hcju/isabelleps huggingface dataset:
axioms_5: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_transformation A B F G \<tau>" shows "natural_transformation_axioms A B F G \<tau>"
scale_right_imp_eq: fixes x :: "'b"   and a :: "real"   and b :: "real" assumes "x \<noteq> (0::'b)"   and "a *\<^sub>R x = b *\<^sub>R x" shows "a = b"
P_lnull_ltl_LCons: assumes "lnull (ltl P)" shows "P = LCons v LNil"
transaction_strand_proj: fixes n :: "'e"   and T :: "('f, 'g, 'h, 'e) prot_transaction" shows "transaction_strand (transaction_proj n T) = proj n (transaction_strand T)"
assms_2: shows "(f ^ n) \<langle>$\<rangle> a = a"
subset2: shows "(\<lambda>(y, f). f(x := y, inv f y := x)) ` ((S - {x}) \<times> derangements (S - {x})) \<subseteq> derangements S"
continuous_on_eq_continuous_within: fixes s :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "continuous_on s f = (\<forall>x\<in>s. continuous (at x within s) f)"
assms_1: shows "t_packing_design V B k t \<Lambda>\<^sub>t"
fact_0: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero 0 = one"
IH: fixes y :: "nat" assumes "y < i'"   and "y cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<And>k' i'. \<lbrakk>k' cd\<^bsup>\<pi>\<^esup>\<rightarrow> k; P k'; i' icd\<^bsup>\<pi>\<^esup>\<rightarrow> k'\<rbrakk> \<Longrightarrow> P i'" shows "P y"
measure_measure_pmf_not_zero: fixes p :: "'c pmf"   and s :: "'c set" assumes "set_pmf p \<inter> s \<noteq> {}" shows "measure_pmf.prob p s \<noteq> 0"
wpo_compat: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term"   and u :: "('f, 'v) Term.term" shows "(s \<succeq> t \<and> t \<succ> u \<longrightarrow> s \<succ> u) \<and> (s \<succ> t \<and> t \<succeq> u \<longrightarrow> s \<succ> u) \<and> (s \<succeq> t \<and> t \<succeq> u \<longrightarrow> s \<succeq> u)"
delete_twist: fixes x :: "'d"   and y :: "'d"   and al :: "('d \<times> 'e) list" shows "delete x (delete y al) = delete y (delete x al)"
m_closed: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier M1"   and "y \<in> carrier M1" shows "x \<oplus>\<^bsub>M1\<^esub> y \<in> carrier M1"
irreflD: fixes x :: "'a" assumes "x \<sqsubset> x" shows "x \<notin> A"
decidable_def: fixes X :: "nat set" shows "decidable X \<equiv> \<exists>f. recfn 1 f \<and> (\<forall>x. eval f [x] \<down>= (if x \<in> X then 1 else 0))"
ck_trms_sound: fixes T :: "'a trm set"   and D :: "'a literal set"   and C :: "'a literal set"   and E :: "'a trm set"   and k :: "inferences" assumes "T = get_trms D (dom_trms C E) k" shows "T \<subseteq> dom_trms C E"
j: shows "k < j \<and> j \<le> n + r \<and> (\<pi> @\<^bsup>n\<^esup> \<pi>') j = ipd (\<pi> k)"
adds_except: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b"   and V :: "'a set" shows "(s adds t) = (except s V adds except t V \<and> except s (- V) adds except t (- V))"
d_IN_def: fixes g :: "'b \<times> 'b \<Rightarrow> ennreal"   and y :: "'b" shows "d_IN g y = (\<Sum>\<^sup>+ x. g (x, y))"
simps_5: fixes \<sigma> :: "(char list \<times> 'b list) Trace.trace"   and v :: "'b list"   and i :: "nat"   and \<phi> :: "'b formula" shows "sat \<sigma> v i (Exists \<phi>) = (\<exists>z. sat \<sigma> (z # v) i \<phi>)"
hom_zero: fixes c :: "'c" shows "c * (0::'c) = (0::'c)"
sum_list_strict_mono: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x < g x" shows "sum_list (map f xs) < sum_list (map g xs)"
bounded: shows "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
b_subset_d: shows "ideal_generated {b} \<subseteq> ideal_generated {d}"
