hcju/isabelleps huggingface dataset:
put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and setter :: "('m \<Rightarrow> 'm) \<Rightarrow> ('c, 'a, 'b, 'k option) RShadowRoot_scheme \<Rightarrow> ('c, 'a, 'b, 'k option) RShadowRoot_scheme"   and v :: "'m" assumes "ShadowRootClass.type_wf h"   and "ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr setter v"
filter_emptyE: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" assumes "filter P xs = []"   and "x \<in> set xs" shows "\<not> P x"
frag_cmul_def: fixes c :: "int"   and a :: "'c \<Rightarrow>\<^sub>0 int" shows "frag_cmul c a = Abs_poly_mapping (\<lambda>x. c * poly_mapping.lookup a x)"
weakenTauChain: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'" shows "\<Psi> \<otimes> \<Psi>' \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"
AsCallExpr: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and f :: "'e"   and b :: "('d, 'e, 'f) com"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and as :: "('d, 'e, 'f) ann list"   and n :: "nat"   and Q :: "'d set"   and A :: "'d set"   and a :: "'d set"   and r :: "'d set" assumes "\<Gamma> f = Some b"   and "\<Theta> f = Some as"   and "n < length as"   and "assertionsR \<Gamma> \<Theta> Q A (as ! n) b a" shows "assertionsR \<Gamma> \<Theta> Q A (AnnCall r n) (Call f) a"
oaodv_init_kD_empty: fixes \<sigma> :: "nat \<Rightarrow> state"   and p :: "(state, msg, pseqp, pseqp label) seqp"   and i :: "nat" assumes "(\<sigma>, p) \<in> \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V'" shows "kD (rt (\<sigma> i)) = {}"
BinOpThrow2: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>2 :: "char list exp"   and e :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and bop :: "bop" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>2\<rangle>"
heapify_snd_tup: fixes n :: "nat"   and xs :: "'b list"   and t :: "'b tree"   and ys :: "'b list" assumes "heapify n xs = (t, ys)"   and "n \<le> length xs" shows "ys = drop n xs"
mpoly_coeff_0: fixes m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff 0 m = (0::'b)"
set_mset_Inf: fixes A :: "'k multiset set" assumes "A \<noteq> {}" shows "set_mset (Inf A) = \<Inter> (set_mset ` A)"
antisympI: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>r x y; r y x\<rbrakk> \<Longrightarrow> x = y" shows "antisymp r"
sw_involutive2: fixes xs :: "'b"   and y :: "'a"   and x :: "'a"   and zs :: "'b"   and z :: "'a" shows "z @zs[x \<and> y]_xs @zs[y \<and> x]_xs = z"
d_restrict_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<le> y \<cdot> ar ar x)"
map_of_is_SomeI: fixes xys :: "('a \<times> 'b) list"   and x :: "'a"   and y :: "'b" assumes "distinct (map fst xys)"   and "(x, y) \<in> set xys" shows "Map.map_of xys x = Some y"
axiom': 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"   and A :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "A \<in> axioms \<Theta>" shows "\<Theta>,\<Gamma> \<turnstile> A"
hyperb_ineq: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and deltaG :: "'b itself \<Rightarrow> real"   and e :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.Gromov_hyperbolic_space dist uniformity open deltaG" shows "min (Gromov_product_at e x y) (Gromov_product_at e y z) - deltaG TYPE('b) \<le> Gromov_product_at e x z"
not_final_thread_conv: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c" shows "final_thread.not_final_thread final s t = (thr s t \<noteq> None \<and> \<not> final_thread.final_thread final s t)"
