hcju/isabelleps huggingface dataset:
strict_mono_add: fixes k :: "'b" shows "strict_mono (\<lambda>n. n + k)"
inverse_distrib_swap: fixes a :: "'a"   and b :: "'a" shows "- (a + b) = - b + - a"
assms_2: shows "\<pi> ` A \<subseteq> ON"
subst_apply_fv_subset_strand_trm2: fixes t :: "('d, 'c) Term.term"   and S :: "('d, 'c) strand_step list"   and V :: "'c set"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "fv t \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V"   and "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "fv (t \<cdot> \<delta>) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<union> fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V)"
mul_eq_add_right: fixes M :: "'a multiset"   and P :: "'a multiset"   and r :: "('a \<times> 'a) set" shows "(M, M + P) \<in> mul_eq r"
Red_I_def: fixes M :: "'a literal multiset set"   and N :: "'a literal multiset set" shows "G.Red_I M N = {\<iota> \<in> G_Inf M. G.redundant_infer N \<iota>}"
consequence: fixes P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool"   and M :: "'a \<Rightarrow> ('b \<times> 'a) option"   and A :: "'b set"   and Q' :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "Collect P \<subseteq> Collect P'"   and "\<lbrace>P'\<rbrace> M \<lbrace>\<lambda>x \<sigma>. x \<in> A \<and> Q' x \<sigma>\<rbrace>"   and "\<forall>x\<in>A. Collect (Q' x) \<subseteq> Collect (Q x)" shows "\<lbrace>P\<rbrace> M \<lbrace>\<lambda>x \<sigma>. x \<in> A \<and> Q x \<sigma>\<rbrace>"
strict_mono_onD: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and r :: "'b"   and s :: "'b" assumes "strict_mono_on f A"   and "r \<in> A"   and "s \<in> A"   and "r < s" shows "f r < f s"
intros_2: fixes T :: "('c, 'd) Term.term list"   and S :: "('c, 'd) Term.term list"   and P :: "'c \<Rightarrow> 'c list"   and g :: "'c"   and f :: "'c" assumes "length T = length S"   and "\<And>i. i < length T \<Longrightarrow> term_variants_pred P (T ! i) (S ! i)"   and "g \<in> set (P f)" shows "term_variants_pred P (Fun f T) (Fun g S)"
ws: shows "\<forall>(w, r', s')\<in>set ws. fold delta (rev w) r = r' \<and> fold delta (rev w) s = s' \<and> wf_word n (rev w) \<and> wf_state r' \<and> wf_state s'"
bij: shows "bij_betw j (S - S' \<inter> S) (T - T' \<inter> T)"
degree_linear_power: fixes a :: "'b"   and n :: "nat" shows "degree ([:a, 1::'b:] ^ n) = n"
add_nth: fixes n :: "nat"   and w :: "'b list" assumes "n < \<^bold>|w\<^bold>|" shows "take n w \<cdot> [w ! n] \<le>p w"
Suffix_def: fixes L :: "'b list set" shows "Suffix L = {w. \<exists>u. u @ w \<in> L}"
not_ev_not: fixes \<phi> :: "'b stream \<Rightarrow> bool" shows "(\<lambda>xs. \<not> ev (\<lambda>xs. \<not> \<phi> xs) xs) = alw \<phi>"
listsE: fixes x :: "'c"   and l :: "'c list"   and A :: "'c set" assumes "x # l \<in> lists A" obtains "x \<in> A" and "l \<in> lists A"
atom_fresh_star_disjoint: fixes bs :: "atom set"   and as :: "atom set" assumes "finite bs" shows "as \<sharp>* bs = (as \<inter> bs = {})"
cnf_of_encode_operator_effect_structure: fixes \<Pi> :: "'b strips_problem"   and t :: "nat"   and op :: "'b strips_operator" shows "cnf (encode_operator_effect \<Pi> t op) = (\<Union>v\<in>set (add_effects_of op). {{(Operator t (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State (Suc t) (index (\<Pi>\<^sub>\<V>) v))\<^sup>+}}) \<union> (\<Union>v\<in>set (delete_effects_of op). {{(Operator t (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State (Suc t) (index (\<Pi>\<^sub>\<V>) v))\<inverse>}})"
