hcju/isabelleps huggingface dataset:
DgrDataRefinement_mono: fixes Q :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and D :: "'a \<times> 'a \<Rightarrow> 'b \<Rightarrow> 'b"   and D' :: "'a \<times> 'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "Q \<sqsubseteq> P"   and "mono_fun R"   and "DgrDataRefinement2 P D R D'" shows "DgrDataRefinement2 Q D R D'"
CanonicalCat'_def: fixes T :: "(ZF, ZF) Axioms" shows "CanonicalCat' T \<equiv> \<lparr>Obj = Ty\<^bsub>aS\<^bsub>T\<^esub>\<^esub>, Mor = {[A,e,B]\<^bsub>T\<^esub> |A e B. Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : A \<turnstile> e : B}, Dom = CLDomain T, Cod = CLCodomain T, Id = \<lambda>A. [A,Vx,A]\<^bsub>T\<^esub>, Comp = CanonicalComp T\<rparr>"
f_opp_chamber_list: fixes Cs :: "'a set list" assumes "order.greater_eq (f \<turnstile> folding.\<C>) (set Cs)" shows "f \<Turnstile> map (\<lambda>C. THE D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = C) Cs = Cs"
hom_power: fixes x :: "'b"   and n :: "nat" shows "to_fract (x ^ n) = to_fract x ^ n"
simps: fixes a1 :: "nat"   and a2 :: "('c, 'd) bintree list" shows "is_bintree_list a1 a2 = (a1 = 0 \<and> a2 = [] \<or> (\<exists>l ts t. a1 = Suc l \<and> a2 = t # ts \<and> is_bintree_list l ts \<and> is_bintree_list l (children t)))"
exec: shows "abc_steps_l (0, lm) B (Suc n) = (sa, lm')"
well_def_wp_healthy: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "well_def prog" shows "healthy (wp prog)"
\<S>_of_mds_def: fixes mds :: "Mode \<Rightarrow> 'Var set" shows "\<S>_of_mds mds \<equiv> (mds AsmNoWrite, mds AsmNoReadOrWrite)"
arg_min_natI: fixes P :: "'c \<Rightarrow> bool"   and k :: "'c"   and m :: "'c \<Rightarrow> nat" assumes "P k" shows "P (arg_min m P)"
set_zero_plus2: fixes A :: "'a set"   and B :: "'a set" assumes "(0::'a) \<in> A" shows "B \<subseteq> A + B"
config_dist_perm: fixes init :: "'a list"   and A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and qs :: "'a list" assumes "distinct init" shows "dist_perm (fst (config\<^sub>p A init qs)) init"
lb: fixes R :: "'s \<Rightarrow> real" assumes "\<lambda>s. \<guillemotleft> G \<guillemotright> s * wp body R s + \<guillemotleft> \<N> G \<guillemotright> s * P s \<tturnstile> R"   and "sound R" shows "Q \<tturnstile> R"
P1: shows "Bet B A A0 \<and> Cong A A0 B A \<and> Bet B C C0 \<and> Cong C C0 B D \<and> Bet B A A1 \<and> Cong A A1 B A \<and> Bet B D D0 \<and> Cong D D0 B C \<and> Cong A0 C0 A1 D0"
basis_left_null_space_in_left_null_space: fixes A :: "(('a, 'c) vec, 'b) vec" shows "basis_left_null_space A \<subseteq> left_null_space A"
lbisimulation_this: shows "lbisimulation (rel_of_pred rel_gc) s1.system_automaton s2.system_automaton"
converse_unfold: fixes r :: "('d \<times> 'c) set" shows "r\<inverse> = {(y, x). (x, y) \<in> r}"
is_PUSH_project_conv: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat" assumes "Network c s t" shows "is_PUSH \<circ> project_operation = (\<lambda>x. is_SAT_PUSH' x \<or> is_NONSAT_PUSH' x)"
filterlim_Pair: fixes f :: "'b \<Rightarrow> 'c"   and G :: "'c filter"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'd"   and H :: "'d filter" assumes "filterlim f G F"   and "filterlim g H F" shows "LIM x F. (f x, g x) :> G \<times>\<^sub>F H"
INFM_E: fixes P :: "'a \<Rightarrow> bool" assumes "\<exists>\<^sub>\<infinity>x. P x" obtains x :: "'a" where "P x"
