hcju/isabelleps huggingface dataset:
c123_1: shows "(fst (fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3), snd (fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3) @ snd (conv_mirror \<gamma>3)) \<in> conv ars"
cong: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" shows "finite_comm_group_adjoin.lift_character G H a \<equiv> finite_comm_group_adjoin.lift_character G H a"
simps_2: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and n :: "nat"   and \<rho> :: "complex mat" shows "denote_while_n_iter M0 M1 DS (Suc n) \<rho> = denote_while_n_iter M0 M1 DS n (DS (M1 * \<rho> * adjoint M1))"
conseq: shows "P \<subseteq> {s. \<exists>Z. s \<in> P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> t \<in> Q) \<and> (\<forall>t. t \<in> A' Z \<longrightarrow> t \<in> A)}"
augment_edge_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a" shows "augment_edge f \<equiv> \<lambda>(u, v) \<Delta>. if (u, v) \<in> E then f((u, v) := f (u, v) + \<Delta>) else if (v, u) \<in> E then f((v, u) := f (v, u) - \<Delta>) else f"
strips_equivalent_to_sas_plus_i_a_IV: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops :: "('c, 'd) sas_plus_operator list"   and I :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"   and "are_all_operators_applicable_in I ops \<and> SAS_Plus_Semantics.are_all_operator_effects_consistent ops" shows "are_all_operators_applicable (\<phi>\<^sub>S \<Psi> I) (map (sasp_op_to_strips \<Psi>) ops) \<and> STRIPS_Semantics.are_all_operator_effects_consistent (map (sasp_op_to_strips \<Psi>) ops)"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>gt. wfP (\<lambda>x y. gt y x) \<Longrightarrow> wfP (\<lambda>xs ys. ext gt ys xs)" shows "ext_wf_axioms ext"
comm_monoid_add_hom_axioms: fixes c :: "'c" shows "comm_monoid_add_hom ((*) c)"
eq: shows "(\<Sum>i\<le>p. if i < k then x i else if i = k then 0 else x (i - 1)) = sum x {..<k} + (\<Sum>i = k..p. if i = k then 0 else x (i - 1))"
assms_1: shows "expectation_gpv' \<equiv> expectation_gpv"
bex_empty: fixes P :: "'b \<Rightarrow> bool" shows "Bex {} P = False"
known_ptrs_new_ptr: fixes h' :: "(_) heap"   and h :: "(_) heap"   and new_ptr :: "(_) object_ptr" assumes "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|new_ptr|}"   and "known_ptr new_ptr"   and "known_ptrs h" shows "known_ptrs h'"
simps_1: shows "SkewBinomialHeapStruc.uniqify [] = []"
ccspan_union: fixes A :: "'a set"   and B :: "'a set" shows "ccspan A \<squnion> ccspan B = ccspan (A \<union> B)"
f3: shows "(if mmu_writable (get_acc_flag (snd (v1_2 (virt_to_phys addr (mmu (add_data_cache s2 addr data bm)) (mem (add_data_cache s2 addr data bm)))))) 10 then Some (mem_mod_w32 10 (fst (v1_2 (virt_to_phys addr (mmu (add_data_cache s2 addr data bm)) (mem (add_data_cache s2 addr data bm))))) bm data (add_data_cache s2 addr data bm)) else None) = (case Some (v1_2 (virt_to_phys addr (mmu (add_data_cache s2 addr data bm)) (mem (add_data_cache s2 addr data bm)))) of None \<Rightarrow> if mmu_writable (get_acc_flag (snd (v1_2 (virt_to_phys addr (mmu (add_data_cache s2 addr data bm)) (mem (add_data_cache s2 addr data bm)))))) 10 then Some (mem_mod_w32 10 (fst (v1_2 (virt_to_phys addr (mmu (add_data_cache s2 addr data bm)) (mem (add_data_cache s2 addr data bm))))) bm data (add_data_cache s1 addr data bm)) else None | Some p \<Rightarrow> if mmu_writable (get_acc_flag (snd p)) 10 then Some (mem_mod_w32 10 (fst p) bm data (add_data_cache s2 addr data bm)) else None)"
