hcju/isabelleps huggingface dataset:
hyps_subset: fixes p :: "'b pl"   and t :: "'b set" shows "hyps p t \<subseteq> (\<Union>v. {#v, #v -> false})"
gfp_def: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.gfp Sup less_eq f = Sup {u. less_eq u (f u)}"
inverse_le_1_iff: fixes x :: "'b" shows "(inverse x \<le> (1::'b)) = (x \<le> (0::'b) \<or> (1::'b) \<le> x)"
simps_1: fixes x1 :: "'b rexp"   and y1 :: "'b rexp" shows "(Lam x1 = Lam y1) = (x1 = y1)"
summable_mult: fixes f :: "nat \<Rightarrow> 'd"   and c :: "'d" assumes "summable f" shows "summable (\<lambda>n. c * f n)"
assms_2: fixes x :: "('dyn, 'b, 'c, 'd, 'opinl) Inca.instr fundef"   and y :: "('dyn, 'b, 'c, 'd, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" assumes "rel_fundef (\<lambda>x y. x = norm_instr y) x y" shows "h x y"
iso_\<phi>\<^sub>0: fixes f :: "'a" assumes "\<guillemotleft>f : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>"   and "ide f" shows "local.iso ((e\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[d\<^sub>1, f \<star> e\<^sub>0, d\<^sub>0]) \<cdot> \<a>[e\<^sub>1, d\<^sub>1, (f \<star> e\<^sub>0) \<star> d\<^sub>0] \<cdot> ((e\<^sub>1 \<star> d\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[f, e\<^sub>0, d\<^sub>0]) \<cdot> (local.inv \<epsilon>\<^sub>1 \<star> f \<star> local.inv \<epsilon>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> src d\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[f])"
assms: shows "\<forall>x\<in>S. \<forall>y\<in>S. x = y"
dom_locals_halloc_mono: fixes G :: "prog"   and s0 :: "state"   and oi :: "obj_tag"   and a :: "loc"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>halloc oi\<succ>a\<rightarrow> s1" shows "dom (locals (snd s0)) \<subseteq> dom (locals (snd s1))"
cblinfun_apply_cblinfun_compose: fixes a :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'a"   and b :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and c :: "'b" shows "cblinfun_apply (a o\<^sub>C\<^sub>L b) c = cblinfun_apply a (cblinfun_apply b c)"
simps: fixes A :: "'b mat" shows "square_mat A = (dim_col A = dim_row A)"
decides: fixes n :: "nat" shows "decides_all (set (map Inf (F n))) (Eps (\<Phi> (F n)))"
dom_eq_and_eq_on_dom_imp_eq: fixes m :: "'a \<Rightarrow> 'b option"   and m' :: "'a \<Rightarrow> 'b option" assumes "dom m = dom m'"   and "\<forall>x\<in>dom m. m x = m' x" shows "m = m'"
le_map_dom_mono: fixes m :: "'a \<Rightarrow> 'b option"   and m' :: "'a \<Rightarrow> 'b option" assumes "m \<le> m'" shows "dom m \<subseteq> dom m'"
i_contract_ideal: fixes R :: "('c, 'd) Ring_scheme"   and R' :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e"   and J :: "'e set" assumes "TwoRings R R'"   and "f \<in> rHom R R'"   and "ideal R' J" shows "ideal R (i_contract f R R' J)"
OneTrans_HPT_SA: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and T :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and SA :: "('d, 'e, 'f) seqauto"   and U :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd" assumes "TS \<in> HPT ST"   and "T \<in> TS"   and "source T \<in> States SA"   and "U \<in> TS"   and "source U \<in> States SA"   and "SA \<in> SAs (HA ST)" shows "T = U"
two_comparisons_into_case_order_3: fixes x :: "'b"   and y :: "'b"   and P :: "'a"   and Q :: "'a"   and R :: "'a" shows "(if tc.le x y then if tc.le y x then P else Q else R) = (case comparator_of x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
star_ref: fixes x :: "'a" shows "reflexive (x\<^sup>\<star>)"
awalk_ends_eqD: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w :: "'a" assumes "awalk u p u"   and "awalk v p w" shows "v = w"