args_to_set_subset3: fixes ps1 :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps2 :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "set ps1 \<subseteq> set ps2" shows "args_to_set (gs, bs, ps1) \<subseteq> args_to_set (gs, bs, ps2)"
i_take_Suc_conv_app_nth: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "f \<Down> Suc n = f \<Down> n @ [f n]"
phiDefs_def: fixes g :: "'g"   and n :: "'node" shows "uninst_code.ssa.phiDefs phis g n \<equiv> {v. (n, v) \<in> dom (Mapping.lookup (phis g))}"
intro: fixes root :: "'a"   and G :: "('a, 'b) Graph_scheme" assumes "root \<in> V\<^bsub>G\<^esub>" shows "RootedTree_axioms G root"
weight_less_imp_le: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" assumes "weight_less s t" shows "weight_le s t"
abs_eq: fixes xa :: "nat"   and x :: "nat" shows "natural_of_nat xa - natural_of_nat x = natural_of_nat (xa - x)"
smap2_szip: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and s1 :: "'d stream"   and s2 :: "'e stream" shows "smap2 f s1 s2 = smap (\<lambda>(x, y). f x y) (s1 ||| s2)"
hd_tl: fixes ys :: "'a list"   and n :: "nat"   and xs :: "'a list" assumes "length ys = Suc n"   and "length xs = Suc n"   and "ext gt ys xs" shows "gt_hd ys xs \<or> gt_tl ys xs"
cases: fixes a :: "'b \<Rightarrow> real"   and X :: "'b set"   and I :: "'b \<Rightarrow> intv"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> intv'"   and r :: "('b \<times> 'b) set" assumes "a \<in> region X I J r" obtains u :: "'b \<Rightarrow> real" and X\<^sub>0 :: "'b set" where "a = u" and "\<forall>x\<in>X. 0 \<le> u x" and "\<forall>x\<in>X. intv_elem x u (I x)" and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d}" and "\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. ((x, y) \<in> r) = (frac (u x) \<le> frac (u y))" and "\<forall>x\<in>X. \<forall>y\<in>X. isGreater (I x) \<or> isGreater (I y) \<longrightarrow> intv'_elem x y u (J x y)"
Suc_4: shows "mod_type_class.to_nat i \<le> Suc n"
i: shows "i < dim_vec (A *\<^sub>v a @\<^sub>v D *\<^sub>v d)"
simps_4: fixes y :: "'a" shows "flat_1 (deny (deny y)) = deny y"
update_id: fixes update :: "nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x :: "nat"   and c :: "'a" assumes "Update update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "let s' = update x c s in \<T> s' = \<T> s \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U> s' = \<U> s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"
group_disjoint_sum_def: fixes G :: "('a, 'b) monoid_scheme"   and A :: "'a set"   and B :: "'a set" shows "group_disjoint_sum G A B \<equiv> Group.group G \<and> subgroup A G \<and> subgroup B G"
z_def: shows "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
T_split_min_ub: fixes t :: "'b tree" assumes "t \<noteq> \<langle>\<rangle>" shows "T_split_min t \<le> height t + 1"
lits_of_cnf_def: fixes cnf :: "lit uprod set" shows "lits_of_cnf cnf = Pos ` vars_of_cnf cnf \<union> Neg ` vars_of_cnf cnf"
assms: fixes x :: "'b"   and s :: "'a" assumes "x \<in> outs_\<I> \<I>1" shows "plossless_gpv \<I>2 (callee s x) \<and> (\<forall>(y, s')\<in>results_gpv \<I>2 (callee s x). y \<in> responses_\<I> \<I>1 x)"
hyps: assumes "k < ncols A" shows "pairwise orthogonal {column i (Gram_Schmidt_upt_k A k) |i. mod_type_class.to_nat i \<le> k}"
lcoset_el_reduce0: fixes G :: "'b set"   and a :: "'b" assumes "Group G"   and "(0::'b) \<in> a +o G" shows "a +o G = G"
