hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "nat \<Rightarrow> 'd"   and a2 :: "'e net_state"   and a3 :: "'f node_action"   and a4 :: "nat \<Rightarrow> 'd"   and a5 :: "'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set" assumes "((a1, a2), a3, a4, a5) \<in> ocnet_sos S" obtains \<sigma> :: "nat \<Rightarrow> 'd" and s :: "'e net_state" and i :: "nat" and i' :: "nat" and \<sigma>' :: "nat \<Rightarrow> 'd" and s' :: "'e net_state" where "a1 = \<sigma>''__" and "a2 = s__" and "a3 = connect(i__, i'__)" and "a4 = \<sigma>'''__" and "a5 = s'__" and "((\<sigma>''__, s__), connect(i__, i'__), \<sigma>'''__, s'__) \<in> S" and "\<forall>j. j \<notin> net_ips s__ \<longrightarrow> \<sigma>'''__ j = \<sigma>''__ j"   | \<sigma> :: "nat \<Rightarrow> 'd" and s :: "'e net_state" and i :: "nat" and i' :: "nat" and \<sigma>' :: "nat \<Rightarrow> 'd" and s' :: "'e net_state" where "a1 = \<sigma>''''__" and "a2 = sa__" and "a3 = disconnect(ia__, i')" and "a4 = \<sigma>'''''__" and "a5 = s'a__" and "((\<sigma>''''__, sa__), disconnect(ia__, i'), \<sigma>'''''__, s'a__) \<in> S" and "\<forall>j. j \<notin> net_ips sa__ \<longrightarrow> \<sigma>'''''__ j = \<sigma>''''__ j"   | \<sigma> :: "nat \<Rightarrow> 'd" and s :: "'e net_state" and R :: "nat set" and m :: "'f" and \<sigma>' :: "nat \<Rightarrow> 'd" and s' :: "'e net_state" where "a1 = \<sigma>''''''__" and "a2 = sb__" and "a3 = \<tau>" and "a4 = \<sigma>'''''''__" and "a5 = s'b__" and "((\<sigma>''''''__, sb__), R:*cast(m), \<sigma>'''''''__, s'b__) \<in> S" and "\<forall>j. j \<notin> net_ips sb__ \<longrightarrow> \<sigma>'''''''__ j = \<sigma>''''''__ j"   | \<sigma> :: "nat \<Rightarrow> 'd" and s :: "'e net_state" and \<sigma>' :: "nat \<Rightarrow> 'd" and s' :: "'e net_state" where "a1 = \<sigma>''''''''__" and "a2 = sc__" and "a3 = \<tau>" and "a4 = \<sigma>'''''''''__" and "a5 = s'c__" and "((\<sigma>''''''''__, sc__), \<tau>, \<sigma>'''''''''__, s'c__) \<in> S" and "\<forall>j. j \<notin> net_ips sc__ \<longrightarrow> \<sigma>'''''''''__ j = \<sigma>''''''''__ j"   | \<sigma> :: "nat \<Rightarrow> 'd" and s :: "'e net_state" and i :: "nat" and d :: "nat" and \<sigma>' :: "nat \<Rightarrow> 'd" and s' :: "'e net_state" where "a1 = \<sigma>''''''''''__" and "a2 = sd__" and "a3 = ib__:deliver(d__)" and "a4 = \<sigma>'''''''''''__" and "a5 = s'd__" and "((\<sigma>''''''''''__, sd__), ib__:deliver(d__), \<sigma>'''''''''''__, s'd__) \<in> S" and "\<forall>j. j \<notin> net_ips sd__ \<longrightarrow> \<sigma>'''''''''''__ j = \<sigma>''''''''''__ j"   | \<sigma> :: "nat \<Rightarrow> 'd" and s :: "'e net_state" and i :: "nat" and K :: "nat set" and d :: "nat" and di :: "nat" and \<sigma>' :: "nat \<Rightarrow> 'd" and s' :: "'e net_state" where "a1 = \<sigma>" and "a2 = s" and "a3 = i:newpkt(d, di)" and "a4 = \<sigma>'" and "a5 = s'" and "((\<sigma>, s), {i}\<not>K:arrive(newpkt (d, di)), \<sigma>', s') \<in> S" and "\<forall>j. j \<notin> net_ips s \<longrightarrow> \<sigma>' j = \<sigma> j"
of_weak_ranking_permute: fixes f :: "'c \<Rightarrow> 'c"   and xs :: "'c set list" assumes "f permutes \<Union> (set xs)" shows "of_weak_ranking (map ((`) f) xs) = map_relation (inv f) (of_weak_ranking xs)"
PP_inverse: fixes y :: "'a \<Rightarrow>\<^sub>0 'b" assumes "y \<in> UNIV" shows "pp.mapping_of (PP y) = y"
all_bound: shows "AE x in M. \<forall>i. norm (s i x) \<le> w x"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "hom.seq g f" shows "hom.comp g f = g \<cdot> f"