le_trans_equiv_rsp_right: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and v :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "equiv_trans t u" shows "le_trans v t = le_trans v u"
assms: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "f i \<subseteq>\<^sub>\<circ> C"
False: shows "fds_nth (f * g) (Suc 0) \<noteq> (0::'a)"
nat_relprime_power_divisors: fixes n :: "nat"   and a :: "nat"   and b :: "nat"   and c :: "nat" assumes "0 < n"   and "a * b = c ^ n"   and "coprime a b" shows "\<exists>k. a = k ^ n"
latestAct_prop_1: fixes n :: "nat"   and nid :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'. latestAct_cond nid t n n'" shows "\<parallel>nid\<parallel>\<^bsub>t \<langle>nid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>\<^esub>"
e: shows "internal_equivalence w' w \<psi> \<phi> \<and> \<guillemotleft>w : src s\<^sub>0 \<rightarrow> src r\<^sub>0\<guillemotright> \<and> \<guillemotleft>w' : src r\<^sub>0 \<rightarrow> src s\<^sub>0\<guillemotright> \<and> \<guillemotleft>\<theta> : r\<^sub>0 \<star> w \<Rightarrow> s\<^sub>0\<guillemotright> \<and> \<guillemotleft>\<nu> : s\<^sub>1 \<Rightarrow> r\<^sub>1 \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> \<sigma> = (s \<star> \<theta>) \<cdot> \<a>[s, r\<^sub>0, w] \<cdot> (\<rho> \<star> w) \<cdot> \<nu> \<and> \<guillemotleft>\<theta>' : s\<^sub>0 \<star> w' \<Rightarrow> r\<^sub>0\<guillemotright> \<and> \<guillemotleft>\<nu>' : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> w'\<guillemotright> \<and> local.iso \<nu>' \<and> \<rho> = (s \<star> \<theta>') \<cdot> \<a>[s, s\<^sub>0, w'] \<cdot> (\<sigma> \<star> w') \<cdot> \<nu>'"
mkarr_def: fixes t :: "'c local.term" shows "mkarr t = {u. local.equiv t u}"
six: shows "set_cod (u B) = F \<^bsub>\<o>\<^esub> B"
leg1_commutes: shows "\<nu>'.cod.leg1 \<cdot> \<nu>'.chine = \<nu>'.dom.leg1"
z_def: shows "z = (1 - v) *\<^sub>R x + v *\<^sub>R y"
4: shows "finite {w. llength w = enat (Suc n)}"
smult_l_null: fixes x :: "'c" assumes "x \<in> carrier V" shows "\<zero>\<^bsub>K\<^esub> \<odot>\<^bsub>V\<^esub> x = \<zero>\<^bsub>V\<^esub>"
distinct_26: fixes x14 :: "Syntax.term" shows "Unroll x14 \<noteq> Unit"
B_2: shows "independent B \<and> S \<subseteq> span B"
set_child_nodes_typess_preserved: fixes w :: "((_) heap, exception, unit) prog"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_child_nodes_locs object_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "DocumentClass.type_wf h = DocumentClass.type_wf h'"
affected_cycle: fixes as :: "'b list" shows "affected \<langle>as\<rangle> \<subseteq> set as"
assms_1: shows "convex (S \<inter> {x. a \<bullet> x = b})"
simps_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and ys :: "'a list" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less (x # ys) = (Ball (list.set ys) (less x) \<and> linorder.strict_sorted less ys)"
distinct_1: fixes x2 :: "'b" shows "ext0.Bot \<noteq> ext0.Val x2"
lderivs_One: fixes \<Sigma> :: "nat \<Rightarrow> 'c set"   and project :: "'c \<Rightarrow> 'c"   and embed :: "'c \<Rightarrow> 'c list"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "Pi_Derivatives.embed \<Sigma> project embed" shows "embed.lderivs lookup embed xs One \<in> {Zero, One}"
transform1_rev: fixes P :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<forall>\<tau>. \<tau> \<Turnstile> P" shows "P = true"
indicator_def: fixes S :: "'e set"   and x :: "'e" shows "indicator S x = (if x \<in> S then 1::'d else (0::'d))"
enabled_imp_UNTIL_alw_discrCf: fixes \<omega> :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream" assumes "enabled (shd \<omega>) (stl \<omega>)" shows "((\<lambda>xs. \<not> holds discrCf xs) until alw (holds discrCf)) \<omega>"
