hcju/isabelleps huggingface dataset:
transpos_vals_nonequiv: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and j :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "j \<le> Suc n" shows "vals_nonequiv K (Suc n) (vv \<circ> \<tau>\<^bsub>0 j\<^esub>)"
LeftDerivationFix_def: fixes \<alpha> :: "symbol list"   and i :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<beta> :: "symbol list" shows "LeftDerivationFix \<alpha> i D j \<beta> = (is_sentence \<alpha> \<and> is_sentence \<beta> \<and> LeftDerivation \<alpha> D \<beta> \<and> i < length \<alpha> \<and> j < length \<beta> \<and> \<alpha> ! i = \<beta> ! j \<and> (\<exists>E F. D = E @ derivation_shift F 0 (Suc j) \<and> LeftDerivation (take i \<alpha>) E (take j \<beta>) \<and> LeftDerivation (drop (Suc i) \<alpha>) F (drop (Suc j) \<beta>)))"
i: shows "{(x, y) |x y. nodes_connected H x y} \<subseteq> V \<times> V"
distrib_4: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'c"   and z :: "'c"   and x :: "'c" assumes "class.distrib_lattice inf less_eq less sup" shows "inf (sup y z) x = sup (inf y x) (inf z x)"
simps_4: fixes p :: "real \<times> real \<times> real" shows "substNegInfinityUni (NeqUni p) = negUni (allZero' p)"
qTermLess_modulo_wf: fixes rel :: "(('a, 'b, 'c, 'd, 'e) qTerm \<times> ('a, 'b, 'c, 'd, 'e) qTerm) set" assumes "\<And>X Y. (X, Y) \<in> rel \<Longrightarrow> qSkel Y = qSkel X" shows "wf (qTermLess_modulo rel)"
abs_mono: fixes G :: "msg set"   and H :: "msg set" assumes "G \<subseteq> H" shows "local.abs G \<subseteq> local.abs H"
label0_def: fixes x :: "'node" shows "label0 x = (if atom0 x then some else none)"
subst_typ_comp_single_left_stronger: fixes single :: "(variable \<times> String.literal set) \<times> typ"   and insts :: "((variable \<times> String.literal set) \<times> typ) list"   and T :: "typ" shows "subst_typ [single] (subst_typ insts T) = subst_typ (map (apsnd (subst_typ [single])) insts @ (if fst single \<in> set (map fst insts) then [] else [single])) T"
is_ub_def: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b" assumes "class.po below" shows "po.is_ub below S x = (\<forall>y\<in>S. below y x)"
simps: fixes a :: "'a"   and g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)" shows "(a \<in> terminates_on g) = ((\<exists>s. a = s \<and> \<not> fst g s) \<or> (\<exists>s. a = s \<and> fst g s \<and> snd (snd g s) \<in> terminates_on g))"
Sim: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> Rel" shows "P \<leadsto>\<^sup>^<Rel> Q"
ring_simprules_19: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> (x \<oplus>\<^bsub>P\<^esub> y) = \<ominus>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> y"
asTerm_equal_iff_alpha: fixes qX :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and qY :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood qX \<or> qGood qY" shows "(asTerm qX = asTerm qY) = (qX #= qY)"
homotopic_paths_refl: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b" shows "homotopic_paths s p p = (path p \<and> path_image p \<subseteq> s)"
det_cols_mul: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m n" shows "Determinant.det (A * B) = (\<Sum>f | (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i). (\<Prod>i = 0..<n. B $$ (f i, i)) * Determinant.det (mat\<^sub>r n n (\<lambda>i. col A (f i))))"
sup_top_left: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and top :: "'d"   and x :: "'d" assumes "class.bounded_lattice_top inf less_eq less sup top" shows "sup top x = top"