unit_in_hom_2: shows "\<guillemotleft>\<phi> : src e \<Rightarrow> e' \<star> e\<guillemotright>"
2: fixes y :: "'b" assumes "y < x" shows "lookup s y = lookup t y"
simps_2: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
meets: fixes i :: "'b"   and j :: "'b" assumes "\<I> i"   and "\<I> j" shows "((i, j) \<in> m) = (END i = BEGIN j)"
sup_bot_right: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and bot :: "'e"   and x :: "'e" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup x bot = x"
poly_rel_times: shows "(poly_rel ===> poly_rel ===> poly_rel) (times_poly_i ops) (*)"
compute_is_float_zero: fixes m :: "int"   and e :: "int" shows "is_float_zero (Float m e) = (0 = m)"
seq_INF_distrib: fixes X :: "'d::type set"   and c :: "'a::refinement_lattice"   and d :: "'d::type \<Rightarrow> 'a::refinement_lattice" assumes "(X::'d::type set) \<noteq> {}" shows "(c::'a::refinement_lattice) ; \<Sqinter> ((d::'d::type \<Rightarrow> 'a::refinement_lattice) ` (X::'d::type set)) = (\<Sqinter>x::'d::type\<in>X. c ; d x)"
prems_2: shows "\<forall>x. c < x \<and> x \<le> d2 \<longrightarrow> (\<forall>q\<in>set (pders p). poly q x \<noteq> 0)"
ok_rctxt_cong: fixes \<phi>s :: "Formula.formula list"   and \<kappa> :: "mregex \<Rightarrow> mregex"   and \<kappa>' :: "Formula.formula Regex.regex \<Rightarrow> Formula.formula Regex.regex"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and r :: "Formula.formula Regex.regex"   and s :: "Formula.formula Regex.regex"   and i :: "nat"   and j :: "nat" assumes "ok_rctxt \<phi>s \<kappa> \<kappa>'"   and "Regex.match (Formula.sat \<sigma> V v) r = Regex.match (Formula.sat \<sigma> V v) s" shows "Regex.match (Formula.sat \<sigma> V v) (\<kappa>' r) i j = Regex.match (Formula.sat \<sigma> V v) (\<kappa>' s) i j"
norm_pre_pure_iff: fixes b :: "bool"   and f :: "'b Heap"   and Q :: "'b \<Rightarrow> assn" shows "<emp * \<up> b> f <Q> = (b \<longrightarrow> <emp> f <Q>)"
R_mult_right_mono: fixes b :: "real"   and a :: "real"   and c :: "real" assumes "b \<le> a"   and "0 \<le> c" shows "b * c \<le> a * c"
zdiv_mono2: fixes a :: "int"   and b' :: "int"   and b :: "int" assumes "0 \<le> a"   and "0 < b'"   and "b' \<le> b" shows "a div b \<le> a div b'"
subst_simps_1: fixes x :: "'e"   and t :: "('d, 'e) Term.term" shows "subst x t x = t"
simps_5: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game"   and \<beta> :: "game" shows "usubstappp \<sigma> U (\<alpha> ;; \<beta>) = (let V\<alpha> = usubstappp \<sigma> U \<alpha>; W\<beta> = usubstappp \<sigma> (fst V\<alpha>) \<beta> in (fst W\<beta>, Composeo (snd V\<alpha>) (snd W\<beta>)))"
not_prime_eq_prod_nat: fixes m :: "nat" assumes "1 < m"   and "\<not> prime m" shows "\<exists>n k. n = m * k \<and> 1 < m \<and> m < n \<and> 1 < k \<and> k < n"
simps_3: fixes S1 :: "com"   and S2 :: "com" shows "well_com (S1;; S2) = (well_com S1 \<and> well_com S2)"
valid_rchild_if_well_valued0: fixes t :: "'b tree"   and as :: "'b list" assumes "well_shaped t"   and "well_valued0 as t" shows "valid as (rchild t)"
rm_vars_img_subset: fixes V :: "'b set"   and s :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "subst_range (rm_vars V s) \<subseteq> subst_range s"
iwlsFSbSwTR_iterSTR_ipresOp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSbSwTR MOD" shows "ipresOp (iterSTR MOD) (iterAbsSTR MOD) MOD"