share_all_until_volatile_write_thread_local: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "ownership_distinct ts"   and "sharing_consis \<S> ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "a \<in> \<O> \<union> all_acquired sb" shows "share_all_until_volatile_write ts \<S> a = share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<S> a"
goedel_first: assumes "\<omega>consistent" shows "\<not> prv \<phi>G \<and> \<not> prv (neg \<phi>G)"
lift_preserves_beta': fixes r :: "dB"   and s :: "dB"   and i :: "nat" assumes "r \<rightarrow>\<^sub>\<beta>\<^sup>* s" shows "lift r i \<rightarrow>\<^sub>\<beta>\<^sup>* lift s i"
in_KeysI: fixes s :: "'a"   and f :: "'a \<Rightarrow>\<^sub>0 'b"   and F :: "('a \<Rightarrow>\<^sub>0 'b) set" assumes "s \<in> keys f"   and "f \<in> F" shows "s \<in> Keys F"
assms_3: shows "component_of_term ` Keys B1 = component_of_term ` Keys B2"
Join_constrains: fixes F :: "'c program"   and G :: "'c program"   and A :: "'c set"   and B :: "'c set" shows "(F \<squnion> G \<in> A co B) = (F \<in> A co B \<and> G \<in> A co B)"
roots_of_complex_main_def: fixes p :: "complex poly" shows "roots_of_complex_main p \<equiv> let n = degree p in if n = 0 then [] else if n = 1 then [roots1 p] else if n = 2 then croots2 p else complex_roots_of_rat_poly (map_poly to_rat p)"
nneg: fixes i :: "nat" shows "0 \<le> f i"
vdist_def: fixes v :: "'a" shows "vdist v = (LEAST n. (\<psi> ^^ n) v \<in> C0)"
min_product_set_finite: fixes X :: "'b fset set set" assumes "\<And>x. x \<in> X \<Longrightarrow> finite x" shows "finite (\<Otimes>\<^sub>m X)"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" assumes "prebicategory_with_units V H \<a> \<i>"   and "prebicategory_with_homs V H \<a> src trg" shows "prebicategory_with_homs_and_units V H \<a> \<i> src trg"
fmlookup_drop_set: fixes A :: "'d set"   and m :: "('d, 'c) fmap"   and x :: "'d" shows "fmdrop_set A m $$ x = (if x \<notin> A then m $$ x else None)"
cs_def: shows "cs = {(a, f a) |a. a \<in> as}"
r'': shows "Catch r'' c\<^sub>2 \<in> redexes c''"
solver: fixes y :: "'a"   and e :: "'a"   and f :: "'a" shows "(y \<in> set (depressed_solver e f)) = (y ^ 3 + e * y + f = (0::'a))"
mC_ZOC_wt_def: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and I0 :: "nat set"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_ZOC_wt c d s t I0 P F \<equiv> sum (wt c s) I0 < 1 \<and> sum (wt d t) (F I0) < 1 \<longrightarrow> (\<forall>I\<in>P - {I0}. sum (wt c s) I / (1 - sum (wt c s) I0) = sum (wt d t) (F I) / (1 - sum (wt d t) (F I0)))"
\<Gamma>'': shows "\<Gamma>' = set_mset \<Gamma>''"
vector_covector: fixes v :: "'c \<times> 'c \<Rightarrow> 'd"   and w :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_bounded_idempotent_semiring.vector v"   and "matrix_bounded_idempotent_semiring.vector w" shows "v \<otimes> w\<^sup>t = v \<odot> w\<^sup>t"
steps_append: fixes A :: "'c \<times> ('d \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)"   and v :: "'d list"   and w :: "'d list" shows "NA.steps A (v @ w) = NA.steps A v O NA.steps A w"
\<nu>_simps_5: shows "cod \<nu>' = g \<star> w'"
ce_rels_def: shows "ce_rels = {R |R. ce_rel_to_set R \<in> ce_sets}"
sub_monom_deg_one: fixes p :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "p \<in> carrier P"   and "a \<in> carrier R" shows "Cring_Poly.compose R (monom P a 1) p = a \<odot>\<^bsub>P\<^esub> p"
less_imp_Interp_subseteq_interp: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "C < D" shows "Interp C \<subseteq> interp D"