simps_2: fixes f :: "'d \<Rightarrow> 'c"   and j :: "nat" shows "app_pair f (Inr j) = Inr j"
indexed_eval_in_carrier: fixes Ps :: "('c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a) list"   and i :: "'c" assumes "list_all M.carrier_coeff Ps" shows "M.carrier_coeff (M.indexed_eval Ps i)"
leg0_is_map: 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"   and leg\<^sub>0 :: "'b"   and leg\<^sub>1 :: "'b" assumes "span_of_maps V H \<a> \<i> src trg leg\<^sub>0 leg\<^sub>1" shows "\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg leg\<^sub>0 g"
eq: fixes x :: "'a" assumes "x \<in> space M"   and "x \<notin> X" shows "f x = g x"
not_outside_1: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "Order_Predicates.preorder_on carrier le"   and "le x y" shows "x \<in> carrier"
prj_simps_14: shows "C.dom \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>1\<^sub>0 = HH\<mu>\<nu>H\<pi>\<rho>.chine"
nonzero_alphaprod: fixes i :: "nat"   and p :: "nat \<Rightarrow> nat" assumes "i \<le> k + 1" shows "(\<Prod>l\<leftarrow>[3..<i]. \<alpha> l ^ p l) \<noteq> 0"
init: fixes M :: "'b dfa" assumes "dfa M" shows "init M \<in> states M"
nodes_to_list_def: fixes g :: "('d, 'e, 'f) graph_scheme" shows "nodes_to_list g \<equiv> it_to_sorted_list (\<lambda>_ _. True) (nodes g)"
x: fixes x :: "'a match_expr"   and as :: "'b negation_type list" assumes "x \<in> set (normalize_match (andfold_MatchExp (map replace_f as)))" shows "normalized_n_primitive (disc2, sel2) f x"
subset_subterms_Union: fixes M :: "('a, 'b) Term.term set" shows "M \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t M"
integral_has_vector_derivative_continuous_at: fixes f :: "real \<Rightarrow> 'b"   and a :: "real"   and b :: "real"   and x :: "real"   and S :: "real set" assumes "f integrable_on {a..b}"   and "x \<in> {a..b} - S"   and "finite S"   and "continuous (at x within {a..b} - S) f" shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b} - S)"
orderI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "Semilat A r f" shows "order r"
notfinal_Suc: fixes lm :: "nat list"   and A :: "abc_inst list"   and n :: "nat" assumes "abc_notfinal (abc_steps_l (0, lm) A (Suc n)) A" shows "abc_notfinal (abc_steps_l (0, lm) A n) A"
equality_def': fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "equality f = (f = (=))"
cmp_simps_4: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.dom (\<Phi>\<^sub>G (f, g)) = G f \<star>\<^sub>D G g"
tab_in_vhom': shows "\<guillemotleft>\<sigma> : s\<^sub>1 \<Rightarrow> s \<star> s\<^sub>0\<guillemotright>"
d_IN_lift_source: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow_attainability \<Delta>" shows "d_IN (flow_attainability.lift \<Delta> f) (Inner (source \<Delta>)) = value_flow \<Delta> f + d_IN f (source \<Delta>)"
assms_10: shows "NQ_neg = newQuery J Q_J_neg (I, t)"
linear_iff: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" shows "linear s1 s2 f = (vector_space s1 \<and> vector_space s2 \<and> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (s1 c x) = s2 c (f x)))"
product_character: fixes x :: "i"   and y :: "i" assumes "O x y" shows "\<forall>w. P w (x \<otimes> y) = (P w x \<and> P w y)"
finite_UnionD: fixes A :: "'d set set" assumes "finite (\<Union> A)" shows "finite A"
VE_prop_asset: shows "\<lparr>nodes = V, edges = E\<rparr> = \<lparr>nodes = V, edges = V \<times> V - (\<Union>m\<in>set M. \<Union> (c_offending_flows m (fully_connected G)))\<rparr>"
iteratei_def: fixes S :: "'m2" shows "m2.iteratei S \<equiv> it_to_it (m2.list_it S)"