ereal_0_le_mult: fixes a :: "ereal"   and b :: "ereal" assumes "0 \<le> a"   and "0 \<le> b" shows "0 \<le> a * b"
contract_conj_inclusion: fixes Cn :: "'b set \<Rightarrow> 'b set"   and contraction :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b"   and \<psi> :: "'b" assumes "AGM_FullContraction Cn contraction true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "K = Cn A"   and "\<phi> \<notin> contraction K (conj_PL \<phi> \<psi>)" shows "contraction K (conj_PL \<phi> \<psi>) \<subseteq> contraction K \<phi>"
Map_cod: fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" assumes "A2_B.arr f" shows "A2_B.Map (A2_B.cod f) = A2_B.Cod f"
P5: shows "B' Out C'' C' \<and> Cong B' C'' B C"
l3_inv4_def: shows "l3_inv4 \<equiv> {s. Tags \<subseteq> ik s}"
simps_1: fixes x :: "Preliminaries.var" shows "TE.tpOf (Var x) = TE.tpOfV x"
fbb_def: shows "bb\<^sub>\<F> = fb\<^sub>\<F> \<circ> op\<^sub>K"
partition_on_quotient: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "equiv A r" shows "partition_on A (A // r)"
r_one: fixes x :: "'a \<Rightarrow> 'd" assumes "x \<in> carrier (DirProds (f \<circ> Gs) I)" shows "x \<otimes>\<^bsub>DirProds (f \<circ> Gs) I\<^esub> \<one>\<^bsub>DirProds (f \<circ> Gs) I\<^esub> = x"
set_iterator_union_def: fixes it_a :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and it_b :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" shows "set_iterator_union it_a it_b \<equiv> \<lambda>c f \<sigma>0. it_b c f (it_a c f \<sigma>0)"
hypsubst: fixes A :: "'g set"   and B :: "'g set"   and x :: "'g" assumes "A = B"   and "x \<in> B" obtains "x \<in> A"
retract_of_space_section_map: fixes S :: "'c set"   and X :: "'c topology" shows "(S retract_of_space X) = (S \<subseteq> topspace X \<and> section_map (subtopology X S) X id)"
helper2: fixes v :: "'a vec"   and ss :: "'a vec list"   and ls :: "'a vec list"   and nr :: "nat" assumes "v \<in> carrier_vec (length ss)"   and "\<And>x. x \<in> set ls \<Longrightarrow> dim_vec x = nr" shows "mat_of_cols nr ss *\<^sub>v v = mat_of_cols nr (ls @ ss) *\<^sub>v (0\<^sub>v (length ls) @\<^sub>v v)"
soundness_graphical_helper: fixes G :: "diagram"   and P :: "interface"   and Q :: "interface"   and C :: "command_gadget"   and A :: "assertion_gadget" assumes "\<And>c. wr_com c = {}" shows "(prov_dia G P Q \<longrightarrow> P = top_dia G \<and> Q = bot_dia G \<and> (\<forall>c. coms_dia G c \<longrightarrow> prov_triple (asn P, c, asn Q))) \<and> (prov_com C P Q \<longrightarrow> (\<forall>c. coms_com C c \<longrightarrow> prov_triple (asn P, c, asn Q))) \<and> (prov_ass A \<longrightarrow> (\<forall>c. coms_ass A c \<longrightarrow> prov_triple (asn (top_ass A), c, asn (bot_ass A))))"
simps: fixes ax :: "name"   and p :: "tm"   and v :: "name"   and w :: "name"   and x :: "name"   and x0 :: "name"   and xw :: "name"   and xevw :: "name"   and allw :: "name"   and allvw :: "name" assumes "atom ax \<sharp> (p, v, w, x, x0, xw, xevw, allw, allvw)"   and "atom allvw \<sharp> (p, v, w, x, x0, xw, xevw, allw)"   and "atom allw \<sharp> (p, v, w, x, x0, xw, xevw)"   and "atom xevw \<sharp> (p, v, w, x, x0, xw)"   and "atom xw \<sharp> (p, v, w, x, x0)"   and "atom x0 \<sharp> (p, v, w, x)"   and "atom x \<sharp> (p, v, w)"   and "atom w \<sharp> (p, v)"   and "atom v \<sharp> p" shows "Induction_axP p = SyntaxN.Ex v (SyntaxN.Ex w (SyntaxN.Ex x (SyntaxN.Ex x0 (SyntaxN.Ex xw (SyntaxN.Ex xevw (SyntaxN.Ex allw (SyntaxN.Ex allvw (SyntaxN.Ex ax (Var v NEQ Var w AND VarNonOccFormP (Var w) (Var x) AND SubstFormP (Var v) Zero (Var x) (Var x0) AND SubstFormP (Var v) (Var w) (Var x) (Var xw) AND SubstFormP (Var v) (Q_Eats (Var v) (Var w)) (Var x) (Var xevw) AND AbstFormP (Var w) Zero (Q_Imp (Var x) (Q_Imp (Var xw) (Var xevw))) (Var allw) AND AbstFormP (Var v) Zero (Q_All (Var allw)) (Var allvw) AND AbstFormP (Var v) Zero (Var x) (Var ax) AND p EQ Q_Imp (Var x0) (Q_Imp (Q_All (Var allvw)) (Q_All (Var ax))))))))))))"
