hcju/isabelleps huggingface dataset:
jvm_trans: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and \<sigma>' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and \<sigma>'' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>'"   and "P \<turnstile> \<sigma>' -jvm\<rightarrow> \<sigma>''" shows "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>''"
split_pol_coeff: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "pol_coeff S c"   and "k \<le> fst c" shows "pol_coeff S (k, snd c)"
double_complement: fixes A :: "'f set" shows "- (- A) = A"
is_arg_min_linorder: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "is_arg_min f P x = (P x \<and> (\<forall>y. P y \<longrightarrow> f x \<le> f y))"
snd_map_help_1: fixes x :: "nat"   and tsi :: "('a \<times> 'b) list" assumes "x \<le> length tsi" shows "\<forall>j<x. snd (tsi ! j) = separators tsi ! j"
empty: assumes "n \<in> set (\<alpha>n g)" shows "P n [n] n"
until_ev_suntil: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and \<psi> :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" assumes "(\<phi> until \<psi>) \<omega>"   and "ev \<psi> \<omega>" shows "(\<phi> suntil \<psi>) \<omega>"
T2_def: shows "T2 \<equiv> \<lparr>verts = {u0, v0}, arcs = {(u0, v0), (v0, u0)}\<rparr>"
ltl_semi_mojmir: fixes \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<psi> :: "'b ltl" assumes "finite \<Sigma>"   and "range w \<subseteq> \<Sigma>" shows "semi_mojmir \<Sigma> \<up>af\<^sub>G (Abs \<psi>) w"
assms_4: shows "I c = Regions.intv.Greater (k c)"
Tn_integral_preserving_1: fixes f :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "integrable M f" shows "integrable M (\<lambda>x. f ((T ^^ n) x))"
invariant_1_root_cond: fixes plr :: "int poly \<times> rat \<times> rat"   and y :: "real" assumes "invariant_1 plr" shows "root_cond plr y = (y = real_of_1 plr)"
IH: fixes n :: "nat" shows "(fps_exp 1 - 1) ^ m $ n * fact n = fact m * real (Stirling n m)"
assms_1: shows "LT1 ars r (s, [\<beta>_step]) (s, [\<alpha>_step]) \<gamma>1 \<gamma>2 \<gamma>3"
isMinimalBackjumpLevelCharacterization: fixes l :: "Literal"   and c :: "Literal list"   and M :: "(Literal \<times> bool) list"   and level :: "nat" assumes "isUIP l c M"   and "clauseFalse c (elements M)"   and "uniq (elements M)" shows "isMinimalBackjumpLevel level l c M = (isBackjumpLevel level l c M \<and> (\<forall>level'<level. \<not> isBackjumpLevel level' l c M))"
share_commute: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and L :: "nat set"   and A :: "nat set"   and R :: "nat set"   and W :: "nat set" assumes "sharing_consistent \<S> \<O> sb"   and "all_shared sb \<inter> L = {}"   and "all_shared sb \<inter> A = {}"   and "all_acquired sb \<inter> R = {}"   and "all_unshared sb \<inter> R = {}"   and "all_shared sb \<inter> R = {}" shows "share sb (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) = share sb \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L"
NP: shows "\<forall>i. u \<le> i \<and> i < u' \<longrightarrow> \<not> P i"
Int_Un_eq_1: fixes S :: "'g set"   and T :: "'g set" shows "S \<inter> T \<union> S = S"
ring_simprules_1: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
simps: fixes A :: "'d atMost \<Rightarrow> 'd atMost \<Rightarrow> 'c" shows "matrix_of_sqmatrix (SqMatrix A) = Matrix A"
r_badblock_prim: fixes s :: "nat \<Rightarrow> nat option" assumes "total_cons s" shows "recfn 2 total_cons.r_badblock"
prj_simps_18: shows "C.cod f_src_src.Prj\<^sub>0\<^sub>0 = src.apex"
eventually_at_left: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and y :: "'c"   and x :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "class.linorder_topology less_eq less open"   and "less y x" shows "eventually P (topological_space.at_within open x (ord.lessThan less x)) = (\<exists>b. less b x \<and> (\<forall>y. less b y \<longrightarrow> less y x \<longrightarrow> P y))"
ccompatible_inv: fixes l :: "'s" shows "ccompatible \<R> (PTA.inv_of A l)"
