hcju/isabelleps huggingface dataset:
Min_insert: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}" shows "Min (insert x A) = min x (Min A)"
abs_eq: fixes x :: "('b, 'c) rbt" assumes "eq_onp (\<lambda>t. ord.is_rbt cless t \<or> ID ccompare = None) x x" shows "RBT_Mapping2.keys (Mapping_RBT x) = set (RBT_Impl.keys x)"
ax_derivs_insertD: fixes G :: "prog"   and A :: "'b triple set"   and t :: "'b triple"   and ts :: "'b triple set" assumes "G,A|\<turnstile>insert t ts" shows "G,A\<turnstile>t \<and> G,A|\<turnstile>ts"
abs_diff_le_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c"   and a :: "'c"   and r :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (abs (minus x a)) r = (less_eq (minus a r) x \<and> less_eq x (plus a r))"
lexext_rev_trans_strong: fixes zs :: "'b list"   and ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>z\<in>set zs. \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "lexext_rev gt zs ys"   and "lexext_rev gt ys xs" shows "lexext_rev gt zs xs"
commute: fixes a :: "'a"   and b :: "'a" shows "ord.max (\<sqsubseteq>) a b = ord.max (\<sqsubseteq>) b a"
assms: shows "\<Omega>1 \<subseteq> \<Omega>2"
assms_1: shows "\<forall>t\<in>M. N \<turnstile>\<^sub>c t"
cases: fixes s :: "Store"   and a1 :: "Location"   and a2 :: "Value" assumes "s\<turnstile> a1 reachable_from a2" obtains l :: "Location" where "a1 = la__" and "a2 = ref la__" and "ref la__ \<noteq> nullV"   | l :: "Location" and k :: "Location" where "a1 = l" and "a2 = ref k" and "s\<turnstile> l reachable_from (s@@k)" and "ref k \<noteq> nullV"
fmdomI: fixes m :: "('d, 'c) fmap"   and x :: "'d"   and y :: "'c" assumes "fmlookup m x = Some y" shows "x |\<in>| fmdom m"
eq_elapsed_def: fixes A :: "('a \<times> ('b, 'd) cconstraint \<times> ('b set \<times> 'a) pmf) set \<times> ('a \<Rightarrow> ('b, 'd) cconstraint)"   and u :: "'b \<Rightarrow> 'e"   and u' :: "'b \<Rightarrow> 'e" assumes "Probabilistic_Timed_Automaton A" shows "Probabilistic_Timed_Automaton.eq_elapsed A u u' \<equiv> (0::'e) < Probabilistic_Timed_Automaton.elapsed A u u' \<longrightarrow> (\<forall>c\<in>Probabilistic_Timed_Automaton.\<X> A. u' c - u c = Probabilistic_Timed_Automaton.elapsed A u u')"
alw_prop_invs: fixes s :: "('b, 't, 'loc) Combined.configuration stream" assumes "FullSpec s" shows "alw (holds (\<lambda>c. \<forall>p. prop_invs (prop_config c p))) s"
induct: fixes P1 :: "'a exp \<Rightarrow> bool"   and P2 :: "'a exp list \<Rightarrow> bool"   and exp_list :: "'a exp list" assumes "\<And>x. P1 (new x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Cast x1 x2)"   and "\<And>x. P1 (Val x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 \<guillemotleft>x2\<guillemotright> x3)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (x1:=x2)"   and "\<And>x1 x2 x3. P1 x1 \<Longrightarrow> P1 (x1\<bullet>x2{x3})"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2{x3} := x4)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P2 x3\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2(x3))"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 {x1:x2; x3}"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (if (x1) x2 else x3)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (while (x1) x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (try x1 catch(x2 x3) x4)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 exp_list"
prob_times: fixes X :: "'a \<Rightarrow> 'b set" assumes "\<And>i. i \<in> I \<Longrightarrow> X i \<in> M.events i" shows "P.prob (Pi\<^sub>E I X) = (\<Prod>i\<in>I. M.prob i (X i))"
