hcju/isabelleps huggingface dataset:
sum_mono_ge: fixes K :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> K \<Longrightarrow> g i \<le> f i" shows "sum g K \<le> sum f K"
closed_vimage: fixes s :: "'b set"   and f :: "'c \<Rightarrow> 'b" assumes "closed s"   and "continuous_on UNIV f" shows "closed (f -` s)"
inverse_arrowsE: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.inverse_arrows f g" obtains "VV.ide (VV.comp g f)" and "VV.ide (VV.comp f g)"
simps_2: fixes f1 :: "'e \<Rightarrow> 'c"   and f2 :: "'f \<Rightarrow> 'd"   and a :: "'f" shows "map_sum f1 f2 (Inr a) = Inr (f2 a)"
recurrence_time_flip_sign_2: fixes T :: "real"   and x :: "'a" assumes "T \<in> existence_ivl0 x"   and "flow0 x T = x" shows "flow0 x (- T) = x"
assms_4: shows "\<forall>\<^sub>F x in at_top. f x \<le> g x"
B: fixes Y :: "'a set" assumes "Y \<subseteq> X"   and "Y \<in> sets A" shows "emeasure B Y \<le> emeasure C Y"
nfa_of_dfa_reach': fixes M :: "nat bdd list \<times> bool list"   and l :: "nat"   and bss :: "bool list list"   and n1 :: "bool list"   and q :: "nat"   and n2 :: "bool list" assumes "wf_dfa M l"   and "list_all (is_alph l) bss"   and "n1 = (replicate (length (fst M)) False)[q := True]"   and "dfa_is_node M q"   and "nfa_reach (nfa_of_dfa M) n1 bss n2" shows "\<exists>p. dfa_reach M q bss p \<and> n2 = (replicate (length (fst M)) False)[p := True]"
continuous_map_axioms: shows "continuous_map X is_open Spec is_zariski_open f"
complement_inf: fixes x :: "'a" shows "x \<sqinter> - x = bot"
distinct_2: fixes x2 :: "'a" shows "Some x2 \<noteq> None"
that: fixes a :: "'b"   and a' :: "'b" assumes "{a, a'} \<subseteq> arcs G - arcs H"   and "a' \<noteq> a"   and "tail G a' = head G a"   and "head G a' = tail G a" shows "thesis"
show_matrix_real_iarrays_def: fixes A :: "real iarray iarray" shows "show_matrix_real_iarrays A = IArray.of_fun (\<lambda>i. show_vec_real_iarrays (A !! i)) (IArray.length A)"
Idl_subset_ideal': fixes a :: "int"   and b :: "int" assumes "a \<in> UNIV"   and "b \<in> UNIV" shows "(Idl\<^bsub>\<Z>\<^esub> {a} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {b}) = (a \<in> Idl\<^bsub>\<Z>\<^esub> {b})"
pgen_lossless_gpvD: fixes fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" assumes "pgen_lossless_gpv fail \<I> gpv" shows "expectation_gpv fail \<I> (\<lambda>_. 1) gpv = 1"
p: shows "p = (\<Sum>x\<in>keys (mapping_of p). MPoly_Type.monom (remove_key x' x) (MPoly_Type.coeff p x) * (if lookup x x' = 0 then 1 else MPoly_Type.monom (monomial (Suc 0) x') 1)) + (\<Sum>x\<in>keys (mapping_of p). MPoly_Type.monom (remove_key x' x) (MPoly_Type.coeff p x) * (if lookup x x' = 0 then 0 else 1) * qr (lookup x x')) * ((MPoly_Type.monom (monomial (Suc 0) x') 1)\<^sup>2 - MPoly_Type.monom (monomial (Suc 0) x') 1)"
riesz_frechet_representation_existence: fixes f :: "'b \<Rightarrow> complex" assumes "bounded_clinear f" shows "\<exists>t. \<forall>x. f x = \<langle>t, x\<rangle>"
A2a: fixes x :: "i" shows "(\<box>x)\<box> \<cong> \<box>x"
Ord_VPow_in_Vset_succI: fixes \<alpha> :: "V"   and a :: "V" assumes "Ord \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>" shows "VPow a \<in>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)"
that: shows "A \<turnstile> l \<longrightarrow>\<^bsup>g,\<mu>,X\<^esup> l'"
inj_on_image_Int: fixes f :: "'c \<Rightarrow> 'd"   and C :: "'c set"   and A :: "'c set"   and B :: "'c set" assumes "inj_on f C"   and "A \<subseteq> C"   and "B \<subseteq> C" shows "f ` (A \<inter> B) = f ` A \<inter> f ` B"