plus_eq_zero: fixes s :: "'a"   and t :: "'a" assumes "s + t = (0::'a)" shows "s = (0::'a)"
g_def: shows "g \<equiv> \<lambda>xs. f (thd xs) (ttl xs) (TCons (thd xs) (ttl xs))"
rep_eq: fixes x :: "'c \<Rightarrow> 'b"   and xa :: "'c pdevs" shows "pdevs_apply (unop_pdevs x xa) = unop_pdevs_raw x (pdevs_apply xa)"
norm_adj: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "norm (A*) = norm A"
agree_UNIV_eq: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<omega> :: "(real, 'a) vec \<times> (real, 'a) vec" assumes "Vagree \<nu> \<omega> UNIV" shows "\<nu> = \<omega>"
hyps_1: fixes I :: "'a set" assumes "n = card I"   and "finite I" shows "sum_list (map f (list_of I)) = sum f I"
"1_2": shows "\<forall>x\<in>fun_vars p. (0 < fun_coeff p x \<longrightarrow> (0::'a) \<le> v x) \<and> (fun_coeff p x < 0 \<longrightarrow> v x \<le> (0::'a))"
cadjoint_is_cadjoint: fixes G :: "'b \<Rightarrow> 'c" assumes "bounded_clinear G" shows "is_cadjoint (G\<^sup>\<dagger>) G"
Bet4_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A1 :: "'a"   and A2 :: "'a"   and A3 :: "'a"   and A4 :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Bet4 Bet A1 A2 A3 A4 \<equiv> Bet A1 A2 A3 \<and> Bet A2 A3 A4 \<and> Bet A1 A3 A4 \<and> Bet A1 A2 A4"
add1: fixes A :: "'b"   and k :: "'b"   and a :: "'b"   and b :: "'b" assumes "A \<equiv> k + a" shows "A + b \<equiv> k + (a + b)"
Fun_canonical: fixes v :: "trm"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty" assumes "[] \<turnstile> v : T\<^sub>1 \<rightarrow> T\<^sub>2"   and "val v" shows "\<exists>x t S. v = (\<lambda>x:S. t)"
alphaAbs_preserves_qGoodAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs"   and A' :: "('a, 'b, 'varSort, 'var, 'c) qAbs" assumes "A $= A'" shows "qGoodAbs A = qGoodAbs A'"
rep_inj: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rep_nat :: "'b \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat"   and "rep_nat x = rep_nat y" shows "x = y"
vrange_atE: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vint_mult" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vint_mult" and "vint_mult\<lparr>a\<rparr> = b"
assms_2: fixes a :: "'a" assumes "a \<in> carrier R" shows "f a \<in> carrier (poly_ring R)"
Snd: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<langle>X, Y\<rangle> \<in> analz H" shows "Y \<in> analz H"
X: shows "dist x\<^sub>0 x\<^sub>1 \<le> \<delta>"
f2: shows "\<pi>\<^sup>* (\<pi>\<^sub>* y) \<sqsubseteq>\<^bsub>\<Y>\<^esub> y"
crb_lem_neg: fixes p :: "real poly"   and x :: "real" assumes "poly p x = 0"   and "p \<noteq> 0" shows "real_of_int (- crb p) < x"
dghm_ArrMap_is_arr: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> : \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>"
mat_of_row_carrier_1: fixes y :: "'a Matrix.vec"   and n :: "nat" assumes "y \<in> carrier_vec n" shows "mat_of_row y \<in> carrier_mat 1 n"
Ai: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> ZFC_in_HOL.set {j}" shows "A\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