WhileLowStepInduct: fixes b :: "expr"   and c' :: "com"   and s1 :: "char list \<Rightarrow> val option"   and s1' :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option"   and s2' :: "char list \<Rightarrow> val option"   and \<Gamma> :: "char list \<Rightarrow> secLevel option" assumes "\<langle>while (b) c',s1\<rangle> \<rightarrow>* \<langle>Skip,s1'\<rangle>"   and "\<langle>while (b) c',s2\<rangle> \<rightarrow>* \<langle>Skip,s2'\<rangle>"   and "\<Gamma> \<turnstile> b : Low"   and "\<And>s1 s1' s2 s2'. \<lbrakk>\<langle>c',s1\<rangle> \<rightarrow>* \<langle>Skip,s1'\<rangle>; \<langle>c',s2\<rangle> \<rightarrow>* \<langle>Skip,s2'\<rangle>; \<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> s1' \<approx>\<^sub>L s2'"   and "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2" shows "\<Gamma> \<turnstile> s1' \<approx>\<^sub>L s2'"
linear_surj_imp_inj: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "surj f" shows "inj f"
fp_cop_F_code: fixes ds :: "'d set" shows "fp_cop_F ds = while (\<lambda>A. cop_F ds A \<noteq> A) (cop_F ds) {}"
isnormNum0: fixes x :: "int \<times> int" assumes "isnormNum x" shows "(INum x = (0::'a)) = (x = 0\<^sub>N)"
Amicable_pair_abundant: fixes n :: "nat"   and m :: "nat" assumes "n < m"   and "m Amic n" shows "abundant_number n"
IH_2: shows "subst_dnf' (list_dnf \<phi>2) m = list_dnf (subst \<phi>2 m)"
arg_subset_TC: fixes a :: "V" shows "a \<le> TC a"
len_lexext_hd_or_tl: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and ys :: "'b list"   and x :: "'b"   and xs :: "'b list" assumes "len_lexext gt (y # ys) (x # xs)" shows "gt y x \<or> len_lexext gt ys xs"
eval_permutation_of_list_2: fixes x :: "'b"   and x' :: "'b"   and y :: "'b"   and xs :: "('b \<times> 'b) list" assumes "x = x'" shows "permutation_of_list ((x', y) # xs) x = y"
\<tau>_p: shows "mgu\<^sub>l\<^sub>s \<tau> (L\<^sub>1 \<union> L\<^sub>2\<^sup>C)"
"3_1": fixes v :: "event_data option list" assumes "ys = map (Formula.eval_trm (map the v)) ts"   and "wf_tuple n (\<Union> (fv_trm ` set ts)) v"   and "\<forall>t\<in>set ts. (\<forall>x\<in>fv_trm t. x < n) \<and> (trm.is_Var t \<or> trm.is_Const t)" shows "\<exists>f. match ts (map (Formula.eval_trm (map the v)) ts) = Some f \<and> v = tabulate f 0 n"
supset_has_bottom_pseudominimal_in_below_inI: fixes P :: "'b set set"   and x :: "'b set"   and y :: "'b set" assumes "dual_order.has_bottom P"   and "x \<in> P"   and "dual_order.pseudominimal_in P y"   and "order.greater_eq y x" shows "dual_order.pseudominimal_in (dual_order.below_in P x) y"
inj_on_cong: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = g a" shows "inj_on f A = inj_on g A"
A: shows "\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2"
ennreal_of_enat_enat: fixes n :: "nat" shows "ennreal_of_enat (enat n) = of_nat n"
add_block_wf: fixes p :: "'a"   and b :: "'a set" shows "incidence_system (add_point p \<union> b) (add_point_sys.add_block b)"
bequiv_up_to_subst: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and s :: "char list \<Rightarrow> int" assumes "P \<Turnstile> b <\<sim>> b'"   and "P s" shows "bval b s = bval b' s"
sup_ge2: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and x :: "'d" assumes "class.semilattice_sup sup less_eq less" shows "less_eq y (sup x y)"
