spec_position stringlengths 41 140 | minilang stringlengths 4 2.76M ⌀ | proof_position stringlengths 43 144 | isar stringlengths 1 726k ⌀ |
|---|---|---|---|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:192 | RULE subset_antisym subsetI
(*goal: "\<I> (\<V> (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)) \<subseteq> \<surd>ideal F" and "\<surd>ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subseteq> \<I> (\<V> F)" *)
RULE subset_antisym subsetI
(*goal: "\<And>x::('a \<Rightarro... | ./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:195:1 | proof (intro subset_antisym subsetI)
fix f
assume "f \<in> \<I> (\<V> F)"
with assms show "f \<in> \<surd>ideal F"
by (rule Nullstellensatz)
qed (metis ideal_ofI variety_ofD variety_of_radical_ideal) |
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:201 | HAVE fact0: "\<I> (\<V> (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)) = \<surd>ideal F"
END WITH strong_Nullstellensatz assms(1,2)
(*introduces facts local.fact0: "\<I> (\<V> (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)) = \<surd>ideal F" *)
END WITH radical_ideal_eq_UNIV_i... | ./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:204:1 | proof -
from assms(1, 2) have "\<I> (\<V> F) = \<surd>ideal F"
by (rule strong_Nullstellensatz)
thus ?thesis
by (simp add: assms(3) flip: radical_ideal_eq_UNIV_iff)
qed |
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:264 | END WITH generates_max_ideal_def assms
| ./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:267:3 | using assms
by (simp add: generates_max_ideal_def) |
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:269 | RULE generates_max_idealI WITH assms(1)
(*goal: "\<And>F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set. \<lbrakk>F' \<subseteq> P[X::'a set]; ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subset> ideal F'\<rbrakk> \<Longrightarrow> ideal F' = UNIV" *)
INTRO VARS F'
(*fixes... | ./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:272:3 | using assms(1)
proof (rule generates_max_idealI)
fix F'
assume "F' \<subseteq> P[X]" and sub: "ideal F \<subset> ideal F'"
from this(2) ideal.span_subset_spanI have "\<not> F' \<subseteq> ideal F"
by blast
then obtain p where "p \<in> F'" and "p \<notin> ideal F"
by blast
from this(1) \<open>F' \<subs... |
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:286 | NEXT WITH generates_max_ideal_def assms
(*goal: "\<lbrakk>(F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subseteq> P[X::'a set]; ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subset> ideal F'\<rbrakk> \<Longrightarrow> ideal F' = UNIV" *)
END WITH generates_max_idea... | ./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:289:3 | using assms
by (simp_all add: generates_max_ideal_def) |
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:291 | END WITH generates_max_ideal_def assms
| ./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:294:3 | using assms
by (auto simp: generates_max_ideal_def) |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneGraphIso.thy:10 | END WITH image_iff
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneGraphIso.thy:11:1 | by (auto simp: image_iff) |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:10 | END
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:12:1 | by simp |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListAux.thy:14 | END
