hcju/isabelleps huggingface dataset:
field_divide_inverse: fixes a :: "'c"   and b :: "'c" shows "a / b = a * inverse b"
The_split_eq: fixes x :: "'e"   and y :: "'f" shows "(THE (x', y'). x = x' \<and> y = y') = (x, y)"
wf_partial_state_def: fixes ps :: "(nat \<times> nat) list" shows "wf_partial_state ps \<equiv> distinct (map fst ps) \<and> (\<forall>(x, v)\<in>set ps. x < numVars \<and> v < numVals x)"
pcr_cr_eq: shows "pcr_scene (=) = cr_scene"
binrelchain_sym_rev: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "\<And>x y. P x y \<Longrightarrow> P y x"   and "binrelchain P xs" shows "binrelchain P (rev xs)"
eval: shows "(cms\<^sub>1, mem\<^sub>1) \<leadsto>\<^bsub>k\<^esub> (cms\<^sub>1', mem\<^sub>1')"
eventually_bot: fixes P :: "'b \<Rightarrow> bool" shows "eventually P bot"
abs_one: 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 sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs one = one"
Data_def: fixes L :: "'b list"   and D :: "'b dataspace" shows "Data L D = (length L = PartNum D \<and> (\<forall>i\<in>{n. n < PartNum D}. L ! i \<in> D !D! i))"
cases: fixes eff :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset \<Rightarrow> bool"   and rules :: "'a stream"   and a :: "('b \<times> 'a) stream" assumes "RuleSystem_Defs.epath eff rules a" obtains steps :: "('b \<times> 'a) stream" and sl :: "'b fset" where "a = steps" and "snd (shd steps) \<in> i.R rules" and "fst (shd (stl steps)) |\<in>| sl" and "eff (snd (shd steps)) (fst (shd steps)) sl" and "RuleSystem_Defs.epath eff rules (stl steps)"
2: fixes a :: "alt"   and b :: "alt"   and a' :: "alt"   and b' :: "alt"   and P :: "indi \<Rightarrow> (alt \<times> alt) set"   and P' :: "indi \<Rightarrow> (alt \<times> alt) set" assumes "a \<noteq> b"   and "a' \<noteq> b'"   and "a \<noteq> b'"   and "b \<noteq> a'"   and "P \<in> Prof"   and "P' \<in> Prof"   and "\<forall>i. (a <\<^bsub>P i\<^esub> b) = (a' <\<^bsub>P' i\<^esub> b')" shows "(a <\<^bsub>F P\<^esub> b) = (a' <\<^bsub>F P'\<^esub> b')"
H_contained_in_set_mult: shows "H \<subseteq> H <#> S"
pr_conv_1_to_3_def: shows "pr_conv_1_to_3 = (\<lambda>f x y z. f (c_pair (c_pair x y) z))"
iSPEC_rule: fixes \<Phi> :: "'b \<Rightarrow> bool"   and \<Psi> :: "'b \<Rightarrow> bool" assumes "\<And>x. \<Phi> x \<Longrightarrow> \<Psi> x" shows "SPEC \<Phi> \<le> SPEC \<Psi>"
deriv_ix_ladder_shift_j: fixes index :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and d :: "nat" assumes "index < length L" shows "deriv_ix (ladder_shift_j d L ! index) = deriv_ix (L ! index)"
option_rel_sv: fixes R :: "('a \<times> 'b) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>option_rel)"
coprime_common_divisor_nat: fixes a :: "nat"   and b :: "nat"   and x :: "nat" assumes "coprime a b"   and "x dvd a"   and "x dvd b" shows "x = 1"
exec_meth_instr: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp :: "'addr option" shows "exec_meth ci P ins xt t h (stk, loc, pc, None) ta h' (stk', loc', pc', xcp) = ((ta, xcp, h', [(stk', loc', undefined, undefined, pc')]) \<in> exec_instr (ins ! pc) P t h stk loc undefined undefined pc [] \<and> pc < length ins \<and> ci_app ci (ins ! pc) P h stk loc undefined undefined pc [])"
cdf_nondecreasing: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "M.C x \<le> M.C y"
assoc: fixes f :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and a :: "'b::type"   and b :: "'b::type"   and c :: "'b::type" assumes "semigroup (f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (f (a::'b::type) (b::'b::type)) (c::'b::type) = f a (f b c)"
