hcju/isabelleps huggingface dataset:
subst_true_false: fixes \<A> :: "'b \<Rightarrow> bool"   and F :: "'b formula"   and n :: "'b" assumes "\<A> \<Turnstile> F" shows "\<A> \<Turnstile> F[\<top>/n] \<^bold>\<or> F[\<bottom>/n]"
continuous_on_avoid: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and a :: "'b" assumes "continuous_on s f"   and "x \<in> s"   and "f x \<noteq> a" shows "\<exists>e>0. \<forall>y\<in>s. dist x y < e \<longrightarrow> f y \<noteq> a"
distinct_trancl_list_impl: fixes L :: "('b \<times> 'b) list"   and ls :: "'b list" assumes "distinct L"   and "distinct ls" shows "distinct (trancl_list_impl L ls)"
monotone_SUP: fixes ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e set"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'a" assumes "monotone ord (\<subseteq>) f"   and "\<And>y. monotone ord (\<le>) (\<lambda>x. g x y)" shows "monotone ord (\<le>) (\<lambda>x. \<Squnion> (g x ` f x))"
bounded_iff: fixes b :: "'b"   and c :: "'b"   and a :: "'b" shows "(max b c \<le> a) = (b \<le> a \<and> c \<le> a)"
comp_cod_arr: fixes f :: "'a1"   and b :: "'a1" assumes "A1.arr f"   and "A1.cod f = b" shows "b \<cdot>\<^sub>A\<^sub>1 f = f"
that: fixes s :: "nat \<Rightarrow> 'a" assumes "\<And>i. P i (s i)"   and "strict_mono s" shows "thesis"
holomorphic_fun_eq_on_ball: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and r :: "real"   and g :: "complex \<Rightarrow> complex"   and w :: "complex" assumes "f holomorphic_on ball z r"   and "g holomorphic_on ball z r"   and "w \<in> ball z r"   and "\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z" shows "f w = g w"
maxBound_less_length: fixes x :: "nat" shows "(x \<le> fromEnum maxBound) = (x < length enum_class.enum)"
zero_le_numeral: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero (numeral.numeral one plus n)"
simps_3: fixes as :: "'a attree list"   and s :: "'a set \<times> 'a set" shows "attack (as \<oplus>\<^sub>\<or>\<^bsup>s\<^esup>) = s"
qp: shows "is_quantum_predicate (adjoint (tensor_P (1\<^sub>m N) (mat_incr K)) * (P' + Q) * tensor_P (1\<^sub>m N) (mat_incr K))"
length_card_dom_map_of: fixes l :: "('a \<times> 'b) list" assumes "distinct (map fst l)" shows "length l = card (dom (map_of l))"
redT_updTs_invariant: fixes final :: "'g \<Rightarrow> bool"   and r :: "('j, 'h, 'g, 'i, 'k, 'l) semantics"   and P :: "'m \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'i \<Rightarrow> bool"   and I :: "'h \<Rightarrow> 'm option"   and ts :: "'h \<Rightarrow> ('g \<times> 'n \<Rightarrow>f nat) option"   and m :: "'i"   and t :: "'h"   and x :: "'g"   and ta :: "('j, 'h, 'g, 'i, 'k, 'l) thread_action"   and x' :: "'g"   and m' :: "'i"   and ln :: "'n \<Rightarrow>f nat"   and ln' :: "'n \<Rightarrow>f nat" assumes "lifting_inv final r P"   and "ts_inv P I ts m"   and "r t (x, m) ta (x', m')"   and "thread_oks ts \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"   and "ts t = \<lfloor>(x, ln)\<rfloor>" shows "ts_inv P (upd_invs I P \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>) (redT_updTs ts \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>(t \<mapsto> (x', ln'))) m'"
run_grun: fixes A :: "('a, 'b) nba"   and k :: "nat"   and w :: "'a stream"   and r :: "'b stream"   and p :: "'b" assumes "run A (sdrop k w ||| r) p" shows "gurun A w (fromN (Suc k) ||| r) (k, p)"
