hcju/isabelleps huggingface dataset:
encode_def: fixes h :: "nat"   and prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "encode h prob \<equiv> if ast_problem.well_formed prob then if \<forall>op\<in>set (ast_problem.ast\<delta> prob). consistent_pres_op op then if \<forall>op\<in>set (ast_problem.ast\<delta> prob). is_standard_operator op then Inl (SASP_to_DIMACS' h prob) else Inr STR ''Error: Conditional effects!'' else Inr STR ''Error: Preconditions inconsistent'' else Inr STR ''Error: Problem malformed!''"
swap: fixes g :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and B :: "'e set"   and A :: "'d set" shows "aggregation.sum_0 (\<lambda>i. aggregation.sum_0 (g i) B) A = aggregation.sum_0 (\<lambda>j. aggregation.sum_0 (\<lambda>i. g i j) A) B"
sub_def: fixes m :: "num"   and n :: "num" shows "Int.sub m n = numeral m - numeral n"
eq: shows "AE x in lborel. indicat_real \<Omega> x * f x = f' x"
rcosets_carrier: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme"   and X :: "'d set" assumes "subgroup H G"   and "Group.group G"   and "X \<in> rcosets\<^bsub>G\<^esub> H" shows "X \<subseteq> carrier G"
Inf_DF_N: fixes k :: "nat"   and p :: "nat" shows "\<Sqinter> (DF k p) \<in> N"
add_diff_assoc: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b" shows "c + (b - a) = c + b - a"
Lindelof_openin: fixes \<F> :: "'c set set"   and U :: "'c set" assumes "\<And>S. S \<in> \<F> \<Longrightarrow> openin (top_of_set U) S" obtains \<F>' :: "'c set set" where "\<F>' \<subseteq> \<F>" and "countable \<F>'" and "\<Union> \<F>' = \<Union> \<F>"
fun_eq_iff: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" shows "(f = g) = (\<forall>x. f x = g x)"
upd_upd_twice: fixes a :: "'a"   and b'' :: "'b"   and b' :: "'b"   and b :: "'b" shows "upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)"
ffold_finsert_fremove: fixes z :: "'b"   and x :: "'a"   and A :: "'a fset" shows "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
substEnv_preserves_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good Y"   and "freshEnv zs z rho"   and "fresh zs z Y"   and "zs \<noteq> ys \<or> z \<noteq> y" shows "freshEnv zs z (rho &[Y / y]_ys)"
expand: fixes enat :: "enat"   and enat' :: "enat" assumes "(enat = 0) = (enat' = 0)"   and "\<lbrakk>enat \<noteq> 0; enat' \<noteq> 0\<rbrakk> \<Longrightarrow> epred enat = epred enat'" shows "enat = enat'"
size_eq_sum_mset: fixes M :: "'k multiset" shows "size M = (\<Sum>a\<in>#M. 1)"
acc_run_sim1: fixes r :: "nat \<Rightarrow> 'u" assumes "is_acc_run r" shows "G'.is_acc_run (f \<circ> r)"
UN_upper: fixes a :: "'e"   and A :: "'e set"   and B :: "'e \<Rightarrow> 'f set" assumes "a \<in> A" shows "B a \<subseteq> \<Union> (B ` A)"
f_classification: fixes f :: "int \<Rightarrow> int" assumes "\<And>a b. f (2 * a) + 2 * f b = f (f (a + b))" shows "(\<forall>x. f x = 0) \<or> (\<forall>x. f x = 2 * x + f 0)"
additive_wp_repeat: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "Transformers.additive (wp a)"   and "well_def a" shows "Transformers.additive (wp (repeat n a))"
main: fixes ia :: "nat"   and ja :: "nat"   and fsa :: "'a vec list"   and la :: "int" assumes "ia < ja"   and "ja < length fsa"   and "set fsa \<subseteq> carrier_vec n" shows "lattice_of fsa \<subseteq> lattice_of (fsa[ia := fsa ! ia + of_int la \<cdot>\<^sub>v fsa ! ja])"
nat_0_le: fixes z :: "int" assumes "0 \<le> z" shows "int (nat z) = z"
prems_1: shows "\<forall>i<Suc k. \<not> (CS (f i) = cs \<and> CSS (f i) = css)"
r0_2: fixes n :: "nat" shows "LINT x|M. norm (u (r0 (Suc n)) x) < (1 / 2) ^ r0 n"