conf_conv_conf: fixes typeof_addr :: "'a \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'b" shows "heap_base.conf (\<lambda>_. typeof_addr) P h = heap_base'.conf TYPE('b) typeof_addr P"
coincides_on_def: fixes Mkt :: "('c, 'd) discrete_market"   and Mkt2 :: "('c, 'd) discrete_market"   and A :: "'d set" shows "coincides_on Mkt Mkt2 A = (stocks Mkt = stocks Mkt2 \<and> (\<forall>x. x \<in> A \<longrightarrow> prices Mkt x = prices Mkt2 x))"
embed_bool_bounded_by_1: fixes P :: "'c \<Rightarrow> bool" shows "bounded_by 1 \<guillemotleft> P \<guillemotright>"
hoare_conj_right: fixes P :: "'e set"   and R :: "('e \<times> 'e) set"   and Q1 :: "'e set"   and Q2 :: "'e set" assumes "{P} R {> Q1}"   and "{P} R {> Q2}" shows "{P} R {> Q1 \<inter> Q2}"
is_red_addsE: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red F p" obtains f :: "'t \<Rightarrow>\<^sub>0 'c" and v :: "'t" where "f \<in> F" and "v \<in> keys p" and "f \<noteq> 0" and "lt f adds\<^sub>t v"
ws: shows "ws = getFrN (y # map snd txs) (map fst txs) [\<phi>] (length txs)"
1: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "(F has_vector_derivative f x) (at x within {a..b})"
degen_run_bound: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat"   and i :: "nat" assumes "num_acc \<noteq> 0"   and "graph_defs.is_run (degeneralize_ext m) r" shows "snd (r i) < num_acc"
from_nat_into_to_nat_on: fixes A :: "'a set"   and a :: "'a" assumes "countable A"   and "a \<in> A" shows "from_nat_into A (to_nat_on A a) = a"
g_approx_def: fixes i :: "nat"   and x :: "real" shows "g_approx i x = x powr p * integral (\<lambda>u. g u / u powr (p + 1)) (bs ! i * x + (hs ! i) x) x"
nhip_is_sip: fixes \<xi> :: "state"   and \<xi>' :: "state"   and ip :: "nat"   and sn :: "nat"   and hops :: "nat"   and sip :: "nat" assumes "\<forall>dip\<in>kD (rt \<xi>). the (nhop (rt \<xi>) dip) \<in> kD (rt \<xi>)"   and "\<xi>' = \<xi>\<lparr>rt := update (rt \<xi>) ip (sn, kno, val, Suc hops, sip, {})\<rparr>"   and "sip \<in> kD (rt \<xi>)" shows "(the (nhop (update (rt \<xi>) ip (sn, kno, val, Suc hops, sip, {})) ip) = ip \<or> the (nhop (update (rt \<xi>) ip (sn, kno, val, Suc hops, sip, {})) ip) \<in> kD (rt \<xi>)) \<and> (\<forall>dip\<in>kD (rt \<xi>). the (nhop (update (rt \<xi>) ip (sn, kno, val, Suc hops, sip, {})) dip) = ip \<or> the (nhop (update (rt \<xi>) ip (sn, kno, val, Suc hops, sip, {})) dip) \<in> kD (rt \<xi>))"
assms: shows "x \<in>\<^sub>\<circ> vfield A"
array_foldr_def: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'b array"   and b :: "'a" shows "array_foldr f a b = foldr (\<lambda>(k, v). f k v) (assoc_list_of_array a) b"
PTrans: shows "P \<longmapsto> \<tau> \<prec> P'"
some_basis_def: fixes V :: "'b set" shows "some_basis V = Eps (is_basis V)"
refl_on_singleton: fixes x :: "'b" shows "refl_on {x} {(x, x)}"
InlI: fixes a :: "'c"   and A :: "'c set"   and B :: "'d set" assumes "a \<in> A" shows "Inl a \<in> A <+> B"
r_e2ls_def: shows "r_e2ls \<equiv> Cn 1 r_pdec22 [r_e2frame]"
assms_4: shows "igSwapAbsIPresIGWlsAbs MOD"
deriv_1: fixes x :: "real"   and A :: "real set" assumes "x \<noteq> a" shows "(F1 has_vector_derivative c / (a - x) ^ n) (at x within A)"
process_prop_1_def: fixes P :: "('b list \<times> 'b set) set \<times> 'b list set" shows "process_prop_1 P \<equiv> ([], {}) \<in> fst P"
assms_3: shows "P \<subseteq> U\<^sub>M"
mute_preserves_triangle: fixes q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "local.mute q k l (r, A) = (r', A')"   and "local.triangular_to l A"   and "l < k"   and "k < dim_row A"   and "l < dim_row A"   and "l < dim_col A" shows "local.triangular_to l A'"