quasinorm_onI: fixes F :: "'b set"   and N :: "'b \<Rightarrow> ennreal"   and C :: "real" assumes "\<And>x y. \<lbrakk>x \<in> F; y \<in> F\<rbrakk> \<Longrightarrow> x + y \<in> F"   and "\<And>x y. \<lbrakk>x \<in> F; y \<in> F\<rbrakk> \<Longrightarrow> N (x + y) \<le> ennreal C * N x + ennreal C * N y"   and "\<And>c x. \<lbrakk>c \<noteq> 0; x \<in> F\<rbrakk> \<Longrightarrow> c *\<^sub>R x \<in> F"   and "\<And>c x. \<lbrakk>c \<noteq> 0; x \<in> F\<rbrakk> \<Longrightarrow> N (c *\<^sub>R x) \<le> ennreal \<bar>c\<bar> * N x"   and "(0::'b) \<in> F"   and "N (0::'b) = 0"   and "1 \<le> C" shows "quasinorm_on F C N"
Lim_dist_ubound: fixes net :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and a :: "'c"   and e :: "real" assumes "net \<noteq> bot"   and "(f \<longlongrightarrow> l) net"   and "\<forall>\<^sub>F x in net. dist a (f x) \<le> e" shows "dist a l \<le> e"
arl_nth_def: shows "arl_nth \<equiv> \<lambda>(a, n). Array.nth a"
HPhase1or2ReadThen_HInv5_q2: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "p \<noteq> q"   and "HInv4a s p"   and "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s q) < mbal (disk s d qq) \<and> \<not> hasRead s q d qq" shows "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s' q) < mbal (disk s' d qq) \<and> \<not> hasRead s' q d qq"
le_st_iff: fixes F :: "'a st"   and G :: "'a st" shows "(F \<le> G) = (\<forall>x. fun F x \<le> fun G x)"
w_defs_5: shows "w5 = snd (bv_chop (snd (bv_chop (snd (bv_chop (snd (bv_chop w (Suc l))) (Suc j))) i)) k)"
next_free_not: fixes \<phi> :: "'b pltl" shows "next_free (not\<^sub>p \<phi>) = next_free \<phi>"
dsj: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "dsj \<phi> \<chi> \<in> fmla"
that: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> F"   and "f \<noteq> 0"   and "lt f adds\<^sub>t t" shows "thesis"
reachable_def: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v \<equiv> (u, v) \<in> rtrancl_on (verts G) (arcs_ends G)"
extended_Gromov_product_at_topological: fixes e :: "'c"   and x :: "'c Gromov_completion"   and y :: "'c Gromov_completion" shows "extended_Gromov_product_at e x y = Inf {liminf (\<lambda>n. ereal (Gromov_product_at e (u n) (v n))) |u v. (\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x \<and> (\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> y}"
rawpsubst_compose: fixes \<phi> :: "'fmla"   and txs1 :: "('trm \<times> 'var) list"   and txs2 :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "snd ` set txs1 \<subseteq> var"   and "fst ` set txs1 \<subseteq> trm"   and "snd ` set txs2 \<subseteq> var"   and "fst ` set txs2 \<subseteq> trm" shows "rawpsubst (rawpsubst \<phi> txs1) txs2 = rawpsubst \<phi> (txs1 @ txs2)"
alphaAbs_ex_distinct_qAFresh_imp_qFresh: 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" assumes "alphaAbs_ex_distinct_qAFresh xs x X xs' x' X'" shows "alphaAbs_ex_distinct_qFresh xs x X xs' x' X'"
abs_eq: fixes x :: "'b \<Rightarrow> 'c" assumes "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'c)}) x x" shows "Poly_Mapping.keys (Abs_poly_mapping x) = {k. x k \<noteq> (0::'c)}"
msubset_of_def: fixes A :: "'d set"   and F :: "('d \<Rightarrow> 'c) set" shows "msubset_of A F = univ (\<lambda>f. image_mset f (mset_set A)) F"
fixf_le_lubH: fixes H :: "'a set"   and x :: "'a" assumes "H = {x \<in> A. (x, f x) \<in> r}"   and "x \<in> fix f A" shows "(x, lub H cl) \<in> r"
intros_7: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P' :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c :: "com"   and Q' :: "'a \<Rightarrow> state \<Rightarrow> bool"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool" assumes "C \<turnstile>\<^sub>t {P'} c {Q'}"   and "(\<forall>s t. (\<forall>z. P' z s \<longrightarrow> Q' z t) \<longrightarrow> (\<forall>z. P z s \<longrightarrow> Q z t)) \<and> (\<forall>s. (\<exists>z. P z s) \<longrightarrow> (\<exists>z. P' z s))" shows "C \<turnstile>\<^sub>t {P} c {Q}"