Gromov_product_at_def: fixes e :: "'b"   and x :: "'b"   and y :: "'b" shows "Gromov_product_at e x y = (dist e x + dist e y - dist x y) / 2"
remove: fixes A :: "'d set"   and x :: "'d"   and g :: "'d \<Rightarrow> 'a" assumes "finite A"   and "x \<in> A" shows "F g A = g x \<^bold>* F g (A - {x})"
eq_iff: fixes a :: "'b::order"   and b :: "'b::order" shows "((a::'b::order) = (b::'b::order)) = (a \<le> b \<and> b \<le> a)"
power2_norm_eq_inner: fixes x :: "'b" shows "(norm x)\<^sup>2 = real_of (inner x x)"
max_torder_ge_conv_disj: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "torder_on A r"   and "x \<in> A"   and "y \<in> A" shows "r z (max_torder r x y) = (r z x \<or> r z y)"
CLCodomain: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF" assumes "ZFAxioms T" shows "CLCodomain T [A,e,B]\<^bsub>T\<^esub> = B"
axioms: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" assumes "monoid_list f z" shows "monoid f z"
mkPoint_img_2: fixes x :: "'s"   and a :: "'s" assumes "\<guillemotleft>x : unity \<rightarrow> a\<guillemotright>" shows "mkPoint a (img x) = x"
closed_map_imp_quotient_map: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d set" assumes "continuous_on S f"   and "T \<subseteq> f ` S"   and "\<And>T. closedin (top_of_set S) T \<Longrightarrow> closedin (top_of_set (f ` S)) (f ` T)" shows "openin (top_of_set S) (S \<inter> f -` T) = openin (top_of_set (f ` S)) T"
obj_at_eq_imp: fixes P :: "('b, 'c, 'd) object \<Rightarrow> bool"   and r :: "'d" shows "eq_imp (\<lambda>_ s. map_option P (sys_heap s r)) (obj_at P r)"
gcd_mod_right: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm"   and "a \<noteq> zero" shows "gcd a (modulo b a) = gcd a b"
runit_char_2: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>\<r>[f] : f \<star> src f \<Rightarrow> f\<guillemotright>"
bounded: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "\<bar>det (matrix (f' x))\<bar> \<le> B"
map_eqI: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "dom f = dom g"   and "\<forall>x\<in>dom f. the (f x) = the (g x)" shows "f = g"
eventually_INF: fixes P :: "'a \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'a filter"   and B :: "'b set" shows "eventually P (Inf (F ` B)) = (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf (F ` X)))"
run_ended_trans: fixes s :: "skl1_state"   and R :: "fid_t"   and s' :: "skl1_state" assumes "run_ended (progress s R)"   and "(s, s') \<in> TS.trans skl1" shows "run_ended (progress s' R)"
a_carrier_vec: fixes l :: "nat"   and i :: "nat" assumes "l \<le> i"   and "i < length (map of_int_hom.vec_hom fs)" shows "gs.a i l \<in> carrier_vec n"
FW_detects_empty_zone: fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c" shows "([FW M n]\<^bsub>v,n\<^esub> = {}) = (\<exists>i\<le>n. FW M n i i < Le (0::'d))"
seq_compact_eq_countably_compact: fixes U :: "'b set" shows "seq_compact U = countably_compact U"
diagonal: fixes A :: "((real, 'n) vec, 'n) vec" assumes "\<And>i j. i \<noteq> j \<Longrightarrow> A $ i $ j = 0" shows "P A"
IH: fixes xs :: "'b list" assumes "distinct ys"   and "(a, b) \<in> set (zip xs ys)"   and "(c, b) \<in> set (zip xs ys)" shows "a = c"
it_to_it_fold': fixes it :: "'a \<Rightarrow> ('b list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> 'b list \<Rightarrow> 'b list"   and it' :: "'a \<Rightarrow> ('c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'a" assumes "proper_it' it it'" shows "it_to_it (it s) = it' s"
1: shows "cone (monomial (1::'a) t', V) \<subseteq> cone (monomial (1::'a) t, U)"