pbij_alt_intro: fixes X :: "'b \<Longrightarrow> 'c" assumes "weak_lens X"   and "\<And>s. s \<in> \<S>\<^bsub>X\<^esub> \<Longrightarrow> create\<^bsub>X\<^esub> (get\<^bsub>X\<^esub> s) = s" shows "pbij_lens X"
1: shows "A \<turnstile>\<^sub>R res1 \<approx> res1'"
X: shows "X (Suc x') = (max (fst (X x')) (nextMax (Range (extend (fst (X x')) R f))), on_triple (extend (fst (X x')) R f) `` edges (snd R) \<union> snd (X x'))"
double_zero_sym: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "(zero = plus a a) = (a = zero)"
plus_of_nat: fixes x :: "'a"   and n :: "nat" shows "f (g x (of_nat n)) = f x"
diff_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V" shows "x - y \<in> V"
sum1Chain: fixes P :: "pi"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "P \<noteq> P'" shows "P \<oplus> Q \<Longrightarrow>\<^sub>\<tau> P'"
cesaro_imp_density_one: fixes u :: "nat \<Rightarrow> real" assumes "\<And>n. 0 \<le> u n"   and "(\<lambda>n. sum u {..<n} / real n) \<longlonglongrightarrow> 0" shows "\<exists>A. lower_asymptotic_density A = 1 \<and> (\<lambda>n. u n * indicat_real A n) \<longlonglongrightarrow> 0"
MinkowskiChain_axioms: shows "MinkowskiChain \<E> \<P> betw"
euclidean_proj_space_def: shows "euclidean = map_topology Proj euclidean"
main_def: fixes us :: "'a vec list" shows "main us = sub2_wit [] us"
lexext_total: fixes B :: "'b set"   and A :: "'b set"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" assumes "\<forall>y\<in>B. \<forall>x\<in>A. gt y x \<or> gt x y \<or> y = x"   and "ys \<in> lists B"   and "xs \<in> lists A" shows "lexext gt ys xs \<or> lexext gt xs ys \<or> ys = xs"
dom_restrict_total: fixes f :: "'a \<Rightarrow> 'b"   and X :: "'a set" shows "dom (to_partial f |` X) = X"
assms_1: shows "x \<cdot> y \<in> t*"
lincomb_union: fixes U :: "'a vec set"   and U' :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "U \<subseteq> carrier_vec (n + 1)"   and "U' \<subseteq> carrier_vec (n + 1)"   and "U \<inter> U' = {}"   and "finite U"   and "finite U'"   and "a \<in> U \<union> U' \<rightarrow> UNIV" shows "next_dim.lincomb a (U \<union> U') = next_dim.lincomb a U + next_dim.lincomb a U'"
less_add_same_cancel2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'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" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less a (plus b a) = less zero b"
removeAll_raw_def: fixes b :: "'e"   and g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f" shows "removeAll_raw b g s = (case g s of Done \<Rightarrow> Done | Skip s' \<Rightarrow> Skip s' | Yield a s' \<Rightarrow> if a = b then Skip s' else Yield a s')"
swaps_inv: fixes sws :: "nat list"   and xs :: "'a list" shows "set (swaps sws xs) = set xs \<and> length (swaps sws xs) = length xs \<and> distinct (swaps sws xs) = distinct xs"
right_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "(y \<oplus> x = z \<oplus> x) = (y = z)"
env_restr_evalHeap_same: fixes h :: "('d \<times> 'f) list"   and eval :: "'f \<Rightarrow> 'e" shows "evalHeap h eval f|` domA h = evalHeap h eval"
closedin_subtopology_empty: fixes U :: "'c topology"   and S :: "'c set" shows "closedin (subtopology U {}) S = (S = {})"
axioms_1: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and basis1 :: "'d set" assumes "finite_dimensional_vector_space_pair_1_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 basis1" shows "finite_dimensional_vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 basis1"