Pmod0_principal_rev1: fixes t :: "'a"   and h :: "'a" assumes "Idomain S"   and "t \<in> carrier S"   and "h \<in> carrier R" shows "P_mod R S X (S \<diamondsuit>\<^sub>p t) (t \<cdot>\<^sub>r h)"
complete_bipartite_digraph_def: fixes m :: "nat"   and n :: "nat"   and G :: "('c, 'd) pre_digraph" shows "K\<^bsub>m,n\<^esub> G \<equiv> graph G \<and> (\<exists>U V. verts G = U \<union> V \<and> U \<inter> V = {} \<and> card U = m \<and> card V = n \<and> arcs_ends G = U \<times> V \<union> V \<times> U)"
is_init_of_def: fixes t :: "nat"   and f :: "nat \<Rightarrow> nat option" shows "is_init_of t f \<equiv> \<forall>i<e_length t. f i \<down>= e_nth t i"
pred_Collect_borel: fixes P :: "'c \<Rightarrow> bool" assumes "Measurable.pred borel P" shows "{x. P x} \<in> sets borel"
ThinChamberComplexFolding_f: shows "ThinChamberComplexFolding X f"
intros_2: fixes y :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a" assumes "y \<in> directed_towards A r"   and "(x, y) \<in> r" shows "x \<in> directed_towards A r"
intro: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "additive_subgroup I R"   and "ring R"   and "ideal_axioms I R" shows "ideal I R"
coprime_iff_gcd_eq_1: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one a b = (gcd a b = one)"
Arcsin_def: shows "Arcsin \<equiv> \<lambda>z. - \<i> * Ln (\<i> * z + csqrt (1 - z\<^sup>2))"
simps_1: fixes \<mu> :: "'c" shows "Src \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
subti_def: shows "subterms_img \<equiv> \<lambda>d. subterms\<^sub>s\<^sub>e\<^sub>t (subst_range d)"
index_of_lookup_1: fixes p :: "'a"   and ls :: "'a list" assumes "p \<in> set ls" shows "index_of ls p < length ls"
indets_dehomogenize: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "indets (dehomogenize x p) \<subseteq> indets p - {x}"
complete_interval: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "less a b"   and "P a"   and "\<not> P b" shows "\<exists>c. less_eq a c \<and> less_eq c b \<and> (\<forall>x. less_eq a x \<and> less x c \<longrightarrow> P x) \<and> (\<forall>d. (\<forall>x. less_eq a x \<and> less x d \<longrightarrow> P x) \<longrightarrow> less_eq d c)"
disj_simps_3: fixes Q :: "fm" shows "Ferrack.disj F Q = Q"
deepR_inv: fixes n :: "nat"   and m :: "('a, 'b) FingerTreeStruc"   and pr :: "('a, 'b) Digit"   and sf :: "('a, 'b) Node list" assumes "FingerTreeStruc.is_leveln_ftree (Suc n) m \<and> FingerTreeStruc.is_measured_ftree m"   and "FingerTreeStruc.is_leveln_digit n pr \<and> FingerTreeStruc.is_measured_digit pr"   and "\<forall>x\<in>set sf. (FingerTreeStruc.is_measured_node x \<and> FingerTreeStruc.is_leveln_node n x) \<and> length sf \<le> 4" shows "FingerTreeStruc.is_leveln_ftree n (FingerTreeStruc.deepR pr m sf) \<and> FingerTreeStruc.is_measured_ftree (FingerTreeStruc.deepR pr m sf)"
mat_inv_transpose: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_transpose (mat_inv M) = mat_inv (mat_transpose M)"
smult_zero: fixes r :: "'f" assumes "r \<in> UNIV" shows "r \<cdot> (0::'v) = (0::'v)"
conc_assoc: fixes A :: "'c list set"   and B :: "'c list set"   and C :: "'c list set" shows "(A @@ B) @@ C = A @@ B @@ C"
b: shows "set t = to_woot_character (M \<union> {Delete m}) ` insert_messages (M \<union> {Delete m})"
map_of_effect_to_assignments_is_none_if: fixes v :: "'a"   and op :: "'a strips_operator" assumes "v \<notin> set (add_effects_of op)"   and "v \<notin> set (delete_effects_of op)" shows "map_of (effect_to_assignments op) v = None"
