hcju/isabelleps huggingface dataset:
resolvingBidBridgingLemmaVariant1: fixes N :: "'c set"   and \<Omega> :: "'d list"   and a :: "('c \<times> 'd set) set"   and b :: "'c \<times> 'd set \<Rightarrow> 'e"   and r :: "integer" assumes "0 < card N"   and "distinct \<Omega>"   and "a \<in> allAllocations N (set \<Omega>)" shows "sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel N (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg N \<Omega>)) r) <| (N \<times> finestpart (set \<Omega>))) N (set \<Omega>)) a = sum (resolvingBidAlg N \<Omega> b r) a"
axioms_2: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme" assumes "two_cats AA BB" shows "category BB"
ord0_leqD: fixes x :: "ord0"   and y :: "ord0"   and a :: "ord0" assumes "(x, y) \<in> ord0_leq"   and "(a, x) \<in> ord0_prec\<^sup>+" shows "(a, y) \<in> ord0_leq O ord0_prec\<^sup>+"
m_comm: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier S"   and "y \<in> carrier S" shows "x \<otimes>\<^bsub>S\<^esub> y = y \<otimes>\<^bsub>S\<^esub> x"
rec_by_index_inj1: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "rec_by_index x1 y1 = rec_by_index x2 y2" shows "x1 = x2"
Carmichael_numberI: fixes n :: "nat" assumes "\<not> prime n"   and "squarefree n"   and "1 < n"   and "\<And>p. p \<in># prime_factorization n \<Longrightarrow> p - 1 dvd n - 1" shows "Carmichael_number n"
is_natural_2: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "\<Phi> (C.VV.cod f) \<cdot>\<^sub>D F.H\<^sub>DoFF.map f = \<Phi> f"
assms: shows "low_equal s1 s2 \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> t1 = snd (fst (read_state_reg_instr instr s1)) \<and> t2 = snd (fst (read_state_reg_instr instr s2))"
content_ff_iff: fixes x :: "'b fract"   and p :: "'b fract poly" shows "divides_ff x (content_ff p) = (\<forall>c\<in>set (coeffs p). divides_ff x c)"
sub_sup_closed: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - y = - - (- x \<squnion> - y)"
sets_Collect_countable_Ex1': fixes I :: "'c set"   and M :: "'d measure"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>i. i \<in> I \<Longrightarrow> {x \<in> space M. P i x} \<in> sets M"   and "countable I" shows "{x \<in> space M. \<exists>!i. i \<in> I \<and> P i x} \<in> sets M"
hyps: fixes state :: "State"   and Q :: "Literal set"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"   and "uniq Wl'"   and "getM state = M @ [(opposite literal, decision)]"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) Q" shows "let state' = notifyWatches_loop literal Wl' newWl state in Let (Q \<union> (set (getQ state') - set (getQ state))) (InvariantGetReasonIsReason (getReason state') (getF state') (getM state'))"
Union_Un_distrib: fixes A :: "'c set set"   and B :: "'c set set" shows "\<Union> (A \<union> B) = \<Union> A \<union> \<Union> B"
dist_eq_0_iff: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.metric_space dist uniformity open" shows "(dist x y = 0) = (x = y)"
to_rat: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_rat :: "'a \<Rightarrow> bool"   and to_rat :: "'a \<Rightarrow> rat"   and x :: "'a" assumes "class.is_rat divide inverse times one plus zero minus uminus is_rat to_rat" shows "to_rat x = (if x \<in> field_char_0.Rats divide one plus zero minus then THE y. x = field_char_0.of_rat divide one plus zero minus y else 0)"
lt_lower_higher: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and u :: "'t" assumes "v \<prec>\<^sub>t lt (lower p u)" shows "lt (lower (higher p v) u) = lt (lower p u)"
hyps: fixes f :: "'a poly" assumes "primitive (prod_list fs)"   and "f \<in> set fs" shows "primitive f"