has_codomain_iff_arr: fixes f :: "(bool, bool list) concrete_category.arr" shows "(J.codomains f \<noteq> {}) = J.arr f"
int_poly_factorization_algorithm_irreducible: fixes alg :: "int_poly_factorization_algorithm"   and f :: "int poly"   and fs :: "int poly list" assumes "int_poly_factorization_algorithm alg f = fs"   and "square_free f"   and "0 < degree f"   and "primitive f" shows "f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible fi \<and> 0 < degree fi \<and> primitive fi)"
pbd_index1_points_imply_unique_block: fixes b1 :: "'a set"   and b2 :: "'a set"   and x :: "'a"   and y :: "'a" assumes "b1 \<in># \<B>"   and "b2 \<in># \<B>"   and "b1 \<noteq> b2"   and "x \<noteq> y"   and "{x, y} \<subseteq> b1"   and "x \<in> b2" shows "y \<notin> b2"
weakCongClosed: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi"   and p :: "(name \<times> name) list" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "env.weakCongruence substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' P Q" shows "env.weakCongruence substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' (p \<bullet> P) (p \<bullet> Q)"
simps_2: fixes t :: "'a FOL_Fitting.term"   and ts :: "'a FOL_Fitting.term list"   and s :: "'a FOL_Fitting.term"   and k :: "nat" shows "(t # ts)[s/k] = t[s/k] # ts[s/k]"
a_rcos_const: fixes h :: "'a" assumes "h \<in> I" shows "I +> h = I"
AE_mp: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a measure"   and Q :: "'a \<Rightarrow> bool" assumes "almost_everywhere M P"   and "AE x in M. P x \<longrightarrow> Q x" shows "almost_everywhere M Q"
hassoc_delete_def: fixes k :: "'a"   and ht :: "('a \<times> 'b) list" shows "hassoc_delete k ht = return (filter (\<lambda>(a, b). a \<noteq> k) ht)"
tendsto_divide: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and b :: "'b" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F"   and "b \<noteq> (0::'b)" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
simps: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and a1 :: "char list exp list"   and a2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a3 :: "char list exp list"   and a4 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" shows "P \<turnstile> \<langle>a1,a2\<rangle> [\<Rightarrow>] \<langle>a3,a4\<rangle> = ((\<exists>s. a1 = [] \<and> a2 = s \<and> a3 = [] \<and> a4 = s) \<or> (\<exists>e s\<^sub>0 v s\<^sub>1 es es' s\<^sub>2. a1 = e # es \<and> a2 = s\<^sub>0 \<and> a3 = Val v # es' \<and> a4 = s\<^sub>2 \<and> P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle> \<and> P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>) \<or> (\<exists>e s\<^sub>0 e' s\<^sub>1 es. a1 = e # es \<and> a2 = s\<^sub>0 \<and> a3 = throw e' # es \<and> a4 = s\<^sub>1 \<and> P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>))"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "signs \<Rightarrow> predicate \<Rightarrow> vbl list \<Rightarrow> 'a"   and f2 :: "signs \<Rightarrow> formula \<Rightarrow> formula \<Rightarrow> 'a"   and f3 :: "signs \<Rightarrow> formula \<Rightarrow> 'a"   and formula :: "formula" shows "P (case formula of FAtom x xa xb \<Rightarrow> f1 x xa xb | FConj x xa xb \<Rightarrow> f2 x xa xb | FAll x xa \<Rightarrow> f3 x xa) = ((\<forall>x11 x12 x13. formula = FAtom x11 x12 x13 \<longrightarrow> P (f1 x11 x12 x13)) \<and> (\<forall>x21 x22 x23. formula = FConj x21 x22 x23 \<longrightarrow> P (f2 x21 x22 x23)) \<and> (\<forall>x31 x32. formula = FAll x31 x32 \<longrightarrow> P (f3 x31 x32)))"
residues_def: fixes m :: "int" shows "residues m \<equiv> 1 < m"
invertible_left_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "invertible x"   and "x \<in> G"   and "y \<in> G"   and "z \<in> G" shows "(x \<cdot> y = x \<cdot> z) = (y = z)"
aff_dim_le_DIM: fixes S :: "'c set" shows "aff_dim S \<le> int DIM('c)"
