hcju/isabelleps huggingface dataset:
h1: shows "set (characterize_consistent_signs_at_roots p (qs1 @ qs2)) \<subseteq> set (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2))))"
list_rel_simp_3: fixes R :: "('a \<times> 'b) set" shows "([], []) \<in> \<langle>R\<rangle>list_rel"
SUP_image: fixes Sup :: "'b set \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "Sup (g ` f ` A) = Sup ((g \<circ> f) ` A)"
tuple_in_hom: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : a \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>HF'.tuple f g : a \<rightarrow> HF'.prod b c\<guillemotright>"
synWhileSC_def: fixes e :: "('c, 'd) Expr" shows "synWhileSC e \<equiv> \<exists>D. \<turnstile>\<^bsub>\<E>\<^esub> e : D \<and> (\<forall>d\<in>D. \<forall>d'. d \<le> d')"
alphaAbs_ex_equal_or_qFresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "alphaAbs_ex_equal_or_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. (y = x \<or> qFresh xs y X) \<and> (y = x' \<or> qFresh xs y X') \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
hyps_3: shows "ideal_generated ((*) u ` S) = ideal_generated S"
stable_rank_succeed: fixes i :: "nat"   and x :: "nat" assumes "infinite (succeed i)"   and "x \<in> succeed i"   and "q\<^sub>0 \<notin> F" shows "\<not> stable_rank x i"
intros_6: shows "CK_nf_pos (I \<circ> K \<circ> I)"
induct: fixes g :: "'g"   and x1 :: "'node list"   and x2 :: "'val \<Rightarrow> 'node option"   and P :: "'node list \<Rightarrow> ('val \<Rightarrow> 'node option) \<Rightarrow> bool" assumes "g \<turnstile> x1\<Down>\<^sub>s x2"   and "P [Entry g] (ssaStep g (Entry g) 0 Map.empty)"   and "\<And>ns s m i. \<lbrakk>g \<turnstile> ns\<Down>\<^sub>s s; P ns s; last ns = old.predecessors g m ! i; m \<in> set (\<alpha>n g); i < length (old.predecessors g m)\<rbrakk> \<Longrightarrow> P (ns @ [m]) (ssaStep g m i s)" shows "P x1 x2"
finite_empty_induct: fixes A :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "finite A"   and "P A"   and "\<And>a A. \<lbrakk>finite A; a \<in> A; P A\<rbrakk> \<Longrightarrow> P (A - {a})" shows "P {}"
sign_def: fixes p :: "'d \<Rightarrow> 'd" shows "sign p = (if evenperm p then 1 else - 1)"
bet_neq12__neq: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "A \<noteq> B" shows "A \<noteq> C"
map_add_Some_iff: fixes m :: "'b \<Rightarrow> 'a option"   and n :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and x :: "'a" shows "((m ++ n) k = Some x) = (n k = Some x \<or> n k = None \<and> m k = Some x)"
is_nextElem_or: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a"   and x :: "'a"   and y :: "'a" assumes "pre_between vs ram1 ram2"   and "is_nextElem vs x y" shows "is_sublist [x, y] (ram1 # between vs ram1 ram2 @ [ram2]) \<or> is_sublist [x, y] (ram2 # between vs ram2 ram1 @ [ram1])"
pre_split_face_def: fixes oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and newVertexList :: "nat list" shows "pre_split_face oldF ram1 ram2 newVertexList \<equiv> distinct (vertices oldF) \<and> distinct newVertexList \<and> \<V> oldF \<inter> set newVertexList = {} \<and> ram1 \<in> \<V> oldF \<and> ram2 \<in> \<V> oldF \<and> ram1 \<noteq> ram2"
x0_le_x1: shows "x\<^sub>0 \<le> x\<^sub>1"
hom_component_def: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" shows "hom_component p n = except p {t. deg_pm t \<noteq> n}"
assms: shows "(add_mset x X, Y) \<in> multpw ns"
vector_matrix_mult_0_right: fixes x :: "('a, 'c) vec" shows "x v* 0 = 0"
pos_val: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and pos :: "'a \<Rightarrow> int"   and z :: "int" assumes "class.spark_enum less_eq less pos"   and "z \<in> range pos" shows "pos (spark_enum.val pos z) = z"
S1: fixes x :: "i" assumes "AxiomsSet6" shows "E x \<^bold>\<leftarrow> E (dom x)"
Diagonalize_Comp_Cod_Arr: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Cod t \<^bold>\<cdot> t\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