mu_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "has_least_fixpoint f" shows "f (\<mu> f) = \<mu> f"
ccApprox_ttree_restr: fixes S :: "Vars.var set"   and t :: "Vars.var ttree" shows "ccApprox (ttree_restr S t) = ccApprox t G|` S"
edom_esing_subset: fixes x :: "'c"   and n :: "'d" shows "edom (esing x\<cdot>n) \<subseteq> {x}"
lang_of_append_rexp_rhs: fixes rhs :: "'b trm set"   and r :: "'b rexp" shows "lang_rhs (Append_rexp_rhs rhs r) = lang_rhs rhs \<cdot> lang r"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "vc (C\<^sub>1;; C\<^sub>2) Q = (vc C\<^sub>1 (pre C\<^sub>2 Q) \<and> vc C\<^sub>2 Q)"
preserves_reflects_arr: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "arr \<a>[fst f, fst (snd f), snd (snd f)] = VVV.arr f"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "\<phi>.A.seq g f" shows "\<phi>.B.seq (G g) (G f)"
image_empty: fixes f :: "'d \<Rightarrow> 'c" shows "bot_pres ((`) f)"
rtrancl_path_appendE: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list"   and z :: "'a" assumes "rtrancl_path r x (xs @ y # ys) z" obtains "rtrancl_path r x (xs @ [y]) y" and "rtrancl_path r y ys z"
reduced_GB_is_auto_reduced_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "punit.is_auto_reduced (punit.reduced_GB F)"
inner_commute: fixes x :: "'b"   and y :: "'b" shows "inner x y = cnj_class.cnj (inner y x)"
dim_col_transfer_rule: fixes A :: "'d mat"   and A' :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" assumes "Mod_Type_Connect.HMA_M A A'" shows "dim_col A = CARD('e)"
eq_is_closed_def: fixes S :: "('d, 'e) eq_object_scheme"   and A :: "'d set" shows "is_closed\<^bsub>S\<^esub> A = (A \<subseteq> carrier S \<and> closure_of\<^bsub>S\<^esub> A = A)"
simps_7: fixes p :: "'e" shows "merge_guards (Call p) = Call p"
gamma2_vec_deriv_within: fixes x :: "real" assumes "x \<in> {0..1} - (\<phi> -` (s' \<inter> {0..1}) \<union> s \<inter> {0..1})" shows "vector_derivative (\<gamma>2 \<circ> \<phi>) (at x) = vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})"
add_eq_conv_ex: fixes a :: "'b"   and M :: "'b multiset"   and b :: "'b"   and N :: "'b multiset" shows "(add_mset a M = add_mset b N) = (M = N \<and> a = b \<or> (\<exists>K. M = add_mset b K \<and> N = add_mset a K))"
simps_2: fixes x :: "char list"   and e :: "aexp"   and S :: "'b"   and x' :: "char list"   and e' :: "aexp"   and S' :: "'b" shows "(x ::= e {S} \<sqsubseteq> x' ::= e' {S'}) = (x = x' \<and> e = e' \<and> S \<sqsubseteq> S')"
wp_while_inv: fixes p :: "('a \<times> 'a) set"   and i :: "('a \<times> 'a) set"   and r :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set" assumes "rel_aka.dom_op p \<subseteq> rel_aka.dom_op i"   and "rel_aka.dom_op i ; rel_ad r \<subseteq> rel_aka.dom_op q"   and "rel_aka.dom_op i ; rel_aka.dom_op r \<subseteq> rel_aka.wp x i" shows "rel_aka.dom_op p \<subseteq> rel_aka.wp (rel_aka.while_inv r i x) q"
disc_nodes: shows "h3 \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"
dagger_simr: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<cdot> x \<le> y \<cdot> z" shows "z \<cdot> x\<^sup>\<dagger> \<le> y\<^sup>\<dagger> \<cdot> z"
less_convert: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a = b"   and "c < b" shows "c < a"
delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr = Heap_Error_Monad.bind get_heap (\<lambda>h. case delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr h of None \<Rightarrow> error HierarchyRequestError | Some h \<Rightarrow> return_heap h)"