assms_1: shows "M #> x \<otimes> inv y = M"
abstract_def: fixes c :: "('d, 'e) conf"   and c\<^sub>a :: "('d, 'e) conf" shows "abstract c c\<^sub>a \<equiv> c \<sqsubseteq> c\<^sub>a"
oalist_inv_list_of_oalist_tc: fixes xs :: "('d, 'e) oalist_tc" shows "tc.oalist_inv_raw (list_of_oalist_tc xs)"
mono_inf: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b"   and B :: "'b" assumes "class.semilattice_inf inf less_eq less"   and "order.mono less_eq f" shows "f (inf A B) \<le> f A \<sqinter> f B"
a_type_wf_def: fixes h :: "(_) heap" shows "l_type_wf_def\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t.a_type_wf h = (CharacterDataClass.type_wf h \<and> (\<forall>document_ptr\<in>fset (document_ptr_kinds h). get document_ptr h \<noteq> None))"
distinct_33: fixes x41 :: "('a, 'b, 'c) formula"   and x42 :: "('a, 'b, 'c) formula"   and x61 :: "('a, 'b, 'c) hp"   and x62 :: "('a, 'b, 'c) formula" shows "(x41 && x42) \<noteq> (\<langle> x61 \<rangle> x62)"
permutesE: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S" obtains "bij p" and "\<forall>x. x \<notin> S \<longrightarrow> p x = x"
simps_1: fixes e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "Nielson_VCG.preT SKIP e = e"
is_SDG_path_slice2: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n'' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n'"   and "n' \<in> sum_SDG_slice2 n''" shows "n \<in> sum_SDG_slice2 n''"
that: fixes a :: "'a" assumes "(a, b) \<in> A\<^sup>*"   and "(a, c) \<in> A\<^sup>*" shows "thesis"
extended_Gromov_product_inside_boundary: fixes e :: "'a"   and x :: "'a"   and y :: "'a Gromov_completion" shows "extended_Gromov_product_at e (to_Gromov_completion x) y = Inf {liminf (\<lambda>n. ereal (Gromov_product_at e x (v n))) |v. (\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> y}"
raise_def: fixes s :: "String.literal" shows "raise s = Heap Map.empty"
not_numeral_le_neg_numeral: 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 m :: "num"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less_eq (numeral.numeral one plus m) (uminus (numeral.numeral one plus n))"
insert_uconnectedV_per_eq: fixes x :: "'a"   and F :: "'a uprod set" assumes "x \<in> verts"   and "F \<subseteq> E" shows "Restr (uconnected (insert (Upair x x) F)) verts = per_union (Restr (uconnected F) verts) x x"
ipset_from_cidr_ipcidr_to_interval: fixes base :: "'b word"   and len :: "nat" shows "ipset_from_cidr base len = {ipcidr_to_interval_start (base, len)..ipcidr_to_interval_end (base, len)}"
npeSum2: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<plusminus> y)^\<^bsup>R n\<^esup> = \<Sigma>\<^sub>e R (\<lambda>i.  \<^bsub>n\<^esub>C\<^bsub>i\<^esub> \<times>\<^bsub>R\<^esub> (x^\<^bsup>R (n - i)\<^esup> \<cdot>\<^sub>r y^\<^bsup>R i\<^esup>)) n"
R_mult_left_mono: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "R a b"   and "0 \<le> c" shows "R (c * a) (c * b)"
hom2_1: shows "homotopic_with (\<lambda>x. True) Y Y (g2 \<circ> f2) id"
C1_def: shows "C1 = Max {B (Suc n) / B n |n. n \<le> N}"
Ass: fixes G :: "'a triple set"   and P :: "'a assn"   and X :: "vname"   and a :: "state \<Rightarrow> nat" shows "G|-{\<lambda>Z s. P Z (s[X::=a s])}. X:==a .{P}"
ZFTriple: fixes a :: "ZF"   and b :: "ZF"   and c :: "ZF"   and a' :: "ZF"   and b' :: "ZF"   and c' :: "ZF" assumes "ZFTriple a b c = ZFTriple a' b' c'" shows "a = a' \<and> b = b' \<and> c = c'"