loose_bvar_incr_bvar: fixes t :: "Core.term"   and lev :: "nat"   and inc :: "nat" shows "loose_bvar t lev = loose_bvar (incr_bv inc lev t) (lev + inc)"
minus_carrier_vec: fixes v\<^sub>1 :: "'a vec"   and n :: "nat"   and v\<^sub>2 :: "'a vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 - v\<^sub>2 \<in> carrier_vec n"
type_wf_preserved: fixes SW :: "((_) heap, 'ab, 'ac) prog set"   and setter :: "((_) heap, 'ab, 'ad) prog"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "\<And>h h' w. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> \<forall>object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'"   and "\<And>h h' w. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> \<forall>node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'" shows "NodeClass.type_wf h = NodeClass.type_wf h'"
DARS3: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "DenyAll \<notin> set p" shows "DenyAll \<notin> set (rm_MT_rules C p)"
fv_not_fresh: fixes x :: "'e"   and e :: "'d" shows "atom x \<sharp> e = (x \<notin> fv e)"
dist_prod_def: fixes x :: "'d \<times> 'e"   and y :: "'d \<times> 'e" shows "dist x y = sqrt ((dist (fst x) (fst y))\<^sup>2 + (dist (snd x) (snd y))\<^sup>2)"
abs_eq: fixes x :: "int" shows "uint_of_int x = Abs_uint (word_of_int x)"
ts_i: shows "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)"
path0: fixes E :: "('e \<times> 'e) set"   and u :: "'e" shows "path E u [] u"
isoI: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" assumes "h \<in> hom G H"   and "bij_betw h (carrier G) (carrier H)" shows "h \<in> Group.iso G H"
basis_reduction_add_rows_loop_2: fixes i :: "nat"   and fs :: "int vec list"   and j :: "nat"   and fs' :: "int vec list" assumes "LLL_invariant True i fs"   and "\<mu>_small_row i fs j"   and "basis_reduction_add_rows_loop i fs j = fs'"   and "i < m"   and "j \<le> i" shows "LLL_measure i fs' = LLL_measure i fs"
submanifold_atlasI: fixes c :: "('b, 'f) chart" assumes "c \<in> dest.atlas" shows "restrict_chart S c \<in> sub.atlas"
execs: shows "\<Gamma>\<turnstile> \<langle>cs,css,s\<rangle> \<Rightarrow> t"
tvar_in_fv_in_tvs: fixes a :: "variable"   and \<tau> :: "typ"   and B :: "Core.term"   and x :: "variable"   and S :: "String.literal set" assumes "(a, \<tau>) \<in> fv B"   and "(x, S) \<in> tvsT \<tau>" shows "(x, S) \<in> tvs B"
delta_gt_SN: fixes \<delta> :: "'c" assumes "(0::'c) < \<delta>" shows "SN {(x, y). (0::'c) \<le> y \<and> delta_gt \<delta> x y}"
HEndPhase1_HInv3: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv3 s" shows "HInv3 s'"
assms_4: fixes p :: "int poly" assumes "p \<in> P" shows "irreducible p"
simps: fixes \<phi> :: "'a ltl" shows "\<A>.\<iota>\<^sub>\<A> \<phi> = (Abs \<phi>, \<iota>\<^sub>\<times> (\<^bold>G \<phi>) (semi_mojmir_def.initial \<circ> Abs \<circ> theG))"
orbit_stab_fun_is_surj: fixes x :: "'c"   and y :: "'c" assumes "x \<in> E"   and "y \<in> orbit G \<phi> x" shows "\<exists>R\<in>rcosets stabilizer G \<phi> x. \<phi> (inv (SOME h. h \<in> R)) x = y"
bliftM2_op: fixes f :: "'c \<rightarrow> 'b \<rightarrow> 'a"   and x :: "'c"   and y :: "'b" shows "bliftM2 f\<cdot>(box\<cdot>x)\<cdot>(box\<cdot>y) = box\<cdot>(f\<cdot>x\<cdot>y)"
bij_betwD_2: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "f ` A = B"
eventually_elim2: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool" assumes "eventually P F"   and "eventually Q F"   and "\<And>i. \<lbrakk>P i; Q i\<rbrakk> \<Longrightarrow> R i" shows "eventually R F"
greaterThan_def: fixes l :: "'b word" shows "signed.greaterThan l \<equiv> {x. l <s x}"