FinGroup_def: fixes G :: "'a set" shows "FinGroup G \<equiv> Group G \<and> FinGroup_axioms G"
obj_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "B.obj (trg\<^sub>B \<mu>)"
semicategory_smc_10: fixes \<alpha> :: "V"   and \<aa> :: "V" assumes "\<Z> \<alpha>"   and "\<aa> \<in>\<^sub>\<circ> Vset \<alpha>" shows "semicategory \<alpha> (smc_10 \<aa>)"
simps: fixes zero :: "'b"   and A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" shows "pivot_positions_main_gen zero A nr nc i j = (if i < nr then if j < nc then if A $$ (i, j) = zero then pivot_positions_main_gen zero A nr nc i (Suc j) else (i, j) # pivot_positions_main_gen zero A nr nc (Suc i) (Suc j) else [] else [])"
typeof_ClassCast: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "preallocated h" shows "typeof\<^bsub>h\<^esub> (Addr (addr_of_sys_xcpt ClassCast)) = \<lfloor>Class ClassCast\<rfloor>"
Stream_snth: fixes x :: "'d"   and s :: "'d stream"   and n :: "nat" shows "(x ## s) !! n = (case n of 0 \<Rightarrow> x | Suc n \<Rightarrow> s !! n)"
decode_exp_shift: fixes e :: "nat"   and M :: "hmultiset" assumes "local.aligned\<^sub>d e M" shows "local.decode 0 M = base ^ e * local.decode e M"
gfp_exp_unitary: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> unitary (t P)" shows "unitary (gfp_exp t)"
d_1: fixes f :: "'b poly"   and g :: "'b poly" shows "karatsuba_mult_poly f g = (let ff = coeffs f; gg = coeffs g; n = length ff; m = length gg in if n \<le> karatsuba_lower_bound \<or> m \<le> karatsuba_lower_bound then if n \<le> m then foldr (\<lambda>a p. smult a g + pCons (0::'b) p) ff 0 else foldr (\<lambda>a p. smult a f + pCons (0::'b) p) gg 0 else if n \<le> m then karatsuba_main gg m ff n else karatsuba_main ff n gg m)"
get_nondet_def: fixes get :: "('i \<Rightarrow> 'h) \<Rightarrow> 'h"   and f :: "'i \<Rightarrow> ('g, 'h) nondetT" shows "nondetM_base.get_nondet get f = NondetT (get (\<lambda>s. run_nondet (f s)))"
update_convs_2: fixes arcs' :: "('c \<times> 'c) set \<Rightarrow> ('c \<times> 'c) set"   and verts :: "'c set"   and arcs :: "('c \<times> 'c) set"   and more :: "'d" shows "arcs_update arcs' \<lparr>verts = verts, arcs = arcs, \<dots> = more\<rparr> = \<lparr>verts = verts, arcs = arcs' arcs, \<dots> = more\<rparr>"
poly_eqI: fixes p :: "'c poly"   and q :: "'c poly" assumes "\<And>n. poly.coeff p n = poly.coeff q n" shows "p = q"
ide_mkIde: fixes A :: "'t set" assumes "A \<subseteq> S'.Univ"   and "|A| <o \<AA>" shows "S'.ide (S'.mkIde A)"
simps: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" shows "Wellfounded.accp r a = (\<exists>x. a = x \<and> (\<forall>xa. r xa x \<longrightarrow> Wellfounded.accp r xa))"
simps_2: fixes P :: "'f \<Rightarrow> bool"   and x :: "'f"   and xs :: "'f list" shows "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
left_commute: fixes b :: "'b multiset"   and a :: "'b multiset"   and c :: "'b multiset" shows "b \<union># (a \<union># c) = a \<union># (b \<union># c)"
3: shows "\<forall>S\<in>\<S>. wf\<^sub>s\<^sub>t (wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t (\<A> @ [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)])) (dual\<^sub>s\<^sub>t S)"
types_wellformed_def: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" shows "types_wellformed \<Gamma> \<equiv> \<forall>x\<in>dom \<Gamma>. type_wellformed (the (\<Gamma> x))"
contraposition_1: fixes p :: "('a, 'b) fm"   and q :: "('a, 'b) fm" shows "\<turnstile> (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> \<^bold>\<not> q \<^bold>\<longrightarrow> \<^bold>\<not> p"