biQuot_inter: fixes a :: "'b"   and b :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "biQuot a b (A \<inter> B) = biQuot a b A \<inter> biQuot a b B"
coplanar_perm_21: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet D B C A"
inject: fixes x1 :: "var\<^sub>C"   and x2 :: "nat"   and y1 :: "var\<^sub>C"   and y2 :: "nat" shows "(bexp\<^sub>C.Eq x1 x2 = bexp\<^sub>C.Eq y1 y2) = (x1 = y1 \<and> x2 = y2)"
simps: fixes callee :: "'e \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'e) spmf"   and c :: "('d, 'f, 'g) gpv"   and s :: "'e" shows "exec_gpv callee c s = the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). exec_gpv callee (c x) y))"
on_stack_is_tree_path: fixes x :: "'v"   and y :: "'v" assumes "x \<in> set (stack s)"   and "y \<in> set (stack s)"   and "\<delta> s x < \<delta> s y" shows "(x, y) \<in> (tree_edges s)\<^sup>+"
config_n_init3: fixes init :: "'b list"   and qs :: "'b list" shows "\<forall>x\<in>set_pmf (Partial_Cost_Model.config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs). snd (snd x) = init"
proj5_addpre: fixes v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and pre :: "nat set" shows "\<pi>\<^sub>5 (addpre v pre) = \<pi>\<^sub>5 v"
j: shows "j = (LEAST n. A $ i $ n \<noteq> (0::'a))"
1: fixes f :: "'a" assumes "hom.arr f" shows "hom'.arr (e\<^sub>1 \<star> f \<star> d\<^sub>0)"
ixc: shows "inv\<^bsub>DirProds G I\<^esub> x \<in> carrier (DirProds G I)"
internal_case_prod_conv: fixes c :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and a :: "'d"   and b :: "'e" shows "??.Product_Type.internal_case_prod c (a, b) = c a b"
is_image_spec_col_cong: fixes P :: "'p"   and P' :: "'p"   and A :: "'p"   and B :: "'p"   and X :: "'p" assumes "P P' ReflectL A B"   and "Col A B X" shows "Cong P X P' X"
raw_match_term_produces_matcher: fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match_term t u subs = Some subs'" shows "tsubst t (convert_subs subs') = u"
Ordinary_def: shows "O! \<equiv> \<^bold>\<lambda>x. \<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>"
r_null: fixes x :: "int" shows "x * 0 = 0"
mask_def2_symmetric: shows "mask (pfxm_length pfx) << LENGTH('a) - pfxm_length pfx = ~~ pfxm_mask pfx"
union_fset: fixes x :: "'a fset"   and xa :: "'a fset" shows "fset (x |\<union>| xa) = fset x \<union> fset xa"
induct: fixes x :: "'a nmultiset"   and P :: "'a nmultiset \<Rightarrow> bool" assumes "no_elem x"   and "\<And>M. \<lbrakk>\<And>X. X \<in># M \<Longrightarrow> no_elem X; \<And>X. X \<in># M \<Longrightarrow> P X\<rbrakk> \<Longrightarrow> P (MSet M)" shows "P x"
bind_assoc: fixes x :: "'d option"   and f :: "'d \<Rightarrow> 'c option"   and g :: "'c \<Rightarrow> 'b option" shows "x \<bind> f \<bind> g = x \<bind> (\<lambda>y. f y \<bind> g)"
cblinfun_apply_cases: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> {f. bounded_clinear f}" obtains x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" where "y = (*\<^sub>V) x"
Join_left_absorb: fixes F :: "'c program"   and G :: "'c program" shows "F \<squnion> (F \<squnion> G) = F \<squnion> G"
safes: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "prop_safe c0"   and "next_change_multiplicity' c0 c1 loc t n" shows "prop_safe c1"
assms: shows "mat_det (A, B, C, D) \<noteq> 0"
simps_1: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and c :: "('a \<times> (Mode \<Rightarrow> 'b set)) list \<times> ('b \<Rightarrow> 'c)"   and c' :: "('a \<times> (Mode \<Rightarrow> 'b set)) list \<times> ('b \<Rightarrow> 'c)" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.meval_k eval 0 c c' = (c = c')"
