s''\<rangle> \<rightarrow>* \<langle>c',s'\<rangle>"
monoD: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f"   and x :: "'e"   and y :: "'e" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" shows "f x \<sqsubseteq> f y"
is_unit_power_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.algebraic_semidom divide plus minus zero times one" shows "dvd.dvd times (power.power one times a n) one = (dvd.dvd times a one \<or> n = 0)"
intros_4: fixes m :: "'a"   and M :: "'b" shows "lformula0 (In m M)"
distinct_disc_15: fixes event :: "('a, 'b, 'c) event" assumes "isRecv event" shows "\<not> isSnapshot event"
simps_11: fixes v :: "'opinl" shows "generalize_instr (IOpInl v) = IOpInl v"
card_of_Sigma_ordLeq_infinite: fixes B :: "'a set"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "infinite B"   and "|I| \<le>o |B|"   and "\<forall>i\<in>I. |A i| \<le>o |B|" shows "|Sigma I A| \<le>o |B|"
list_nonempty_induct: fixes xs :: "'g list"   and P :: "'g list \<Rightarrow> bool" assumes "xs \<noteq> []"   and "\<And>x. P [x]"   and "\<And>x xs. \<lbrakk>xs \<noteq> []; P xs\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P xs"
sum_upto_asymptotics_lift_nat_real_aux: fixes f :: "nat \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and h :: "real \<Rightarrow> real" assumes "(\<lambda>n. sum f {1..n} - g (real n)) \<in> O(\<lambda>n. h (real n))"   and "(\<lambda>n. g (real n) - g (real (Suc n))) \<in> O(\<lambda>n. h (real n))"   and "(\<lambda>n. h (real n)) \<in> O(\<lambda>n. h (real (Suc n)))"   and "\<And>x. 1 \<le> x \<Longrightarrow> 0 < h x"   and "mono_on g {1..} \<or> mono_on (\<lambda>x. - g x) {1..}"   and "mono_on h {1..} \<or> mono_on (\<lambda>x. - h x) {1..}" shows "\<exists>c>0. \<forall>x\<ge>1. sum_upto f x - g x \<le> c * h x"
lhd_ldropn: fixes n :: "nat"   and xs :: "'c llist" assumes "enat n < llength xs" shows "lhd (ldropn n xs) = xs $ n"
homogenize_def: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "homogenize x p = (\<Sum>t\<in>keys p. monomial (lookup p t) (monomial (poly_deg p - deg_pm t) x + t))"
next_plane0_vertices_subset: fixes g' :: "graph"   and p :: "nat"   and g :: "graph" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "minGraphProps g" shows "\<V> g \<subseteq> \<V> g'"
types_preserved_e1: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and \<S> :: "s_context"   and tvs :: "t list"   and \<C> :: "t_context"   and arb_return :: "t list option"   and arb_labs :: "t list list"   and ts :: "t list"   and ts' :: "t list" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"   and "store_typing s \<S>"   and "tvs = map typeof vs"   and "i < length (inst s)"   and "\<C> = (s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ tvs, label := arb_labs, return := arb_return\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> es : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> es' : ts _> ts' \<and> map typeof vs = map typeof vs'"
option2set_def: fixes n :: "'a option" shows "option2set n \<equiv> case n of None \<Rightarrow> {} | Some s \<Rightarrow> {s}"
islimpt_EMPTY: fixes x :: "'a" shows "\<not> x islimpt {}"
continuous_on_cmult_ereal: fixes c :: "ereal"   and A :: "'a set"   and f :: "'a \<Rightarrow> ereal" assumes "\<bar>c\<bar> \<noteq> \<infinity>"   and "continuous_on A f" shows "continuous_on A (\<lambda>x. c * f x)"
cring_simprules_8: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "cring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
exec_gpv_resource_of_oracle: fixes gpv :: "('a, 'b, 'c) gpv"   and "oracle" :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and s :: "'d" shows "exec_gpv run_resource gpv (RES oracle s) = map_spmf (map_prod id (RES oracle)) (exec_gpv oracle gpv s)"
hoare_strip_\<Gamma>: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and p :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and F' :: "'f set" assumes "\<Gamma>,{}\<turnstile>\<^bsub>/F \<^esub>P p Q,A"   and "F' \<subseteq> - F" shows "strip F' \<Gamma>,{}\<turnstile>\<^bsub>/F \<^esub>P p Q,A"
simps_2: fixes x :: "Runs.var" shows "in_progress None x = False"
ffun_lens_mwb: fixes i :: "'b" shows "mwb_lens (ffun_lens i)"