bounded_measure: fixes M :: "'b measure"   and A :: "'b set" assumes "finite_measure M" shows "Sigma_Algebra.measure M A \<le> Sigma_Algebra.measure M (space M)"
quorum_def: fixes Q :: "'a set" shows "i2.proj.quorum Q \<equiv> \<forall>p\<in>Q \<inter> W. \<exists>Sl\<in>project slices I\<^sub>2 p. Sl \<subseteq> Q"
setcomp_dot2: fixes P :: "real \<Rightarrow> bool"   and i :: "'b" shows "{z. P (z \<bullet> (0::'a, i))} = {(x, y). P (y \<bullet> i)}"
is_lasso_pre_prpl_of_lasso: fixes L :: "'Q lasso" shows "is_lasso_prpl_pre (prpl_of_lasso L) = is_lasso_pre L"
that: fixes as' :: "'edge list"   and as'' :: "'edge list" assumes "n -as'\<rightarrow>* n'"   and "length as' < k"   and "n' -as''\<rightarrow>* n''"   and "as = as' @ as''" shows "thesis"
intro: fixes S :: "'a set"   and ccw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>p q r s t. \<lbrakk>distinct5 p q r s t; in5 S p q r s t; ccw t s p; ccw t s q; ccw t s r; ccw t p q; ccw t q r\<rbrakk> \<Longrightarrow> ccw t p r" shows "ccw_system1235_axioms ccw S"
insert_3: fixes f :: "(nat \<Rightarrow> 'a) \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (Pi\<^sub>M V N)" shows "integral\<^sup>N (distr (Pi\<^sub>M (Suc ` V) (\<lambda>a. case a of 0 \<Rightarrow> M | Suc a \<Rightarrow> N a)) (Pi\<^sub>M V N) (\<lambda>\<sigma>. \<sigma> \<circ> Suc)) f = integral\<^sup>N (Pi\<^sub>M V N) f"
unat_arith_simps_7: fixes x :: "'a word"   and y :: "'a word" shows "unat (x mod y) = unat x mod unat y"
pmat_extension_def: fixes m :: "'a mat" shows "ps2_P.pmat_extension m = tensor_P m (1\<^sub>m ps2_P.d2)"
FrameD: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and F :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "Frame P Q F"   and "P s" shows "(F \<and>* Q) s"
rep_eq: fixes x :: "'state"   and xa :: "('b, 'pred, 'act) Tree\<^sub>\<alpha>" shows "valid_Tree\<^sub>\<alpha> x xa = valid_Tree x (rep_Tree\<^sub>\<alpha> xa)"
rbt_sorted_rbtreeify: fixes kvs :: "('a \<times> 'c) list" assumes "sorted (map fst kvs)"   and "distinct (map fst kvs)" shows "rbt_sorted (rbtreeify kvs)"
hmem_rel_def: shows "hmem_rel = {(x, y). x \<^bold>\<in> y}\<^sup>+"
cpx_abelian_monoid: shows "abelian_monoid cpx_rng"
finite_reach_product: fixes \<iota>\<^sub>m :: "'a \<Rightarrow> 'b option"   and \<Sigma> :: "'c set"   and \<delta>\<^sub>m :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b" assumes "finite (dom \<iota>\<^sub>m)"   and "\<And>x. x \<in> dom \<iota>\<^sub>m \<Longrightarrow> finite (reach \<Sigma> (\<delta>\<^sub>m x) (the (\<iota>\<^sub>m x)))" shows "finite (reach \<Sigma> (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m)"
assms_1: shows "set (vwalk_arcs p) \<subseteq> E"
execute_instr_sub1_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t1 :: "'b sparc_state"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> \<not> snd (execute_instr_sub1 instr s1) \<and> \<not> snd (execute_instr_sub1 instr s2) \<and> t1 = snd (fst (execute_instr_sub1 instr s1)) \<and> t2 = snd (fst (execute_instr_sub1 instr s2))" shows "low_equal t1 t2"
basis_left_null_space_def: fixes A :: "(('d, 'f) vec, 'e) vec" shows "basis_left_null_space A = {row i (P_Gauss_Jordan A) |i. rank A \<le> mod_type_class.to_nat i}"