rComm2: fixes \<Psi>\<^sub>Q :: "'b"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* C" shows "Prop (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')"
antidom_test_comp: fixes x :: "'a" shows "a x = tc (x ; 1)"
rot_diamond_cube_is_type_II: shows "typeII_twoCube rot_diamond_cube"
test_s_prod_aux2: fixes a :: "'a"   and A :: "'a set"   and R :: "('a \<times> 'a set) set"   and P :: "('a \<times> 'a set) set" assumes "(a, A) \<in> R"   and "(a, {a}) \<in> P" shows "(a, A) \<in> P \<cdot> R"
enc_preserves_pred_iff_indRelL_reflects_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_preserves_pred Pred = rel_reflects_pred indRelL Pred"
comp_hd_def: fixes \<xi> :: "('s, 'v) hd"   and \<zeta> :: "('s, 'v) hd" shows "(\<xi> \<le>\<ge>\<^sub>h\<^sub>d \<zeta>) = (\<xi> = \<zeta> \<or> \<xi> >\<^sub>h\<^sub>d \<zeta> \<or> \<zeta> >\<^sub>h\<^sub>d \<xi>)"
P1: shows "Bet A D C' \<and> Cong D C' C D"
unit_simps_1: fixes a :: "'a" assumes "S.obj a" shows "S.arr \<i>[a]"
exhaust: fixes y :: "Formula.formula" obtains x11 :: "String.literal" and x12 :: "Formula.trm list" where "y = formula.Pred x11 x12"   | x21 :: "String.literal" and x22 :: "Formula.formula" and x23 :: "Formula.formula" where "y = formula.Let x21 x22 x23"   | x31 :: "Formula.trm" and x32 :: "Formula.trm" where "y = formula.Eq x31 x32"   | x41 :: "Formula.trm" and x42 :: "Formula.trm" where "y = formula.Less x41 x42"   | x51 :: "Formula.trm" and x52 :: "Formula.trm" where "y = formula.LessEq x51 x52"   | x6 :: "Formula.formula" where "y = formula.Neg x6"   | x71 :: "Formula.formula" and x72 :: "Formula.formula" where "y = formula.Or x71 x72"   | x81 :: "Formula.formula" and x82 :: "Formula.formula" where "y = formula.And x81 x82"   | x9 :: "Formula.formula list" where "y = formula.Ands x9"   | x10 :: "Formula.formula" where "y = formula.Exists x10"   | x111 :: "nat" and x112 :: "Formula.agg_type \<times> event_data" and x113 :: "nat" and x114 :: "Formula.trm" and x115 :: "Formula.formula" where "y = formula.Agg x111 x112 x113 x114 x115"   | x121 :: "\<I>" and x122 :: "Formula.formula" where "y = formula.Prev x121 x122"   | x131 :: "\<I>" and x132 :: "Formula.formula" where "y = formula.Next x131 x132"   | x141 :: "Formula.formula" and x142 :: "\<I>" and x143 :: "Formula.formula" where "y = formula.Since x141 x142 x143"   | x151 :: "Formula.formula" and x152 :: "\<I>" and x153 :: "Formula.formula" where "y = formula.Until x151 x152 x153"   | x161 :: "\<I>" and x162 :: "Formula.formula Regex.regex" where "y = formula.MatchF x161 x162"   | x171 :: "\<I>" and x172 :: "Formula.formula Regex.regex" where "y = formula.MatchP x171 x172"
arr_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" shows "S.arr (S.mkArr A B F) = (A \<subseteq> S.Univ \<and> |A| <o \<AA> \<and> B \<subseteq> S.Univ \<and> |B| <o \<AA> \<and> F \<in> A \<rightarrow> B)"
simps_2: fixes s :: "('a, 'b) tm"   and uu :: "('a, 'b) tm" shows "head0 (App s uu) = head0 s"
sorted_list_of_fmap_def: fixes m :: "('c, 'd) fmap" shows "sorted_list_of_fmap m = map (\<lambda>k. (k, the (fmlookup m k))) (sorted_list_of_fset (fmdom m))"
ordering_top_axioms: shows "ordering_top (\<le>) (<) top"
l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_val set_val_locs type_wf create_character_data known_ptr"
card_Diff_subset_Int: fixes A :: "'a set"   and B :: "'a set" assumes "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
wt_ParT_WtFT_notPickFT_0: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and i :: "nat"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" assumes "PL cval"   and "properL cl"   and "n < length cl"   and "i < PL.brn (cl ! n)"   and "WtFT cl = 1"   and "n = pickFT cl \<longrightarrow> i \<noteq> 0" shows "PL.wt aval tval cval (ParT cl) s (PL.brnL cl n + i) = 0"