no_multi_arcs: fixes e1 :: "'b"   and e2 :: "'b" assumes "e1 \<in> pre_digraph.arcs G"   and "e2 \<in> pre_digraph.arcs G"   and "arc_to_ends G e1 = arc_to_ends G e2" shows "e1 = e2"
CDHq: shows "Polynomial.smult q (sdiv_poly (C - D * H) q) = C - D * H"
simps: fixes a1 :: "'b tag_const"   and a2 :: "'b tag_const" shows "(a1 \<le> a2) = (\<exists>n m. a1 = \<tau>\<^sub>c\<^sub>s\<^sub>t n \<and> a2 = \<tau>\<^sub>c\<^sub>s\<^sub>t m \<and> n \<le> m)"
next_not: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<circle>\<not> F = \<not> \<circle>F"
vpairs_mono: fixes r :: "V"   and s :: "V" assumes "r \<subseteq>\<^sub>\<circ> s" shows "vpairs r \<subseteq>\<^sub>\<circ> vpairs s"
inverse_equality: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c"   and v :: "'c" assumes "monoid M composition unit"   and "composition u v = unit"   and "composition v u = unit"   and "u \<in> M"   and "v \<in> M" shows "monoid.inverse M composition unit u = v"
r: shows "approx_slp p ras (a # VS) = Some X"
pos_poly_total: fixes p :: "'b poly" shows "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
almost_diff_const: shows "valid (Equals (Differential (Const 0)) (Const 0))"
ApproximationTr3: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and x :: "nat \<Rightarrow> 'a"   and j :: "nat"   and m :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "\<forall>l\<le>Suc n. x l \<in> carrier K"   and "j \<le> Suc n" shows "\<exists>L. \<forall>N>L. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>k\<in>{h. h \<le> Suc n}. x k \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) k^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j)"
is_unit_polyE: fixes p :: "'b poly" assumes "p dvd 1" obtains c :: "'b" where "p = [:c:]" and "c dvd (1::'b)"
singular_subdivision_zero: fixes X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_chain 0 X c" shows "singular_subdivision 0 c = c"
distinct_map_fst_remove1: fixes xs :: "('c \<times> 'd) list"   and x :: "'c \<times> 'd" assumes "distinct (map fst xs)" shows "distinct (map fst (remove1 x xs))"
ij_3: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a < b" shows "c \<notin> {i a b c..j__ a b c}"
simps_4: fixes test :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e"   and r :: "'d regex"   and s :: "'d regex" shows "lpd test i (Times r s) = TimesR (lpd test i r) s \<union> (if eps test i r then lpd test i s else {})"
eval_exec_max: fixes s1 :: "state"   and c :: "stmt"   and n1 :: "nat"   and t1 :: "state"   and s2 :: "state"   and e :: "expr"   and v :: "val"   and n2 :: "nat"   and t2 :: "state" assumes "s1 -c-n1\<rightarrow> t1"   and "s2 -e\<succ>v-n2\<rightarrow> t2" shows "s1 -c-max n1 n2\<rightarrow> t1 \<and> s2 -e\<succ>v-max n1 n2\<rightarrow> t2"
initial_error: shows "norm s0 \<le> B / L * (exp 1 - 1) * stepsize 0 ^ p"
cases: fixes x :: "'c list" obtains "x = []"   | xa :: "'c" where "x = [xa__]"   | xa :: "'c" and y :: "'c" and xs :: "'c list" where "x = xa # y # xs"
fresh_prod: fixes a :: "'d"   and x :: "'e"   and y :: "'f" shows "a \<sharp> (x, y) = (a \<sharp> x \<and> a \<sharp> y)"
nat_distrib_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b + c) = a * b + a * c"
index_mat_addrow_4: fixes ad :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mul :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "dim_row (mat_addrow_gen ad mul a k l A) = dim_row A"
lp_dehomogenize: fixes x :: "'x"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "is_hom_ord x"   and "homogeneous p" shows "lpp (dehomogenize x p) = except (lpp p) {x}"
ipurge_tr_idem: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list" shows "ipurge_tr I D u (ipurge_tr I D u xs) = ipurge_tr I D u xs"
valid_append_valid_pref: fixes as1 :: "'a list"   and as2 :: "'a list"   and PROB :: "'a set" assumes "as1 @ as2 \<in> valid_plans PROB" shows "as1 \<in> valid_plans PROB"