inverse_fract: fixes a :: "'a"   and b :: "'a" shows "inverse (Fract a b) = Fract b a"
cinsertE: fixes a :: "'d"   and b :: "'d"   and A :: "'d cset" assumes "cin a (cinsert b A)" obtains "a = b"   | "cin a A"
Integral_eq: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and k :: "real" shows "Integral (a, b) f k = (\<forall>e>0. \<exists>\<delta>. gauge {a..b} \<delta> \<and> (\<forall>D. fine \<delta> (a, b) D \<longrightarrow> \<bar>rsum D f - k\<bar> < e))"
exec_gpv_restrict_gpv_invariant: fixes callee :: "'b \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'b) spmf"   and I :: "'b \<Rightarrow> bool"   and \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and s :: "'b" assumes "callee_invariant_on callee I \<I>"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "exec_gpv callee (restrict_gpv \<I> gpv) s = exec_gpv callee gpv s"
s_has_real_derivative_at: fixes x :: "real" assumes "0 < x"   and "0 \<le> v"   and "a < 0" shows "(s has_real_derivative s' x) (at x)"
assms_2: fixes e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option" shows "R e h l e h l"
IH: fixes x :: "'a" assumes "x \<in> A" shows "merge c pc ls x = (if \<forall>a\<in>set ls. case a of (pc', s') \<Rightarrow> pc' \<noteq> pc + 1 \<longrightarrow> s' \<sqsubseteq>\<^bsub>r\<^esub> c ! pc' then map snd (filter (\<lambda>a. case a of (p', t') \<Rightarrow> p' = pc + 1) ls) \<Squnion>\<^bsub>f\<^esub> x else \<top>)"
simps: fixes prec :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and t1 :: "taylor_model"   and t2 :: "taylor_model" shows "tm_union prec I a t1 t2 = (let b1 = compute_bound_tm prec I a t1; b2 = compute_bound_tm prec I a t2; b_combined = sup b1 b2 in TaylorModel (mid b_combined)\<^sub>p (centered b_combined))"
s: shows "B = Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> (strip_all_single_body B)"
assms_2: fixes P :: "pi" assumes "(P, Q) \<in> Rel" shows "F (!P)"
assms_4: shows "\<D>\<^sub>\<circ> a = I"
lessThan_Suc_eq_insert_0: fixes n :: "nat" shows "{..<Suc n} = insert 0 (Suc ` {..<n})"
list_rel_sv: fixes R :: "('e \<times> 'f) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>list_rel)"
has_default_policy: fixes rs :: "'a simple_rule list"   and p :: "('a, 'b) simple_packet_scheme" assumes "has_default_policy rs" shows "simple_fw rs p = Decision FinalAllow \<or> simple_fw rs p = Decision FinalDeny"
rule_charn4: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and r :: "((int \<times> int) set set, int) Combinators" assumes "wellformed_policy1 p"   and "allNetsDistinct p"   and "singleCombinators p"   and "wellformed_policy3 p"   and "DenyAll \<notin> set p"   and "applied_rule_rev C x p = \<lfloor>DenyAllFromTo a b\<rfloor>"   and "r \<in> set p"   and "x \<in> dom (C r)" shows "r = DenyAllFromTo a b"
fst: shows "\<forall>x\<in>set_pmf p. pmf (mix_pmf a p q) x = pmf (mix_pmf (1 - a) q p) x"
DBM_val_bounded_def: fixes v :: "'d \<Rightarrow> nat"   and u :: "'d \<Rightarrow> 'e"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry"   and n :: "nat" shows "u \<turnstile>\<^bsub>v,n\<^esub> m \<equiv> Le (0::'e) \<preceq> m 0 0 \<and> (\<forall>c. v c \<le> n \<longrightarrow> dbm_entry_val u None (Some c) (m 0 (v c)) \<and> dbm_entry_val u (Some c) None (m (v c) 0)) \<and> (\<forall>c1 c2. v c1 \<le> n \<and> v c2 \<le> n \<longrightarrow> dbm_entry_val u (Some c1) (Some c2) (m (v c1) (v c2)))"
isometry_inverse_2: fixes f :: "'c \<Rightarrow> 'd" assumes "isometry f" shows "bij f"