reachable_path: fixes Q :: "'a set"   and b :: "'a"   and a :: "'a" assumes "Q \<in> \<P>"   and "b \<in> \<E>"   and "a \<in> Q - \<emptyset> Q b" shows "\<exists>R\<in>\<P>. a \<in> R \<and> b \<in> R"
of_nat_lt_\<omega>\<^sub>z: fixes n :: "nat" shows "of_nat n < \<omega>\<^sub>z"
agop_gop: fixes A :: "('e, 'f) aGroup_scheme" assumes "aGroup A" shows "(\<cdot>\<^bsub>b_ag A\<^esub>) = (\<plusminus>\<^bsub>A\<^esub>)"
simps_18: fixes f1 :: "'a"   and f2 :: "qtname \<Rightarrow> 'a"   and f3 :: "qtname \<Rightarrow> 'a"   and f4 :: "ty \<Rightarrow> 'a"   and x3 :: "qtname" shows "(case ClassT x3 of NullT \<Rightarrow> f1 | IfaceT x \<Rightarrow> f2 x | ClassT x \<Rightarrow> f3 x | ArrayT x \<Rightarrow> f4 x) = f3 x3"
Pi_iff: fixes f :: "'b \<Rightarrow> 'c"   and I :: "'b set"   and X :: "'b \<Rightarrow> 'c set" shows "(f \<in> Pi I X) = (\<forall>i\<in>I. f i \<in> X i)"
cycle_free_def: fixes g :: "'b ugraph" shows "cycle_free g \<equiv> \<nexists>p u. p \<noteq> [] \<and> simple p \<and> path g u p u"
GMVT': fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real" assumes "a < b"   and "\<And>z. \<lbrakk>a \<le> z; z \<le> b\<rbrakk> \<Longrightarrow> isCont f z"   and "\<And>z. \<lbrakk>a \<le> z; z \<le> b\<rbrakk> \<Longrightarrow> isCont g z"   and "\<And>z. \<lbrakk>a < z; z < b\<rbrakk> \<Longrightarrow> (g has_real_derivative g' z) (at z)"   and "\<And>z. \<lbrakk>a < z; z < b\<rbrakk> \<Longrightarrow> (f has_real_derivative f' z) (at z)" shows "\<exists>c>a. c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
gB: shows "\<forall>\<^sub>F a in net. f a = 0 \<or> norm (g a) \<le> B"
linear_fun_on_zero: fixes S :: "'c set" assumes "subspace S" shows "linear_fun_on S 0"
assms_6: fixes z :: "complex" assumes "z \<in> S" shows "g (f z) = z"
diff_gt_0_iff_gt: fixes a :: "'b"   and b :: "'b" shows "((0::'b) < a - b) = (b < a)"
match_def: fixes \<sigma> :: "char list \<Rightarrow> nat option"   and \<Gamma> :: "char list set" shows "match \<sigma> \<Gamma> \<equiv> dom \<sigma> = \<Gamma>"
Max_Un_absorb: fixes X :: "'a set"   and Y :: "'a set" assumes "finite X"   and "X \<noteq> {}"   and "finite Y"   and "\<And>x y. \<lbrakk>y \<in> Y; x \<in> X\<rbrakk> \<Longrightarrow> y \<le> x" shows "Max (X \<union> Y) = Max X"
sortTermCase_def: fixes fVar :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and fOp :: "'opSym \<Rightarrow> ('index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option) \<Rightarrow> ('bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option) \<Rightarrow> 'b"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" shows "sortTermCase fVar fOp s X \<equiv> if isVar s X then fVar (fst (castVar s X)) (snd (castVar s X)) else if isOp s X then fOp (fst3 (castOp s X)) (snd3 (castOp s X)) (trd3 (castOp s X)) else undefined"
rev_predicate2D: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "P x y"   and "P \<le> Q" shows "Q x y"
good_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" shows "good P f = (\<exists>i j. i < j \<and> P (f i) (f j))"
Rep_formula_inject: fixes x :: "('a, 'b, 'c) Formula.formula"   and y :: "('a, 'b, 'c) Formula.formula" shows "(Formula.formula.Rep_formula x = Formula.formula.Rep_formula y) = (x = y)"
ent_trans: fixes P :: "assn"   and Q :: "assn"   and R :: "assn" assumes "P \<Longrightarrow>\<^sub>A Q"   and "Q \<Longrightarrow>\<^sub>A R" shows "P \<Longrightarrow>\<^sub>A R"
sum_Stirling_binomial: fixes n :: "nat"   and m :: "nat" shows "Stirling (Suc n) (Suc m) = (\<Sum>i = 0..n. Stirling i m * (n choose i))"