sg_i_induced: fixes L :: "'a set"   and a :: "'a" assumes "G \<guillemotright> L "   and "a \<in> L" shows "\<rho>\<^bsub>\<natural>L\<^esub> a = \<rho> a"
holomorphic_pre_zeta_aux': fixes a :: "real"   and U :: "complex set"   and \<sigma> :: "real"   and n :: "nat" assumes "0 < a"   and "bounded U"   and "open U"   and "U \<subseteq> {s. \<sigma> < Re s}"   and "- 2 * real n < \<sigma>" shows "pre_zeta_aux n a holomorphic_on U"
dom_lunit: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and a :: "'a" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "partial_magma.dom C (monoidal_category.lunit C T \<alpha> \<iota> a) = T (monoidal_category.unity C \<iota>, a)"
P_def: fixes j :: "nat"   and i :: "nat" shows "P j i = (i < a + b \<and> j = Suc i)"
dom_lhs: shows "\<D>\<^sub>\<circ> (op_dg (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)) = 4\<^sub>\<nat>"
order_ifex_ite_invar: fixes i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex" assumes "ifex_ordered i"   and "ifex_ordered t"   and "ifex_ordered e" shows "ifex_ordered (ifex_ite i t e)"
array_grow_def: shows "array_grow \<equiv> rec_array (\<lambda>A inc x. Array (A @ replicate inc x))"
system_needed_asses_submap: fixes PROB :: "(('a, 'b) fmap \<times> 'c) set"   and s :: "('a, 'b) fmap" shows "system_needed_asses PROB s \<subseteq>\<^sub>f s"
abs_eq: fixes y :: "'b" shows "(K$ y) = Abs_finfun (\<lambda>x. y)"
sort_eqI: fixes ys :: "'b list"   and xs :: "'b list"   and cmp :: "'b comparator" assumes "mset ys = mset xs"   and "Sorting_Algorithms.sorted cmp xs"   and "\<And>y. y \<in> set ys \<Longrightarrow> filter (\<lambda>ya. compare cmp y ya = Equiv) ys = filter (\<lambda>ya. compare cmp y ya = Equiv) xs" shows "Sorting_Algorithms.sort cmp ys = xs"
rep_eq: fixes x :: "'b coreflexive"   and xa :: "'b coreflexive" shows "Rep_coreflexive (x \<squnion> xa) = Rep_coreflexive x \<squnion> Rep_coreflexive xa"
E_closed: fixes i :: "'i" assumes "i \<in> I" shows "E i \<subseteq> Pow (space (M i))"
assms_1: shows "M-lipschitz_on {A..B} c"
assms: shows "\<exists>subs. raw_match' T2 T1 Map.empty = Some subs"
loose_linkable: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "loose (quotient_web \<Gamma> f)" shows "linkable (quotient_web \<Gamma> f)"
bounded_plus_comp: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "bounded (f ` S)"   and "bounded (g ` S)" shows "bounded ((\<lambda>x. f x + g x) ` S)"
split_random_bst: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "map_pmf (split_bst x) (random_bst A) = pair_pmf (random_bst {y \<in> A. y < x}) (random_bst {y \<in> A. x < y})"
cases: fixes get :: "'a \<Rightarrow> ('b, 'c, 'a, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr fundef option"   and a1 :: "('a, 'dyn) frame list"   and a2 :: "('a, ('dyn, 'ubx1, 'ubx2) unboxed) frame list"   and a3 :: "'a option" assumes "rel_stacktraces get a1 a2 a3" obtains opt :: "'a option" where "a1 = []" and "a2 = []" and "a3 = opt__"   | st1 :: "('a, 'dyn) frame list" and st2 :: "('a, ('dyn, 'ubx1, 'ubx2) unboxed) frame list" and f :: "'a" and \<Sigma>1 :: "'dyn list" and \<Sigma>2 :: "('dyn, 'ubx1, 'ubx2) unboxed list" and fd2 :: "('b, 'c, 'a, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr fundef" and n :: "nat" and opt :: "'a option" where "a1 = Frame f n \<Sigma>1 # st1" and "a2 = Frame f n \<Sigma>2 # st2" and "a3 = opt" and "rel_stacktraces get st1 st2 (Some f)" and "\<Sigma>1 = norm_stack \<Sigma>2" and "get f = Some fd2" and "sp_fundef get fd2 (take n (body fd2)) = Ok (map typeof \<Sigma>2)" and "pred_option (is_valid_fun_call get fd2 n \<Sigma>2) opt"