typed_model_def: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term" shows "typed_model arity public Ana \<Gamma> \<equiv> intruder_model arity Ana \<and> typed_model_axioms arity public \<Gamma>"
int_sum: fixes f :: "'a \<Rightarrow> nat"   and A :: "'a set" shows "int (sum f A) = (\<Sum>x\<in>A. int (f x))"
relcompE: fixes xz :: "'b \<times> 'c"   and r :: "('b \<times> 'd) set"   and s :: "('d \<times> 'c) set" assumes "xz \<in> r O s" obtains x :: "'b" and y :: "'d" and z :: "'c" where "xz = (x, z)" and "(x, y) \<in> r" and "(y, z) \<in> s"
bfs_spec_def: fixes src :: "nat"   and dst :: "nat" shows "bfs_spec src dst \<equiv> SPEC (\<lambda>r. case r of None \<Rightarrow> \<not> conn src dst | Some d \<Rightarrow> conn src dst \<and> min_dist src dst = d)"
NS_Var_imp_eq_least: fixes x :: "'b"   and t :: "('f, 'b) Term.term" assumes "NS (Var x) t" shows "t = Var x \<or> (\<exists>f. t = Fun f [] \<and> least f)"
epiE: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "B.epi f"   and "B.seq g f"   and "B.seq g' f"   and "g \<cdot>\<^sub>B f = g' \<cdot>\<^sub>B f" shows "g = g'"
a_d_closed: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.antidomain_pre_dioid antidomain_op plus times one zero less_eq less" shows "antidomain_pre_dioid.apd_d antidomain_op (antidomain_op x) = antidomain_op x"
kronecker_factorization_rat_1: fixes p :: "rat poly"   and q :: "rat poly" assumes "kronecker_factorization_rat p = Some q" shows "1 \<le> degree q \<and> degree q < degree p \<and> q dvd p"
bernoulli'_conv_bernoulli: fixes n :: "nat" assumes "n \<noteq> 1" shows "bernoulli' n = bernoulli n"
is_alignedE: fixes w :: "'b word"   and n :: "nat" assumes "is_aligned w n" obtains q :: "nat" where "w = 2 ^ n * word_of_nat q" and "q < 2 ^ (LENGTH('b) - n)"
gtr_empty_conf_s_2: fixes c :: "'c multiset"   and w :: "'d"   and S :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'e \<times> 'e multiset) set" shows "(c, w, {#}) \<notin> gtr S"
half_gt_zero_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide a (numeral.numeral one plus (num.Bit0 num.One))) = less zero a"
mset_merge: fixes cmp :: "'b comparator"   and xs :: "'b list"   and ys :: "'b list" shows "mset (Sorting_Algorithms.merge cmp xs ys) = mset xs + mset ys"
Obj_par_red: fixes f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and z :: "sterm" assumes "Obj f T => z" obtains lz :: "Label \<Rightarrow> sterm option" where "dom lz = dom f" and "z = Obj lz T"
ptr_kinds_M_ptr_kinds: fixes h :: "(_) heap"   and xa :: "(_) element_ptr list" shows "h \<turnstile> element_ptr_kinds_M \<rightarrow>\<^sub>r xa = (xa = sorted_list_of_set (fset (element_ptr_kinds h)))"
degree_prod_sum: shows "degree (prod m A) = (\<Sum>i\<in>A. degree (m i))"
code: fixes r :: "real"   and x :: "complex" shows "r *\<^sub>R x = Complex (r * Re x) (r * Im x)"
ChiRel_ChiStar_notmem: 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 S :: "'d"   and T :: "'d" assumes "TS \<in> HPT ST"   and "(S, T) \<in> ChiRel (HA ST)"   and "S \<in> ChiStar (HA ST) `` Source TS" shows "T \<notin> Source TS"
sum_mult_mem1: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier R"   and "B \<subseteq> carrier R"   and "x \<in> sum_mult R A B" shows "\<exists>n. \<exists>f\<in>{j. j \<le> n} \<rightarrow> set_mult R A B. \<Sigma>\<^sub>e R f n = x"
doesnt_read_annos: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var"   and annos :: "'Var ModeUpd list" assumes "doesnt_read c x" shows "doesnt_read (c \<otimes> annos) x"
unwrapB_def: shows "unwrapB \<equiv> \<Lambda> f. unbox oo f oo box"
extensional_funcset_def: fixes S :: "'b set"   and T :: "'c set" shows "S \<rightarrow>\<^sub>E T = (S \<rightarrow> T) \<inter> extensional S"