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListAux.thy:15:1 | by auto |
./contrib/afp-2025-02-12/thys/Formal_SSA/While_Combinator_Exts.thy:11 | APPLY (insert assms)
(*goal: "\<lbrakk>while_option (b::'a \<Rightarrow> bool) (c::'a \<Rightarrow> 'a) (s::'a) = None; wf (r::('a \<times> 'a) set); (I::'a \<Rightarrow> bool) s; \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> I (c s); \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> ... | ./contrib/afp-2025-02-12/thys/Formal_SSA/While_Combinator_Exts.thy:16:3 | using assms
by -(drule wf_rel_while_option_Some [of r I b c], auto) |
./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:14 | END WITH three_phase_def
| ./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:15:1 | by (simp add: three_phase_def) |
./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:12 | END WITH two_phase_def
| ./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:13:1 | by (simp add: two_phase_def) |
./contrib/afp-2025-02-12/thys/AutoCorres2/SimplConv.thy:31 | NEXT
(*goal: "switch (v::'e \<Rightarrow> 'h) ((a::'h set, b::('e, 'f, 'g) com) # (vs::('h set \<times> ('e, 'f, 'g) com) list)) \<equiv> IF \<acute>v \<in> a THEN
b
ELSE
switch v vs
FI" *)
END
| ./contrib/afp-2025-02-12/thys/AutoCorres2/SimplConv.thy:34:3 | by auto |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:14 | END
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:15:1 | by simp |
./contrib/afp-2025-02-12/thys/Dependent_SIFUM_Refinement/CompositionalRefinement.thy:10 | END WITH card_inj_on_le
| ./contrib/afp-2025-02-12/thys/Dependent_SIFUM_Refinement/CompositionalRefinement.thy:12:3 | by (blast intro: card_inj_on_le) |
./contrib/afp-2025-02-12/thys/Consensus_Refined/Consensus_Misc.thy:17 | RULE option.expand
(*goal: "\<lbrakk>((option::'a option) = None) = ((option'::'a option) = None); \<And>(x::'a) y::'a. \<lbrakk>option = Some x; option' = Some y\<rbrakk> \<Longrightarrow> x = y\<rbrakk> \<Longrightarrow> (option = None) = (option' = None)" and "\<lbrakk>((option::'a option) = None) = ((option'::'a op... | ./contrib/afp-2025-02-12/thys/Consensus_Refined/Consensus_Misc.thy:20:3 | by (rule option.expand, auto) |
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Trace.thy:14 | APPLY (coinduction arbitrary: n)
(*goal: "\<And>n::'a. \<forall>n::'a. n \<le> (f::'a \<Rightarrow> 'a) n \<Longrightarrow> \<exists>s::'a stream. siterate f n = s \<and> shd s \<le> shd (stl s) \<and> ((\<exists>n::'a. stl s = siterate f n \<and> (\<forall>n::'a. n \<le> f n)) \<or> ssorted (stl s))" *)
END
| ./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Trace.thy:15:3 | by (coinduction arbitrary: n) auto |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Quasi_Order.thy:22 | END WITH subseteq_qle_def in_qle_def
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Quasi_Order.thy:23:1 | by (auto simp add: subseteq_qle_def in_qle_def) |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ScoreProps.thy:10 | END WITH deleteAround_def
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ScoreProps.thy:11:1 | by (simp add: deleteAround_def) |
./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:24 | END WITH ogets_def spec_monad_eqI runs_to_iff(1-31)
| ./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:25:3 | apply (rule spec_monad_eqI)
apply (auto simp add: runs_to_iff ogets_def)
done |