simps_1: fixes p :: "linear_poly"   and c :: "rat"   and q :: "linear_poly"   and d :: "QDelta" shows "compatible_cs (Le_Constraint Leq_Rel p c) (Le_Constraint Leq_Rel q d) = (q = p \<and> d = QDelta c 0)"
rel_fun_def: fixes A :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "A ===> B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
unwrapH_strict: shows "unwrapH\<cdot>\<bottom> = \<bottom>"
assms_10: shows "length xvec5 = residualLength cRs"
fact_prod_Suc: fixes n :: "nat" shows "fact n = of_nat (prod Suc {0..<n})"
sympE: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a" assumes "symp r"   and "r b a" obtains "r a b"
map_cong: fixes xs :: "'c list"   and ys :: "'c list"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "map f xs = map g ys"
distinct_29: fixes x3 :: "('a, 'b, 'c) formula"   and x71 :: "'b"   and x72 :: "('a, 'b, 'c) formula" shows "! x3 \<noteq> InContext x71 x72"
to_vec_diag: fixes x :: "'c" shows "to_vec (diag x) = mat x"
rev_sub_to_poly: fixes R :: "('c, 'd) ring_scheme"   and g :: "nat \<Rightarrow> 'c"   and a :: "'c" assumes "UP_cring R"   and "g \<in> carrier (UP R)"   and "a \<in> carrier R" shows "rev_compose R g (to_polynomial R a) = to_polynomial R a"
leq_trans: fixes x :: "'f option"   and y :: "'f option"   and z :: "'f option" assumes "option_ord x y"   and "option_ord y z" shows "option_ord x z"
Cauchy_altdef: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" assumes "class.metric_space dist uniformity open" shows "uniform_space.Cauchy uniformity f = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e)"
g: shows "monotone (ord_spmf (=)) (ord_spmf (=)) G"
from_list_correct'_1: fixes \<alpha> :: "'c \<Rightarrow> 'd \<times> 'd \<Rightarrow> enat"   and invar :: "'c \<Rightarrow> bool"   and succ :: "'c \<Rightarrow> 'd \<Rightarrow> (nat \<times> 'd) list"   and empty_graph :: "'c"   and add_edge :: "'d \<times> 'd \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'c"   and l :: "('d \<times> 'd \<times> nat) list" assumes "wgraph_from_list_algo \<alpha> invar succ empty_graph add_edge"   and "valid_graph_rep l" shows "invar (wgraph_from_list_algo.from_list empty_graph add_edge l)"
Ball_set_Cons: fixes x :: "'b set"   and y :: "'b list set"   and P :: "'b list \<Rightarrow> bool" shows "(\<forall>a\<in>set_Cons x y. P a) = (\<forall>a\<in>x. \<forall>b\<in>y. P (a # b))"
axioms_2: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>"
alternation: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "d p \<cdot>  |x\<^sup>\<star>] ((p \<leadsto> x \<leadsto> q) \<cdot> (q \<leadsto> x \<leadsto> p)) =  |(x \<cdot> x)\<^sup>\<star>] (d p \<cdot> (q \<leadsto> x \<leadsto> p)) \<cdot>  |x \<cdot> (x \<cdot> x)\<^sup>\<star>] (d q \<cdot> (p \<leadsto> x \<leadsto> q))"
object_ptr_kinds_M_eq_h: fixes ptrs :: "(_) object_ptr list" shows "h \<turnstile> object_ptr_kinds_M \<rightarrow>\<^sub>r ptrs = h2 \<turnstile> object_ptr_kinds_M \<rightarrow>\<^sub>r ptrs"
primes_1000: shows "primes_1000 = filter prime [0..<1001]"
invertible_inverse_inverse: fixes u :: "'a" assumes "invertible u"   and "u \<in> G" shows "inverse (inverse u) = u"
assms_2: shows "[M]\<^bsub>v,n\<^esub> \<noteq> {}"
admits_SNF: shows "\<forall>A. isDiagonal A \<longrightarrow> (\<exists>P Q. invertible P \<and> invertible Q \<and> Smith_normal_form (P ** A ** Q))"