axioms_3: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "partially_ordered_set A less_eq" shows "transitive A less_eq"
invar_def: fixes bq :: "('c, 'd) SkewBinomialTree list" shows "invar bq = (queue_invar bq \<and> rank_skew_invar bq)"
star_1l: fixes x :: "'a" shows "x \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
assoc_unit_l: fixes a :: "'a"   and b :: "'a" assumes "a \<sim> b"   and "b \<in> Units G"   and "a \<in> carrier G" shows "a \<in> Units G"
rep_eq: fixes x :: "'c \<Rightarrow> bool"   and xa :: "'c fset" shows "fset (ffilter x xa) = Set.filter x (fset xa)"
pid_def: fixes x :: "'a" shows "\<delta> x = (if x \<in> \<xi> then 1::'c else \<bottom>)"
simps_2: fixes r :: "'a language"   and s :: "'a language" shows "\<dd> (Plus r s) = (\<lambda>a. Plus (\<dd> r a) (\<dd> s a))"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (store_sub2 instr curr_win rd 10 addr s1)) \<and> t2 = snd (fst (store_sub2 instr curr_win rd 10 addr s2))"
fmempty_lookup: fixes x :: "'b" shows "{$$} $$ x = None"
id_outside_supp: fixes x :: "'b"   and cs :: "'b list" assumes "x \<notin> set cs" shows "cycle_of_list cs x = x"
subset_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and B :: "'a set"   and A :: "'a set" assumes "inj_on f B"   and "A \<subseteq> B" shows "inj_on f A"
SigmaD1: fixes a :: "'e"   and b :: "'f"   and A :: "'e set"   and B :: "'e \<Rightarrow> 'f set" assumes "(a, b) \<in> Sigma A B" shows "a \<in> A"
eq_\<I>_gpv_into_rel_gpv: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and gpv :: "('a, 'c, 'd) gpv"   and gpv' :: "('b, 'c, 'd) gpv" assumes "eq_\<I>_gpv A \<I>_full gpv gpv'" shows "rel_gpv A (=) gpv gpv'"
individually_rational_on_def: fixes ds :: "D2 set"   and X :: "X4 set" shows "StableNoDecomp.individually_rational_on ds X = (StableNoDecomp.CD_on ds X = X \<and> StableNoDecomp.CH X = X)"
\<delta>\<^sub>h_le_\<epsilon>\<^sub>h: shows "\<delta>\<^sub>h \<le> \<epsilon>\<^sub>h"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat"   and y :: "rat" assumes "gs.\<sigma> x xa xb = y" obtains i :: "nat" and j :: "nat" where "x = 0" and "xa = ib__" and "xb = jb__" and "y = 0"   | l :: "nat" and i :: "nat" and j :: "nat" where "x = Suc l" and "xa = i" and "xb = j" and "y = (gs.d (Suc l) * gs.\<sigma> l i j + gs.\<mu>' i l * gs.\<mu>' j l) / gs.d l"
not_simple_elim: fixes S :: "('a, 'b) strand_step list" assumes "\<not> simple S" shows "(\<exists>A B f T. S = A @ send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t # B \<and> simple A) \<or> (\<exists>A B a t t'. S = A @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # B \<and> simple A) \<or> (\<exists>A B X F. S = A @ \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # B \<and> (\<nexists>\<I>. ineq_model \<I> X F))"
compatible_iff_seq: fixes g :: "'s"   and f :: "'s" shows "(domains g \<inter> codomains f \<noteq> {}) = seq g f"
sigma_sets_mono: fixes A :: "'b set set"   and X :: "'b set"   and B :: "'b set set" assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
smallsteps_ideal_deterministic: fixes t :: "Syntax.term"   and i :: "nat"   and u :: "Syntax.term"   and u' :: "Syntax.term" assumes "\<lless>[], t\<ggreater> I\<rightarrow>i \<lless>[], u\<ggreater>"   and "\<lless>[], t\<ggreater> I\<rightarrow>i \<lless>[], u'\<ggreater>" shows "u = u'"
Abinds_env_cong: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae' :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "\<And>x. x \<in> domA \<Delta> \<Longrightarrow> ae x = ae' x" shows "ABinds \<Delta>\<cdot>ae = ABinds \<Delta>\<cdot>ae'"
commutative_squares_7: shows "C.commutative_square \<mu>.leg0 (\<nu>.leg1 \<cdot> \<nu>_\<pi>\<rho>.prj\<^sub>1) \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1\<^sub>1 \<langle>\<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0\<^sub>1 \<lbrakk>\<nu>.leg0, \<pi>\<rho>.leg1\<rbrakk> \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0\<rangle>"
smooth_on_minus: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x - g x)"
prod_simps_2: fixes f0 :: "'a"   and f1 :: "'a" assumes "arr f0"   and "arr f1" shows "local.dom (ECC.prod f0 f1) = ECC.prod (local.dom f0) (local.dom f1)"