Utrans_P_is_tensor_P1: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat"   and A :: "complex mat" assumes "grover_state_sig n f dims R K" shows "state_sig.Utrans_P dims (grover_state_sig.vars1 n) A = Utrans (partial_state2.ptensor_mat dims (grover_state_sig.vars1 n) (grover_state_sig.vars2 n) A (1\<^sub>m K))"
simps_2: fixes x :: "nat"   and v :: "val"   and \<rho> :: "(nat \<times> val) list" shows "Tenv ((x, v) # \<rho>) = (x, T v) # Tenv \<rho>"
f_sf: shows "(\<lambda>i. \<integral>\<^sup>+ x. ennreal (norm (f x - sf i x)) \<partial>M) \<longlonglongrightarrow> 0"
sat_Since_rec: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula" shows "Formula.sat \<sigma> V v i (formula.Since \<phi> I \<psi>) = (mem 0 I \<and> Formula.sat \<sigma> V v i \<psi> \<or> 0 < i \<and> enat (\<Delta> \<sigma> i) \<le> right I \<and> Formula.sat \<sigma> V v i \<phi> \<and> Formula.sat \<sigma> V v (i - 1) (formula.Since \<phi> (subtract (\<Delta> \<sigma> i) I) \<psi>))"
assms_1: shows "[a = e + b - q * c - d] (mod q)"
restrict_relpI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "R x y"   and "P x"   and "Q y" shows "(R \<upharpoonleft> P \<otimes> Q) x y"
comp_reduce: fixes f :: "'c"   and g :: "'c"   and k :: "'c"   and h :: "'c" assumes "f \<cdot>\<^sub>C g = k"   and "C.seq f g"   and "C.seq g h" shows "f \<cdot>\<^sub>C g \<cdot>\<^sub>C h = k \<cdot>\<^sub>C h"
sort_edges_def: shows "sort_edges \<equiv> quicksort_by_rel edges_less_eq []"
signed_real_of_word_plus_numeral_eq_signed_real_of_word_iff: fixes v :: "'a Word.word"   and n :: "num"   and w :: "'b Word.word" shows "(signed_real_of_word v + numeral n = signed_real_of_word w) = (sint v + numeral n = sint w)"
set_disconnected_nodes_impl: shows "set_disconnected_nodes = a_set_disconnected_nodes"
sinh_field_def: fixes z :: "'a" shows "sinh z = (exp z - exp (- z)) / (2::'a)"
cones_map_ide: fixes \<chi> :: "'j \<Rightarrow> 'c"   and a :: "'c" assumes "\<chi> \<in> D.cones a" shows "D.cones_map a \<chi> = \<chi>"
nMTSortQ: fixes p :: "((int \<times> int) set set, int) Combinators list"   and l :: "(int \<times> int) set set set list" assumes "none_MT_rules C p" shows "none_MT_rules C (qsort p l)"
I: fixes a :: "'a"   and b :: "'a" shows "a * b l\<rightarrow> b = (1::'a)"
wits': shows "wits' = map (\<lambda>i. map (\<mu> i) [0..<i]) [Suc i..<m]"
method_spec_def: shows "method_spec = SPEC (\<lambda>m. True)"
wf_graph_def: fixes G :: "'a graph" shows "wf_graph G \<equiv> (fst ` edges G \<subseteq> nodes G \<and> snd ` edges G \<subseteq> nodes G) \<and> finite (edges G) \<and> finite (nodes G)"
K4_imp_K2: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "\<exists>Ta. Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs"
FG_advice_af: fixes w' :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and w :: "'b set list"   and X :: "'b ltln set" assumes "w' \<Turnstile>\<^sub>n af \<phi> w[X]\<^sub>\<mu>" shows "(w \<frown> w') \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<mu>"
of_nat_Suc: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (Suc m) = plus one (semiring_1.of_nat one plus zero m)"
obv: fixes a :: "'a \<times> 'b" shows "a = (fst a, snd a)"
assms: fixes x :: "hf" assumes "\<And>y. (y, x) \<in> hmem_rel \<Longrightarrow> P y" shows "P x"
relpow_1: fixes R :: "('c \<times> 'c) set" shows "R ^^ 1 = R"
mult_dvd_mono: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a b"   and "dvd.dvd times c d" shows "dvd.dvd times (times a c) (times b d)"