Block_def: shows "e2.Class = (\<lambda>a\<in>Sigma I \<FF>2. THE A. a \<in> A \<and> A \<in> e2.Partition)"
psubsetI: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater_eq B A"   and "A \<noteq> B" shows "order.greater B A"
single_leadsTo_I: fixes A :: "'c set"   and F :: "'c program"   and B :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> F \<in> {x} \<longmapsto> B" shows "F \<in> A \<longmapsto> B"
align_lem_or: fixes xs :: "bool list"   and n :: "nat"   and m :: "nat"   and ys :: "bool list" assumes "length xs = n + m"   and "length ys = n + m"   and "drop m xs = replicate n False"   and "take m ys = replicate m False" shows "map2 (\<or>) xs ys = take m xs @ drop m ys"
map_simp: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and f :: "'a" assumes "constant_functor A B b"   and "partial_magma.arr A f" shows "constant_functor.map A B b f = b"
third_point: fixes A :: "'p"   and B :: "'p"   and P :: "'p" assumes "Col A B P" shows "Bet P A B \<or> Bet A P B \<or> Bet A B P"
effect_ifE: fixes c :: "bool"   and t :: "'b Heap"   and e :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'b" assumes "effect (if c then t else e) h h' r" obtains "c" and "effect t h h' r"   | "\<not> c" and "effect e h h' r"
smult_pCons: fixes a :: "'a"   and b :: "'a"   and p :: "'a poly" shows "Polynomial.smult a (pCons b p) = pCons (a * b) (Polynomial.smult a p)"
order_subst1: fixes a :: "'b::order"   and f :: "'c::order \<Rightarrow> 'b::order"   and b :: "'c::order"   and c :: "'c::order" assumes "(a::'b::order) \<le> (f::'c::order \<Rightarrow> 'b::order) (b::'c::order)"   and "(b::'c::order) \<le> (c::'c::order)"   and "\<And>(x::'c::order) y::'c::order. x \<le> y \<Longrightarrow> (f::'c::order \<Rightarrow> 'b::order) x \<le> f y" shows "(a::'b::order) \<le> (f::'c::order \<Rightarrow> 'b::order) (c::'c::order)"
Joints_def: fixes XS :: "('a \<times> 'a pdevs) list" shows "Joints XS = valuate (\<lambda>e. map (aform_val e) XS)"
distinct_13: fixes x2 :: "nat"   and x3 :: "nat" shows "Number x2 \<noteq> Nonce x3"
Inter_insert: fixes a :: "'b set"   and B :: "'b set set" shows "\<Inter> (insert a B) = a \<inter> \<Inter> B"
invariant3p_correct_jvm_state_mexecdT: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" shows "invariant3p (mexecdT P) (correct_jvm_state \<Phi>)"
blinding_of_on_axioms: shows "blinding_of_on B rhb bob"
init_fin_simulation1: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and t :: "'b"   and s1 :: "(status \<times> 'a) \<times> 'c"   and s2 :: "(status \<times> 'g) \<times> 'h"   and tl1 :: "('d, 'b, status \<times> 'a, 'c, 'e, 'f action) thread_action"   and s1' :: "(status \<times> 'a) \<times> 'c" assumes "FWbisimulation final1 r1 final2 r2 bisim"   and "FWbisimulation_base.init_fin_bisim final1 final2 bisim t s1 s2"   and "multithreaded_base.init_fin final1 r1 t s1 tl1 s1'" shows "\<exists>s2' tl2. multithreaded_base.init_fin final2 r2 t s2 tl2 s2' \<and> FWbisimulation_base.init_fin_bisim final1 final2 bisim t s1' s2' \<and> ta_bisim (FWbisimulation_base.init_fin_bisim final1 final2 bisim) tl1 tl2"
add_ac_2: fixes a :: "'m"   and b :: "'m" shows "a + b = b + a"
completely_multiplicative_function_axioms: shows "completely_multiplicative_function (fds_nth f)"
wf_atom_constT_imp_objT: fixes P :: "ast_problem"   and Q :: "variable \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and a :: "PDDL_STRIPS_Semantics.term atom" assumes "ast_domain.wf_atom (domain P) (ty_term Q (ast_domain.constT (domain P))) a" shows "ast_domain.wf_atom (domain P) (ty_term Q (ast_problem.objT P)) a"
ldropn_eq_LNil: fixes n :: "nat"   and xs :: "'a llist" shows "(ldropn n xs = []\<^sub>l) = (llength xs \<le> enat n)"