PTrans: shows "P \<Longrightarrow>\<^sub>ly in P''\<rightarrow>a<x> \<prec> P'"
consistent_drop_block: fixes A :: "'b set"   and block :: "('c, 'b) fm list \<times> 'b"   and S :: "(('c, 'b) fm list \<times> 'b) set" assumes "consistent A ({block} \<union> S)" shows "consistent A S"
conv_terminating: fixes x :: "'a" shows "terminating x = terminating (x\<^sup>T)"
that: fixes q :: "'a set set" assumes "q division_of cbox a b \<union> \<Union> p" shows "thesis"
strip_pfp: fixes g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'a"   and x0 :: "'a"   and x :: "'a" assumes "\<And>x. g (f x) = g x"   and "pfp f x0 = Some x" shows "g x = g x0"
soundness': fixes s :: "(nat \<times> form) list"   and m :: "nat"   and h :: "nat" assumes "init s"   and "finite (deriv s)"   and "m \<in> fst ` deriv s"   and "\<forall>y u. (y, u) \<in> deriv s \<longrightarrow> y \<le> m" shows "\<forall>n t. h = m - n \<and> (n, t) \<in> deriv s \<longrightarrow> Svalid (s_of_ns t)"
par_symmetry: fixes A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "A B Par C D" shows "C D Par A B"
rule_charnDAFT: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and r :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "wellformed_policy1_strong p"   and "allNetsDistinct p"   and "singleCombinators p"   and "wellformed_policy3Pr p"   and "applied_rule_rev Cp x p = \<lfloor>DenyAllFromTo a b\<rfloor>"   and "r \<in> set (tl p)"   and "x \<in> dom (Cp r)" shows "r = DenyAllFromTo a b"
mult_scalar_zero_left: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and p :: "'b \<Rightarrow>\<^sub>0 'e" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_scalar pair_of_term term_of_pair 0 p = 0"
InitState_CompFun_InitConf: fixes S :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and SA :: "('d, 'e, 'f) seqauto" assumes "S \<in> HAStates A"   and "SA \<in> the (CompFun A S)"   and "S \<in> InitConf A" shows "InitState SA \<in> InitConf A"
nrows_notzero: fixes A :: "'a matrix"   and m :: "nat"   and n :: "nat" assumes "Rep_matrix A m n \<noteq> (0::'a)" shows "m < nrows A"
curryE: fixes f :: "'a \<times> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" assumes "curry f a b" obtains "f (a, b)"
h3: assumes "Av = 0"   and "Bv = 0" shows "(Av + Bv * sqrt Cv < 0) = (Av < 0 \<and> Bv\<^sup>2 * Cv < Av\<^sup>2 \<or> Bv \<le> 0 \<and> (Av < 0 \<or> Av\<^sup>2 < Bv\<^sup>2 * Cv))"
map_default_delete: fixes b :: "'b"   and f :: "'a \<Rightarrow> 'b option"   and a :: "'a" shows "map_default b (f(a := None)) = (map_default b f)(a := b)"
equiv_def: fixes x :: "'a"   and y :: "'a" shows "(x \<approx> y) = (x \<le> y \<and> y \<le> x)"
law: fixes X :: "'a" assumes "P X" shows "f (g X) = h (f X) \<and> P (g X)"
simps_4: fixes S :: "'c ltl set" shows "eval\<^sub>G S true = true"
upd_def: fixes i :: "nat"   and x :: "'b"   and a :: "'b array" shows "(a.(i) \<leftarrow> x) = Heap_Monad.guard (\<lambda>h. i < Array.length h a) (\<lambda>h. (a, Array.update a i x h))"
l_m: fixes a :: "'a" assumes "a \<in> carrier A" shows "-\<^sub>a a \<plusminus> a = \<zero>"
FG_advice_inter_subformulas: fixes \<phi> :: "'b ltln"   and Y :: "'b ltln set" shows "\<phi>[Y \<inter> subformulas\<^sub>\<nu> \<phi>]\<^sub>\<mu> = \<phi>[Y]\<^sub>\<mu>"
simps_4: fixes a :: "'b"   and uw :: "'b tree list" shows "cost (Insert a) uw = 1"
minus_1_power_ring_neq_zero: fixes n :: "nat" shows "(- (1::'b)) ^ n \<noteq> (0::'b)"
degree_smult_le: fixes a :: "'b"   and p :: "'b poly" shows "degree (smult a p) \<le> degree p"
CM2_S2TTP_analz_knows_Spy: fixes R :: "agent"   and A :: "agent"   and B :: "agent"   and em :: "msg"   and AO :: "nat"   and cleartext :: "nat"   and q :: "nat"   and S2TTP :: "msg"   and evs :: "event list" assumes "Gets R \<lbrace>Agent A, Agent B, em, Number AO, Number cleartext, Nonce q, S2TTP\<rbrace> \<in> set evs"   and "evs \<in> certified_mail" shows "S2TTP \<in> analz (knows Spy evs)"