pochhammer_prod_rev: fixes a :: "'a"   and n :: "nat" shows "pochhammer a n = (\<Prod>i = 1..n. a + of_nat (n - i))"
type_aexpr_elim: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and e :: "'AExp"   and t :: "Sec" assumes "\<Gamma> \<turnstile>\<^sub>a e \<in> t" obtains "t = max_dom (to_total \<Gamma> ` aexp_vars e)"
object_ptr_kinds_M_defs: shows "object_ptr_kinds_M = Heap_Error_Monad.bind get_heap (\<lambda>h. return (sorted_list_of_set (fset (object_ptr_kinds h))))"
Nmlize_Nml: fixes t :: "'a E.term" assumes "E.Nml t" shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> = t"
cnf: shows "\<forall>F\<in>#image_mset (cnf_form_of \<circ> nnf) (\<^bold>\<not> F, S'). is_cnf F"
delete_bal: fixes k :: "nat"   and t :: "'a btree"   and x :: "'a" assumes "0 < k"   and "bal t"   and "root_order k t" shows "bal (delete k x t)"
map_psubst_new: fixes c :: "'a"   and z :: "('a, 'b) form list"   and x :: "'a" assumes "news c z" shows "map (psubst (id(c := x))) z = z"
valid_path_circlepath: fixes z :: "complex"   and r :: "real" shows "valid_path (circlepath z r)"
IH: shows "subterms\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t D) \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t (M \<union> N)"
fully_connected_def: fixes G :: "'a graph" shows "fully_connected G \<equiv> \<lparr>nodes = nodes G, edges = nodes G \<times> nodes G\<rparr>"
minimalize_offending_overapprox_spec_impl: fixes G :: "'a list_graph"   and sinvar_spec :: "'a graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and sinvar_impl :: "'a list_graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and nP :: "'a \<Rightarrow> 'b"   and fs :: "('a \<times> 'a) list"   and keeps :: "('a \<times> 'a) list" assumes "wf_list_graph G"   and "\<And>G nP. wf_list_graph G \<Longrightarrow> sinvar_spec (list_graph_to_graph G) nP = sinvar_impl G nP" shows "TopoS_Interface_impl.minimalize_offending_overapprox (\<lambda>G. sinvar_impl G nP) fs keeps G = TopoS_withOffendingFlows.minimalize_offending_overapprox (\<lambda>G. sinvar_spec G nP) fs keeps (list_graph_to_graph G)"
eq_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "ordering less_eq less" shows "(a = b) = (less_eq a b \<and> less_eq b a)"
incl_Rel_components_2: fixes A :: "V"   and B :: "V" shows "incl_Rel A B\<lparr>ArrDom\<rparr> = A"
Sup_subset_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "Sup A \<le> Sup B"
c_manifolds_def: fixes k :: "enat"   and charts1 :: "('c, 'd) chart set"   and charts2 :: "('e, 'f) chart set" shows "c_manifolds k charts1 charts2 \<equiv> c_manifold charts1 k \<and> c_manifold charts2 k"
Rep_fin_string_cases: fixes y :: "char list" assumes "y \<in> {s. length s \<le> MAX_STR}" obtains x :: "fin_string" where "y = Rep_fin_string x"
cts_wp_SetPC: fixes p :: "'b \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>x s. x \<in> supp (p s) \<Longrightarrow> bd_cts (wp (a x))"   and "\<And>x s. x \<in> supp (p s) \<Longrightarrow> healthy (wp (a x))"   and "\<And>s. unitary (p s)"   and "\<And>s. sum (p s) (supp (p s)) \<le> 1"   and "\<And>s. finite (supp (p s))" shows "bd_cts (wp (SetPC a p))"
qring_pOp: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<uplus>\<^bsub>R\<^esub> I \<plusminus>\<^bsub>R /\<^sub>r I\<^esub> b \<uplus>\<^bsub>R\<^esub> I = (a \<plusminus> b) \<uplus>\<^bsub>R\<^esub> I"
a_Hcarr: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme"   and h :: "'e" assumes "additive_subgroup H G"   and "h \<in> H" shows "h \<in> carrier G"