local_lipschitz_subset: fixes T :: "'b set"   and X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'b set"   and Y :: "'c set" assumes "local_lipschitz T X f"   and "S \<subseteq> T"   and "Y \<subseteq> X" shows "local_lipschitz S Y f"
ACI_refl: fixes r :: "'b rexp" shows "ACI r r"
insertE: fixes a :: "'f"   and b :: "'f"   and A :: "'f set" assumes "a \<in> insert b A" obtains "a = b"   | "a \<in> A"
that: fixes g :: "(nat \<Rightarrow> real) \<times> (nat \<Rightarrow> real) \<Rightarrow> (nat \<Rightarrow> real) \<times> (nat \<Rightarrow> real)" assumes "homeomorphic_map (prod_topology (Euclidean_space m) (Euclidean_space n)) (prod_topology (Euclidean_space n) (Euclidean_space m)) g"   and "\<And>x. x \<in> S \<Longrightarrow> g (x, \<lambda>i. 0) = (f x, \<lambda>i. 0)" shows "thesis"
poly_mult_semiassoc: fixes p :: "'a list"   and q :: "'a list"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "set q \<subseteq> carrier R"   and "a \<in> carrier R" shows "poly_mult (poly_mult [a] p) q = poly_mult [a] (poly_mult p q)"
l2wi_intersect_def: shows "l2wi_intersect = foldl (\<lambda>acc (s, e). wordinterval_intersection (WordInterval s e) acc) wordinterval_UNIV"
e1: shows "fst (trunc_bound_eucl p (fst (fst X) * fst (fst Y))) = fst (fst X) * fst (fst Y) + e1"
closed_segment_eq_real_ivl1: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "{a--b} = {a..b}"
subseqs_powset: fixes xs :: "'d list" shows "set ` set (subseqs xs) = Pow (set xs)"
Neg_def: fixes n :: "num" shows "Int.Neg n = - Int.Pos n"
one_disjoint: fixes ks :: "'c list"   and ls :: "'c list" assumes "disjoint ks ls" shows "ks \<langle>mone\<rangle> ls = mbot"
L1_ControlSystem: fixes s :: "nat \<Rightarrow> nat list" assumes "ControlSystemArch s" shows "ts s"
prems: shows "list_all (\<lambda>A. A \<noteq> {}) (A # XS)"
extend_for_chain: fixes g :: "nat \<Rightarrow> ('e \<times> 'f) set"   and f :: "('e \<times> 'f) set"   and S :: "nat \<Rightarrow> ('g, 'e) labeled_graph"   and C :: "('g, 'f) labeled_graph" assumes "g 0 = f"   and "\<And>i. graph_homomorphism (S i) C (g i)"   and "\<And>i. agree_on (S i) (g i) (g (i + 1))"   and "chain S" shows "extensible (S 0, chain_sup S) C f"
Phi_Psi: fixes n :: "nat"   and l :: "nat" shows "\<Phi> (n, l) = \<Psi> (int l0 \<le> \<lfloor>real l / c\<rfloor>) ai ad (f1' * real l) (f2' * real l) n"
length_eq_append: fixes y :: "'a list"   and b :: "'a list"   and x :: "'a list"   and a :: "'a list" assumes "length y = length b" shows "(x @ y = a @ b) = (x = a \<and> y = b)"
surjective: fixes r :: "'b prog_scheme" shows "r = \<lparr>ifaces = ifaces r, classes = classes r, \<dots> = prog.more r\<rparr>"
lfp_fixpoint: fixes f :: "'g \<Rightarrow> 'g" assumes "mono f" shows "f (lfp f) = lfp f"
add_ac_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + b + c = a + (b + c)"
maximal_path_lappend: fixes P' :: "'a llist"   and P :: "'a llist" assumes "\<not> lnull P'"   and "maximal_path P'" shows "maximal_path (lappend P P')"
lucas_lehmer_mult_in_carrier: fixes m :: "nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "0 < m" shows "lucas_lehmer_mult m x y \<in> {..<m} \<times> {..<m}"
simps_1: fixes t :: "trm" shows "t \<star> stack.Id = t"
b: shows "\<phi> \<notin> K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> (\<phi> .\<and>. \<psi>)"
f_inv_into_f: fixes y :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "y \<in> f ` A" shows "f (inv_into A f y) = y"