aprimedivisor_primepow_factors_conv_prime_factorization: fixes n :: "'a" assumes "n \<noteq> (0::'a)" shows "image_mset aprimedivisor (mset_set (primepow_factors n)) = prime_factorization n"
add_smult_distrib_right_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and k :: "'b"   and l :: "'b" assumes "A \<in> carrier_mat nr nc" shows "(k + l) \<cdot>\<^sub>m A = k \<cdot>\<^sub>m A + l \<cdot>\<^sub>m A"
IH: fixes \<Delta>s :: "'a list" assumes "\<Delta> \<in> set \<Delta>s'"   and "\<Delta>s = \<Delta>s' ## \<Delta>1" shows "\<exists>\<Delta>2. (\<Delta>, \<Delta>2) \<in> allConsec \<Delta>s"
ereal_le_add_mono1: fixes x :: "ereal"   and y :: "ereal"   and z :: "ereal" assumes "x \<le> y"   and "0 \<le> z" shows "x \<le> y + z"
ext: fixes g :: "('a, 'a set) arr"   and f :: "('a, 'a set) arr" assumes "g \<odot> f \<noteq> null" shows "seq g f"
g_list_to_map_def: fixes l :: "('k \<times> 'v) list" shows "g_list_to_map l \<equiv> foldl (\<lambda>m (k, v). update k v m) (local.empty ()) (rev l)"
ltl_equivalence_axioms: shows "ltl_equivalence (\<sim>\<^sub>Q)"
topology_generated_by_Basis: fixes s :: "'c set"   and S :: "'c set set" assumes "s \<in> S" shows "openin (topology_generated_by S) s"
induct: fixes P :: "name list \<Rightarrow> name list \<Rightarrow> bool"   and a0 :: "name list"   and a1 :: "name list" assumes "P [] []"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>v va. P (v # va) []"   and "\<And>v va. P [] (v # va)" shows "P a0 a1"
COUPLING: shows "coupling (Q_lookup Q) (M_lookup V) D S"
LID\<^sub>SE: fixes l :: "'d"   and \<sigma> :: "'d \<Rightarrow> ('e, 'd, 'f) val option" assumes "l \<in> LID\<^sub>S \<sigma>" obtains "l \<in> dom \<sigma>"   | l' :: "'d" and v :: "('e, 'd, 'f) val" where "\<sigma> l' = Some v" and "l \<in> LID\<^sub>V v"
infiter_inf_top: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "infinite_iteration.infiter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) = seq (finite_or_infinite_iteration.iter seq (nil::'b::refinement_lattice) c) \<top>"
simps_2: fixes x21 :: "'a"   and x22 :: "'a"   and x23 :: "'b" shows "J_C.Null \<noteq> J_C.MkArr x21 x22 x23"
reflp_def: fixes r :: "'f \<Rightarrow> 'f \<Rightarrow> bool" shows "reflp r = (\<forall>x. r x x)"
onorm_bound: fixes b :: "real"   and f :: "'d \<Rightarrow> 'e" assumes "0 \<le> b"   and "\<And>x. norm (f x) \<le> b * norm x" shows "onorm f \<le> b"
simps: fixes a :: "'a"   and outPorts :: "'b \<Rightarrow> 'c fset"   and vertices :: "'a fset"   and nodeOf :: "'a \<Rightarrow> 'b"   and edges :: "(('a \<times> 'c) \<times> 'a \<times> 'd) set"   and ps :: "'a \<times> 'd" shows "(a \<in> Pre_Port_Graph.scope outPorts vertices nodeOf edges ps) = (\<exists>v. a = v \<and> v |\<in>| vertices \<and> (\<forall>x xa. Pre_Port_Graph.path edges v xa x \<longrightarrow> Vertex_Graph.terminal_vertex outPorts vertices nodeOf xa \<longrightarrow> ps \<in> snd ` set x))"
in_rtrancl_on_in_F_1: fixes a :: "'c"   and b :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set" assumes "(a, b) \<in> rtrancl_on F r" shows "a \<in> F"
has_suc_notempty: fixes n :: "nat"   and S :: "'a set" assumes "has (Suc n) S" shows "{} \<noteq> S"
inext_nth_surj_on: fixes I :: "nat set" shows "surj_on (inext_nth I) UNIV I"
assms_1: shows "fmrestrict_set vs s = fmrestrict_set vs s'"
remdups_list_of_dlist: fixes dxs :: "'b dlist" shows "remdups (list_of_dlist dxs) = list_of_dlist dxs"