no_vs_change_obtain_snapshot_bound_3rd_step: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list"   and vs :: "'a set"   and s :: "('a, bool) fmap" assumes "finite PROB"   and "vars_change as vs s = []"   and "no_effectless_act as"   and "sat_precond_as s as"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) (exec_plan s as) = fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) (exec_plan s as') \<and> subseq as' as \<and> length as' \<le> problem_plan_bound (snapshot PROB (fmrestrict_set vs s))"
lang_dual_lderivs_dual: fixes n :: "nat"   and r :: "'b rexp_dual"   and ws :: "'a list" assumes "wf_dual n r"   and "wf_word n ws" shows "lang_dual n (lderivs_dual ws r) = lQuots ws (lang_dual n r)"
path_def: fixes x :: "bool"   and y :: "bool"   and p :: "'arr list" shows "G.path x y p \<equiv> p = [] \<and> x = y \<and> x \<in> {False, True} \<or> p \<noteq> [] \<and> x = False \<and> y = True \<and> (\<forall>n. 0 \<le> n \<and> n < length p \<longrightarrow> p ! n \<in> Arr) \<and> (\<forall>n. 0 \<le> n \<and> n < length p - 1 \<longrightarrow> True = False)"
Gcd_eucl: shows "normalization_euclidean_semiring_class.Gcd = Gcd"
minSetOfComponentsTestL1p3: shows "minSetOfComponents level1 {data1, data10, data11} = {sA12, sA11, sA21}"
Tsmult2_def: shows "(\<star>\<cdot>) \<equiv> \<lambda>a T v. a \<sharp>\<star> T v"
ivl_integral_combine: fixes f :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real"   and c :: "real" assumes "f integrable_on closed_segment a b"   and "f integrable_on closed_segment b c"   and "f integrable_on closed_segment a c" shows "ivl_integral a b f + ivl_integral b c f = ivl_integral a c f"
a_coset_eq: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "abelian_subgroup H G" shows "\<forall>x\<in>carrier G. H +>\<^bsub>G\<^esub> x = x <+\<^bsub>G\<^esub> H"
simps: fixes xp :: "real"   and yp :: "real"   and xq :: "real"   and yq :: "real"   and xr :: "real"   and yr :: "real" shows "det3 (xp, yp) (xq, yq) (xr, yr) = xp * yq + yp * xr + xq * yr - yq * xr - yp * xq - xp * yr"
simps: fixes x :: "nat"   and n :: "nat" shows "older_seniors x n = {s. \<exists>y. s = senior y n \<and> s < senior x n \<and> \<not> sink (token_run s n)}"
borel_eq_lessThan: shows "borel = sigma UNIV (range (\<lambda>a. {x. eucl_less x a}))"
Val_abs_axioms: shows "Val_abs \<gamma>_parity num_parity plus_parity"
da_elim_cases_36: fixes Env :: "env"   and B :: "lname set"   and binop :: "binop"   and e1 :: "expr"   and e2 :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l (BinOp binop e1 e2)\<guillemotright> A" obtains E1 :: "assigned" and E2 :: "assigned" where "binop = CondAnd" and "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1__" and "Env\<turnstile> B \<union> assigns_if True e1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2__" and "nrm A = B \<union> assigns_if True (BinOp CondAnd e1 e2) \<inter> assigns_if False (BinOp CondAnd e1 e2)" and "brk A = (\<lambda>l. UNIV)"   | E1 :: "assigned" and E2 :: "assigned" where "binop = CondOr" and "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1a__" and "Env\<turnstile> B \<union> assigns_if False e1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2" and "nrm A = B \<union> assigns_if True (BinOp CondOr e1 e2) \<inter> assigns_if False (BinOp CondOr e1 e2)" and "brk A = (\<lambda>l. UNIV)"   | E1 :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1" and "Env\<turnstile> nrm E1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> A" and "binop \<noteq> CondAnd" and "binop \<noteq> CondOr"
inputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "a<b>.P \<longmapsto> Rs" obtains x :: "name" and aa :: "name" and u :: "name" and P :: "pi" where "a = aa \<and> [b].Pb__ = [x].P" and "Rs = aa<u> \<prec> P[x::=u]" and "x \<noteq> aa" and "x \<noteq> u"
set2_G_bound: fixes x :: "('t, 'n, 'o, 'p, 'q, 'r, 's) G" shows "|set2_G x| \<le>o bd_G"
continuous_on_finite: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite S" shows "continuous_on S f"
basis_row_space_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "basis_row_space A = {row i (Gauss_Jordan A) |i. row i (Gauss_Jordan A) \<noteq> 0}"
assms_1: fixes i :: "'a" assumes "i \<in> I" shows "continuous_map T1 (T i) (\<lambda>x. f x i)"
formal_in_THE: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and ins :: "'var list"   and outs :: "'var list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs" shows "(THE ins. \<exists>outs. (p, ins, outs) \<in> set procs) = ins"
ideal_generated_in: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "a \<in> ideal_generated A"
assms_4: fixes k :: "nat" assumes "k \<in> {a'<..b'}" shows "\<not> prime k"
Rep_matrix: fixes x :: "'b matrix" shows "Rep_matrix x \<in> matrix"
alist_merge: fixes t :: "('d, 'c) bintree"   and r :: "('d, 'c) bintree" assumes "distinct (bt_dfs val t)"   and "distinct (bt_dfs val r)"   and "set (bt_dfs val t) \<inter> set (bt_dfs val r) = {}" shows "set (bt_dfs alist (merge t r)) = set (bt_dfs alist t) \<union> set (bt_dfs alist r)"
lincomb_list_smult: fixes Vs :: "'a vec list"   and a :: "'a"   and c :: "nat \<Rightarrow> 'a" assumes "set Vs \<subseteq> carrier_vec n" shows "lincomb_list (\<lambda>i. a * c i) Vs = a \<cdot>\<^sub>v lincomb_list c Vs"
l3_inv9_derived: shows "l3_inv2 \<inter> l3_inv3 \<inter> l3_inv4 \<subseteq> l3_inv9"
purge_def: fixes G :: "'a set"   and ucl :: "('a \<times> 'b) list" shows "GM_sec_model.purge G ucl \<equiv> filter (\<lambda>(u, c). u \<notin> G) ucl"
simps: fixes a1 :: "'a list"   and a2 :: "'a list" shows "(a1 <~~> a2) = (a1 = [] \<and> a2 = [] \<or> (\<exists>y x l. a1 = y # x # l \<and> a2 = x # y # l) \<or> (\<exists>xs ys z. a1 = z # xs \<and> a2 = z # ys \<and> xs <~~> ys) \<or> (\<exists>xs ys zs. a1 = xs \<and> a2 = zs \<and> xs <~~> ys \<and> ys <~~> zs))"
obtain_ideal_generated_pair: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "c \<in> ring.ideal_generated plus zero uminus times {a, b}" obtains p :: "'b" and q :: "'b" where "plus (times p a) (times q b) = c"
prems: shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (g # G \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>) \<subseteq> subst_domain \<sigma>"
testN_mult_testN: fixes k :: "nat" shows "testN k * testN k = testN k"
unreach_3: shows "Q\<^sub>x \<noteq> Q\<^sub>z"
measurable_vimage_algebra1: fixes f :: "'c \<Rightarrow> 'd"   and X :: "'c set"   and M :: "'d measure" assumes "f \<in> X \<rightarrow> space M" shows "f \<in> vimage_algebra X f M \<rightarrow>\<^sub>M M"
bidel_arc_biconnected_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" shows "bidel_arc_biconnected G M a \<equiv> bidel_arc G M a \<and> bidel_arc_biconnected_axioms G M a"
assms_2: shows "P (vinsert a (set (insert b B)))"
assms_2: shows "xs \<in>\<^sub>\<circ> vproduct (vcard xs) A"
add_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "D x y \<and> D (x \<oplus> y) z" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
cInf_lower: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and X :: "'a set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> X"   and "preorder.bdd_below less_eq X" shows "less_eq (Inf X) x"
a1: shows "s' = snd (memory_read asi addr s) \<and> get_S (cpu_reg_val PSR s) = 0"