field_inverse_zero: shows "inverse (0::'e) = (0::'e)"
r_neg2: fixes x :: "'a vec"   and y :: "'a vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "x + (\<ominus>\<^bsub>V\<^esub> x + y) = y"
ldrop_eq_LConsD: fixes n :: "enat"   and xs :: "'b llist"   and y :: "'b"   and ys :: "'b llist" assumes "ldrop n xs = LCons y ys" shows "n < llength xs"
ordering_axioms: shows "ordering dual_order.greater_eq dual_order.greater"
simps: fixes a1 :: "(test, atom) com \<times> (Concrete.var \<Rightarrow> nat)"   and a2 :: "Concrete.var \<Rightarrow> nat" shows "Example_PL.MtransT a1 a2 = (\<exists>cf cf' s''. a1 = cf \<and> a2 = s'' \<and> Example_PL.MtransC cf cf' \<and> Example_PL.transT cf' s'')"
simps_4: fixes f :: "'a"   and args :: "'b \<Rightarrow> ('a, 'b) trm"   and \<sigma> :: "('a, 'c, 'b) subst" shows "Tsubst ($f f args) \<sigma> = (case SFunctions \<sigma> f of None \<Rightarrow> $f f | Some f' \<Rightarrow> TsubstFO f') (\<lambda>i. Tsubst (args i) \<sigma>)"
slim_giapath_if_giapath: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "gen_iapath (verts3 (with_proj G)) u p v" shows "\<exists>p. pre_digraph.gen_iapath (with_proj slim) (verts3 (with_proj G)) u p v"
objT_alt: shows "objT = map_of (consts D @ objects P)"
path_empty_simp: fixes v :: "'v"   and v' :: "'v" shows "path v v' [] = (v' = v)"
nlen_suc: fixes m :: "nat"   and n :: "nat" assumes "Suc m = nlen n" shows "m = nlen (n div 10)"
\<epsilon>_props_2: shows "\<forall>x. x \<noteq> x\<^sub>0 \<and> \<bar>x - x\<^sub>0\<bar> < \<epsilon> \<longrightarrow> poly p x \<noteq> 0 \<and> poly q x \<noteq> 0 \<and> sgn (poly (pderiv p) x) = sgn (poly q x)"
simps_5: fixes I :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc ({I} WHILE b DO C) Q = ((\<forall>s. (I s \<and> bval b s \<longrightarrow> pre C I s) \<and> (I s \<and> \<not> bval b s \<longrightarrow> Q s)) \<and> vc C I)"
le_imp_less_or_eq: fixes y :: "'d"   and x :: "'d" assumes "y \<le> x" shows "y < x \<or> x = y"
assms_1: shows "ordermap A r y \<le> ordermap A r x"
seqI': fixes f :: "'s"   and a :: "'s"   and b :: "'s"   and g :: "'s"   and c :: "'s" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>S c\<guillemotright>" shows "S.seq g f"
topological_basis_iff: fixes B :: "'a set set" assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'" shows "topological_basis B = (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
subgroup_singular_relboundary_relcycle: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "subgroup (singular_relboundary_set p X S) (relcycle_group p X S)"
std_Red_I_eq: shows "std.Red_I = Red_I_\<G>"
False: shows "\<not> sum_list (map abs ks) \<le> \<bar>l\<bar>"
lepoll_iff_leqpoll: fixes A :: "'a set"   and B :: "'b set" shows "(A \<lesssim> B) = (A \<prec> B \<or> A \<approx> B)"
empty_subset_add_mset: fixes x :: "'a"   and M :: "'a multiset" shows "{#} \<subset># add_mset x M"
GuardsCons: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and f :: "'f"   and g :: "'e set"   and gs :: "('f \<times> 'e set) list"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Guard f g (guards gs c) Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P guards ((f, g) # gs) c Q,A"
int_lsb_numeral_1: shows "lsb 0 = False"
sel_1: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "'c"   and x4 :: "'a" shows "srcOf (Trans x1 x2 x3 x4) = x1"