sorted_insort_key: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and xs :: "'d list" shows "sorted (map f (insort_key f a xs)) = sorted (map f xs)"
RealPairVal_def: shows "RealPairVal \<equiv> \<lambda>(x, y). <|RealVal x, RealVal y|>"
explosion_not_valid: shows "\<not> valid (Imp' (Con' (Pro ''p'') (Neg' (Pro ''p''))) (Pro ''q''))"
hb_consistent_elim_2: fixes xs :: "'a list"   and y :: "'a" assumes "hb_consistent (xs @ [y])" obtains "hb_consistent xs" and "\<forall>x\<in>set xs. \<not> y \<prec> x"
Disj_mono: fixes A :: "fm"   and H :: "fm set"   and B :: "fm"   and C :: "fm"   and D :: "fm" assumes "insert A H \<turnstile> B"   and "insert C H \<turnstile> D" shows "insert (A OR C) H \<turnstile> B OR D"
subset_code_1: fixes xs :: "'x list"   and B :: "'x set" shows "(set xs \<subseteq> B) = (\<forall>x\<in>set xs. x \<in> B)"
fold_append_queue_rep: fixes xs :: "'a list"   and q :: "'a queue" shows "linearize (fold append_queue xs q) = linearize q @ xs"
pure_I_conv: fixes x :: "'d af" shows "pure (\<lambda>x. x) \<diamondop> x = x"
strong_induct: fixes P :: "'a \<Rightarrow> Lambda_Z.term \<Rightarrow> bool"   and c :: "'a"   and "term" :: "Lambda_Z.term" assumes "\<And>x c. P c (Var x)"   and "\<And>x1a x2 c. \<lbrakk>\<And>c. P c x1a; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (App x1a x2)"   and "\<And>x1a x2 c. \<lbrakk>{atom x1a} \<sharp>* c; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (Abs x1a x2)" shows "P c term"
normal_vector_1: fixes W :: "'a vec set" assumes "finite W"   and "Suc (card W) = n"   and "lin_indpt W"   and "W \<subseteq> carrier_vec n" shows "normal_vector W \<in> carrier_vec n"
F1map_comp_id: fixes g1 :: "'g \<Rightarrow> 'd"   and g2 :: "'h \<Rightarrow> 'e"   and g3 :: "'i \<Rightarrow> 'f"   and f2 :: "'j \<Rightarrow> 'h"   and f3 :: "'k \<Rightarrow> 'i"   and x :: "('g, 'j, 'k) F1" shows "F1map g1 g2 g3 (F1map id f2 f3 x) = F1map g1 (g2 \<circ> f2) (g3 \<circ> f3) x"
lr_of_tran_def: fixes rt :: "32 routing_rule list"   and fw :: "32 simple_rule list"   and ifs :: "char list list" shows "lr_of_tran rt fw ifs \<equiv> if \<not> (no_oif_match fw \<and> has_default_policy fw \<and> simple_fw_valid fw \<and> valid_prefixes rt \<and> has_default_route rt \<and> distinct ifs) then Inl ''Error in creating OpenFlow table: prerequisites not satisifed'' else let nrd = lr_of_tran_fbs rt fw ifs; ard = map (apfst word_of_nat) (annotate_rlen nrd) in if length nrd < unat (- 1) then Inr (pack_OF_entries ifs ard) else Inl ''Error in creating OpenFlow table: priority number space exhausted''"
takedropWhile_queue_snd: fixes f :: "'a \<Rightarrow> bool"   and q :: "'a queue" shows "snd (takedropWhile_queue f q) = takeWhile f (linearize q)"
is_path_undir_split_distinct: fixes v :: "'v"   and p :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v"   and a :: "'v"   and w :: "'w"   and b :: "'v" assumes "is_path_undir G v p v'"   and "(a, w, b) \<in> set p \<or> (b, w, a) \<in> set p" shows "\<exists>p' p'' u u'. is_path_undir G v p' u \<and> is_path_undir G u' p'' v' \<and> length p' < length p \<and> length p'' < length p \<and> (u \<in> {a, b} \<and> u' \<in> {a, b}) \<and> (a, w, b) \<notin> set p' \<and> (b, w, a) \<notin> set p' \<and> (a, w, b) \<notin> set p'' \<and> (b, w, a) \<notin> set p''"
transp_on_strict: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "transp_on P A" shows "transp_on (strict P) A"
IVT: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and y :: "'b"   and b :: "'c" assumes "f a \<le> y"   and "y \<le> f b"   and "a \<le> b"   and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x" shows "\<exists>x\<ge>a. x \<le> b \<and> f x = y"