poly_eqI: fixes p :: "'a poly"   and q :: "'a poly" assumes "\<And>n. coeff p n = coeff q n" shows "p = q"
C_def: shows "C = {uu_. \<exists>a b z. uu_ = box a b \<and> f analytic_on box a b \<and> z \<in> box a b \<and> z \<in> S}"
shiftl_eq_push_bit: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.semiring_bit_syntax plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_bit_syntax.shiftl push_bit a n = push_bit n a"
trancl_domain: fixes r :: "('a \<times> 'a) set" shows "Domain (r\<^sup>+) = Domain r"
assms_13: shows "InvariantVarsQ (getQ state) F0 Vbl"
Ord_pred: fixes \<beta> :: "V" assumes "Ord \<beta>" shows "\<Squnion> (insert \<beta> (elts \<beta>)) = \<beta>"
locally_path_connected_quotient_image: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "locally path_connected S"   and "\<And>T. T \<subseteq> f ` S \<Longrightarrow> openin (top_of_set S) (S \<inter> f -` T) = openin (top_of_set (f ` S)) T" shows "locally path_connected (f ` S)"
SD_inefficientI': fixes q :: "'alt pmf"   and p :: "'alt pmf" assumes "q \<in> lotteries_on alts"   and "\<And>i. i \<in> agents \<Longrightarrow> SD (R i) p q"   and "\<exists>i\<in>agents. p \<prec>[SD (R i)] q" shows "\<not> SD_efficient R p"
replicate_mset_eq_iff: fixes m :: "nat"   and a :: "'a"   and n :: "nat"   and b :: "'a" shows "(replicate_mset m a = replicate_mset n b) = (m = 0 \<and> n = 0 \<or> m = n \<and> a = b)"
asymp_equivI: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "((\<lambda>x. if f x = (0::'c) \<and> g x = (0::'c) then 1::'c else f x / g x) \<longlongrightarrow> (1::'c)) F" shows "f \<sim>[F] g"
cases: fixes x :: "('d, 'c) Term.term" obtains xa :: "'c" where "x = Var xa"   | f :: "'d" and T :: "('d, 'c) Term.term list" where "x = Fun f T"
case_sum_o_inj_2: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "case_sum f g \<circ> Inr = g"
divide_real_alg_def: fixes x :: "real_alg"   and y :: "real_alg" shows "x / y = x * inverse y"
bij_betwE: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "bij_betw f A B" shows "\<forall>a\<in>A. f a \<in> B"
gb_red_aux_not_zero: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "0 \<notin> set (gb_red_aux bs ps)"
elims: fixes x :: "nat"   and xa :: "bit list"   and y :: "bit list" assumes "nat_to_bv_helper x xa = y" obtains n :: "nat" and bs :: "bit list" where "x = n" and "xa = bs" and "y = (if n = 0 then bs else nat_to_bv_helper (n div 2) ((if n mod 2 = 0 then \<zero> else \<one>) # bs))"
fw_innermost_id: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and j' :: "nat"   and i' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "i \<le> n"   and "j \<le> n"   and "j' \<le> n"   and "i' < i" shows "fw m n 0 i' j' i j = m i j"
fg: fixes x :: "'a" assumes "x \<in> X frontier_of {x. P x}" shows "f x = g x"
common_facet: fixes z :: "'b set"   and x :: "'b set"   and y :: "'b set" assumes "z \<lhd> x"   and "z \<lhd> y"   and "x \<noteq> y" shows "z = x \<inter> y"
Zn_cyclic_group: fixes n :: "nat" obtains x :: "int" where "Finite_And_Cyclic_Groups.cyclic_group (Z n) x"
T3_deRe: shows "serial (\<lambda>w v. w r v \<and> (\<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa w \<longrightarrow> xa x)))"
supp_Abs_formula: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "supp (Abs_formula t\<^sub>\<alpha>) = supp t\<^sub>\<alpha>"
b: shows "\<forall>(k2, \<gamma>2)\<in>one_chain2. valid_path \<gamma>2"
compute_SCC_impl_refine: fixes G :: "('a, 'b) graph_rec_scheme" assumes "fr_graph G" shows "fr_graph.compute_SCC_impl G \<le> \<Down> Id (fr_graph.compute_SCC G)"