finite_init_pred_symvars: shows "\<forall>e\<in>pred (init_conf prb). finite (Bexp.vars e)"
RedInitDone: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "sh C = \<lfloor>(sfs, Done)\<rfloor>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>INIT C' (Cs,True) \<leftarrow> e,(h, l, sh),b\<rangle>"
sturm_adjacent_root_not_squarefree: fixes i :: "nat"   and p :: "real poly"   and x :: "real" assumes "i < length (sturm p) - 1"   and "poly (sturm p ! i) x = 0"   and "poly (sturm p ! (i + 1)) x = 0" shows "\<not> rsquarefree p"
smap_singleton_def: fixes k :: "'a"   and vs :: "'b set" shows "{ k := vs}. = {}.(k := vs)"
rep_eq: fixes x :: "'b ratfps" shows "ratfps_subdegree x = poly_subdegree (numerator (Rep_ratfps x))"
Rf_indep: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set"   and N' :: "'b literal multiset set"   and N :: "'b literal multiset set" assumes "redundancy_criterion \<Gamma> Rf Ri"   and "N' \<subseteq> Rf N" shows "Rf N \<subseteq> Rf (N - N')"
cancel_times_left: fixes a :: "'a N"   and b :: "'a N"   and c :: "'a N" assumes "a * b = a * c" shows "b = c"
loc_connectivity_implies_confluence: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "relation_order r ord A"   and "is_loc_connective_on A ord r"   and "relation.dw_closed r A" shows "relation.is_confluent_on r A"
simps_4: fixes p :: "('psym, 'tp) epsym"   and Tl :: "'a trm list" shows "nv2L (Neg (Pr p Tl)) = {x \<in> \<Union> (set (map nv2T Tl)). pol (GE.tpOfV x) p = Text}"
mult_le_one: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less_eq a one"   and "less_eq zero b"   and "less_eq b one" shows "less_eq (times a b) one"
equiv_transD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" assumes "equiv_trans t u"   and "sound P" shows "t P = u P"
evaldjf_bound0: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> fm" assumes "\<forall>x\<in>set xs. bound0 (f x)" shows "bound0 (evaldjf f xs)"
in_carrier2: shows "\<^bold>g [^] \<alpha>1 \<in> carrier \<G>"
jozsa_transform_is_unitary_index_even: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row U\<^sub>f"   and "j < dim_col U\<^sub>f"   and "even i" shows "(U\<^sub>f * U\<^sub>f) $$ (i, j) = 1\<^sub>m (dim_col U\<^sub>f) $$ (i, j)"
Lim_cong_within: fixes a :: "'c"   and b :: "'c"   and x :: "'d"   and y :: "'d"   and S :: "'c set"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "a = b"   and "x = y"   and "S = T"   and "\<And>x. \<lbrakk>x \<noteq> b; x \<in> T\<rbrakk> \<Longrightarrow> f x = g x" shows "(f \<longlongrightarrow> x) (at a within S) = (g \<longlongrightarrow> y) (at b within T)"
b_in: fixes b :: "'a set" assumes "b \<in># remove1_mset bl \<B>"   and "ps \<subseteq> b" shows "ps \<subseteq> b \<inter> bl"
BinOp_exec2D: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and e1 :: "'addr expr1"   and bop :: "bop"   and e2 :: "'addr expr1"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and v1 :: "'addr val"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "exec_meth ci (compP2 P) (compE2 (e1 \<guillemotleft>bop\<guillemotright> e2)) (compxE2 (e1 \<guillemotleft>bop\<guillemotright> e2) 0 0) t h (stk @ [v1], loc, length (compE2 e1) + pc, xcp) ta h' (stk', loc', pc', xcp')"   and "pc < length (compE2 e2)" shows "exec_meth ci (compP2 P) (compE2 e2) (stack_xlift (length [v1]) (compxE2 e2 0 0)) t h (stk @ [v1], loc, pc, xcp) ta h' (stk', loc', pc' - length (compE2 e1), xcp') \<and> length (compE2 e1) \<le> pc'"