maximalAllactionWithoutSeller: fixes b :: "integer \<times> 'c set \<Rightarrow> 'd"   and \<Omega> :: "'c set"   and N :: "integer set" shows "Outside' {seller} ` maximalStrictAllocations N \<Omega> b \<subseteq> soldAllocations N \<Omega>"
evalD_evalDdb_open: fixes e :: "expr"   and \<Gamma> :: "nat list"   and \<rho>' :: "nat \<rightarrow> ValD"   and \<rho> :: "nat \<rightarrow> ValD" assumes "set (free e) \<subseteq> set \<Gamma>"   and "\<forall>v\<in>set \<Gamma>. \<rho>'\<cdot>(index \<Gamma> v 0) = \<rho>\<cdot>v" shows "\<lbrakk>e\<rbrakk>\<rho> = evalDdb (transdb e \<Gamma>)\<cdot>\<rho>'"
SigmaD1: fixes a :: "'b"   and b :: "'c"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "(a, b) \<in> Sigma A B" shows "a \<in> A"
D: fixes \<nu> :: "'a" assumes "\<guillemotleft>\<nu> : v \<Rightarrow> u \<star> f\<guillemotright>" shows "trnr\<^sub>\<eta> v (trnr\<^sub>\<epsilon> u \<nu>) = \<nu>"
HOM_is_ag: fixes N :: "('g, 'b, 'h) Module_scheme" assumes "R module N" shows "aGroup (HOM\<^bsub>R\<^esub> M N)"
select_convs_1: fixes vertex :: "'b Point"   and slope :: "'b"   and more :: "'c" shows "vertex \<lparr>vertex = vertex, slope = slope, \<dots> = more\<rparr> = vertex"
gOpPresGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gOpPresGWls MOD \<equiv> \<forall>delta inp' inp binp' binp. wlsInp delta inp' \<and> gWlsInp MOD delta inp \<and> wlsBinp delta binp' \<and> gWlsBinp MOD delta binp \<longrightarrow> gWls MOD (stOf delta) (gOp MOD delta inp' inp binp' binp)"
env_restr_below_self: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "f f|` S \<sqsubseteq> f"
DERIV_inverse: fixes x :: "'a"   and s :: "'a set" assumes "x \<noteq> (0::'a)" shows "(inverse has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
not_eq_extremum: fixes a :: "'a" shows "(a \<noteq> top_class.top) = (a < top_class.top)"
iso_iff_section_and_retraction: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "category.iso C f = (category.section C f \<and> category.retraction C f)"
P1_security_rsa: fixes n :: "nat"   and m1 :: "bool \<times> bool"   and m2 :: "bool" shows "etp.OT_12.perfect_sec_P1 n m1 m2"
ne_lists_1: fixes xs :: "'b list"   and A :: "'b set" assumes "xs \<noteq> []"   and "xs \<in> lists A" shows "hd xs \<in> A"
FvarsT_rawpsubstT: fixes r :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm"   and "distinct (map snd txs)"   and "\<forall>x\<in>snd ` set txs. \<forall>t\<in>fst ` set txs. x \<notin> FvarsT t" shows "FvarsT (rawpsubstT r txs) = FvarsT r - snd ` set txs \<union> \<Union> {if x \<in> FvarsT r then FvarsT t else {} |t x. (t, x) \<in> set txs}"
conc_fun_FAIL: fixes R :: "('c \<times> 'd) set" shows "\<Down> R FAIL = FAIL"
top_left_normalize: fixes a :: "'c" shows "gcd (0::'c) a = normalize a"
monomial_is_monomial: fixes c :: "'b"   and t :: "'c" assumes "c \<noteq> (0::'b)" shows "is_monomial (monomial c t)"
f: shows "arr f \<and> local.dom f = a \<and> cod f = b"
is_aligned_iff_take_bit_eq_0: fixes w :: "'b word"   and n :: "nat" shows "is_aligned w n = (take_bit n w = 0)"
unordered_pair_equality: fixes p :: "'b"   and q :: "'b" shows "{p, q} = {q, p}"
eqvt_lambda: fixes p :: "perm"   and f :: "'d \<Rightarrow> 'e" shows "p \<bullet> f \<equiv> \<lambda>x. p \<bullet> f (unpermute p x)"
fNext_geq_not_member: fixes S :: "'a fset"   and s :: "'a" assumes "fNext S \<le> s" shows "s |\<notin>| S"