Limsup_def: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> 'b" shows "Limsup F f = (INF P\<in>{P. eventually P F}. Sup (f ` {x. P x}))"
additiveD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" assumes "Transformers.additive t"   and "sound P"   and "sound Q" shows "t (\<lambda>s. P s + Q s) = (\<lambda>s. t P s + t Q s)"
sum_of_naturals: fixes n :: "nat" shows "2 * \<Sum> {0..n} = n * (n + 1)"
range_vreal_of_real_impl_vlepoll_VPow_omega: shows "ZFC_in_HOL.set (range vreal_of_real_impl) \<lesssim>\<^sub>\<circ> VPow \<omega>"
image_these: fixes f :: "'c \<Rightarrow> 'b"   and X :: "'c option set" shows "f ` Option.these X = Option.these (map_option f ` X)"
simps_1: fixes x :: "quat"   and y :: "quat" shows "quat.Re (x * y) = quat.Re x * quat.Re y - Im1 x * Im1 y - Im2 x * Im2 y - Im3 x * Im3 y"
simps: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "sfirst P \<omega> = (if P \<omega> then 0 else eSuc (sfirst P (stl \<omega>)))"
sin_eq_0_pi: fixes x :: "real" assumes "- pi < x"   and "x < pi"   and "sin x = 0" shows "x = 0"
less_real_def: fixes x :: "real"   and y :: "real" shows "(x < y) = positive (y - x)"
LEX_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "LEX P f g = (\<exists>i. P (f i) (g i) \<and> (\<forall>j<i. f j = g j))"
gbm_add_node_impl: shows "graph_add_node gbm_\<alpha> gbm_invar gbm_add_node"
pos_imp_ne: fixes xs :: "real list" assumes "pos xs" shows "xs \<noteq> []"
set_restrictionD_spec: fixes f :: "'a set \<Rightarrow> 'a set"   and A :: "'a set" assumes "set_restriction f" shows "\<exists>P. f A = {x \<in> A. P x}"
nchotomy: shows "\<forall>ExtendedBool. ExtendedBool = TRUE \<or> ExtendedBool = FALSE \<or> ExtendedBool = UNDEF"
Inf_greatest: fixes A :: "'b set"   and z :: "'b" assumes "\<And>x. x \<in> A \<Longrightarrow> z \<le> x" shows "z \<le> Inf A"
build_lin_independent_seq: fixes us :: "'v list"   and n :: "nat" assumes "set us \<subseteq> V"   and "R_lin_independent us" shows "\<exists>ws. set ws \<subseteq> V \<and> R_lin_independent (ws @ us) \<and> (RSpan (ws @ us) = V \<or> length ws = n)"
MaxR_maxR: fixes d :: "'d" shows "MaxR.maxR (Pd d) = maxR d"
create_residualEqvt: fixes p :: "(name \<times> name) list"   and \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi" shows "p \<bullet> \<alpha> \<prec> P = p \<bullet> \<alpha> \<prec> p \<bullet> P"
starlike_imp_connected: fixes S :: "'a set" assumes "starlike S" shows "connected S"
real_hyp2_C_hyp2_cltn2: fixes J :: "cltn2"   and a :: "hyp2"   and b :: "hyp2" assumes "is_K2_isometry J" shows "a b \<congruent>\<^sub>K hyp2_cltn2 a J hyp2_cltn2 b J"
that: fixes root :: "nat \<Rightarrow> complex" assumes "smult (lead_coeff p) (\<Prod>i<degree p. [:- root i, 1:]) = p" shows "thesis"
opposite_monoidal_category_def: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>C :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>C :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d" shows "opposite_monoidal_category C T\<^sub>C \<alpha>\<^sub>C \<iota> \<equiv> monoidal_category C T\<^sub>C \<alpha>\<^sub>C \<iota>"
iff1: shows "(0 < poly (prod_list (retrieve_polys qs (a # xa))) x) = (0 < poly (qs ! a) x \<and> 0 < poly (prod_list (retrieve_polys qs xa)) x \<or> poly (qs ! a) x < 0 \<and> poly (prod_list (retrieve_polys qs xa)) x < 0)"
sum_delta'': fixes s :: "'c set"   and y :: "'c"   and f :: "'c \<Rightarrow> real" assumes "finite s" shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y \<in> s then f y *\<^sub>R y else (0::'c))"
left_unique_cr_spmf_gpv: shows "left_unique cr_spmf_gpv"