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:13 | INDUCT n arbitrary: x
(*fixes variables x :: nat *)
(*goal: "\<And>x::nat. tabulate (f::nat \<Rightarrow> 'a) x (0::nat) = map f [x..<x + (0::nat)]" and "\<And>(n::nat) x::nat. (\<And>x::nat. tabulate (f::nat \<Rightarrow> 'a) x n = map f [x..<x + n]) \<Longrightarrow> tabulate f x (Suc n) = map f [x..<x + Suc n]" *)
N... | ./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:14:3 | by (induct n arbitrary: x) (auto simp: not_le Suc_le_eq upt_rec) |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Computation/ArchComp.thy:11 | APPLY (eval)
(*no goal remains!*)
END
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Computation/ArchComp.thy:12:1 | by eval |
./contrib/afp-2025-02-12/thys/AutoCorres2/LocalVarExtract.thy:18 | END WITH set_eqI
| ./contrib/afp-2025-02-12/thys/AutoCorres2/LocalVarExtract.thy:20:3 | by (fastforce intro: set_eqI) |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListSum.thy:18 | INDUCT V
(*goal: "(\<Sum>\<^bsub>v::'a\<in>[]\<^esub> (0::nat)) = (0::nat)" and "\<And>(a::'a) V::'a list. (\<Sum>\<^bsub>v::'a\<in>V\<^esub> (0::nat)) = (0::nat) \<Longrightarrow> (\<Sum>\<^bsub>v::'a\<in>a # V\<^esub> (0::nat)) = (0::nat)" *)
NEXT
(*fixes variables a :: 'a and V :: "'a list" *)
(*introduces facts... | ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListSum.thy:18:64 | by (induct V) simp_all |
./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:5 | END WITH AB AP BP
| ./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:10:1 | proof -
from AB AP BP show ?thesis
by blast
qed |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/EnumeratorProps.thy:9 | INDUCT xs
(*fixes variables x :: 'a *)
(*goal: "\<And>x::'a. |hideDupsRec x []| = |[]|" and "\<And>(a::'a) (xs::'a list) x::'a. (\<And>x::'a. |hideDupsRec x xs| = |xs|) \<Longrightarrow> |hideDupsRec x (a # xs)| = |a # xs|" *)
NEXT
(*fixes variables a :: 'a and xs :: "'a list" and x :: 'a *)
(*introduces facts Cons... | ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/EnumeratorProps.thy:10:1 | by (induct xs) auto |
./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/Memcpy.thy:18 | END WITH intvl_Suc ptr_aligned_def c_null_guard_def c_guard_def
| ./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/Memcpy.thy:20:3 | unfolding c_guard_def c_null_guard_def ptr_aligned_def
by (clarsimp simp: intvl_Suc) |
./contrib/afp-2025-02-12/thys/AutoCorres2/L1Peephole.thy:17 | END WITH L1_seq_def bind_assoc
| ./contrib/afp-2025-02-12/thys/AutoCorres2/L1Peephole.thy:18:3 | apply (clarsimp simp: L1_seq_def bind_assoc)
done |
./contrib/afp-2025-02-12/thys/Formal_SSA/RBT_Mapping_Exts.thy:12 | APPLY (transfer)
(*goal: "\<And>(f::'a \<Rightarrow> 'b) r::('a, unit) RBT.rbt. (Some \<circ> f) |` RBT_Set.Set r = RBT.lookup (RBT.map (\<lambda>(a::'a) _::unit. f a) r)" *)
END WITH restrict_map_def RBT_Set.Set_def
| ./contrib/afp-2025-02-12/thys/Formal_SSA/RBT_Mapping_Exts.thy:14:3 | by transfer (auto simp: RBT_Set.Set_def restrict_map_def) |
./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:17 | END WITH three_step_def
| ./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:18:1 | by (simp add: three_step_def) |
./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:15 | END WITH two_step_def
| ./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:16:1 | by (simp add: two_step_def) |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneGraphIso.thy:36 | END WITH Iso_def