1: shows "{v \<in> fmdom' (fmrestrict_set (system_needed_vars PROB s) s). v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup (fmrestrict_set (system_needed_vars PROB s) s) v} \<subseteq> {v \<in> fmdom' s. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s v}"
nc: shows "\<not> m' cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"
subst_bvs_Abs: fixes args :: "Core.term list"   and T :: "typ"   and t :: "Core.term" assumes "\<forall>x\<in>set args. is_closed x" shows "subst_bvs args (Abs T t) = Abs T (subst_bvs1' t 1 args)"
connected_component_eq_self: fixes S :: "'b set"   and x :: "'b" assumes "connected S"   and "x \<in> S" shows "connected_component_set S x = S"
cont_def: fixes f :: "'a \<Rightarrow> 'b" shows "cont f = (\<forall>Y. chain Y \<longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i))"
map_zip_map: fixes f :: "'e \<times> 'f \<Rightarrow> 'd"   and g :: "'g \<Rightarrow> 'e"   and xs :: "'g list"   and ys :: "'f list" shows "map f (map g xs || ys) = map2 (\<lambda>x y. f (g x, y)) xs ys"
no_fail_def: fixes P :: "'d \<Rightarrow> bool"   and m :: "'d \<Rightarrow> ('e \<times> 'd) \<times> bool" shows "no_fail P m \<equiv> \<forall>s. P s \<longrightarrow> \<not> snd (m s)"
AE_emeasure_singleton: fixes M :: "'a measure"   and x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "emeasure M {x} \<noteq> 0"   and "almost_everywhere M P" shows "P x"
hausdorff_distanceI2: fixes A :: "'b set"   and B :: "'b set"   and D :: "real" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. dist x y \<le> D"   and "\<And>x. x \<in> B \<Longrightarrow> \<exists>y\<in>A. dist x y \<le> D"   and "0 \<le> D" shows "hausdorff_distance A B \<le> D"
H_cond: fixes p :: "'a"   and r :: "'a"   and x :: "'a"   and q :: "'a"   and y :: "'a" assumes "H (d p \<cdot> d r) x q"   and "H (d p \<cdot> ad r) y q" shows "H p (if r then x else y fi) q"
simps: fixes p :: "'d spmf" shows "the_gpv (lift_spmf p) = map_spmf Pure p"
safe_distance_2r_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>" shows "safe_distance_normal.safe_distance_2r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta> = v\<^sub>e * \<delta> - v\<^sub>e\<^sup>2 / 2 / a\<^sub>e + v\<^sub>o\<^sup>2 / 2 / a\<^sub>o"
real_sqrt_ge_1_iff: fixes y :: "real" shows "(1 \<le> sqrt y) = (1 \<le> y)"
prod_mset_primes_irreducible_imp_prime: fixes A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" assumes "irreducible (\<Prod>\<^sub># A)"   and "\<And>x. x \<in># A \<Longrightarrow> prime x"   and "\<And>x. x \<in># B \<Longrightarrow> prime x"   and "\<And>x. x \<in># C \<Longrightarrow> prime x"   and "\<Prod>\<^sub># A dvd \<Prod>\<^sub># B * \<Prod>\<^sub># C" shows "\<Prod>\<^sub># A dvd \<Prod>\<^sub># B \<or> \<Prod>\<^sub># A dvd \<Prod>\<^sub># C"
not_transC_transT: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and cf :: "('a, 'c) com \<times> 'b"   and cf' :: "('a, 'c) com \<times> 'b"   and s' :: "'b"   and phi :: "bool" assumes "PL.transC tval aval cf cf'"   and "PL.transT tval aval cf s'" shows "phi"
assms: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>)"
VInter_Inter: fixes A :: "V" shows "\<Inter>\<^sub>\<circ> A = set (\<Inter> (elts ` elts A))"
fls_conv_base_factor_to_fps_shift_subdegree: fixes f :: "'a fls" shows "f = fls_shift (- fls_subdegree f) (fls_regpart_as_fls (fls_base_factor f))"