graph_pred_it_axioms_def: fixes \<alpha> :: "'d \<Rightarrow> ('e, 'f) graph"   and invar :: "'d \<Rightarrow> bool"   and pred_list_it :: "'d \<Rightarrow> 'e \<Rightarrow> (('e \<times> 'f) list \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> ('e \<times> 'f) list \<Rightarrow> ('e \<times> 'f) list) \<Rightarrow> ('e \<times> 'f) list \<Rightarrow> ('e \<times> 'f) list" shows "graph_pred_it_axioms \<alpha> invar pred_list_it \<equiv> \<forall>g v. invar g \<longrightarrow> set_iterator (pred_list_it g v) (pred (\<alpha> g) v)"
dX_def: fixes X :: "'x set"   and d :: "'d" shows "ManiPref.dX X d = {x \<in> X. Xd x = d}"
left_total_fun: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "left_unique A"   and "left_total B" shows "left_total (rel_fun A B)"
period_posD: fixes C :: "'s set"   and x :: "'s" assumes "C \<in> UNIV // communicating"   and "0 < period C"   and "x \<in> C" shows "\<exists>n>0. 0 < p x x n"
surjec_def: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e" shows "surjec\<^bsub>F,G\<^esub> f = (f \<in> aHom F G \<and> surj_to f (carrier F) (carrier G))"
refine_list_1: fixes R :: "('a \<times> 'b) set" shows "([], []) \<in> \<langle>R\<rangle>list_rel"
alphaAbs_qAbs_iff_ex_distinct_qAFresh: fixes xs :: "'h"   and x :: "'i"   and X :: "('f, 'g, 'h, 'i, 'j) qTerm"   and xs' :: "'h"   and x' :: "'i"   and X' :: "('f, 'g, 'h, 'i, 'j) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
simps_2: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and p :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" shows "gb_sel gs bs (p # ps) data = [p]"
compose: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h' :: "'d \<Rightarrow> 'b"   and G :: "'d filter" assumes "f \<in> O[F](g)"   and "filterlim h' F G" shows "(\<lambda>x. f (h' x)) \<in> O[G](\<lambda>x. g (h' x))"
simps: fixes gpv :: "('d, 'out, 'e) gpv" shows "local.interaction_bound gpv = (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. local.interaction_bound (c input)) else \<Squnion>input. local.interaction_bound (c input))"
diameter_compact_attained: fixes S :: "'b set" assumes "compact S"   and "S \<noteq> {}" shows "\<exists>x\<in>S. \<exists>y\<in>S. dist x y = diameter S"
eq: shows "p = smult (lead_coeff p) (\<Prod>i<length roots. [:- roots ! i, 1:])"
PiE_iff: fixes f :: "'b \<Rightarrow> 'c"   and I :: "'b set"   and X :: "'b \<Rightarrow> 'c set" shows "(f \<in> Pi\<^sub>E I X) = ((\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I)"
bmod_bres_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and x :: "'f" shows "\<lhd> R f g x = \<Sqinter>{f y \<rightarrow> g z |y z. R z y x}"
le_approx3: fixes P :: "'c process"   and Q :: "'c process" assumes "P \<sqsubseteq> Q" shows "min_elems (D P) \<subseteq> T Q"
one_strongly_connected_implies_7_eq: fixes x :: "'a" assumes "one_strongly_connected x" shows "x ; 1 ; x = x\<^sup>+"
case_prod_twice: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and h :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x :: "'d \<times> 'e" shows "(case case x of (a, b) \<Rightarrow> (g a b, h a b) of (x, xa) \<Rightarrow> f x xa) = (case x of (a, b) \<Rightarrow> f (g a b) (h a b))"
tl_append2: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<noteq> \<epsilon>" shows "tl (xs \<cdot> ys) = tl xs \<cdot> ys"
h_plus_mono: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "h_plus i e \<le> h_plus (Suc i) e"
meet_le2: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and join :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Top :: "'b"   and meet :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.L_top_bot le join Top meet bot" shows "le (meet x y) y"