pinf_2: fixes P :: "'b \<Rightarrow> bool"   and P' :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and Q' :: "'b \<Rightarrow> bool" assumes "\<exists>z. \<forall>x>z. P x = P' x"   and "\<exists>z. \<forall>x>z. Q x = Q' x" shows "\<exists>z. \<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
simps_1: fixes x :: "real"   and ya :: "real" shows "(Rmin x = Rmin ya) = (x = ya)"
lincomb_is_linear: fixes S :: "'c set" assumes "finite S"   and "S \<subseteq> carrier V" shows "linear_map K (func_space S) V (\<lambda>a. lincomb a S)"
mgf_methds: fixes n :: "nat" shows "\<forall>C sig. G,A\<turnstile>{=:n} In1l (Methd C sig)\<succ> {G\<rightarrow>}"
that: assumes "vsv f"   and "n \<in>\<^sub>\<circ> \<omega>"   and "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"   and "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A" shows "thesis"
intro: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and list_of_oalistx :: "'c \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and oalist_of_listx :: "('b \<times> 'd) list \<times> 'a \<Rightarrow> 'c"   and list_of_oalisty :: "'e \<Rightarrow> ('b \<times> 'f) list \<times> 'a"   and oalist_of_listy :: "('b \<times> 'f) list \<times> 'a \<Rightarrow> 'e"   and list_of_oalistz :: "'g \<Rightarrow> ('b \<times> 'h) list \<times> 'a"   and oalist_of_listz :: "('b \<times> 'h) list \<times> 'a \<Rightarrow> 'g" assumes "oalist_abstract rep_key_order list_of_oalistx oalist_of_listx"   and "oalist_abstract rep_key_order list_of_oalisty oalist_of_listy"   and "oalist_abstract rep_key_order list_of_oalistz oalist_of_listz" shows "oalist_abstract3 rep_key_order list_of_oalistx oalist_of_listx list_of_oalisty oalist_of_listy list_of_oalistz oalist_of_listz"
fun_respects: shows "(\<lambda>U. {freefun U}) respects exprel"
nconst: shows "\<forall>c. q1 \<noteq> smult c q2"
UnCI: fixes c :: "'b"   and B :: "'b set"   and A :: "'b set" assumes "(c::'b) \<notin> (B::'b set) \<Longrightarrow> c \<in> (A::'b set)" shows "(c::'b) \<in> (A::'b set) \<union> (B::'b set)"
comp_in_homI: fixes f :: "'d"   and a :: "'d"   and b :: "'d"   and g :: "'d"   and c :: "'d" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>D c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>D f : a \<rightarrow>\<^sub>D c\<guillemotright>"
WhileTReds: fixes b :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and c :: "expr"   and v\<^sub>1 :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e :: "expr"   and s\<^sub>3 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>b,s\<^sub>0\<rangle> \<rightarrow>* \<langle>true,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>c,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>2\<rangle>"   and "P,E \<turnstile> \<langle>while (b) c,s\<^sub>2\<rangle> \<rightarrow>* \<langle>e,s\<^sub>3\<rangle>" shows "P,E \<turnstile> \<langle>while (b) c,s\<^sub>0\<rangle> \<rightarrow>* \<langle>e,s\<^sub>3\<rangle>"
add_right_imp_eq: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and b :: "'f"   and a :: "'f"   and c :: "'f" assumes "class.cancel_semigroup_add plus"   and "plus b a = plus c a" shows "b = c"
sym_symcl: fixes Rel :: "('b \<times> 'b) set" shows "sym (symcl Rel)"
p_has_vector_derivative: fixes a :: "real"   and v :: "real"   and s0 :: "real"   and t :: "real"   and s :: "real set" shows "(movement.p a v s0 has_vector_derivative movement.p' a v t) (at t within s)"
set_iterator_sng_foldli_conv: fixes x :: "'e" shows "set_iterator_sng x = foldli [x]"
comm_semiring_0_cancel_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.comm_semiring_0_cancel plus minus zero times \<equiv> class.cancel_comm_monoid_add plus minus zero \<and> class.comm_semiring plus times"
order_degree: fixes p :: "'a poly"   and a :: "'a" assumes "p \<noteq> 0" shows "order a p \<le> degree p"
simps_4: fixes x :: "'a rexp"   and y :: "'a rexp" shows "lang (x +\<^sub>r y) = lang x + lang y"
unless_alwaysI: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "(\<box>P) \<sigma>" shows "(P \<W> Q) \<sigma>"