Derivation_Derives1: fixes a :: "symbol list"   and S :: "(nat \<times> symbol \<times> symbol list) list"   and y :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and z :: "symbol list" assumes "Derivation a S y"   and "Derives1 y i r z" shows "Derivation a (S @ [(i, r)]) z"
set_iterator_linord_def: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "'a set" shows "set_iterator_linord it S0 \<equiv> set_iterator_genord it S0 (\<le>)"
red_induct: fixes x1a :: "cmd"   and x1b :: "char list \<Rightarrow> val option"   and x2a :: "cmd"   and x2b :: "char list \<Rightarrow> val option"   and P :: "cmd \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> cmd \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> bool" assumes "\<langle>x1a,x1b\<rangle> \<rightarrow> \<langle>x2a,x2b\<rangle>"   and "\<And>V e s. P (V:=e) s Skip (s(V := interpret e s))"   and "\<And>c\<^sub>1 s c\<^sub>1' s' c\<^sub>2. \<lbrakk>\<langle>c\<^sub>1,s\<rangle> \<rightarrow> \<langle>c\<^sub>1',s'\<rangle>; P c\<^sub>1 s c\<^sub>1' s'\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) s (c\<^sub>1';; c\<^sub>2) s'"   and "\<And>c\<^sub>2 s. P (Skip;; c\<^sub>2) s c\<^sub>2 s"   and "\<And>b s c\<^sub>1 c\<^sub>2. interpret b s = Some true \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) s c\<^sub>1 s"   and "\<And>b s c\<^sub>1 c\<^sub>2. interpret b s = Some false \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) s c\<^sub>2 s"   and "\<And>b s c. interpret b s = Some true \<Longrightarrow> P (while (b) c) s (c;; while (b) c) s"   and "\<And>b s c. interpret b s = Some false \<Longrightarrow> P (while (b) c) s Skip s" shows "P x1a x1b x2a x2b"
existence_ivl_subset: fixes x0 :: "'a" shows "existence_ivl0 x0 \<subseteq> UNIV"
assms_1: shows "A' \<in> set (tr\<^sub>p\<^sub>c A [])"
star_slide_var: fixes x :: "'a list set" shows "p_prod (p_star x) x = p_prod x (p_star x)"
op_eq: shows "map_spmf (\<lambda>S0. S0 \<oplus> S3 \<oplus> m1) coin_spmf = map_spmf ((\<oplus>) (S3 \<oplus> m1)) coin_spmf"
comp_Ex2: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "f \<in> Ex2"   and "g \<in> Ex2" shows "f \<circ> g \<in> Ex2"
semi_mojmir_def: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b" shows "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w \<equiv> finite \<Sigma> \<and> finite (reach \<Sigma> \<delta> q\<^sub>0) \<and> range w \<subseteq> \<Sigma>"
Init_eq: fixes init :: "'b set"   and acts :: "('b \<times> 'b) set set"   and allowed :: "('b \<times> 'b) set set" shows "Init (mk_program (init, acts, allowed)) = init"
commute: fixes a :: "'b word"   and b :: "'b word" shows "signed.max a b = signed.max b a"
remdups_fwd_acc_empty: fixes xs :: "'b list"   and Acc :: "'b set" shows "(set xs \<subseteq> Acc) = (remdups_fwd_acc Acc xs = [])"
phl_export1: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "x \<cdot> y \<le> y \<cdot> x \<cdot> y"   and "x \<cdot> y \<cdot> z \<le> z \<cdot> w" shows "x \<cdot> (y \<cdot> z) \<le> y \<cdot> z \<cdot> w"
HComplex_mult: fixes x1 :: "real star"   and y1 :: "real star"   and x2 :: "real star"   and y2 :: "real star" shows "HComplex x1 y1 * HComplex x2 y2 = HComplex (x1 * x2 - y1 * y2) (x1 * y2 + y1 * x2)"
weak_partial_order_def: fixes L :: "('d, 'e) gorder_scheme" shows "weak_partial_order L \<equiv> equivalence L \<and> weak_partial_order_axioms L"
not_less_zero: fixes n :: "'k multiset" shows "\<not> n \<subset># {#}"