abs_in_Basis_absE: fixes y :: "'c" assumes "\<bar>y\<bar> \<in> Basis" obtains "\<bar>y\<bar> = y"   | "\<bar>y\<bar> = - y"
ex: fixes c :: "real"   and d :: "real" assumes "a \<le> c"   and "c \<le> b"   and "g2 c \<le> d"   and "d \<le> g1 c" shows "\<exists>z\<in>unit_cube. (c, d) = (case z of (x, y) \<Rightarrow> (a - x * a + x * b, g2 (a - x * a + x * b) - y * g2 (a - x * a + x * b) + y * g1 (a - x * a + x * b)))"
openin_trans: fixes T :: "'a set"   and S :: "'a set"   and U :: "'a set" assumes "openin (top_of_set T) S"   and "openin (top_of_set U) T" shows "openin (top_of_set U) S"
assms_3: assumes "\<not> b"   and "\<not> b'" shows "S2 \<le> \<Down> R S2'"
c_sum_def: fixes u :: "nat" shows "c_sum u = (LEAST z. u < sf (z + 1))"
wf_formula_def: fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>" shows "formula.wf_formula \<Sigma> n \<phi> \<equiv> formula.pre_wf_formula \<Sigma> n \<phi> \<and> FOV \<phi> \<inter> SOV \<phi> = {}"
abs_range_map_def: shows "abs_range_map \<equiv> map_of (zip abs_ast_variable_section (map ((\<lambda>vals. [0..<length vals]) \<circ> snd \<circ> snd) astDom))"
that: fixes i :: "'a" shows "f i \<in> topspace (Y i) \<rightarrow> X"
low_proc_no_input_change_def: fixes s\<^sub>L :: "stmt" shows "low_proc_no_input_change s\<^sub>L \<equiv> \<forall>\<sigma> \<sigma>'. match \<sigma> \<Gamma>\<^sub>0 \<and> s\<^sub>L \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<longrightarrow> the (\<sigma>' ''lowIn'') = the (\<sigma> ''lowIn'') \<and> the (\<sigma>' ''highIn'') = the (\<sigma> ''highIn'')"
SN_on_subset1: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and s :: "('b \<times> 'b) set" assumes "SN_on r A"   and "s \<subseteq> r" shows "SN_on s A"
one_more_paths_exists: shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = Suc sep_size"
lm086: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "Graph f ,, x = f x"
atomic_step_ev_signal_def: fixes tid :: "thread_id_t"   and partner :: "thread_id_t"   and s :: "state_t" shows "atomic_step_ev_signal tid partner s = s\<lparr>thread := (thread s)(partner := thread s partner\<lparr>ev_counter := Suc (ev_counter (thread s partner))\<rparr>)\<rparr>"
closure_conj_distr: fixes P :: "'a upred"   and Q :: "'a upred" shows "([P]\<^sub>u \<and> [Q]\<^sub>u) = [P \<and> Q]\<^sub>u"
bind_lunit: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b option" shows "Some x \<bind> f = f x"
top_neq_ennreal: fixes r :: "real" shows "top_class.top \<noteq> ennreal r"
elims_2: fixes x :: "nat"   and xa :: "nat"   and xb :: "cell list \<times> cell list" assumes "wadjust_goon_left_moving x xa xb" obtains m :: "nat" and rs :: "nat" and l :: "cell list" and r :: "cell list" where "x = m" and "xa = rs" and "xb = (l, r)" and "wadjust_goon_left_moving_B m rs (l, r) \<or> wadjust_goon_left_moving_O m rs (l, r)"
div_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {arith_type UDIV, arith_type UDIVcc, arith_type SDIV}" shows "snd (div_instr i s) = False"
totient_imp_prime: fixes p :: "nat" assumes "totient p = p - 1"   and "0 < p" shows "prime p"
mult_ac_3: fixes b :: "'d"   and a :: "'d"   and c :: "'d" shows "b * (a * c) = a * (b * c)"
assms: shows "low_equal s1 s2 \<and> Some t1 = memory_write 10 addr bm data s1"
Abs_finfun_inverse: fixes y :: "'d \<Rightarrow> 'e" assumes "y \<in> finfun" shows "($) (Abs_finfun y) = y"
Quotient_cr_rel: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "Quotient R Abs Rep T" shows "T = (\<lambda>x y. R x x \<and> Abs x = y)"
sum_weights_append: fixes ys :: "'sum list"   and xs :: "'sum list" shows "sum_weights (ys @ xs) = followed_by (sum_weights ys) (sum_weights xs)"