assms_2: shows "a \<le> b \<or> b \<le> a"
InitNonObjectSuperRed: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and P :: "(char list list \<times> char list exp) prog"   and D :: "char list"   and r :: "fdecl list \<times> (char list list \<times> char list exp) mdecl list"   and sh' :: "sheap"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool" assumes "sh C = \<lfloor>(sfs, Prepared)\<rfloor>"   and "C \<noteq> Object"   and "class P C = \<lfloor>(D, r)\<rfloor>"   and "sh' = sh(C \<mapsto> (sfs, Processing))" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>INIT C' (D # C # Cs,False) \<leftarrow> e,(h, l, sh'),b\<rangle>"
map_definedness: fixes f :: "'c \<Rightarrow> 'b option"   and x :: "'c"   and y :: "'b" assumes "f x = Some y" shows "x \<in> dom f"
simps_2: fixes x :: "char list"   and e :: "aexp"   and P :: "'c" shows "strip (x ::= e {P}) = x ::= e"
replicate_upd_inj: fixes q :: "nat"   and n :: "nat"   and p :: "nat" assumes "q < n"   and "(replicate n False)[q := True] = (replicate n False)[p := True]" shows "q = p"
ring_tOp_assoc: fixes R :: "('d, 'e) Ring_scheme"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<cdot>\<^sub>r\<^bsub>R\<^esub> z = x \<cdot>\<^sub>r\<^bsub>R\<^esub> (y \<cdot>\<^sub>r\<^bsub>R\<^esub> z)"
strict_implies_order: fixes b :: "'g"   and a :: "'g" assumes "b < a" shows "b \<le> a"
in_Hom_iff: fixes f :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V" shows "(f \<in>\<^sub>\<circ> Hom \<CC> a b) = f : a \<mapsto>\<^bsub>\<CC>\<^esub> b"
selectlike_equal_cond: fixes X :: "(variable \<Rightarrow> real) set"   and \<nu> :: "variable \<Rightarrow> real"   and V :: "variable set"   and Y :: "(variable \<Rightarrow> real) set" shows "(selectlike X \<nu> V = selectlike Y \<nu> V) = (\<forall>\<mu>. Uvariation \<mu> \<nu> (- V) \<longrightarrow> (\<mu> \<in> X) = (\<mu> \<in> Y))"
mat_leqb_eqc_split_correct1: fixes b :: "rat vec"   and A :: "rat mat"   and c :: "rat vec"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec"   and l :: "nat" assumes "dim_vec b = dim_row A"   and "\<langle>X\<rangle> \<Turnstile>\<^sub>c\<^sub>s set (mat_leqb_eqc A b c)"   and "(x, y) = split_i_j_x (dim_col A) l X" shows "[A*\<^sub>vx]\<le>b"
IH: fixes x :: "nat"   and e :: "cexpr" shows "free_vars (cexpr_subst x e e') \<subseteq> (\<lambda>y. if x \<le> y then y + 1 else y) -` free_vars e' \<union> free_vars e"
simps_7: fixes opt :: "atom fm \<Rightarrow> atom fm"   and step :: "nat \<Rightarrow> (atom list \<times> atom fm list \<times> nat) list \<Rightarrow> atom fm"   and a :: "atom" shows "QE_dnf' opt step (fm.Atom a) = simp_atom a"
m0: shows "tensor_P (1\<^sub>m N) M0 * tensor_P (1\<^sub>m N) M0 = tensor_P (1\<^sub>m N) M0"
bit_push_bit_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "nat"   and a :: "'a"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "bit (push_bit m a) n = (m \<le> n \<and> power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> bit a (n - m))"
rcD_4: shows "\<exists>!x. root_cond (p, l, r) x"
step_lem2a: fixes list :: "'a set list"   and x1 :: "'a set"   and a :: "'a set"   and aa :: "'a attree"   and listb :: "'a attree list" assumes "0 < length list"   and "map (\<lambda>i. \<N>\<^bsub>((x1 # a # list) ! i, (a # list) ! i)\<^esub>) [0..<length list] @ [\<N>\<^bsub>((x1 # a # list) ! length list, (a # list) ! length list)\<^esub>] = aa # listb" shows "\<N>\<^bsub>(x1, a)\<^esub> = aa"
mono: fixes c :: "'c"   and a :: "'c"   and d :: "'c"   and b :: "'c" assumes "c \<le> a"   and "d \<le> b" shows "c \<squnion> d \<le> a \<squnion> b"