retraction_maps_def: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "retraction_maps X Y f g \<equiv> continuous_map X Y f \<and> continuous_map Y X g \<and> (\<forall>x\<in>topspace Y. f (g x) = x)"
paths_withoutI: fixes xs :: "'b list"   and t :: "'b ttree"   and x :: "'b" assumes "xs \<in> paths t"   and "x \<notin> set xs" shows "xs \<in> paths (without x t)"
wls_swap_compose: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and zs' :: "'varSort"   and x' :: "'var"   and y' :: "'var"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "wls s X" shows "X #[x \<and> y]_zs #[x' \<and> y']_zs' = X #[x' \<and> y']_zs' #[x @zs[x' \<and> y']_zs' \<and> y @zs[x' \<and> y']_zs']_zs"
llist_inf_simps_2: fixes xs :: "'a llist" shows "xs \<sqinter> LNil = LNil"
igSwapBinpIPresIGWlsBinpSTR_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSwapBinpIPresIGWlsBinpSTR MOD \<equiv> \<forall>zs z1 z2 delta binp. igWlsBinp MOD delta (igSwapBinp MOD zs z1 z2 binp) = igWlsBinp MOD delta binp"
valid_plans_def: fixes prob :: "'b set" shows "valid_plans prob \<equiv> {as. set as \<subseteq> prob}"
max_bot2: fixes x :: "'a list" shows "prefix_order.max x [] = x"
mat_gt_ge_trans: fixes sd :: "nat"   and n :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat" assumes "sd \<le> n"   and "mat_gt (\<succ>) sd A B"   and "B \<ge>\<^sub>m C"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "mat_gt (\<succ>) sd A C"
exhaust: fixes y :: "'a acom" obtains x1 :: "'a" where "y = SKIP {x1}"   | x21 :: "char list" and x22 :: "aexp" and x23 :: "'a" where "y = x21 ::= x22 {x23}"   | x31 :: "'a acom" and x32 :: "'a acom" where "y = x31;; x32"   | x41 :: "bexp" and x42 :: "'a acom" and x43 :: "'a acom" and x44 :: "'a" where "y = IF x41 THEN x42 ELSE x43 {x44}"   | x51 :: "'a" and x52 :: "bexp" and x53 :: "'a acom" and x54 :: "'a" where "y = {x51} WHILE x52 DO x53 {x54}"
lookup_alt_def: fixes x :: "'b" shows "SFP_lookup x = get_state (\<lambda>s. case apply s x of None \<Rightarrow> fail_state (fail_option return_pmf) | Some y \<Rightarrow> return_state (return_option return_pmf) y)"
binrelchain_cong: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "\<And>x y. P x y \<Longrightarrow> Q x y"   and "binrelchain P xs" shows "binrelchain Q xs"
intereqt12: fixes t :: "real" assumes "t1 \<le> t"   and "t \<le> t2"   and "flow0 x t \<in> {flow0 x t1--flow0 x t2}" shows "t = t1 \<or> t = t2"
treeEquation: fixes subs :: "'b \<Rightarrow> 'b set"   and gamma :: "'b" shows "tree subs gamma = insert (0, gamma) (\<Union>delta\<in>subs gamma. incLevel ` tree subs delta)"
map_cong0: fixes x :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'e" assumes "\<And>z. s2p (range x) z \<Longrightarrow> f z = g z" shows "f \<circ> x = g \<circ> x"
pref_profile_wf_axioms: shows "is_pref_profile R"
SN_relto_imp_SN_rel: fixes S :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set" assumes "SN (S\<^sup>* O R O S\<^sup>*)" shows "SN_rel R S"
l_get_shadow_root_component\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_shadow_root_component\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf get_shadow_root get_shadow_root_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_tag_name ShadowRootClass.known_ptr Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs ShadowRootClass.known_ptrs Shadow_DOM.to_tree_order Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM_DOM_Components.get_dom_component Shadow_DOM_DOM_Components.is_strongly_dom_component_safe Shadow_DOM_DOM_Components.is_weakly_dom_component_safe Shadow_DOM.get_root_node Shadow_DOM.get_root_node_locs Shadow_DOM.get_ancestors Shadow_DOM.get_ancestors_locs Shadow_DOM.get_element_by_id Shadow_DOM.get_elements_by_tag_name remove_shadow_root_locs"