rcp_specI: fixes rcp :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" assumes "\<And>gs bs ps sps data. 0 \<notin> fst ` set (fst (rcp gs bs ps sps data))"   and "\<And>gs bs ps sps h b data. \<lbrakk>h \<in> set (fst (rcp gs bs ps sps data)); b \<in> set gs \<union> set bs; fst b \<noteq> 0\<rbrakk> \<Longrightarrow> \<not> local.punit.adds_term (punit.lt (fst b)) (punit.lt (fst h))"   and "\<And>gs bs ps sps d data. dickson_grading d \<Longrightarrow> punit.dgrad_p_set_le d (fst ` set (fst (rcp gs bs ps sps data))) (punit.args_to_set (gs, bs, sps))"   and "\<And>gs bs ps sps data. local.punit.component_of_term ` Keys (fst ` set (fst (rcp gs bs ps sps data))) \<subseteq> local.punit.component_of_term ` Keys (punit.args_to_set (gs, bs, sps))"   and "\<And>gs bs ps sps data. \<lbrakk>punit.is_Groebner_basis (fst ` set gs); unique_idx (gs @ bs) data\<rbrakk> \<Longrightarrow> fst ` set (fst (rcp gs bs ps sps data)) \<subseteq> local.punit.pmdl (punit.args_to_set (gs, bs, sps)) \<and> (\<forall>(p, q)\<in>set sps. set sps \<subseteq> set bs \<times> (set gs \<union> set bs) \<longrightarrow> (punit.red (fst ` (set gs \<union> set bs) \<union> fst ` set (fst (rcp gs bs ps sps data))))\<^sup>*\<^sup>* (punit.spoly (fst p) (fst q)) 0)" shows "punit.rcp_spec rcp"
induct: fixes x1 :: "('d, 'e, 'f, 'g, 'h) qTerm"   and x2 :: "('d, 'e, 'f, 'g, 'h) qTerm"   and P :: "('d, 'e, 'f, 'g, 'h) qTerm \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm \<Rightarrow> bool" assumes "(x1, x2) \<in> qSwapped"   and "\<And>X. P X X"   and "\<And>X Y Z. \<lbrakk>(X, Y) \<in> qSwapped; P X Y; (Y, Z) \<in> qSwapped; P Y Z\<rbrakk> \<Longrightarrow> P X Z"   and "\<And>X Y x y zs. \<lbrakk>(X, Y) \<in> qSwapped; P X Y\<rbrakk> \<Longrightarrow> P X (Y #[[x \<and> y]]_zs)" shows "P x1 x2"
dense: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.dense_order less_eq less"   and "less x y" shows "\<exists>z. less x z \<and> less z y"
simps_1: fixes uu :: "char list \<Rightarrow> bool"   and uv :: "'b fm \<Rightarrow> bool" shows "eval uu uv \<^bold>\<bottom> = False"
proj_set_dist_le: fixes y :: "'b"   and A :: "'b set"   and p :: "'b"   and x :: "'b" assumes "y \<in> A"   and "p \<in> proj_set x A" shows "dist x p \<le> dist x y"
gcd_Suc_0: fixes m :: "nat" shows "gcd m (Suc 0) = Suc 0"
1: shows "Tokens k (\<T> k u) (\<J> k u) = Sel (\<T> k u) (TokensAt k (\<J> k u))"
list_rel_pres_neq_nil: fixes x' :: "'a list"   and x :: "'b list"   and A :: "('a \<times> 'b) set" assumes "(x', x) \<in> \<langle>A\<rangle>list_rel" shows "(x' \<noteq> []) = (x \<noteq> [])"
drop_stake: fixes n :: "nat"   and m :: "nat"   and s :: "'d stream" shows "drop n (stake m s) = stake (m - n) (sdrop n s)"
r_OO_conversep_into_equivclp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "r\<^sup>*\<^sup>* OO r\<inverse>\<inverse>\<^sup>*\<^sup>* \<le> equivclp r"
odd_numeral_BitM: 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 w :: "num" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (numeral.numeral one plus (Num.BitM w))"
inv_i_iso_\<iota>_1: fixes f :: "'a" assumes "\<guillemotleft>f : \<I> \<rightarrow> \<I>\<^sub>1\<guillemotright>"   and "local.iso f"   and "f \<cdot> \<iota> = \<iota>\<^sub>1 \<cdot> (f \<otimes>\<^sub>1 f)" shows "\<guillemotleft>local.inv i \<cdot> f : \<I> \<rightarrow> \<I>\<guillemotright>"
Inv_preserves_Ide: fixes t :: "'c local.term" assumes "Ide t" shows "Ide (Inv t)"
ah_interleavable1: fixes w :: "'c list"   and w1 :: "'c list"   and \<alpha> :: "'c \<Rightarrow> 'd set \<times> 'd set"   and w2 :: "'c list" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" shows "\<alpha>ah (map \<alpha> w1) [*] \<alpha>ah (map \<alpha> w2)"