CBlinfun_inverse: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> {f. bounded_clinear f}" shows "(*\<^sub>V) (CBlinfun y) = y"
urel_integer_inverse: fixes pp :: "integer"   and x :: "integer"   and x' :: "int" assumes "p = int_of_integer pp"   and "urel_integer x x'" shows "urel_integer (inverse_p_integer pp x) (inverse_p p x')"
emeasure_eq_measure: fixes A :: "'a set" shows "emeasure (lebesgue_on (cbox x y)) A = ennreal (Sigma_Algebra.measure (lebesgue_on (cbox x y)) A)"
Y_simp: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and S :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and \<phi> :: "'ag \<times> 'ag \<Rightarrow> 'ag \<Rightarrow> 'ah"   and f :: "'ag" assumes "yoneda_functor C S \<phi>"   and "partial_magma.arr C f" shows "yoneda_functor.Y C S \<phi> f = (\<lambda>g. hom_functor.map C S \<phi> (g, f))"
sub_assoc: fixes f :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and r :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "q \<in> carrier P"   and "r \<in> carrier P" shows "Cring_Poly.compose R f (Cring_Poly.compose R q r) = Cring_Poly.compose R (Cring_Poly.compose R f q) r"
surjE: fixes f :: "'d \<Rightarrow> 'c"   and y :: "'c" assumes "surj f" obtains x :: "'d" where "y = f x"
multi_union_self_other_eq: fixes A :: "'b multiset"   and X :: "'b multiset"   and Y :: "'b multiset" assumes "A + X = A + Y" shows "X = Y"
fd_def: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "relfdia x y = rdom (x ; y)"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "y \<le> x"
Units_inv_closed: fixes x :: "'a" assumes "x \<in> Units \<G>" shows "inv x \<in> carrier \<G>"
linear_plus_1_le_power: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x" shows "real n * x + 1 \<le> (x + 1) ^ n"
deriv: fixes z :: "real" assumes "z \<in> {y..x} - X" shows "(f has_vector_derivative f' z) (at z)"
absorb2: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "a \<squnion> b = b"
assms_4: shows "val_process Mkt pf n w \<le> 0"
locally_path_connected_connected_component: fixes S :: "'b set"   and x :: "'b" assumes "locally path_connected S" shows "locally path_connected (connected_component_set S x)"
noFaultn_def': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and f :: "'f" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>{Fault f} = (\<not> \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> Fault f)"
surjective: fixes r :: "('a, 'b) det_while_algo_scheme" shows "r = \<lparr>dwa_cond = dwa_cond r, dwa_step = dwa_step r, dwa_initial = dwa_initial r, dwa_invar = dwa_invar r, \<dots> = det_while_algo.more r\<rparr>"
assms_1: shows "chain (\<leadsto>RP) Sts"
Int_space_eq2: fixes x :: "'a set" assumes "x \<in> A" shows "x \<inter> \<Omega> = x"
dist_fst_le: fixes x :: "'d \<times> 'e"   and y :: "'d \<times> 'e" shows "dist (fst x) (fst y) \<le> dist x y"
wcode_halt_case_le_def: shows "wcode_halt_case_le \<equiv> inv_image Recursive.lex_pair wcode_halt_case_measure"
isShortestPath_alt: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "Graph.isShortestPath c u p v = (Graph.isSimplePath c u p v \<and> length p = Graph.min_dist c u v)"
exec_c: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
leg1_simps_3: shows "src t\<^sub>0u\<^sub>1.p\<^sub>1 = src t\<^sub>0u\<^sub>1.p\<^sub>0"
cA: shows "convex_hull A \<subseteq> carrier_vec n"
polynomial_in_carrier: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "polynomial\<^bsub>K [X]\<^esub> K p" shows "set p \<subseteq> carrier (K [X])"