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneGraphIso.thy:37:1 | by (simp add:Iso_def) |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Worklist.thy:20 | END WITH worklist_aux_def while_option_unfold
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Worklist.thy:21:1 | by (simp add: worklist_aux_def while_option_unfold) |
./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Nat_Mod_Helper.thy:7 | END WITH add_diff_cancel_left' mod_if mod_mult_div_eq mod_mult_self1_is_0
| ./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Nat_Mod_Helper.thy:9:3 | by (metis add_diff_cancel_left' mod_if mod_mult_div_eq mod_mult_self1_is_0) |
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Trace.thy:17 | INDUCT i arbitrary: s
(*fixes variables s :: "'a stream" *)
(*introduces facts 0.prems: "ssorted (sa__::'a stream)" *)
(*goal: "\<And>s::'a stream. ssorted s \<Longrightarrow> s !! (0::nat) \<le> stl s !! (0::nat)" and "\<And>(i::nat) s::'a stream. \<lbrakk>\<And>s::'a stream. ssorted s \<Longrightarrow> s !! i \<le> s... | ./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Trace.thy:18:3 | by (induct i arbitrary: s) (auto elim: ssorted.cases) |
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:16 | INDUCT n arbitrary: x
(*fixes variables x :: nat *)
(*goal: "\<And>x::nat. length (tabulate (f::nat \<Rightarrow> 'a) x (0::nat)) = (0::nat)" and "\<And>(n::nat) x::nat. (\<And>x::nat. length (tabulate (f::nat \<Rightarrow> 'a) x n) = n) \<Longrightarrow> length (tabulate f x (Suc n)) = Suc n" *)
NEXT
(*fixes variabl... | ./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:17:3 | by (induction n arbitrary: x) simp_all |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneProps.thy:11 | HAVE fact0: "(f::face) \<in> \<F> (g::graph)"
END WITH inv_mgp minGraphProps assms(1,3,4)
(*introduces facts local.fact0: "(f::face) \<in> \<F> (g::graph)" *)
END WITH finalGraph_face assms(2) fact0
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneProps.thy:13:1 | proof -
from assms(1,3,4) have "f \<in> \<F> g"
by (blast intro: minGraphProps inv_mgp)
with assms(2) show ?thesis
by (rule finalGraph_face)
qed |
./contrib/afp-2025-02-12/thys/Green/Derivs.thy:5 | END WITH has_real_derivative_iff_has_vector_derivative
| ./contrib/afp-2025-02-12/thys/Green/Derivs.thy:7:3 | by (simp add: has_real_derivative_iff_has_vector_derivative) |
./contrib/afp-2025-02-12/thys/AutoCorres2/LocalVarExtract.thy:22 | END WITH set_eqI
| ./contrib/afp-2025-02-12/thys/AutoCorres2/LocalVarExtract.thy:24:3 | by (fastforce intro: set_eqI) |
./contrib/afp-2025-02-12/thys/AutoCorres2/ExceptionRewrite.thy:16 | END WITH top_post_state_def always_fail_def
| ./contrib/afp-2025-02-12/thys/AutoCorres2/ExceptionRewrite.thy:17:3 | by (clarsimp simp: always_fail_def top_post_state_def) |
./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:29 | END WITH ogets_def spec_monad_eqI runs_to_iff(1-31)
| ./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:30:3 | apply (rule spec_monad_eqI)
apply (auto simp add: runs_to_iff ogets_def)
done |
./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:14 | END WITH AB AP BP CP
| ./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:20:1 | proof -
from AB AP BP CP show ?thesis
by blast
qed |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:17 | END
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:19:1 | by simp |