a_inv_closed: fixes G :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "abelian_group G"   and "x \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x \<in> carrier G"
top_mult_top: shows "matrix_bounded_idempotent_semiring.surjective mtop"
restrict_codomain_to_image: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "ChamberComplexMorphism X (f \<turnstile> X) f"
get_elements_by_tag_name_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and name :: "char list"   and results :: "(_) element_ptr list"   and result :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_elements_by_tag_name ptr' name \<rightarrow>\<^sub>r results"   and "result \<in> set results" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r result \<in> set c) = (ptr' \<in> set c)"
ereal_infty_less_eq_2: fixes x :: "ereal" shows "(x \<le> - \<infinity>) = (x = - \<infinity>)"
redT_updWs_def: fixes t :: "'c" shows "redT_updWs t = rtrancl3p (redT_updW t)"
intros_1: fixes xs :: "'c option list"   and P :: "'c option process"   and X :: "'c option set"   and Q :: "'c option process" assumes "xs \<notin> sentences P"   and "(xs, X) \<in> failures P"   and "None \<notin> set xs" shows "seq_comp_failuresp P Q xs X"
simple_blocks_eq_sup: shows "mset_set design_support = \<B>"
divisors_fun_def: fixes df :: "'b \<Rightarrow> 'b list" shows "divisors_fun df \<equiv> (\<forall>x. x \<noteq> (0::'b) \<longrightarrow> set (df x) = {d. d dvd x}) \<and> (\<forall>x. distinct (df x))"
AssumeH_6: fixes A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and H :: "fm set" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 H))))) \<turnstile> A6"
Polys_empty: shows "P[{}] = range (Poly_Mapping.single 0)"
erlX: shows "distributed M lborel X (\<lambda>x. ennreal (erlang_density k l x))"
enforce_spmf_def: fixes P :: "'c \<Rightarrow> bool" shows "enforce_spmf P = map_pmf (enforce_option P)"
ChamberComplexWithApartmentSystem_axioms: shows "ChamberComplexWithApartmentSystem X \<A>"
eq_refl: fixes x :: "'a" assumes "eq\<cdot>x\<cdot>x \<noteq> \<bottom>" shows "eq\<cdot>x\<cdot>x = TT"
\<section>: shows "exp (s * Ln z) * inverse z = exp ((s - 1) * Ln z)"
IH: shows "\<exists>y\<in>complex_rotations. y \<circ> s = id"
has_fps_expansion_minus: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a fps" assumes "f has_fps_expansion F" shows "(\<lambda>x. - f x) has_fps_expansion - F"
geometric_spmf_fixp_induct: fixes P :: "(real \<Rightarrow> nat spmf) \<Rightarrow> bool" assumes "spmf.admissible P"   and "P (\<lambda>geometric_spmf. return_pmf None)"   and "\<And>geometric_spmf'. P geometric_spmf' \<Longrightarrow> P (\<lambda>p. bernoulli p \<bind> (\<lambda>b. if b then return_spmf 0 else map_spmf ((+) 1) (geometric_spmf' p)))" shows "P geometric_spmf"
assms: shows "(a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>*"
descartes_sign: fixes p :: "real poly" assumes "p \<noteq> 0" shows "int (proots_count p {x. 0 < x}) \<le> changes (coeffs p) \<and> even (changes (coeffs p) - int (proots_count p {x. 0 < x}))"
moebius_pt_bilinear: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex"   and z :: "complex_homo" assumes "mat_det (a, b, c, d) \<noteq> 0" shows "moebius_pt (mk_moebius a b c d) z = (if z \<noteq> \<infinity>\<^sub>h then (of_complex a *\<^sub>h z +\<^sub>h of_complex b) :\<^sub>h (of_complex c *\<^sub>h z +\<^sub>h of_complex d) else of_complex a :\<^sub>h of_complex c)"
domIff: fixes a :: "'a"   and m :: "'a \<Rightarrow> 'b option" shows "(a \<in> dom m) = (m a \<noteq> None)"