horiz_vector_helper_pos: fixes p :: "real poly"   and qs :: "real poly list"   and x :: "real"   and sign :: "rat list"   and I :: "nat list" assumes "p \<noteq> 0"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec_copr qs x"   and "list_constr I (length qs)" shows "(0 < poly (prod_list (retrieve_polys qs I)) x) = (z I sign = 1)"
dim_def: fixes V :: "('b, 'c) vec set" shows "vec.dim V = (if \<exists>b. vec.independent b \<and> vec.span b = vec.span V then card (SOME b. vec.independent b \<and> vec.span b = vec.span V) else 0)"
nths_empty: fixes xs :: "'b list" shows "nths xs {} = []"
index_from_with3: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set" assumes "[[a .. b .. c]X]" shows "\<exists>f. (f 0 = a \<or> f 0 = c) \<and> ch_by_ord f X"
Rep_game_inverse: fixes x :: "game" shows "Abs_game (Rep_game x) = x"
distinct_5: shows "X4.Xd1 \<noteq> Xd2'"
cong_scalar_right: 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 b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo (times b d) (times c d) a"
right_neutral: fixes a :: "'c" shows "a AND - (1::'c) = a"
m_prefers_def: fixes m :: "'m"   and \<mu> :: "('m \<times> 'w) set" shows "m_prefers m \<mu> = {w' \<in> Field (Pm m). \<forall>w\<in>{w. (m, w) \<in> \<mu>}. (w, w') \<in> Pm m}"
assms_18: shows "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"
simps_1: fixes a :: "'b set"   and b :: "'b set" shows "trivial_series2 a b 0 = a"
PV: shows "pathVertices s p = pv1 @ u # pv2"
sel_2: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a stream" shows "stl (sfilter P s) = sfilter P (stl (sdrop_while (Not \<circ> P) s))"
f: fixes x :: "'a" assumes "x \<in> carrier L" shows "f x \<in> carrier L"
index_mat_addrow_mat_2: fixes n :: "nat"   and a :: "'a"   and k :: "nat"   and l :: "nat" shows "dim_row (addrow_mat n a k l) = n"
distinct_53: fixes x6 :: "(char list \<times> trm) list"   and x81 :: "pat"   and x82 :: "trm"   and x83 :: "trm" shows "Rcd x6 \<noteq> (LET x81 = x82 IN x83)"
pointfree_idE: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "f \<circ> g = id" shows "f (g x) = x"
complex_cnj_minus: fixes x :: "complex" shows "cnj (- x) = - cnj x"
af_identity: fixes x :: "'d af" shows "pure id \<diamondop> x = x"
vbrelation_vinE: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> r" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>" and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
58: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> y) \<squnion> - (x \<sqinter> - - y) = - (x \<sqinter> - - y)"
conf_trpl_ex: fixes m :: "nat"   and lm :: "'b"   and stp :: "nat" shows "\<exists>p q r. conf m (bl2wc (<lm>)) stp = trpl p q r"
k_def_1: shows "inst.tab (inst s ! i) = Some k'"
Abss_Cons: fixes x :: "char list \<times> Unary_PCF.type"   and xs :: "(char list \<times> Unary_PCF.type) list"   and b :: "expr" shows "\<Lambda>[(x # xs)] b = \<Lambda>\<langle>snd x\<rangle> close0_Var x (\<Lambda>[xs] b)"
c: fixes S :: "('a, 'b) Term.term list"   and P :: "'a \<Rightarrow> 'a list"   and T :: "('a, 'b) Term.term list" assumes "S \<in> set (U P T)" shows "length T = length S"
fresh_pgwt_has_fresh_const: fixes S :: "'fun set"   and t :: "('fun, 'var) Term.term" assumes "finite S"   and "wf\<^sub>t\<^sub>r\<^sub>m t" obtains c :: "'fun" where "Fun c [] \<sqsubseteq> fresh_pgwt S (\<Gamma> t)" and "c \<notin> S"
one_dim_iso_times: fixes \<psi> :: "'b"   and \<phi> :: "'b" shows "one_dim_iso (\<psi> * \<phi>) = one_dim_iso \<psi> * one_dim_iso \<phi>"