2: fixes b :: "'a" assumes "b \<in> B" shows "f (g b) = b"
int_of_fbint_inject: fixes x :: "fbint"   and y :: "fbint" shows "(int_of_fbint x = int_of_fbint y) = (x = y)"
tendsto_componentwise_iff: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and F :: "'c filter" shows "(f \<longlongrightarrow> l) F = (\<forall>i\<in>Basis. ((\<lambda>x. f x \<bullet> i) \<longlongrightarrow> l \<bullet> i) F)"
pullback_commutes: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "commutative_square f g \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]"
conjr2_iff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<rhd> y \<le> z) = (x \<le> - (y \<lhd> - z))"
FI_def: fixes m :: "(assn \<times> assn) list"   and p :: "assn"   and q :: "assn"   and up :: "assn"   and uq :: "assn"   and f :: "assn" shows "FI m p q up uq f \<equiv> fi_m_match m \<longrightarrow> (fi_m_fst m * p * up \<Longrightarrow>\<^sub>A fi_m_snd m * q * uq * f)"
card_of_Field_ordIso: fixes r :: "('a \<times> 'a) set" assumes "Card_order r" shows "|Field r| =o r"
igSubstIGAbsSTR_def: fixes MOD :: "('h, 'i, 'j, 'k, 'l, 'm, 'n, 'o, 'p) model_scheme" shows "igSubstIGAbsSTR MOD \<equiv> \<forall>ys y Y xs x X. (xs \<noteq> ys \<or> x \<noteq> y) \<and> igFresh MOD xs x Y \<longrightarrow> igSubstAbs MOD ys Y y (igAbs MOD xs x X) = igAbs MOD xs x (igSubst MOD ys Y y X)"
order_Rep_N: fixes a :: "'a N"   and b :: "'a N" shows "(a \<le> b) = (Rep_N a \<le> Rep_N b)"
lcm_rat_def: shows "lcm = normalization_euclidean_semiring_class.lcm"
nth_append_length: fixes xs :: "'g list"   and x :: "'g"   and ys :: "'g list" shows "(xs @ x # ys) ! length xs = x"
runs_ne: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "runs xs \<noteq> []"
cInf_lower2: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and x :: "'b::type"   and X :: "'b::type set"   and y :: "'b::type" assumes "class.conditionally_complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)"   and "(x::'b::type) \<in> (X::'b::type set)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type)"   and "preorder.bdd_below (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (X::'b::type set)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((Inf::'b::type set \<Rightarrow> 'b::type) (X::'b::type set)) (y::'b::type)"
limit_in_range_suffixD: fixes c :: "'b"   and x :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "c \<in> limit x" shows "c \<in> range (suffix k x)"
gcd_condition: fixes G :: "('c, 'd) monoid_scheme" assumes "factorial_monoid G" shows "gcd_condition_monoid G"
lcm_mult_distrib: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and k :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd_mult_normalize one zero times gcd lcm divide plus minus unit_factor normalize" shows "times k (lcm a b) = times (lcm (times k a) (times k b)) (unit_factor k)"
eval: shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>"
create_graph_eq_create_graph\<^sub>T: fixes \<phi> :: "'b ltlr" shows "create_graph \<phi> = create_graph\<^sub>T \<phi>"
poly_eq_fold_coeffs: fixes p :: "'b poly" shows "poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0::'b)"
degree_pseudo_mod_less: fixes q :: "'a poly"   and p :: "'a poly" assumes "q \<noteq> 0"   and "pseudo_mod p q \<noteq> 0" shows "degree (pseudo_mod p q) < degree q"
iT_drule_5: fixes x :: "nat"   and r :: "nat"   and m :: "nat" assumes "x \<in> [ r, mod m ]" shows "x mod m = r mod m"