strip_guards_simps_9: fixes F :: "'f set"   and f :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "strip_guards F (Guard f g c) = (if f \<in> F then strip_guards F c else Guard f g (strip_guards F c))"
fps_compose_power: fixes c :: "'b fps"   and a :: "'b fps"   and n :: "nat" assumes "c $ 0 = (0::'b)" shows "(a oo c) ^ n = a ^ n oo c"
vec_nth_cases: fixes y :: "'d \<Rightarrow> 'e" assumes "y \<in> UNIV" obtains x :: "('e, 'd) Finite_Cartesian_Product.vec" where "y = ($h) x"
bin_lcp_short: shows "\<^bold>|\<alpha>\<^bold>| < \<^bold>|u\<^sub>0\<^bold>| + \<^bold>|u\<^sub>1\<^bold>|"
l_set_mode_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_mode_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<equiv> l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs \<and> l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
weakFreshDerivative: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"   and "x \<sharp> P"   and "x \<sharp> \<alpha>"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "distinct (bn \<alpha>)" shows "x \<sharp> P'"
not_strong_postdominate_successor_set: fixes n :: "'node"   and a :: "'edge"   and N :: "'node set" assumes "\<not> n strongly-postdominates sourcenode a"   and "valid_node n"   and "valid_edge a"   and "\<forall>nx\<in>N. \<exists>a'. valid_edge a' \<and> sourcenode a' = sourcenode a \<and> targetnode a' = nx \<and> n strongly-postdominates nx" obtains a' :: "'edge" where "valid_edge a'" and "sourcenode a' = sourcenode a" and "targetnode a' \<notin> N"
dist_eqs_2: fixes a :: "'a"   and slist :: "[:'a:]" shows "a :# slist \<noteq> [::]"
simps_1: fixes x :: "hmultiset multiset"   and ya :: "hmultiset multiset" shows "(HMSet x = HMSet ya) = (x = ya)"
chart_last_domainP_chart_last_inv: fixes x :: "'b" shows "chart_last_domainP (chart_last_inv x)"
pred_map: fixes Q :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d option" shows "pred_option Q (map_option f x) = pred_option (Q \<circ> f) x"
elims: fixes p :: "integer"   and x :: "integer"   and xa :: "integer"   and y :: "integer" assumes "power_p_integer p x xa = y" obtains xb :: "integer" and n :: "integer" where "x = xb" and "xa = n" and "y = (if n \<le> 0 then 1 else let rec = power_p_integer p (mult_p_integer p xb xb) (n >> 1) in if n AND 1 = 0 then rec else mult_p_integer p rec xb)"
aaba: shows "(aa, ba) = mk_v I (OVar vid1) (sol 0, b) (sol t)"
insert_sort_by_perm: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a list" shows "insert_sort_by cmp l <~~> l"
zero_prefix_match_all: fixes m :: "'h prefix_match"   and ip :: "'h word" assumes "valid_prefix m"   and "pfxm_length m = 0" shows "prefix_match_semantics m ip"
aa: shows "take l (snd (Strat ! x)) = take l (snd ((take x Strat @ (fst (Strat ! x), take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) # drop (Suc x) Strat) ! x))"
WR_iff_eq_W: fixes x :: "hf"   and y :: "hf"   and e :: "(name, hf) finfun" assumes "Ord x" shows "WR x y = (y = \<lbrakk>W x\<rbrakk>e)"
simps: fixes a1 :: "('fun, 'var) Term.term set"   and a2 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and a3 :: "('fun, 'var) extstrand_step list" shows "sem\<^sub>e\<^sub>s\<^sub>t_d a1 a2 a3 = ((\<exists>M\<^sub>0 \<I>. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = []) \<or> (\<exists>M\<^sub>0 \<I> S t. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = S @ [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)] \<and> sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S \<and> ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>) \<or> (\<exists>M\<^sub>0 \<I> S t. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = S @ [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)] \<and> sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S) \<or> (\<exists>M\<^sub>0 \<I> S t t' a. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = S @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t] \<and> sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S \<and> Unifier \<I> t t') \<or> (\<exists>M\<^sub>0 \<I> S X F. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = S @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t] \<and> sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S \<and> ineq_model \<I> X F) \<or> (\<exists>M\<^sub>0 \<I> S t K M. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = S @ [Decomp t] \<and> sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S \<and> ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I> \<and> Ana t = (K, M) \<and> (\<forall>x. x \<in> set K \<longrightarrow> ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> x \<cdot> \<I>)))"
simps_9: fixes l :: "formula list" shows "future_bounded (Ands l) = list_all future_bounded l"
dual_adjoint_equivalence: shows "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg e' e (local.inv \<psi>') (local.inv \<phi>)"
UN_equiv_class: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'c set"   and a :: "'b" assumes "equiv A r"   and "f respects r"   and "a \<in> A" shows "\<Union> (f ` r `` {a}) = f a"
succg_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>succg_rel \<equiv> (R \<rightarrow> \<langle>R\<rangle>list_set_rel) O br E_of_succ (\<lambda>_. True)"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "gcd b (gcd a c) = gcd a (gcd b c)"
div_int_unique: fixes k :: "int"   and l :: "int"   and q :: "int"   and r :: "int" assumes "eucl_rel_int k l (q, r)" shows "k div l = q"
shortest_augmenting_path_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity" shows "shortest_augmenting_path f = (SOME p. Graph.isShortestPath (cf_of f) s p t)"
sg1: shows "split_time s (Suc k) (Suc k * i) = s i"
assms: fixes x1 :: "'a1"   and x2 :: "'a2" assumes "\<And>y1 y2. h1 y1 < h1 x1 \<Longrightarrow> phi y1 y2"   and "\<And>y1 y2. \<lbrakk>h1 y1 \<le> h1 x1; h2 y2 < h2 x2\<rbrakk> \<Longrightarrow> phi y1 y2" shows "phi x1 x2"
allDefs_narrows: fixes v :: "'val"   and n :: "'node" assumes "v \<in> step.allDefs g n" shows "v \<in> allDefs g n"
step: shows "\<Gamma>\<turnstile> (cs, pcss @ css, s) \<rightarrow> (cs', pcss' @ css, t)"
D: shows "ExtAgrK (S (Card n, n, run)) = Some e"
A_def: shows "A = {x \<in> space M. f x < f' x}"
arithmetic_simps_78: fixes a :: "'b" shows "(1::'b) * a = a"
const: fixes name :: "name"   and C_info :: "(name, dt_def) fmap"   and \<Gamma> :: "(name, value) fmap"   and val :: "value" assumes "name |\<notin>| constructors.C C_info"   and "fmlookup \<Gamma> name = Some val" shows "special_constants.veval' C_info \<Gamma> (Sconst name) val"
psubset_imp_subset: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subset> B" shows "A \<subseteq> B"
continuous_at_imp_continuous_within: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "isCont f x" shows "continuous (at x within s) f"
opp_chamberD2: fixes C :: "'a set" assumes "C \<in> f \<turnstile> folding.\<C>" shows "f ` (THE D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = C) = C"
sum_list_zeroI': fixes xs :: "'b list" assumes "\<And>i. i < length xs \<Longrightarrow> xs ! i = (0::'b)" shows "sum_list xs = (0::'b)"
Inr_dec_interp_length: fixes P :: "nat set"   and n :: "nat"   and FO :: "nat set"   and w :: "('a \<times> bool list) list"   and p :: "nat" assumes "Inr P \<in> set (dec_interp n FO w)"   and "p \<in> P" shows "p < length w"
Base_accessible: fixes P :: "pname" shows "tprg \<turnstile> Class Base accessible_in P"
simps_1: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and s :: "nat \<Rightarrow> nat option" shows "simulate_plan' problem [] s = Some s"