induct: fixes P1 :: "state \<Rightarrow> stmt \<Rightarrow> nat \<Rightarrow> assn"   and P2 :: "state \<Rightarrow> expr \<Rightarrow> val \<Rightarrow> nat \<Rightarrow> assn"   and x1 :: "state"   and x2 :: "stmt"   and x3 :: "nat"   and x4 :: "state"   and x5 :: "state"   and x6 :: "expr"   and x7 :: "val"   and x8 :: "nat"   and x9 :: "state" assumes "\<And>s n. P1 s Skip n s"   and "\<And>s0 c1 n s1 c2 s2. \<lbrakk>s0 -c1-n\<rightarrow> s1; P1 s0 c1 n s1; s1 -c2-n\<rightarrow> s2; P1 s1 c2 n s2\<rbrakk> \<Longrightarrow> P1 s0 (c1;; c2) n s2"   and "\<And>s0 e v n s1 c1 c2 s2. \<lbrakk>s0 -e\<succ>v-n\<rightarrow> s1; P2 s0 e v n s1; s1 -(if v \<noteq> Null then c1 else c2)-n\<rightarrow> s2; P1 s1 (if v \<noteq> Null then c1 else c2) n s2\<rbrakk> \<Longrightarrow> P1 s0 (If (e) c1 Else c2) n s2"   and "\<And>s0 x c n. s0<x> = Null \<Longrightarrow> P1 s0 (While (x) c) n s0"   and "\<And>s0 x c n s1 s2. \<lbrakk>s0<x> \<noteq> Null; s0 -c-n\<rightarrow> s1; P1 s0 c n s1; s1 -While (x) c-n\<rightarrow> s2; P1 s1 (While (x) c) n s2\<rbrakk> \<Longrightarrow> P1 s0 (While (x) c) n s2"   and "\<And>s x n. P2 s (LAcc x) (s<x>) n s"   and "\<And>s e v n s' x. \<lbrakk>s -e\<succ>v-n\<rightarrow> s'; P2 s e v n s'\<rbrakk> \<Longrightarrow> P1 s (x :== e) n (lupd(x\<mapsto>v) s')"   and "\<And>s e a n s' f. \<lbrakk>s -e\<succ>Addr a-n\<rightarrow> s'; P2 s e (Addr a) n s'\<rbrakk> \<Longrightarrow> P2 s (e..f) (get_field s' a f) n s'"   and "\<And>s0 e1 a n s1 e2 v s2 f. \<lbrakk>s0 -e1\<succ>Addr a-n\<rightarrow> s1; P2 s0 e1 (Addr a) n s1; s1 -e2\<succ>v-n\<rightarrow> s2; P2 s1 e2 v n s2\<rbrakk> \<Longrightarrow> P1 s0 (e1..f:==e2) n (upd_obj a f v s2)"   and "\<And>s a C n. new_Addr s = Addr a \<Longrightarrow> P2 s (new C) (Addr a) n (new_obj a C s)"   and "\<And>s e v n s' C. \<lbrakk>s -e\<succ>v-n\<rightarrow> s'; P2 s e v n s'; case v of Null \<Rightarrow> True | Addr a \<Rightarrow> obj_class s' a \<preceq>C C\<rbrakk> \<Longrightarrow> P2 s (Cast C e) v n s'"   and "\<And>s0 e1 a n s1 e2 p s2 C m s3. \<lbrakk>s0 -e1\<succ>a-n\<rightarrow> s1; P2 s0 e1 a n s1; s1 -e2\<succ>p-n\<rightarrow> s2; P2 s1 e2 p n s2; lupd(This\<mapsto>a) (lupd(Par\<mapsto>p) (del_locs s2)) -Meth (C, m)-n\<rightarrow> s3; P1 (lupd(This\<mapsto>a) (lupd(Par\<mapsto>p) (del_locs s2))) (Meth (C, m)) n s3\<rbrakk> \<Longrightarrow> P2 s0 ({C}e1..m(e2)) (s3<Res>) n (set_locs s2 s3)"   and "\<And>s a D C m n s'. \<lbrakk>s<This> = Addr a; D = obj_class s a; D \<preceq>C C; init_locs D m s -Impl (D, m)-n\<rightarrow> s'; P1 (init_locs D m s) (Impl (D, m)) n s'\<rbrakk> \<Longrightarrow> P1 s (Meth (C, m)) n s'"   and "\<And>s Cm n s'. \<lbrakk>s -body Cm-n\<rightarrow> s'; P1 s (body Cm) n s'\<rbrakk> \<Longrightarrow> P1 s (Impl Cm) (Suc n) s'" shows "(x1 -x2-x3\<rightarrow> x4 \<longrightarrow> P1 x1 x2 x3 x4) \<and> (x5 -x6\<succ>x7-x8\<rightarrow> x9 \<longrightarrow> P2 x5 x6 x7 x8 x9)"
nlcons_invmeas: fixes t :: "('c, 'd) FingerTreeStruc"   and nd :: "('c, 'd) Node" assumes "is_measured_ftree t"   and "is_measured_node nd" shows "is_measured_ftree (nlcons nd t)"
less_mod_ge_imp_div_less: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a < b"   and "b mod m \<le> a mod m" shows "a div m < b div m"
Dflow_trans: fixes h :: "real"   and x0 :: "'a"   and i :: "real" assumes "h \<in> existence_ivl0 x0"   and "i \<in> existence_ivl0 (flow0 x0 h)" shows "Dflow x0 (h + i) = Dflow (flow0 x0 h) i o\<^sub>L Dflow x0 h"
resTy_mthd: fixes m :: "'c \<times> 'b" shows "resTy (mthd m) = resTy m"