assms: fixes x :: "Preliminaries.var" assumes "x \<in> varsL l" shows "\<xi>1 x = \<xi>2 x"
discounted_value_real_cond_exp: fixes M :: "'b measure"   and X :: "'b \<Rightarrow> real"   and r :: "real"   and G :: "'b measure"   and n :: "nat" assumes "prob_space M"   and "integrable M X"   and "- 1 < r"   and "subalgebra M G" shows "AEeq M (real_cond_exp M G (discounted_value r (\<lambda>m. X) n)) (discounted_value r (\<lambda>m. real_cond_exp M G X) n)"
R3_equiv_entailment: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and \<Gamma>'' :: "'Var \<Rightarrow> 'BExp set option"   and P'' :: "'BExp set" assumes "\<langle>c, mds, mem\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c', mds', mem'\<rangle>"   and "\<Gamma>' =:\<^sub>P' \<Gamma>''"   and "P' \<turnstile> P''"   and "\<forall>mds. tyenv_wellformed mds \<Gamma>' \<S>' P' \<longrightarrow> tyenv_wellformed mds \<Gamma>'' \<S>' P''" shows "\<langle>c, mds, mem\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>'',\<S>',P''\<^esub> \<langle>c', mds', mem'\<rangle>"
preserves_cod: fixes f :: "'a" assumes "A.arr f" shows "B.cod (A_B.Dom (fst Fg) f) = A_B.Dom (fst Fg) (A.cod f)"
keys_of_def: fixes M :: "('b, 'c) Mapping_Str.map" shows "keys_of M = {x. M\<langle>x\<rangle> \<noteq> None}"
red_rtrancl_subset: fixes G :: "('a \<Rightarrow>\<^sub>0 'd) set"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and F :: "('a \<Rightarrow>\<^sub>0 'd) set" assumes "(punit.red G)\<^sup>*\<^sup>* p q"   and "G \<subseteq> F" shows "(punit.red F)\<^sup>*\<^sup>* p q"
butlast_conv_take: fixes xs :: "'e list" shows "butlast xs = take (length xs - 1) xs"
compare_strict_1: fixes compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering"   and y :: "'b" assumes "class.Ord_strict compare" shows "compare\<cdot>\<bottom>\<cdot>y = \<bottom>"
singleton_conv: fixes a :: "'c" shows "{x. x = a} = \<eta> a"
simps_15: fixes x1 :: "'a" shows "set_literal (Pos x1) = {x1}"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "comm_semiring_hom hom"   and "inj_semiring_hom hom" shows "inj_comm_semiring_hom hom"
lNAct_ex: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "dynamic_component.lNAct active c t n = n \<or> dynamic_component.lNAct active c t n < n \<and> \<not> (\<exists>k\<ge>dynamic_component.lNAct active c t n. k < n \<and> active c (t k))"
index_of_def: fixes xs :: "'a list"   and y :: "'a" shows "index_of xs y = index_of_aux xs y 0"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) (\<mu> \<star> \<nu>)"
is_GB_cofactor_boundE: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) set"   and b :: "nat" assumes "is_GB_cofactor_bound F b" obtains G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) set" where "punit.is_Groebner_basis G" and "ideal G = ideal F" and "\<Union> (indets ` G) \<subseteq> \<Union> (indets ` F)" and "\<And>g. g \<in> G \<Longrightarrow> \<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f. indets (q f) \<subseteq> \<Union> (indets ` F) \<and> poly_deg (q f * f) \<le> b \<and> (f \<notin> F' \<longrightarrow> q f = 0))"
arr_mkArr: fixes A :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set"   and B :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set"   and F :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr \<Rightarrow> (('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" shows "S.arr (S.mkArr A B F) = (A \<subseteq> S.Univ \<and> B \<subseteq> S.Univ \<and> F \<in> A \<rightarrow> B)"
distinct_3: fixes x31 :: "tm"   and x32 :: "tm" shows "Zero \<noteq> Eats x31 x32"