./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/FactorialTest.thy:23 | END WITH fact.simps
| ./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/FactorialTest.thy:24:3 | by (subst fact.simps) simp |
./contrib/afp-2025-02-12/thys/AutoCorres2/SimplConv.thy:36 | APPLY (subst signed.less_le)
(*goal: "\<lbrakk>(a::'a word) < (n::'a word); n \<le> (2::'a word) ^ (LENGTH('a) - (1::nat)) - (1::'a word)\<rbrakk> \<Longrightarrow> a \<le>s n \<and> a \<noteq> n" *)
APPLY (safe)
(*goal: "\<lbrakk>(a::'a word) < (n::'a word); n \<le> (2::'a word) ^ (LENGTH('a) - (1::nat)) - (1::'a word... | ./contrib/afp-2025-02-12/thys/AutoCorres2/SimplConv.thy:38:3 | apply (subst signed.less_le)
apply safe
apply (subst word_sle_msb_le)
apply safe
apply (force simp: not_msb_from_less)
apply simp
apply simp
done |
./contrib/afp-2025-02-12/thys/Ordered_Resolution_Prover/Lazy_List_Liminf.thy:24 | END
| ./contrib/afp-2025-02-12/thys/Ordered_Resolution_Prover/Lazy_List_Liminf.thy:25:3 | by simp |
./contrib/afp-2025-02-12/thys/LocalLexing/TheoremD10.thy:7 | END WITH admissible_wellformed_tokens "\<P>_are_admissible"
| ./contrib/afp-2025-02-12/thys/LocalLexing/TheoremD10.thy:8:1 | using \<P>_are_admissible admissible_wellformed_tokens
by blast |
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_VCG.thy:8 | END WITH HT'_def BB_PROTECT_def assms[unfolded "BB_PROTECT_def" "HT'_def"]
| ./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_VCG.thy:11:5 | using assms
unfolding BB_PROTECT_def HT'_def
by auto |
./contrib/afp-2025-02-12/thys/BurrowsWheeler/counting/Count_Util.thy:12 | END WITH gr0I count_list_0_iff
| ./contrib/afp-2025-02-12/thys/BurrowsWheeler/counting/Count_Util.thy:14:3 | by (meson count_list_0_iff gr0I) |
./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Rotated_Substring.thy:21 | NEXT WITH inc_one_bounded_def
(*goal: "\<lbrakk>inc_one_bounded (n::nat) (xs::nat list); (i::nat) < length xs\<rbrakk> \<Longrightarrow> xs ! i < n" *)
END WITH inc_one_bounded_def
| ./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Rotated_Substring.thy:24:3 | using inc_one_bounded_def
by blast+ |
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_Var_Postprocessor.thy:8 | END WITH RENAMING_def
| ./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_Var_Postprocessor.thy:8:59 | unfolding RENAMING_def
by auto |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/EnumeratorProps.thy:12 | CASE_SPLIT xs
(*introduces facts Nil.prem0: "(xs::'a list) = []" *)
(*goal: "(xs::'a list) = [] \<Longrightarrow> |hideDups xs| = |xs|" and "\<And>(a::'a) list::'a list. (xs::'a list) = a # list \<Longrightarrow> |hideDups xs| = |xs|" *)
NEXT WITH Nil
(*fixes variables a :: 'a and list :: "'a list" *)
(*introduces ... | ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/EnumeratorProps.thy:13:1 | by (cases xs) simp_all |
./contrib/afp-2025-02-12/thys/AutoCorres2/L1Peephole.thy:21 | NEXT WITH L1_skip_def L1_seq_def
(*goal: "L1_seq L1_skip (A::(unit option, unit, 'a) spec_monad) = A" *)
END WITH L1_skip_def L1_seq_def
| ./contrib/afp-2025-02-12/thys/AutoCorres2/L1Peephole.thy:24:3 | apply (clarsimp simp: L1_seq_def L1_skip_def)+
done |
./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:20 | END WITH three_step_def three_phase_def
| ./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:21:3 | by (auto simp add: three_phase_def three_step_def) |