in_dom: fixes i :: "'agent" assumes "i \<in> agents'" shows "Order_Predicates.preorder_on alts' (R i)"
Span_base_incl: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K (L\<lparr>carrier := A\<rparr>)"   and "set Us \<subseteq> carrier (L\<lparr>carrier := A\<rparr>)" shows "set Us \<subseteq> Hom.Span K Us"
fminus_finter_distrib2: fixes A :: "'a fset"   and B :: "'a fset"   and C :: "'a fset" shows "A |-| B |\<inter>| C = A |\<inter>| C |-| (B |\<inter>| C)"
some_vertices: fixes c :: "'form"   and i :: "nat"   and "is" :: "nat list" shows "((c, i # is) |\<in>| vertices) = (c \<in> set conclusions \<and> i = 0 \<and> is \<in> it_paths (it' c))"
OrI2: fixes G :: "('a, 'b) form list"   and b :: "('a, 'b) form"   and a :: "('a, 'b) form" assumes "G \<turnstile> b" shows "G \<turnstile> Or a b"
c4: shows "pi \<bullet> a' \<sharp> (pi \<bullet> h) (pi \<bullet> a')"
cInf_lower: fixes x :: "'a multiset"   and X :: "'a multiset set" assumes "x \<in> X"   and "subset_mset.bdd_below X" shows "\<Sqinter> X \<subseteq># x"
lift_eq_lift: fixes x :: "V"   and y :: "V"   and z :: "V" assumes "lift x y = lift x z" shows "y = z"
Gromov_product_e_x_x: fixes e :: "'a"   and x :: "'a" shows "Gromov_product_at e x x = dist e x"
simps_1: fixes c :: "color" shows "paint c \<langle>\<rangle> = \<langle>\<rangle>"
Leaf_def: shows "Leaf \<equiv> Atom \<circ> Inl"
coreflexive_commutative: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv"   and "less_eq x one"   and "less_eq y one" shows "times x y = times y x"
stateT_induct: fixes P :: "'b\<cdot>('c, 'a) stateT \<Rightarrow> bool"   and y :: "'b\<cdot>('c, 'a) stateT" assumes "\<And>k. P (StateT\<cdot>k)" shows "P y"
l_set_child_nodes_get_child_nodes_axioms: shows "l_set_child_nodes_get_child_nodes ShadowRootClass.type_wf ShadowRootClass.known_ptr Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs"
image_mset_eq_implies_permutes: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c" assumes "finite A"   and "image_mset f (mset_set A) = image_mset f' (mset_set A)" obtains p :: "'b \<Rightarrow> 'b" where "p permutes A" and "\<forall>x\<in>A. f x = f' (p x)"
resolution_def: fixes C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set"   and L\<^sub>1 :: "fterm literal set"   and L\<^sub>2 :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "resolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma> = C\<^sub>1 - L\<^sub>1 \<union> (C\<^sub>2 - L\<^sub>2) \<cdot>\<^sub>l\<^sub>s \<sigma>"
wpQ_is_weakestprePotential1: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<Turnstile>\<^sub>2\<^sub>' {P} c {Q}" shows "\<exists>k>0. \<forall>s. wp\<^sub>Q c (\<lambda>s. enat k * Q s) s \<le> enat k * P s"
E_Abs: fixes x :: "vrs"   and v\<^sub>2 :: "trm"   and t\<^sub>12 :: "trm"   and T\<^sub>11 :: "ty" assumes "x \<sharp> v\<^sub>2"   and "val v\<^sub>2" shows "(\<lambda>x:T\<^sub>11. t\<^sub>12) \<cdot> v\<^sub>2 \<longmapsto> t\<^sub>12[x \<mapsto> v\<^sub>2]"
ABind_eq: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and v :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ArityAnalysis.ABind Aexp v e\<cdot>ae = fup\<cdot>(Aexp e)\<cdot>(ae v)"
env_delete_mono: fixes m :: "'a \<Rightarrow> 'b"   and m' :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "m \<sqsubseteq> m'" shows "env_delete x m \<sqsubseteq> env_delete x m'"
flip_altsublist_chain_sum_list: fixes x :: "'w list"   and zs :: "'w list list"   and y :: "'w list" assumes "x \<in> lists S"   and "flip_altsublist_chain (x # zs @ [y])" shows "sum_list y = sum_list x"