grover_partial_deduct: shows "\<turnstile>\<^sub>p {tensor_P pre (proj_k 0)} Grover {tensor_P post (1\<^sub>m K)}"
smult_l_null: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<zero> \<odot>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>"
upd_nested_insert': fixes m :: "('e, ('f, nat + nat) mapping) mapping"   and x :: "'e \<times> 'f"   and A :: "('e \<times> 'f) set" shows "upd_nested m d (max_tstp d) (insert x A) = upd_nested_step d (max_tstp d) x (upd_nested m d (max_tstp d) A)"
cont: shows "isCont (\<lambda>x. interpret_floatarith fa (list_of_eucl x @ params)) x"
pr_swap: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "f \<in> PrimRec2" shows "(\<lambda>x y. f y x) \<in> PrimRec2"
simps_3: fixes v :: "prim" shows "benv_in_d (DP v) = {}"
zero_pnat_top: fixes x :: "pnat" shows "x \<le> 1"
monom_eq_const_iff: fixes c :: "'a"   and n :: "nat"   and d :: "'a" shows "(Polynomial.monom c n = [:d:]) = (c = d \<and> (c = (0::'a) \<or> n = 0))"
g: fixes F :: "'a filter"   and G :: "('a \<Rightarrow> real) set"   and H :: "('a \<Rightarrow> real) set"   and g :: "'a \<Rightarrow> real" assumes "landau_function_family_pair F G H g" shows "g \<in> G"
free_unfree: fixes u :: "'a"   and name :: "name" assumes "unfree u = Some name" shows "u = free name"
SpecStuck: fixes s :: "'a"   and r :: "('a \<times> 'a) set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and n :: "nat" assumes "\<forall>t. (s, t) \<notin> r" shows "\<Gamma>\<turnstile> \<langle>Spec r,Normal s\<rangle> =n\<Rightarrow> Stuck"
enc_set_\<sigma>: fixes x :: "('a \<times> bool list) list"   and w :: "'a list"   and I :: "(nat + nat set) list"   and n :: "nat"   and a :: "'a \<times> bool list" assumes "x \<in> enc (w, I)"   and "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"   and "length I = n"   and "a \<in> set x"   and "set w \<subseteq> set \<Sigma>" shows "a \<in> set (\<sigma> \<Sigma> n)"
find_closest_eq_val_find_closest_tm: fixes p :: "int \<times> int"   and \<delta> :: "real"   and ps :: "(int \<times> int) list" shows "Time_Monad.val (find_closest_tm p \<delta> ps) = find_closest p \<delta> ps"
pfp_inv: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "pfp f x = Some y"   and "\<And>x. P x \<Longrightarrow> P (f x)"   and "P x" shows "P y"
some_eq_intro: fixes X :: "('b \<times> 'c) set"   and x :: "'b"   and y :: "'c" assumes "single_valued X"   and "(x, y) \<in> X" shows "(SOME y. (x, y) \<in> X) = y"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter" shows "ptr |\<in>| element_ptr_kinds h"
blocksEval: fixes ps :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option" assumes "length ps = length Ts"   and "length ps = length vs"   and "P \<turnstile> \<langle>blocks (ps, Ts, vs, e),(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>" shows "\<exists>l''. P \<turnstile> \<langle>e,(h, l(ps [\<mapsto>] vs))\<rangle> \<Rightarrow> \<langle>e',(h', l'')\<rangle>"
unit_prod: fixes S :: "'a set"   and U :: "(('b, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" assumes "finite S" shows "is_unit (\<Prod>i\<in>S. U $h i $h i) = (\<forall>i\<in>S. is_unit (U $h i $h i))"
l8_20_1: 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"   and P :: "'a"   and C' :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong P C' D"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A C' C"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong B D C" shows "Tarski_neutral_dimensionless.Per Bet Cong B A P"