./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:18 | END WITH two_step_def two_phase_def
| ./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:19:3 | by (auto simp add: two_phase_def two_step_def) |
./contrib/afp-2025-02-12/thys/AutoCorres2/Split_Heap.thy:20 | END WITH array_fields_def
| ./contrib/afp-2025-02-12/thys/AutoCorres2/Split_Heap.thy:21:3 | by (auto simp add: array_fields_def) |
./contrib/afp-2025-02-12/thys/MDP-Rewards/Blinfun_Util.thy:11 | NEXT WITH blinfun_eqI
(*goal: "(f::'a \<Rightarrow>\<^sub>L 'b) o\<^sub>L id_blinfun = f" *)
END WITH blinfun_eqI
| ./contrib/afp-2025-02-12/thys/MDP-Rewards/Blinfun_Util.thy:14:3 | by (auto intro!: blinfun_eqI) |
./contrib/afp-2025-02-12/thys/Bernoulli/Bernoulli.thy:14 | END WITH power_Suc numeral_eq_Suc
| ./contrib/afp-2025-02-12/thys/Bernoulli/Bernoulli.thy:15:3 | by (simp only: numeral_eq_Suc power_Suc) |
./contrib/afp-2025-02-12/thys/Ordered_Resolution_Prover/Map2.thy:22 | END WITH zip_Nil zip_Cons_Cons zip.simps list.simps Nil_is_map_conv
| ./contrib/afp-2025-02-12/thys/Ordered_Resolution_Prover/Map2.thy:23:3 | by (metis Nil_is_map_conv list.exhaust list.simps(3) zip.simps(1) zip_Cons_Cons zip_Nil) |
./contrib/afp-2025-02-12/thys/MDP-Rewards/MDP_reward_Util.thy:8 | END WITH mult.commute assms
| ./contrib/afp-2025-02-12/thys/MDP-Rewards/MDP_reward_Util.thy:12:3 | using assms
by (auto simp: mult.commute) |
./contrib/afp-2025-02-12/thys/AutoCorres2/AbstractArrays.thy:26 | CASE_SPLIT n
(*introduces facts 0.prem0: "(n::nat) = (0::nat)" *)
(*goal: "(n::nat) = (0::nat) \<Longrightarrow> ((x::'a ptr) \<in> set (array_addrs x n)) = ((0::nat) < n)" and "\<And>nat::nat. (n::nat) = Suc nat \<Longrightarrow> ((x::'a ptr) \<in> set (array_addrs x n)) = ((0::nat) < n)" *)
NEXT WITH array_addrs.simp... | ./contrib/afp-2025-02-12/thys/AutoCorres2/AbstractArrays.thy:28:3 | by (cases n, auto simp: array_addrs.simps(2)) |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/FaceDivisionProps.thy:13 | INDUCT g
(*fixes variables x1a :: "face list" and x2a :: nat and x3a :: "face list list" and x4a :: "nat list" *)
(*goal: "\<And>(x1a::face list) (x2a::nat) (x3a::face list list) x4a::nat list. vertices (makeFaceFinal (f::face) (Graph x1a x2a x3a x4a)) = vertices (Graph x1a x2a x3a x4a)" *)
END WITH makeFaceFinal_def v... | ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/FaceDivisionProps.thy:14:1 | by (induct g)(simp add:vertices_graph_def makeFaceFinal_def) |
./contrib/afp-2025-02-12/thys/Amicable_Numbers/Amicable_Numbers.thy:18 | END WITH assms
| ./contrib/afp-2025-02-12/thys/Amicable_Numbers/Amicable_Numbers.thy:22:3 | using assms
by linarith |
./contrib/afp-2025-02-12/thys/HOL-CSP/Stop.thy:56 | END WITH FAILURES_def Failures.rep_eq STOP.rep_eq
| ./contrib/afp-2025-02-12/thys/HOL-CSP/Stop.thy:57:3 | by (simp add: FAILURES_def Failures.rep_eq STOP.rep_eq) |
./contrib/afp-2025-02-12/thys/Formal_SSA/Serial_Rel.thy:21 | INDUCT m
(*goal: "iterated_serial_on (A::'a set) (r::('a \<times> 'a) set) (x::'a) ((n::nat) + (0::nat)) = iterated_serial_on A r (iterated_serial_on A r x n) (0::nat)" and "\<And>m::nat. iterated_serial_on (A::'a set) (r::('a \<times> 'a) set) (x::'a) ((n::nat) + m) = iterated_serial_on A r (iterated_serial_on A r x n... | ./contrib/afp-2025-02-12/thys/Formal_SSA/Serial_Rel.thy:22:1 | by (induction m) auto |
./contrib/afp-2025-02-12/thys/Combinatorics_Words_Graph_Lemma/Glued_Codes.thy:16 | END WITH last_append if_split