rel_nabla_unfold: fixes X :: "('a \<times> 'a) set" shows "rel_nabla X \<subseteq> relfdia X (rel_nabla X)"
methd_def: fixes G :: "prog"   and C :: "qtname" shows "methd G C = class_rec G C Map.empty (\<lambda>C c subcls_mthds. filter_tab (\<lambda>sig m. G \<turnstile> C inherits method sig m) subcls_mthds ++ table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)))"
higher_differentiable_on_congI: fixes S :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and n :: "nat"   and f :: "'b \<Rightarrow> 'c" assumes "open S"   and "higher_differentiable_on S g n"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "higher_differentiable_on S f n"
suffix_snoc: fixes xs :: "'a list"   and ys :: "'a list"   and y :: "'a" shows "suffix xs (ys @ [y]) = (xs = [] \<or> (\<exists>zs. xs = zs @ [y] \<and> suffix zs ys))"
elims_2: fixes wtFsym :: "'e \<Rightarrow> bool"   and wtPsym :: "'f \<Rightarrow> bool"   and arOf :: "'e \<Rightarrow> 'g list"   and resOf :: "'e \<Rightarrow> 'g"   and intT :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and intF :: "'e \<Rightarrow> 'h list \<Rightarrow> 'h"   and intP :: "'f \<Rightarrow> 'h list \<Rightarrow> bool"   and x :: "Preliminaries.var \<Rightarrow> 'h"   and xa :: "('e, 'f) lit" assumes "Struct wtFsym wtPsym arOf resOf intT intF intP"   and "Struct.satL intF intP x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'h" and a :: "('e, 'f) atm" where "x = \<xi>'__" and "xa = Pos a__" and "Struct.satA intF intP \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'h" and a :: "('e, 'f) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> Struct.satA intF intP \<xi> a"
liftable_set_extensionI_weak: fixes alts' :: "'a set"   and set_ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and alts :: "'a set" assumes "\<And>R A B. \<lbrakk>total_preorder_on alts' R; A \<noteq> {}; B \<noteq> {}; A \<subseteq> alts'; B \<subseteq> alts'\<rbrakk> \<Longrightarrow> set_ext R A B = set_ext (lift_prefs alts' alts R) A B" shows "liftable_set_extension alts' alts set_ext"
l1: fixes s :: "'a" assumes "s \<in> S" shows "measure_pmf.expectation (K s) l1 + c s \<le> l1 s"
update_def: fixes destr :: "'f \<Rightarrow> 'g"   and constr :: "'h \<Rightarrow> 'f"   and n :: "'e"   and f :: "'g \<Rightarrow> 'h"   and s :: "'e \<Rightarrow> 'f" shows "update destr constr n f s = s(n := constr (f (destr (s n))))"
add_0_left: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e" assumes "class.monoid_add plus zero" shows "plus zero a = a"
runit_commutes_with_R: fixes f :: "'a" assumes "Right.ide f" shows "Right.runit (Right.R f) = Right.R (Right.runit f)"
of_int_less_of_int_power_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "int"   and w :: "nat"   and x :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (power.power one times (ring_1.of_int one plus zero minus b) w) (ring_1.of_int one plus zero minus x) = (b ^ w < x)"
simps_2: fixes n :: "nat"   and l :: "real" shows "T Del (n, l) = (if f1 * l \<le> real (n - 1) then 1 else if l0 \<le> l / c then real n else 1)"
less_le_not_le: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f" assumes "class.preorder less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
Standard_subset_HFinite: shows "Standard \<subseteq> HFinite"
simps: fixes a :: "'a"   and p :: "'a poly" shows "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons (0::'a) (pderiv p))"