cases: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and a :: "('b, 'c, 'a, 'd) seqp" assumes "ctermsp \<Gamma> a" obtains p :: "('b, 'c, 'a, 'd) seqp" and pn :: "'a" where "a = p__" and "p__ \<in> sterms \<Gamma> (\<Gamma> pn)"   | pp :: "('b, 'c, 'a, 'd) seqp" and p :: "('b, 'c, 'a, 'd) seqp" where "a = p" and "ctermsp \<Gamma> pp" and "p \<in> dterms \<Gamma> pp"
IH_1: assumes "j \<le> ia" shows "reachable (execConf j) (execConf ia)"
sharing_consistent_mono_shared: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "dom \<S> \<subseteq> dom \<S>'"   and "sharing_consistent \<S> \<O> sb" shows "sharing_consistent \<S>' \<O> sb"
poly_inf_0_iff_2: fixes p :: "'a poly" shows "(poly_neg_inf p = (0::'a)) = (p = 0)"
compatible_def: fixes G :: "('a, 'b) pre_digraph"   and H :: "('a, 'b) pre_digraph" shows "compatible G H \<equiv> tail G = tail H \<and> head G = head H"
axioms_1: fixes get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool" assumes "l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf" shows "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"
finite_pos_order: assumes "finite (carrier G)" shows "0 < order G"
decseq_SucD: fixes A :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "decseq A" shows "A (Suc i) \<le> A i"
assms: shows "subst_domain \<theta> \<inter> range_vars \<sigma> = {}"
step_3_c_inner_result: fixes A :: "'a mat"   and k :: "nat"   and lb :: "nat"   and l :: "nat"   and i_begin :: "nat"   and i_end :: "nat" assumes "local.inv_upto local.jb A k"   and "local.inv_from local.uppert A k"   and "local.inv_at local.one_zero A k"   and "local.ev_block n A"   and "k < n"   and "A \<in> carrier_mat n n"   and "(lb, l) \<in> set (identify_blocks A k)"   and "(i_begin, i_end) \<in> set (identify_blocks A k)"   and "i_end \<noteq> l"   and "i_end - i_begin \<le> l - lb"   and "A $$ (l, k) \<noteq> (0::'a)" shows "step_3_c_inner_loop (A $$ (i_end, k) / A $$ (l, k)) l i_end (Suc i_end - i_begin) A = mat n n (\<lambda>(i, j). if (i, j) = (i_end, k) then 0::'a else if i_begin \<le> i \<and> i \<le> i_end \<and> k < j then A $$ (i, j) - A $$ (i_end, k) / A $$ (l, k) * A $$ (l + i - i_end, j) else A $$ (i, j))"
dfa_isomorphism_def: fixes M :: "'b dfa"   and N :: "'b dfa"   and h :: "hf \<Rightarrow> hf" shows "dfa_isomorphism M N h \<equiv> dfa M \<and> dfa N \<and> dfa_isomorphism_axioms M N h"
map_cong0: fixes x :: "('l, 'k, 'm) F3"   and f1 :: "'k \<Rightarrow> 'n"   and g1 :: "'k \<Rightarrow> 'n"   and f2 :: "'m \<Rightarrow> 'o"   and g2 :: "'m \<Rightarrow> 'o" assumes "\<And>z1. z1 \<in> F3set1 x \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> F3set2 x \<Longrightarrow> f2 z2 = g2 z2" shows "F3map f1 f2 x = F3map g1 g2 x"
signed_modulo_int_def: fixes k :: "int"   and l :: "int" shows "k smod l = k - k sdiv l * l"
inverse_zero_imp_zero: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "inverse a = zero" shows "a = zero"
diag_mat_def: fixes f :: "'e \<Rightarrow> 'd" shows "diag_mat f = (\<chi>i j. if i = j then f i else (0::'d))"
t_stop_zero_not_moving: fixes t :: "real" assumes "ego2.t_stop = 0" shows "ego.q t = s\<^sub>e"
Units_closed: fixes x :: "cltn2" assumes "x \<in> Units \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "x \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"
two: shows "(fs ! i $ j)\<^sup>2 \<le> (if i \<noteq> k \<or> outside then int (N * m) else int (4 ^ (m - 1) * N ^ m * m * m))"
catch_gpv_def: fixes gpv :: "('d, 'e, 'f) gpv" shows "catch_gpv gpv = TRY map_gpv Some id gpv ELSE Generative_Probabilistic_Value.Done None"
rcs_Unit2: fixes H :: "'a set"   and h :: "'a" assumes "G \<guillemotright> H "   and "h \<in> H" shows "H \<bullet> h = H"
Abs_freeletter_in_FreeGroup_iff: fixes s :: "'a"   and S :: "'a set" shows "(Abs_freeletter s \<in> FreeGroup S) = (s \<in> S)"
cases: fixes x :: "('fsym, 'tp) efsym" obtains f :: "'fsym" where "x = Oldf f"   | \<sigma> :: "'tp" where "x = Tag \<sigma>__"   | \<sigma> :: "'tp" where "x = Wit \<sigma>"