literalElClause_def: fixes literal :: "Literal"   and clause :: "Literal list" shows "literal el clause \<equiv> literal \<in> set clause"
b: shows "abc_steps_l (0, lm) A na = (sa, lma)"
strict_trans2: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" assumes "strict_prefix a b"   and "prefix b c" shows "strict_prefix a c"
IH: assumes "a \<in> set (insert_spec xs (e, i))"   and "a \<notin> set xs" shows "a = e"
hp_1: shows "tms \<Gamma> \<Rightarrow> F, tms \<Delta>"
UN_simps_1: fixes a :: "'a"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "(\<Union>x\<in>C. insert a (B x)) = (if C = {} then {} else insert a (\<Union> (B ` C)))"
continuous_at_sequentially: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "isCont f a = (\<forall>x. x \<longlonglongrightarrow> a \<longrightarrow> (f \<circ> x) \<longlonglongrightarrow> f a)"
new_character_data_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and ptr :: "(_) document_ptr"   and getter :: "('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> 'w" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
min_None_trans: fixes x :: "('c, 'd) bintree option"   and xs :: "('c, 'd) bintree option list" assumes "Binomial_Queue.min (x # xs) = None" shows "Binomial_Queue.min xs = None"
assms: shows "\<forall>x\<in>set c. Pratt_Certificate.size_pratt x \<le> d"
res: fixes q :: "'b"   and r :: "'c"   and s' :: "'a"   and s :: "'a" assumes "q \<in> outs_\<I> \<I>"   and "(r, s') \<in> results_gpv \<I>' (callee s q)" shows "r \<in> responses_\<I> \<I> q"
id_stake_snth_sdrop: fixes s :: "'b stream"   and i :: "nat" shows "s = stake i s @- s !! i ## sdrop (Suc i) s"
disc_36: fixes x1 :: "('d, 'e) Term.term" shows "\<not> is_InSet (send\<langle>x1\<rangle>)"
d2: shows "v3 = Matrix.mat 8 1 (\<lambda>(i, j). [[0, 0, 0, 0, \<phi> $$ (0, 0), - \<phi> $$ (1, 0), 0, 0]] ! j ! i)"
typing_alt_simps_4: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and c :: "unat"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E Literal (UnlimitedNaturalLiteral c) : \<tau>) = (\<tau> = UnlimitedNatural[1])"
prefs_wf'_2: fixes i :: "'agent" assumes "i \<in> agents" shows "Order_Predicates.preorder_on alts (R i)"
aeT_def: fixes c :: "('test, 'atom, 'choice) cmd" shows "aeT c = (\<forall>s. AE bT in T (c, s). qsend ((c, s) ## bT) < \<infinity>)"
one: assumes "y \<sqinter> nc = (0::'a)" shows "(x \<sqinter> nc) \<cdot> nc \<cdot> ((y \<sqinter> nc) \<cdot> nc) \<cdot> nc = (x \<sqinter> nc) \<cdot> nc \<cdot> ((y \<sqinter> nc) \<cdot> nc)"
hyps_1: shows "p' \<in> grid b (ds' \<union> ds)"
Pd'_range: fixes d :: "'b" shows "Field (Pd' d) \<subseteq> {x. Xd x = d}"
prems: shows "\<Gamma>\<turnstile> \<langle>merge_guards (Catch c1 c2),Normal s\<rangle> =n\<Rightarrow> t"
finprod_insert: fixes F :: "'a set"   and a :: "'a"   and f :: "'a \<Rightarrow> int" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier R"   and "f a \<in> carrier R" shows "finsum R f (insert a F) = f a \<oplus> finsum R f F"
hyp1: shows "matrix_to_iarray (Gauss_Jordan_upt_k A k) = Gauss_Jordan_upt_k_iarrays (matrix_to_iarray A) k"
inf_extended_case: fixes x1a :: "'a extended"   and x2a :: "'a extended" shows "inf x1a x2a = (case x1a of Fin xda \<Rightarrow> case x2a of Fin x2da \<Rightarrow> Fin (inf xda x2da) | \<infinity> \<Rightarrow> Fin xda | -\<infinity> \<Rightarrow> -\<infinity> | \<infinity> \<Rightarrow> case x2a of Fin xca \<Rightarrow> Fin xca | \<infinity> \<Rightarrow> \<infinity> | -\<infinity> \<Rightarrow> -\<infinity> | -\<infinity> \<Rightarrow> case x2a of Fin xba \<Rightarrow> -\<infinity> | _ \<Rightarrow> -\<infinity>)"