radical_sqrt_rule_division: fixes x :: "real"   and y :: "real" assumes "x \<in> radical_sqrt"   and "y \<in> radical_sqrt"   and "y \<noteq> 0" shows "x / y \<in> radical_sqrt"
distinct_7: fixes x2 :: "'b"   and x3 :: "'b" shows "right x2 \<noteq> mid x3"
spn_def: fixes \<mu> :: "'a" shows "spn \<mu> \<equiv> arrow_of_tabulations_in_maps.chine (\<cdot>) (\<star>) \<a> \<i> src trg (tab_of_ide (local.dom \<mu>)) (tab\<^sub>0 (local.dom \<mu>)) (cod \<mu>) (tab_of_ide (cod \<mu>)) (tab\<^sub>0 (cod \<mu>)) (tab\<^sub>1 (cod \<mu>)) \<mu>"
gfp_def: 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"   and f :: "'a \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.gfp Sup less_eq f = Sup {u. less_eq u (f u)}"
some_geodesic_segment_between_exists: shows "\<exists>f. \<forall>x y S. f x S y = f y S x \<and> (if \<exists>G. geodesic_segment_between G x y \<and> G \<subseteq> S then geodesic_segment_between (f x S y) x y \<and> f x S y \<subseteq> S else f x S y = {x, y})"
cross3_simps_9: fixes b :: "'d"   and a :: "'d"   and c :: "'d" shows "b + (a + c) = a + (b + c)"
prime_product_factor_main: fixes n :: "nat"   and limit :: "nat"   and m :: "nat"   and factor_sq :: "nat"   and factor_pr :: "nat"   and i :: "nat"   and sq :: "nat"   and p :: "nat" assumes "\<nexists>s. s * s = n"   and "limit = nat (root_nat_floor 3 n)"   and "m = factor_sq * factor_sq * factor_pr * n"   and "prime_product_factor_main factor_sq factor_pr limit n i = (sq, p)"   and "2 \<le> i"   and "\<And>j. \<lbrakk>2 \<le> j; j < i\<rbrakk> \<Longrightarrow> \<not> j dvd n"   and "\<And>j. \<lbrakk>prime j; j < i\<rbrakk> \<Longrightarrow> multiplicity j factor_pr \<le> 1"   and "\<And>j. \<lbrakk>prime j; i \<le> j\<rbrakk> \<Longrightarrow> multiplicity j factor_pr = 0"   and "0 < factor_pr" shows "m = sq * sq * p \<and> prime_product p"
hyps: fixes n :: "'node" assumes "n -as'\<rightarrow>* n'"   and "as' \<noteq> []" shows "sourcenodes as' = n # butlast (targetnodes as')"
assoc_list_of_array_code_induct: fixes a :: "'a array"   and P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "\<And>n. (n < array_length a \<Longrightarrow> P (Suc n)) \<Longrightarrow> P n" shows "P n"
disc_rfr_proc_positive: fixes r :: "real"   and n :: "nat"   and w :: "'c" assumes "- 1 < r" shows "0 < disc_rfr_proc r n w"
merge_conv': fixes xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" shows "map_of (AList.merge xs ys) = map_of xs ++ map_of ys"
simplex_likeD_iso: fixes P :: "'b set" assumes "codomain.simplex_like P" shows "\<exists>f A. OrderingSetIso (\<^bold>\<le>*) (\<^bold><*) dual_order.greater_eq dual_order.greater P f \<and> f ` P = Pow A"
induct: fixes P :: "nat \<Rightarrow> int \<Rightarrow> int poly \<Rightarrow> int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "int"   and a2 :: "int poly"   and a3 :: "int"   and a4 :: "nat"   and a5 :: "nat" assumes "\<And>dd c g mm k kk. (\<And>x xa. \<lbrakk>x = mahler_landau_graeffe_approximation kk dd g; xa = (if k = 0 then x else min mm x); \<not> bnd \<le> k\<rbrakk> \<Longrightarrow> P (dd * dd) c (graeffe_one_step c g) xa (Suc k) (2 * kk)) \<Longrightarrow> P dd c g mm k kk" shows "P a0 a1 a2 a3 a4 a5"
closed_segment: fixes a :: "'a"   and b :: "'a" shows "closed {a--b}"
asAbs_equal_iff_alphaAbs: fixes qA :: "('c, 'd, 'varSort, 'var, 'e) qAbs"   and qB :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "qGoodAbs qA \<or> qGoodAbs qB" shows "(asAbs qA = asAbs qB) = (qA $= qB)"
Not_Rall: fixes F :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "\<turnstile> \<not> (\<forall>x. F x) = (\<exists>x. \<not> F x)"