dens': fixes M :: "(int \<times> int) measure"   and \<delta> :: "int \<times> int \<Rightarrow> ennreal" assumes "has_density M (count_space UNIV) \<delta>" shows "has_density (distr M (count_space UNIV) (\<lambda>(x, y). f x y)) (count_space UNIV) (g \<delta>)"
exhaust_disc: fixes step :: "('e, 'f) step" obtains "step = Done"   | "is_Skip step"   | "is_Yield step"
lift2_def: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c err"   and e\<^sub>1 :: "'d err"   and e\<^sub>2 :: "'e err" shows "lift2 f e\<^sub>1 e\<^sub>2 = (case e\<^sub>1 of Err \<Rightarrow> Err | OK x \<Rightarrow> case e\<^sub>2 of Err \<Rightarrow> Err | OK y \<Rightarrow> f x y)"
CompDefComp_1: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "f \<approx>> g"   and "g \<approx>> h" shows "f \<approx>> g ;; h"
fst_row_add_iterate_PA_eq_fst_Gauss_Jordan_in_ij_PA: fixes P :: "(('a, 'b) vec, 'b) vec"   and i :: "'b"   and A :: "(('a, 'c) vec, 'b) vec"   and j :: "'c" shows "fst (row_add_iterate_PA (mult_row (interchange_rows P i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n)) i ((1::'a) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ i $ j), mult_row (interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n)) i ((1::'a) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ i $ j)) (nrows A - 1) i j) = fst (Gauss_Jordan_in_ij_PA (P, A) i j)"
cbox_Pair_eq: fixes a :: "'a"   and c :: "'b"   and b :: "'a"   and d :: "'b" shows "cbox (a, c) (b, d) = cbox a b \<times> cbox c d"
that: fixes ys' :: "'a list" assumes "ys = x # ys'"   and "prefix xs ys'" shows "thesis"
less_divide_eq_numeral1_1: fixes a :: "'a"   and b :: "'a"   and w :: "num" shows "(a < b / numeral w) = (a * numeral w < b)"
lless_cong_l: fixes x :: "'a"   and x' :: "'a"   and y :: "'a" assumes "x .=\<^bsub>division_rel G\<^esub> x'"   and "x' \<sqsubset>\<^bsub>division_rel G\<^esub> y"   and "x \<in> carrier (division_rel G)"   and "x' \<in> carrier (division_rel G)"   and "y \<in> carrier (division_rel G)" shows "x \<sqsubset>\<^bsub>division_rel G\<^esub> y"
union_diff2: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a measure" assumes "finite A"   and "finite B" shows "sup_measure.F g (A \<union> B) = sup (sup (sup_measure.F g (A - B)) (sup_measure.F g (B - A))) (sup_measure.F g (A \<inter> B))"
game\<^sub>1_def: fixes x :: "nat"   and y :: "nat" shows "game\<^sub>1 x y = coin_spmf \<bind> (\<lambda>b. exec_gpv oracle' (\<A>\<^sub>1 (\<^bold>g [^] x)) ({}, empty) \<bind> (\<lambda>(((m\<^sub>1, m\<^sub>2), \<sigma>), s, s_h). TRY assert_spmf (valid_plains m\<^sub>1 m\<^sub>2) \<bind> (\<lambda>_. oracle s_h (\<^bold>g [^] (x * y)) \<bind> (\<lambda>(h, s_h'). let c = (\<^bold>g [^] y, h [\<oplus>] (if b then m\<^sub>1 else m\<^sub>2)) in exec_gpv oracle' (\<A>\<^sub>2 c \<sigma>) (s, s_h') \<bind> (\<lambda>(b', s', s_h''). return_spmf (b' = b, \<^bold>g [^] (x * y) \<in> s')))) ELSE (coin_spmf \<bind> (\<lambda>b. return_spmf (b, \<^bold>g [^] (x * y) \<in> s)))))"
bar1: fixes s :: "'a \<Rightarrow> 'b" shows "s\<langle>B := True\<rangle>\<cdot>C = s\<cdot>C"
lunit_char_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and b :: "'b"   and f :: "'b" assumes "left_hom_with_unit V H \<a> \<iota> b"   and "partial_magma.ide (subcategory.comp V (weak_composition.left V H b)) f" shows "partial_magma.in_hom (subcategory.comp V (weak_composition.left V H b)) (left_hom_with_unit.lunit V H \<a> \<iota> b f) (weak_composition.H\<^sub>L H b f) f"