lzip_lmap: fixes f :: "'e \<Rightarrow> 'c"   and xs :: "'e llist"   and g :: "'f \<Rightarrow> 'd"   and ys :: "'f llist" shows "lzip (lmap f xs) (lmap g ys) = lmap (\<lambda>(x, y). (f x, g y)) (lzip xs ys)"
hb_irrefl: fixes m1 :: "'a"   and m2 :: "'a" assumes "hb m1 m2" shows "m1 \<noteq> m2"
statClosed: fixes \<Psi> :: "'b"   and \<phi> :: "'c"   and p :: "(name \<times> name) list" assumes "\<Psi> \<turnstile> \<phi>" shows "(p \<bullet> \<Psi>) \<turnstile> (p \<bullet> \<phi>)"
n_def: shows "Rep_nat_int (res ts c) \<union> Rep_nat_int (clm ts c) = {n, n + 1}"
post_lfp: fixes c :: "com"   and f :: "'a set acom \<Rightarrow> 'a set acom" shows "post (Collecting.lfp c f) = \<Inter> {post C |C. strip C = c \<and> f C \<le> C}"
has_bochner_integral_simple_bochner_integrable: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "Bochner_Integration.simple_bochner_integrable M f" shows "has_bochner_integral M f (Bochner_Integration.simple_bochner_integral M f)"
simps_2: fixes P :: "'c \<Rightarrow> ('a, 'b, bool) prog"   and x :: "'c"   and xs :: "'c list" shows "filter_M P (x # xs) = P x \<bind> (\<lambda>p. filter_M P xs \<bind> (\<lambda>ys. return (if p then x # ys else ys)))"
card_g: shows "(k', roomk (s\<lparr>isin := (isin s)(r1 := isin s r1 \<union> {g1}), roomk := (roomk s)(r1 := k1), safe := (safe s)(r1 := owns s r1 = \<lfloor>g1\<rfloor> \<and> isin s r1 = {} \<and> k1 = currk s r1 \<or> safe s r1)\<rparr>) r) \<in> cards (s\<lparr>isin := (isin s)(r1 := isin s r1 \<union> {g1}), roomk := (roomk s)(r1 := k1), safe := (safe s)(r1 := owns s r1 = \<lfloor>g1\<rfloor> \<and> isin s r1 = {} \<and> k1 = currk s r1 \<or> safe s r1)\<rparr>) g"
store_sub2_low_equal_sub2: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and addr :: "word32"   and instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and y :: "'b sparc_state"   and ya :: "'b sparc_state"   and yb :: "'b sparc_state" assumes "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 = None \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) ya) ya = Some yb" shows "False"
write_cpu_result: fixes w :: "word32"   and r :: "CPU_register"   and s :: "'b sparc_state" shows "snd (write_cpu w r s) = False"
HSem_ignores_fresh_restr'': fixes ESem :: "'d \<Rightarrow> ('e \<Rightarrow> 'f) \<rightarrow> 'f"   and \<Gamma> :: "('e \<times> 'd) list"   and S :: "'e set"   and \<rho> :: "'e \<Rightarrow> 'f" assumes "has_ignore_fresh_ESem ESem"   and "fv \<Gamma> \<subseteq> S" shows "has_ESem.HSem ESem \<Gamma>\<cdot>\<rho> f|` S = has_ESem.HSem ESem \<Gamma>\<cdot>(\<rho> f|` S)"
satisfies_atom_set_def: fixes v :: "nat \<Rightarrow> 'c"   and as :: "'c atom set" shows "v \<Turnstile>\<^sub>a\<^sub>s as \<equiv> \<forall>a\<in>as. v \<Turnstile>\<^sub>a a"
contrapos_nn: fixes Q :: "bool"   and P :: "bool" assumes "\<not> Q"   and "P \<Longrightarrow> Q" shows "\<not> P"
l3_inv5D: fixes x :: "l3_state" assumes "x \<in> l3_inv5" shows "synth (analz (ik x)) \<subseteq> dy_fake_msg (l3_state.bad x) (ik x \<inter> payload) (local.abs (ik x)) \<union> - payload"
to_st_append: fixes A :: "('fun, 'var) extstrand_step list"   and B :: "('fun, 'var) extstrand_step list" shows "to_st (A @ B) = to_st A @ to_st B"