open_inv_chart_image: fixes X :: "'c set"   and c :: "('d, 'c) chart" assumes "open X"   and "X \<subseteq> codomain c" shows "open (inv_chart c ` X)"
ideal_decompE: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and f0 :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "finite F"   and "F \<subseteq> P[X]"   and "f0 \<in> P[X]"   and "\<And>f. f \<in> F \<Longrightarrow> poly_deg f \<le> poly_deg f0" obtains T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" where "valid_decomp X ps" and "standard_decomp (poly_deg f0) ps" and "cone_decomp T ps" and "(\<And>f. f \<in> F \<Longrightarrow> homogeneous f) \<Longrightarrow> hom_decomp ps" and "direct_decomp (ideal (insert f0 F) \<inter> P[X]) [ideal {f0} \<inter> P[X], T]"
le_supE: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and x :: "'b" assumes "class.semilattice_sup sup less_eq less"   and "less_eq (sup a b) x" obtains "less_eq a x" and "less_eq b x"
ContractsWithUnilateralSubstitutesAndIRC_axioms: shows "ContractsWithUnilateralSubstitutesAndIRC Xd Xh Pd Ch"
eventually_at_within_mono: fixes t :: "'b"   and T :: "'b set"   and S :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "t \<in> interior T"   and "T \<subseteq> S"   and "eventually P (at t within T)" shows "eventually P (at t within S)"
utp_bottom: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel"   and P :: "'a hrel" assumes "utp_theory_lattice hcond"   and "P is hcond" shows "\<bottom>\<^bsub>utp_order hcond\<^esub> \<sqsubseteq> P"
preserves_arr: fixes f :: "'a" assumes "C.arr f" shows "C.arr (C.R f)"
fds_converges_altdef: fixes f :: "'a fds"   and s :: "'a" shows "fds_converges f s = (\<lambda>n. fds_nth f (Suc n) / nat_power (Suc n) s) sums eval_fds f s"
assms: shows "(yps, Y) \<in> futures (c_process step out s\<^sub>0) xps"
FAss1Red1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and F :: "String.literal"   and D :: "String.literal"   and e2 :: "'addr expr1" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>e\<bullet>F{D} := e2,s\<rangle> -ta\<rightarrow> \<langle>e'\<bullet>F{D} := e2,s'\<rangle>"
card_of_Un_ordLeq_infinite_Field: fixes r :: "('a \<times> 'a) set"   and A :: "'b set"   and B :: "'b set" assumes "infinite (Field r)"   and "|A| \<le>o r"   and "|B| \<le>o r"   and "Card_order r" shows "|A \<union> B| \<le>o r"
CFG_edge_Uses_pred_equal: fixes a :: "'edge"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "valid_edge a"   and "pred (kind a) s"   and "snd (hd s) = snd (hd s')"   and "\<forall>V\<in>Use (sourcenode a). state_val s V = state_val s' V"   and "length s = length s'" shows "pred (kind a) s'"
sublens_def: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" shows "X \<subseteq>\<^sub>L Y = (\<exists>Z. vwb_lens Z \<and> X = Z ;\<^sub>L Y)"
T'_def: shows "T' = (\<lambda>a. (\<lambda>x. x - of_int_vec a) ` T a)"
dep_var_set_def: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs1 :: "'d set"   and vs2 :: "'d set" shows "dep_var_set PROB vs1 vs2 \<equiv> disjnt vs1 vs2 \<and> (\<exists>v1 v2. v1 \<in> vs1 \<and> v2 \<in> vs2 \<and> dep PROB v1 v2)"
lin_dep_rows_imp_det_0: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "lin_dep (set (rows A))" shows "det A = (0::'a)"
755: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> - y \<squnion> (z \<squnion> x) = z \<squnion> x"
ccApprox_either: fixes t :: "Vars.var ttree"   and t' :: "Vars.var ttree" shows "ccApprox (t \<oplus>\<oplus> t') = ccApprox t \<squnion> ccApprox t'"
lead_coeff_pCons_2: fixes p :: "'a poly"   and a :: "'a" assumes "p = 0" shows "lead_coeff (pCons a p) = a"