bv_extend_shorter: fixes w :: "bit list"   and n :: "nat"   and b :: "bit" assumes "length w < n" shows "bv_extend n b w = bv_extend n b (b # w)"
dom_ab: shows "\<D>\<^sub>\<circ> ab = 2\<^sub>\<nat>"
finfun_All_except_const_finfun_UNIV_code: fixes A :: "'a list"   and b :: "bool" shows "finfun_All_except A (finfun_const b) = (b \<or> is_list_UNIV A)"
FAccRedsVal: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs' :: "char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and Ds :: "char list list"   and Cs :: "char list list"   and fs :: "char list \<Rightarrow> val option"   and F :: "char list"   and v :: "val" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>ref (a, Cs'),s'\<rangle>"   and "hp s' a = \<lfloor>(D, S)\<rfloor>"   and "Ds = Cs' @\<^sub>p Cs"   and "(Ds, fs) \<in> S"   and "fs F = \<lfloor>v\<rfloor>" shows "P,E \<turnstile> \<langle>e\<bullet>F{Cs},s\<rangle> \<rightarrow>* \<langle>Val v,s'\<rangle>"
geometric_observation_1: shows "(dist S M)\<^sup>2 = (dist C S)\<^sup>2 - (dist M C)\<^sup>2"
HSem_edom_subset: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "edom (\<N>\<lbrace>\<Gamma>\<rbrace>\<rho>) \<subseteq> edom \<rho> \<union> domA \<Gamma>"
matrix_vector_mult_0_right: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "A *v 0 = 0"
R_def: shows "R = (\<lambda>x. S x + ln (ln x) + (B + meissel_mertens))"
rel_stacktraces_generalize: fixes F :: "'fenv_ubx"   and st1 :: "('fun, 'dyn) frame list"   and st2 :: "('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame list"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat"   and \<Sigma>2 :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and \<Sigma>1 :: "'dyn list"   and d :: "'dyn" assumes "rel_stacktraces (Fubx_get F) st1 st2 (Some f)"   and "Fubx_get F f = Some fd"   and "sp_fundef (Fubx_get F) fd (take pc (body fd)) = Ok (None # map typeof \<Sigma>2)"   and "\<Sigma>1 = norm_stack \<Sigma>2"   and "pc < length (body fd)"   and "Subx.sp_instr (Fubx_get F) (Subx.generalize_instr (body fd ! pc)) (None # map Map.empty \<Sigma>2) = Ok (None # map (typeof \<circ> Subx.box_operand) \<Sigma>2)" shows "rel_stacktraces (Fubx_get (Fubx_add F f (Subx.generalize_fundef fd))) (Frame f (Suc pc) (d # \<Sigma>1) # st1) (Frame f (Suc pc) (OpDyn d # map Subx.box_operand \<Sigma>2) # Subx.box_stack f st2) None"
that: assumes "rel_option R x y"   and "pred_option P x"   and "pred_option Q y" shows "thesis"
\<phi>_def: shows "\<phi> = vector_to_cblinfun x"
sub_X: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "Cring_Poly.compose R (X_poly R) p = p"
Rep_comp_inverse: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a" assumes "Rep \<circ> f = g" shows "Abs \<circ> g = f"
simps_2: fixes p :: "perm"   and t :: "('b, 'c, 'd, 'e) Tree" shows "p \<bullet> tNot t = tNot (p \<bullet> t)"
non_infE: fixes r :: "('b \<times> 'b) set"   and f :: "nat \<Rightarrow> 'b"   and a :: "'b" assumes "non_inf r" obtains i :: "nat" where "(f i, f (Suc i)) \<notin> r \<or> (f i, a) \<notin> r"
wfE_subst: fixes \<Delta> :: "binding list"   and B :: "binding"   and \<Gamma> :: "binding list"   and U :: "POPLmark.type" assumes "\<Delta> @ B \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f U" shows "\<Delta>[0 \<mapsto>\<^sub>\<tau> U]\<^sub>e @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f"
lookup_pair_update_by_fun_pair: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and k1 :: "'a"   and f :: "'b \<Rightarrow> 'b"   and k2 :: "'a" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "comparator.lookup_pair comp (comparator.update_by_fun_pair comp k1 f xs) k2 = (if k1 = k2 then f else id) (comparator.lookup_pair comp xs k2)"
P_maximal: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "vm_path G P v0" shows "Digraph.maximal_path G P"
induct: fixes P :: "'c dual \<Rightarrow> bool"   and dual :: "'c dual" assumes "\<And>xa. P (dual xa)" shows "P dual"
d'_def: shows "d' = Sup ({a..b} \<inter> g -` {c..d})"