fixp_induct: fixes P :: "(('c, 'out, 'd) gpv \<Rightarrow> enat) \<Rightarrow> bool" assumes "lfp.admissible P"   and "P (\<lambda>interaction_bound. \<Squnion> {})"   and "\<And>gpv. \<lbrakk>lfp.le_fun gpv local.interaction_bound; lfp.le_fun gpv (\<lambda>gpva. \<Squnion>generat\<in>set_spmf (the_gpv gpva). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. gpv (c input)) else \<Squnion>input. gpv (c input)); P gpv\<rbrakk> \<Longrightarrow> P (\<lambda>gpva. \<Squnion>generat\<in>set_spmf (the_gpv gpva). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. gpv (c input)) else \<Squnion>input. gpv (c input))" shows "P local.interaction_bound"
ring_hom_zero: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme" assumes "h \<in> ring_hom R S"   and "ring R"   and "ring S" shows "h \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>S\<^esub>"
can_mkcan_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set"   and b :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a"   and "partial_magma.ide (free_monoidal_category.comp C) b"   and "monoidal_language.Diagonalize C (monoidal_language.Dom C (free_monoidal_category.rep C a)) = monoidal_language.Diagonalize C (monoidal_language.Cod C (free_monoidal_category.rep C b))" shows "partial_magma.in_hom (free_monoidal_category.comp C) (free_monoidal_category.mkcan C a b) a b"
is_real_div: fixes b :: "complex"   and a :: "complex" assumes "b \<noteq> 0" shows "is_real (a / b) = (a * cnj b = b * cnj a)"
J_rel_coind_coind2: fixes R2 :: "'d JF1 \<Rightarrow> 'e JF1 \<Rightarrow> bool"   and f :: "'d JF1 \<Rightarrow> 'e JF1 \<Rightarrow> ('d \<times> 'e, 'd JF1 \<times> 'e JF1, 'd JF2 \<times> 'e JF2) F1"   and R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R3 :: "'d JF2 \<Rightarrow> 'e JF2 \<Rightarrow> bool"   and g :: "'d JF2 \<Rightarrow> 'e JF2 \<Rightarrow> ('d \<times> 'e, 'd JF1 \<times> 'e JF1, 'd JF2 \<times> 'e JF2) F2"   and y1 :: "'e JF1"   and y1' :: "'e JF1"   and y2 :: "'e JF2"   and y2' :: "'e JF2" assumes "\<forall>x y. R2 x y \<longrightarrow> f x y \<in> F1in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y} \<and> F1map fst fst fst (f x y) = dtor1 x \<and> F1map snd snd snd (f x y) = dtor1 y"   and "\<forall>x y. R3 x y \<longrightarrow> g x y \<in> F2in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y} \<and> F2map fst fst fst (g x y) = dtor2 x \<and> F2map snd snd snd (g x y) = dtor2 y" shows "((\<exists>x. R2 x y1 \<and> y1' = JF1map snd (unfold1 (\<lambda>(x, y). f x y) (\<lambda>(x, y). g x y) (x, y1))) \<longrightarrow> y1' = y1) \<and> ((\<exists>x. R3 x y2 \<and> y2' = JF2map snd (unfold2 (\<lambda>(x, y). f x y) (\<lambda>(x, y). g x y) (x, y2))) \<longrightarrow> y2' = y2)"
merge_append: fixes xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" shows "map_of (xs @ ys) = map_of (AList.merge ys xs)"
lt_minus_lessI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p - q \<noteq> 0"   and "lt q = lt p"   and "lc q = lc p" shows "lt (p - q) \<prec>\<^sub>t lt p"
omit_set: fixes xs :: "nat set"   and ps :: "('c, 'd) fm list" shows "set (omit xs ps) \<subseteq> set ps"
cyclic_group_def: fixes G :: "('b, 'c) cyclic_group_scheme" shows "cyclic_group G \<equiv> Group.group G \<and> cyclic_group_axioms G"
initmissing_def: fixes \<sigma> :: "(nat \<Rightarrow> 'g option) \<times> 'b" shows "initmissing \<sigma> = (\<lambda>i. case fst \<sigma> i of None \<Rightarrow> someinit i | Some s \<Rightarrow> s, snd \<sigma>)"
frontier_not_empty: fixes S :: "'a set" assumes "S \<noteq> {}"   and "S \<noteq> UNIV" shows "frontier S \<noteq> {}"