poly_div_gcd_squarefree_1: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "coprime (p div gcd p (pderiv p)) (pderiv (p div gcd p (pderiv p)))"
prems_3: shows "(ab_, a'b_) \<in> R \<rightarrow> Rv"
rep_real_alg_3: fixes x :: "real_alg_3" shows "rep_real_alg_3 x \<in> Collect invariant_2"
stuck_sim: fixes a :: "'a"   and b :: "'b" assumes "(a, b) \<in> R"   and "b \<notin> Domain b.E" shows "a \<notin> Domain a.E"
\<Delta>_same_shift: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'c" assumes "\<Delta> f = \<Delta> g" shows "\<exists>l. (+) l \<circ> f \<circ> of_int = g \<circ> of_int"
rep_eq: fixes x :: "('f, 'e) generator" shows "generator (drop_trans x) = drop_raw (generator x)"
right_neutral: fixes a :: "'b multiset" shows "a \<union># \<Empt> = a"
Zm_to_M_0: fixes M :: "('f, 'a, 'g) Module_scheme"   and f :: "'h \<Rightarrow> 'f"   and e :: "'h" assumes "R module M"   and "f \<in> mHom R (Zm R e) M" shows "f e = \<zero>\<^bsub>M\<^esub>"
cet_sub_int: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : INTEG"   and "\<Gamma> \<turnstile>\<^sub>c e2 : INTEG" shows "\<Gamma> \<turnstile>\<^sub>c e1 -\<^sub>c e2 : INTEG"
Q2: shows "P \<noteq> X \<and> S \<noteq> X \<and> R \<noteq> Y"
lemma_2_reachability_s: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and s :: "('a, bool) fmap"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "card (reachable_s PROB s) - 1 < length as" shows "\<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []"
domain_eq: shows "Domainp (pcr_dnelist (=)) = (\<lambda>xs. distinct xs \<and> xs \<noteq> [])"
truncate_down_nonneg: fixes y :: "real"   and prec :: "nat" assumes "0 \<le> y" shows "0 \<le> truncate_down prec y"
oInv_decreasing: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal" assumes "normal F"   and "F 0 \<le> x" shows "oInv F x \<le> x"
is_natural_1: fixes f :: "'a" assumes "arr f" shows "trg f \<cdot> trg (local.dom f) = trg f"
f_eq_iff: fixes x :: "'a"   and y :: "'b" shows "(hom x = y) = (x = inv hom y)"
hurwitz_zeta_nat_conv_infsetsum_2: fixes a :: "nat"   and s :: "complex" assumes "0 < a"   and "1 < Re s" shows "hurwitz_zeta (real a) s = (\<Sum>\<^sub>an\<in>{a..}. of_nat n powr - s)"
to_from_enum: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and x :: "'b" assumes "class.enum enum enum_all enum_ex" shows "enum.toEnum enum (enum.fromEnum enum x) = x"
path_connected_open_diff_countable: fixes S :: "'b set"   and T :: "'b set" assumes "2 \<le> DIM('b)"   and "open S"   and "connected S"   and "countable T" shows "path_connected (S - T)"
monadic_WHILEIT_def: fixes I :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool nres"   and f :: "'c \<Rightarrow> 'c nres"   and s :: "'c" shows "monadic_WHILEIT I b f s \<equiv> REC\<^sub>T (\<lambda>D s. ASSERT (I s) \<bind> (\<lambda>_. b s \<bind> (\<lambda>bv. if bv then f s \<bind> D else RETURN s))) s"
genideal_ideal: fixes S :: "'a set" assumes "S \<subseteq> carrier R" shows "ideal (Idl S) R"
nz: shows "pochhammer c n \<noteq> (0::'a)"
w_simps_4: shows "trg f.chine = src (tab\<^sub>0 f)"
longer_falsifies\<^sub>g: fixes C :: "fterm literal set"   and ds :: "bool list"   and d :: "bool list" assumes "falsifies\<^sub>g ds C" shows "falsifies\<^sub>g (ds @ d) C"
nths_empty: fixes xs :: "'a list" shows "nths xs {} = []"