csum_mono2: fixes p2 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set"   and q :: "('f \<times> 'f) set" assumes "p2 \<le>o r2" shows "q +c p2 \<le>o q +c r2"
subdistl_eq: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z ; x + z ; (x + y) = z ; (x + y)"
map_disj_None_left': fixes h\<^sub>0 :: "'d \<Rightarrow> 'c option"   and x :: "'d"   and y :: "'c"   and h\<^sub>1 :: "'d \<Rightarrow> 'c option" assumes "h\<^sub>0 x = Some y"   and "h\<^sub>1 \<bottom> h\<^sub>0" shows "h\<^sub>1 x = None"
exhaust: fixes y :: "pdf_type" obtains "y = UNIT"   | "y = BOOL"   | "y = INTEG"   | "y = REAL"   | x51 :: "pdf_type" and x52 :: "pdf_type" where "y = PRODUCT x51 x52"
intros_4: fixes i :: "init_state" shows "i \<le>\<^sub>i i"
lm085: fixes x :: "'c"   and X :: "'c set"   and Y :: "'d set" assumes "x \<notin> X" shows "injections ({x} \<union> X) Y \<subseteq> (\<Union>f\<in>injections X Y. {f \<union> {(x, y)} |y. y \<in> Y - Range f})"
ocircline_01inf: fixes H :: "ocircline" assumes "0\<^sub>h \<in> ocircline_set H \<and> 1\<^sub>h \<in> ocircline_set H \<and> \<infinity>\<^sub>h \<in> ocircline_set H" shows "H = o_x_axis \<or> H = opposite_ocircline o_x_axis"
\<sigma>2l: shows "length (labels \<sigma>2') \<le> 1"
inverse_exists: fixes R :: "('g, 'h) ring_scheme"   and a :: "'g" assumes "field R"   and "a \<in> carrier R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>" shows "inv\<^bsub>R\<^esub> a \<in> carrier R"
Inc_base: shows "basevars (x, y, sem, pc1, pc2)"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b \<sqsubseteq> a"
rv_Y: fixes i :: "'b" shows "random_variable borel (Y i)"
scaleR_scaleR: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "real"   and b :: "real"   and x :: "'c" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR a (scaleR b x) = scaleR (a * b) x"
linearityD2: fixes E :: "(node fset \<times> command_gadget \<times> node fset) list" assumes "linearity E" shows "\<forall>e f. e \<in> set E \<and> f \<in> set E \<and> e \<noteq> f \<longrightarrow> fst3 e |\<inter>| fst3 f = {||} \<and> thd3 e |\<inter>| thd3 f = {||}"
minus_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 a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus a) b = less_eq (uminus b) a"
DERIV_Si: fixes x :: "real" shows "(Si has_real_derivative sinc x) (at x)"
rComm1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P (M\<lparr>N\<rparr>) P' A\<^sub>P \<Psi>\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q (K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle>) Q' A\<^sub>Q \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q') (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
rem_unPath_valid: fixes g :: "('a, 'b) graph"   and ps :: "('a \<times> 'b \<times> 'a) list" assumes "valid_unMultigraph g" shows "valid_unMultigraph (rem_unPath ps g)"
intros_1: fixes n :: "('form, 'rule) graph_node"   and p :: "'form"   and fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and s :: "'subst"   and i :: "nat"   and \<Gamma> :: "'form fset"   and ants :: "('form, 'rule, 'subst, 'var) itnode rose_tree list"   and c :: "'form" assumes "n \<in> sset nodes"   and "Reg p |\<in>| outPorts n"   and "list_all2 (\<lambda>ip t. iwf fc t ((\<lambda>h. subst s (freshen i (labelsOut n h))) |`| hyps_for n ip |\<union>| \<Gamma> \<turnstile> subst s (freshen i (labelsIn n ip)))) (inPorts' n) ants"   and "fc n i s (\<Gamma> \<turnstile> c)"   and "c = subst s (freshen i p)" shows "iwf fc (INode n p i s ants) (\<Gamma> \<turnstile> c)"
val_Zp_def: shows "val_Zp = (\<lambda>x. if x = \<zero> then \<infinity> else eint (padic_val p x))"