"case": fixes f :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a"   and x :: "'b \<Rightarrow> 'c" shows "(case EnvT x of EnvT x \<Rightarrow> f x) = f x"
MakeNT: fixes \<eta> :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" assumes "NatTransP \<eta>" shows "NatTrans (MakeNT \<eta>)"
le_INF_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and u :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq u (Inf (f ` A)) = (\<forall>i\<in>A. less_eq u (f i))"
rec: assumes "phiDefNodes_aux g v unvisited n = fold (\<union>) (map (phiDefNodes_aux g v (removeAll n unvisited)) (predecessors g n)) (if length (predecessors g n) = 1 then {} else {n})"   and "n \<in> set unvisited"   and "v \<notin> defs g n" shows "thesis"
map_gL_satC: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "('fsym, 'psym) lit list" shows "GE.satC \<xi> (map gL c) = Ik.satC \<xi> c"
ABC: shows "\<forall>t\<in>A. \<forall>s\<in>B. P t s \<longrightarrow> Q t \<and> Q s \<or> R t \<and> R s"
\<Phi>_def: fixes a :: "presb" shows "\<Phi> a = (case a of presb.Eq is n z \<Rightarrow> {FBase (presb.Eq is i 0) |i. i \<in> {min (- h is) n..max (- l is) n}} \<union> {FBool False})"
code: fixes sys :: "'b \<Rightarrow> ('a, 'b) process_term"   and T :: "('a, 'b) process_term" shows "solution sys T = (if isACT sys T then Action (PREF sys T) (solution sys (CONT sys T)) else Choice (solution sys (CH1 sys T)) (solution sys (CH2 sys T)))"
run_state_return: fixes x :: "'a"   and s :: "'b" shows "run_state (State_Monad.return x) s = (x, s)"
1: shows "injective (p \<sqinter> - (1::'a))"
open_segment_def: fixes a :: "'b"   and b :: "'b" shows "open_segment a b \<equiv> closed_segment a b - {a, b}"
SUP_le_iff: 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 bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and u :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup (f ` A)) u = (\<forall>i\<in>A. less_eq (f i) u)"
n\<^sub>1_def: shows "n\<^sub>1 = size (mset_heap ts\<^sub>1)"
hyps: fixes t :: "'a list" assumes "n + 1 < length t" shows "swap_events n (n + 1) t = t[n := t ! (n + 1), n + 1 := t ! n]"
bit_set_bit_iff: 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 "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and m :: "nat"   and a :: "'c"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "bit (ring_bit_operations.set_bit one push_bit or m a) n = (bit a n \<or> m = n \<and> power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero)"
rept_def: fixes s :: "'s \<times> ('c \<Rightarrow> real)"   and \<mu>_abs :: "('s \<times> ('c \<Rightarrow> real) set) pmf" shows "rept s \<mu>_abs \<equiv> let (l, u) = s in if \<exists>R'. (l, u) \<in> S \<and> \<mu>_abs = return_pmf (l, R') \<and> [u]\<^sub>\<R> = R' \<and> (\<forall>c\<in>\<X>. real (k c) < u c) then return_pmf (l, u \<oplus> 5 / 10) else if \<exists>R'. (l, u) \<in> S \<and> \<mu>_abs = return_pmf (l, R') \<and> R' \<in> Succ \<R> ([u]\<^sub>\<R>) \<and> [u]\<^sub>\<R> \<noteq> R' \<and> (\<forall>u\<in>R'. \<forall>c\<in>\<X>. \<not> (\<exists>d\<le>k c. u c = real d)) then return_pmf (l, delayedR (SOME R'. \<mu>_abs = return_pmf (l, R')) u) else SOME \<mu>. \<mu> \<in> K s \<and> abst \<mu> = \<mu>_abs"
LSC_reassociate_Ands: fixes C :: "('b literal list \<times> 'b literal list) multiset"   and \<Gamma> :: "'b formula multiset" assumes "{#disj_of_clause c \<^bold>\<or> disj_of_clause d. (c, d) \<in># C#} + \<Gamma> \<Rightarrow>\<^sub>n"   and "is_nnf_mset \<Gamma>" shows "{#disj_of_clause (c @ d). (c, d) \<in># C#} + \<Gamma> \<Rightarrow>\<^sub>n"