indecomposable_ordertype_eq: fixes \<alpha> :: "V"   and A :: "V set"   and B :: "V set" assumes "indecomposable \<alpha>"   and "ordertype A VWF = \<alpha>"   and "B \<subseteq> A"   and "small A" shows "ordertype B VWF = \<alpha> \<or> ordertype (A - B) VWF = \<alpha>"
freeword_funlift_diff: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b freeword"   and g' :: "'b freeword" shows "freeword_funlift f (g - g') = freeword_funlift f g - freeword_funlift f g'"
Match_1: fixes P :: "pi"   and b :: "name"   and x :: "name"   and P' :: "pi"   and a :: "name" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^b<\<nu>x> \<prec> P'" shows "[a\<frown>a]P \<Longrightarrow>\<^sub>l\<^sup>^b<\<nu>x> \<prec> P'"
Cons_1: fixes m :: "'c multiset" assumes "list_all carrier_coeff Ps"   and "list_all (\<lambda>P. index_free P i) Ps"   and "indexed_eval_aux Ps i m \<noteq> \<zero>" shows "count m i \<le> degree Ps"
comp_min_basis_nonzero: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "p \<in> set (comp_min_basis xs)" shows "p \<noteq> 0"
wf_sub_rel_def: fixes subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "wf_sub_rel subs \<equiv> (\<forall>sub. sub \<in> subs \<longrightarrow> fst (subsumee sub) = fst (subsumer sub)) \<and> (\<forall>v\<in>subsumees subs. \<exists>!v'. (v, v') \<in> subs) \<and> subsumers subs \<inter> subsumees subs = {}"
K_def: shows "K = (\<lambda>t. fst (index t))"
SN_on_conv_acc: fixes r :: "('b \<times> 'b) set"   and x :: "'b" shows "SN_on {(y, z). (z, y) \<in> r} {x} = (x \<in> Wellfounded.acc r)"
PiE_restrict: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'e \<Rightarrow> 'f set" assumes "f \<in> Pi\<^sub>E A B" shows "restrict f A = f"
zero_not_one: fixes R :: "('e, 'f) ring_scheme" assumes "domain R" shows "\<zero>\<^bsub>R\<^esub> \<noteq> \<one>\<^bsub>R\<^esub>"
assms_2: shows "approx_tm prec ord I a f env = Some t"
Ball_rev_wrap_def: fixes A :: "'b list set"   and P :: "'b list \<Rightarrow> bool" shows "Ball_rev_wrap A P = (\<forall>x\<in>rev ` A. P (rev x))"
trans: shows "\<forall>z y x. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"
eq_\<I>_converter_eq: fixes conv1 :: "('a, 'b, 'c, 'd) converter"   and conv2 :: "('a, 'b, 'c, 'd) converter" assumes "\<I>_full,\<I>_full \<turnstile>\<^sub>C conv1 \<sim> conv2" shows "conv1 = conv2"
set_diff_diff_left: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A - B - C = A - (B \<union> C)"
aug_commute: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set"   and x :: "'e"   and y :: "'e" assumes "commute_bpp f (aug_pm_set z i A)"   and "x \<in> addition_set f (aug_pm_set z i A)"   and "y \<in> addition_set f (aug_pm_set z i A)" shows "x \<^sub>f+ y = y \<^sub>f+ x"
exhaust: fixes y :: "memberdecl" obtains x1 :: "fdecl" where "y = fdecl x1"   | x2 :: "mdecl" where "y = mdecl x2"
normalize_eq: fixes a :: "int"   and b :: "int"   and p :: "int"   and q :: "int" assumes "Rat.normalize (a, b) = (p, q)" shows "Fract p q = Fract a b"
prod_pOp_assoc: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and a :: "'d \<Rightarrow> 'e"   and b :: "'d \<Rightarrow> 'e"   and c :: "'d \<Rightarrow> 'e" assumes "\<forall>k\<in>I. aGroup (A k)"   and "a \<in> carr_prodag I A"   and "b \<in> carr_prodag I A"   and "c \<in> carr_prodag I A" shows "prod_pOp I A (prod_pOp I A a b) c = prod_pOp I A a (prod_pOp I A b c)"