n_2: shows "\<forall>m. 0 \<le> m \<and> m dvd n \<longrightarrow> m = 1 \<or> m = n"
RSubmodule_RSpan: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list" assumes "RModule R smult M"   and "set ms \<subseteq> M" shows "RModule R smult (R_scalar_mult.RSpan R smult ms) \<and> R_scalar_mult.RSpan R smult ms \<subseteq> M"
execl_sub: fixes P :: "instr list"   and P' :: "instr list"   and P'' :: "instr list"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list"   and m :: "nat"   and n :: "nat" assumes "P @ P' @ P'' \<Turnstile> cfs"   and "\<forall>k\<in>{m..<n}. size P \<le> fst (cfs ! k) \<and> fst (cfs ! k) - size P < size P'" shows "P' \<Turnstile> map (off P) (drop m (take (Suc n) cfs))"
minus_eq: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_group G" shows "x \<ominus>\<^bsub>G\<^esub> y = x \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> y"
subst_cls_list_id_subst: fixes Cs :: "'a literal multiset list" shows "Cs \<cdot>cl id_subst = Cs"
factorization_def: fixes Factors :: "'a poly set"   and f :: "'a poly"   and cfs :: "'a \<times> 'a poly multiset" shows "factorization Factors f cfs \<equiv> case cfs of (c, fs) \<Rightarrow> f = smult c (\<Prod>\<^sub># fs) \<and> set_mset fs \<subseteq> Factors"
c': fixes na :: "nat \<times> 'a"   and k :: "nat" assumes "na \<in> set n_as" shows "Q (fst na) (snd na) k (c' na)"
pa01_Un: shows "path_image (subpath 0 t q) \<union> path_image (subpath 1 t q) = path_image q"
wf_action_schema_sound_inst: fixes act :: "ast_action_schema"   and args :: "object list" assumes "action_params_match act args"   and "wf_action_schema act" shows "sound_opr (instantiate_action_schema act args) (pddl_opr_to_act (instantiate_action_schema act args))"
NMPrm: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and p :: "'b list" assumes "not_MT C p" shows "not_MT C (rm_MT_rules C p)"
inv_image_def: fixes r :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "inv_image r f = {(x, y). (f x, f y) \<in> r}"
cong_exp_iff_simps_3: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and n :: "num"   and q :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit1 n)) (numeral.numeral one plus (num.Bit0 q)) = zero) = False"
replace_in_dgrad_p_set: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "G \<subseteq> dgrad_p_set d m" obtains n :: "nat" where "q \<in> dgrad_p_set d n" and "G \<subseteq> dgrad_p_set d n" and "insert q (G - {p}) \<subseteq> dgrad_p_set d n"
leg_simps_1: shows "C.arr \<nu>.cod.leg0"
trans_par_comp: fixes s :: "('c, 'e seq_action) automaton"   and t :: "('d, 'e seq_action) automaton" shows "automaton.trans (s \<langle>\<langle> t) = parp_sos (automaton.trans s) (automaton.trans t)"
cases: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and a1 :: "'b list"   and a2 :: "'b literal multiset" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.eligible S a1 a2" obtains DA :: "'b literal multiset" and As :: "'b list" where "a1 = As" and "a2 = DA" and "S DA = negs (mset As) \<or> S DA = {#} \<and> length As = 1 \<and> ground_resolution_with_selection.maximal_wrt (As ! 0) DA"
sub_composition_closed: fixes a :: "'a \<Rightarrow> 'a"   and b :: "'a \<Rightarrow> 'a" assumes "a \<in> G"   and "b \<in> G" shows "compose S a b \<in> G"
less_wrong: fixes v :: "val" assumes "v \<sqsubseteq> Wrong" obtains "v = Wrong"
ahm_empty_def: shows "ahm_empty \<equiv> \<lambda>_. ahm_empty_const"