problem_plan_bound_S_bound_1st_step: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and vs :: "'a set"   and lss :: "('a, bool) fmap list"   and s :: "('a, bool) fmap"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list" assumes "finite PROB"   and "top_sorted_abs (\<lambda>x y. y \<in> state_successors (prob_proj PROB vs) x) lss"   and "set lss = valid_states (prob_proj PROB vs)"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "no_effectless_act as"   and "sat_precond_as s as" shows "\<exists>as'. exec_plan s as' = exec_plan s as \<and> subseq as' as \<and> length as' \<le> S vs lss PROB (fmrestrict_set vs s)"
Seq: fixes P\<^sub>1 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c\<^sub>1 :: "com"   and P\<^sub>2 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c\<^sub>2 :: "com"   and P\<^sub>3 :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2 {P\<^sub>1} c\<^sub>1 {P\<^sub>2}"   and "\<turnstile>\<^sub>2 {P\<^sub>2} c\<^sub>2 {P\<^sub>3}" shows "\<turnstile>\<^sub>2 {P\<^sub>1} c\<^sub>1;; c\<^sub>2 {P\<^sub>3}"
parallel_is_affine: fixes S :: "'a set"   and T :: "'a set" assumes "affine S"   and "affine_parallel S T" shows "affine T"
a1: shows "low_equal s1 s2 \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s1 = Some y \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s2 = Some ya \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) y) y = Some yb \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) ya) ya = None"
prod_mset_empty: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one {#} = one"
det_swaprows: fixes k :: "nat"   and n :: "nat"   and l :: "nat"   and A :: "'b mat" assumes "k < n"   and "l < n"   and "k \<noteq> l"   and "A \<in> carrier_mat n n" shows "det (swaprows k l A) = - det A"
fstI: fixes x :: "'f \<times> 'g"   and y :: "'f"   and z :: "'g" assumes "x = (y, z)" shows "fst x = y"
hyps_2: shows "w \<notin> subst_domain \<theta>"
aug_pm_set_def: fixes z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set" shows "aug_pm_set z i A = {z} \<union> A \<union> minus_set i A"
SeqAbstForm_def: fixes v :: "hf"   and i :: "hf"   and x :: "hf"   and x' :: "hf"   and s :: "hf"   and k :: "hf" shows "SeqAbstForm v i x x' s k \<equiv> BuildSeq3 (AbstAtomic v) AbstMakeForm s k i x x'"
schroeder_3_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> - z) = (id x \<sqinter> z \<le> - y)"
rec_add_def: shows "rec_add \<equiv> Pr 1 (recf.id 1 0) (Cn 3 s [recf.id 3 2])"
action_proj_idempot: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and vs :: "'d set" shows "action_proj (action_proj a vs) vs = action_proj a vs"
span_subset_spanI: fixes A :: "('b \<Rightarrow>\<^sub>0 'c) set"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "A \<subseteq> phull B" shows "phull A \<subseteq> phull B"
g: fixes j :: "nat"   and r :: "nat"   and i :: "nat"   and e :: "nat" shows "eval g [j, r, i, e] = (if eval r_psi [i, j] \<up> then None else if eval r_psi [i, j] \<down>= e_nth e j then Some r else Some 1)"
valid_prefix_fw_def: fixes m :: "'b word \<times> nat" shows "valid_prefix_fw m = valid_prefix (uncurry PrefixMatch m)"
dims0_def: shows "st2.dims0 = nths dims st2.vars0"
IH: assumes "is_ascii (upChar Y)" shows "is_ascii (fresh_string (Xs - {Y}) (upChar Y))"
sig_trd_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'b" shows "sig_trd p = (if rep_list p = 0 then p else sig_trd_aux (punit.lt (rep_list p), p))"
CallUndefined: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and p :: "'b"   and s :: "'a" assumes "\<Gamma> p = None" shows "\<Gamma>\<turnstile>Call p \<down> Normal s"
untime_eq_filter: fixes s :: "'b message_af list" shows "map Msg (untime s) = filter (\<lambda>x. x \<noteq> NoMsg) s"
Fun_mapsto: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "Fun f \<in> extensional (Dom f) \<inter> (Dom f \<rightarrow> Cod f)"