simps: fixes G :: "('c, 'd) ParityGame_scheme"   and a1 :: "Player"   and a2 :: "'c llist"   and a3 :: "'c \<Rightarrow> 'c" assumes "ParityGame G" shows "ParityGame.path_conforms_with_strategy G a1 a2 a3 = ((\<exists>p \<sigma>. a1 = p \<and> a2 = LNil \<and> a3 = \<sigma>) \<or> (\<exists>p v \<sigma>. a1 = p \<and> a2 = LCons v LNil \<and> a3 = \<sigma>) \<or> (\<exists>v p w \<sigma> Ps. a1 = p \<and> a2 = LCons v (LCons w Ps) \<and> a3 = \<sigma> \<and> v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<and> w = \<sigma> v \<and> ParityGame.path_conforms_with_strategy G p (LCons w Ps) \<sigma>) \<or> (\<exists>v p Ps \<sigma>. a1 = p \<and> a2 = LCons v Ps \<and> a3 = \<sigma> \<and> v \<notin> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<and> ParityGame.path_conforms_with_strategy G p Ps \<sigma>))"
monotone_extreme_imp_extreme_bound: shows "extreme_bound A (\<sqsubseteq>) (f ` I) (f e)"
map_of_zip_nth: fixes xs :: "'d list"   and ys :: "'e list"   and i :: "nat" assumes "length xs = length ys"   and "distinct xs"   and "i < length ys" shows "map_of (zip xs ys) (xs ! i) = Some (ys ! i)"
lam_simps_2: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>f : local.prod a b \<rightarrow> c\<guillemotright>" shows "local.dom (\<Lambda> a b c f) = a"
\<omega>consistent_impliesStd2: assumes "\<omega>consistent" shows "\<omega>consistentStd2"
NatTransComp_defs_5: fixes \<eta> :: "('j, 'g, 'k, 'h, 'l, 'i, 'm) NatTrans_scheme" shows "NTCatCod \<eta> \<equiv> CatCod (NTCod \<eta>)"
nonneg: fixes x :: "'a" assumes "x \<in> \<Omega>" shows "0 \<le> f x"
intro: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" assumes "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs"   and "l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed parent_child_rel get_disconnected_nodes"   and "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"   and "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs" shows "l_remove_child_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel"
decs_unclocked_2: fixes mn :: "char list list"   and env :: "v sem_env"   and count1 :: "nat"   and s :: "'b state"   and ds :: "dec list"   and s' :: "'b state"   and r :: "(v sem_env, v) result"   and count2 :: "nat" assumes "evaluate_decs False mn env (update_clock (\<lambda>_. count1) s) ds (update_clock (\<lambda>_. count1) s', r)" shows "evaluate_decs False mn env (update_clock (\<lambda>_. count2) s) ds (update_clock (\<lambda>_. count2) s', r)"
enabled_iff: fixes s :: "'a cfg"   and \<omega> :: "'a cfg stream" shows "MC.enabled s \<omega> = (MC.enabled (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K_cfg s))"
list_spec_def: fixes X :: "'a set" shows "list_spec X = SPEC (\<lambda>xs. set xs = X)"
all: fixes qs :: "'a list" assumes "set qs \<subseteq> set init" shows "T\<^sub>p_on_rand' A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs \<le> c * real (T\<^sub>p_opt init qs) + b * real (length init * (length init - 1)) / 2"
trunc_closed: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "f \<in> carrier (UP R)" shows "Cring_Poly.truncate R f \<in> carrier (UP R)"
minus_mod_self2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus a b) b = modulo a b"
induct: fixes P :: "('c, 'd) bintree option \<Rightarrow> ('c, 'd) bintree option list \<Rightarrow> bool"   and a0 :: "('c, 'd) bintree option"   and a1 :: "('c, 'd) bintree option list" assumes "\<And>xs. P None xs"   and "\<And>t. P (Some t) []"   and "\<And>t xs. P (Some t) (None # xs)"   and "\<And>t r xs. P (Some (merge t r)) xs \<Longrightarrow> P (Some t) (Some r # xs)" shows "P a0 a1"
finsert_fminus: fixes a :: "'a"   and A :: "'a fset" assumes "a |\<in>| A" shows "finsert a (A |-| {|a|}) = A"
pathstart_reversepath: fixes g :: "real \<Rightarrow> 'c" shows "pathstart (reversepath g) = pathfinish g"