ts_invD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and I :: "'b \<Rightarrow> 'a option"   and ts :: "'b \<Rightarrow> ('c \<times> 'e \<Rightarrow>f nat) option"   and m :: "'d"   and t :: "'b"   and x :: "'c"   and ln :: "'e \<Rightarrow>f nat" assumes "ts_inv P I ts m"   and "ts t = \<lfloor>(x, ln)\<rfloor>" shows "\<exists>i. I t = \<lfloor>i\<rfloor> \<and> P i t x m"
dense_up_closed: fixes x :: "'c"   and y :: "'c" assumes "brouwer.dense x"   and "y \<le> x" shows "brouwer.dense y"
residual_def: fixes f :: "'a \<Rightarrow> 'a" shows "residual f \<equiv> THE g. residuated_pair f g"
dvd_imp_degree_le: fixes p :: "'a poly"   and q :: "'a poly" assumes "p dvd q"   and "q \<noteq> 0" shows "degree p \<le> degree q"
X_def: shows "X = codomain d1 \<inter> inv_chart d1 -` (src.carrier \<inter> f -` domain d2)"
A_1: fixes z :: "complex" assumes "\<bar>Re z\<bar> < 1" shows "(h has_field_derivative h2' z) (at z)"
sorted_wrt_rev: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" shows "sorted_wrt P (rev xs) = sorted_wrt (\<lambda>x y. P y x) xs"
zero_lens_def: shows "0\<^sub>L = \<lparr>lens_get = \<lambda>_. (), lens_put = \<lambda>\<sigma> x. \<sigma>\<rparr>"
not_Cons_self: fixes xs :: "'b list"   and x :: "'b" shows "xs \<noteq> x # xs"
algebraic_int_power: fixes x :: "'b"   and n :: "nat" assumes "algebraic_int x" shows "algebraic_int (x ^ n)"
commit_correct: shows "AND_\<Sigma>_commit.abstract_com.correct"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "C.arr f"   and "C.cod f = b" shows "b \<cdot> f = f"
case_prodI2: fixes p :: "'g \<times> 'h"   and c :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
prems_4: shows "A \<union> \<Union> (block_nominals ` S) \<subseteq> used"
induct: fixes x1 :: "com"   and x2 :: "state"   and x3 :: "state"   and P :: "com \<Rightarrow> state \<Rightarrow> state \<Rightarrow> bool" assumes "<x1,x2> -c-> x3"   and "\<And>s. P SKIP s s"   and "\<And>X a s. P (X:==a) s (s[X::=a s])"   and "\<And>c s0 Y a s1. \<lbrakk><c,s0[Loc Y::=a s0]> -c-> s1; P c (s0[Loc Y::=a s0]) s1\<rbrakk> \<Longrightarrow> P (LOCAL Y:=a IN c) s0 (s1[Loc Y::=s0<Y>])"   and "\<And>c0 s0 s1 c1 s2. \<lbrakk><c0,s0> -c-> s1; P c0 s0 s1; <c1,s1> -c-> s2; P c1 s1 s2\<rbrakk> \<Longrightarrow> P (c0;; c1) s0 s2"   and "\<And>b s c0 s1 c1. \<lbrakk>b s; <c0,s> -c-> s1; P c0 s s1\<rbrakk> \<Longrightarrow> P (IF b THEN c0 ELSE c1) s s1"   and "\<And>b s c1 s1 c0. \<lbrakk>\<not> b s; <c1,s> -c-> s1; P c1 s s1\<rbrakk> \<Longrightarrow> P (IF b THEN c0 ELSE c1) s s1"   and "\<And>b s c. \<not> b s \<Longrightarrow> P (WHILE b DO c) s s"   and "\<And>b s0 c s1 s2. \<lbrakk>b s0; <c,s0> -c-> s1; P c s0 s1; <WHILE b DO c,s1> -c-> s2; P (WHILE b DO c) s1 s2\<rbrakk> \<Longrightarrow> P (WHILE b DO c) s0 s2"   and "\<And>pn s0 s1. \<lbrakk><the (body pn),s0> -c-> s1; P (the (body pn)) s0 s1\<rbrakk> \<Longrightarrow> P (BODY pn) s0 s1"   and "\<And>pn s0 a s1 X. \<lbrakk><BODY pn,setlocs s0 newlocs[Loc Arg::=a s0]> -c-> s1; P (BODY pn) (setlocs s0 newlocs[Loc Arg::=a s0]) s1\<rbrakk> \<Longrightarrow> P (X:=CALL pn(a)) s0 (setlocs s1 (loc s0)[X::=s1<Res>])" shows "P x1 x2 x3"
mod_minus_minus: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (uminus a) (uminus b) = uminus (modulo a b)"
PiE_choice: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'e set"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(\<exists>f\<in>Pi\<^sub>E I F. \<forall>i\<in>I. P i (f i)) = (\<forall>i\<in>I. \<exists>x\<in>F i. P i x)"