| ./contrib/afp-2025-02-12/thys/Combinatorics_Words_Graph_Lemma/Glued_Codes.thy:18:3 | by (auto simp only: last_append split: if_split) |
./contrib/afp-2025-02-12/thys/Euler_Partition/Euler_Partition.thy:17 | END WITH assms nonzero_mult_div_cancel_right le_add_diff_inverse2 less_not_refl2 power_add power_not_zero
| ./contrib/afp-2025-02-12/thys/Euler_Partition/Euler_Partition.thy:21:1 | by (metis assms nonzero_mult_div_cancel_right le_add_diff_inverse2 less_not_refl2 power_add power_not_zero) |
./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Nat_Mod_Helper.thy:11 | END
| ./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Nat_Mod_Helper.thy:13:3 | using order_le_less_trans
by blast |
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:19 | INDUCT n arbitrary: x
(*fixes variables x :: nat *)
(*goal: "\<And>x::nat. map (f::'b \<Rightarrow> 'a) (tabulate (g::nat \<Rightarrow> 'b) x (0::nat)) = tabulate (\<lambda>x::nat. f (g x)) x (0::nat)" and "\<And>(n::nat) x::nat. (\<And>x::nat. map (f::'b \<Rightarrow> 'a) (tabulate (g::nat \<Rightarrow> 'b) x n) = tab... | ./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:20:3 | by (induction n arbitrary: x) simp_all |
./contrib/afp-2025-02-12/thys/Green/DiamExample.thy:7 | END WITH dual_order.order_iff_strict abs_if
| ./contrib/afp-2025-02-12/thys/Green/DiamExample.thy:10:3 | by (simp add: abs_if dual_order.order_iff_strict) |
./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/Memcpy.thy:24 | CASE_SPLIT x
(*fixes variables x :: "32 word" *)
(*introduces facts Ptr.prem0: "(x::'b ptr) = PTR('b) (xa__::32 word)" *)
(*goal: "\<And>xa::32 word. (x::'b ptr) = PTR('b) xa \<Longrightarrow> ptr_val (PTR_COERCE('b \<rightarrow> 'a) x +\<^sub>p (a::int)) = ptr_val x + word_of_int a * of_nat_addr (size_of TYPE('a))" *)... | ./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/Memcpy.thy:25:3 | apply (case_tac x)
apply (clarsimp simp: CTypesDefs.ptr_add_def)
done |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Worklist.thy:23 | END WITH worklist_aux_def while_option_unfold
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Worklist.thy:25:1 | by (simp add: worklist_aux_def while_option_unfold) |
./contrib/afp-2025-02-12/thys/CoreC++/Auxiliary.thy:19 | END
| ./contrib/afp-2025-02-12/thys/CoreC++/Auxiliary.thy:21:2 | by arith |
./contrib/afp-2025-02-12/thys/AutoCorres2/L2Peephole.thy:17 | END
| ./contrib/afp-2025-02-12/thys/AutoCorres2/L2Peephole.thy:18:3 | by simp |
./contrib/afp-2025-02-12/thys/Formal_SSA/SSA_Transfer_Rules.thy:16 | END WITH right_totalE rel_funE
| ./contrib/afp-2025-02-12/thys/Formal_SSA/SSA_Transfer_Rules.thy:17:3 | by (auto 4 4 elim: right_totalE rel_funE) |
./contrib/afp-2025-02-12/thys/CoreC++/WWellForm.thy:16 | END WITH wwf_mdecl_def
| ./contrib/afp-2025-02-12/thys/CoreC++/WWellForm.thy:19:1 | by (simp add:wwf_mdecl_def) |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListSum.thy:20 | INDUCT xs
(*goal: "ListSum (filter (\<lambda>x::'a. \<not> (P::'a \<Rightarrow> bool) x) []) (f::'a \<Rightarrow> nat) + ListSum (filter P []) f = ListSum [] f" and "\<And>(a::'a) xs::'a list. ListSum (filter (\<lambda>x::'a. \<not> (P::'a \<Rightarrow> bool) x) xs) (f::'a \<Rightarrow> nat) + ListSum (filter P xs) f =... | ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListSum.thy:22:2 | by (induct xs) simp_all |
./contrib/afp-2025-02-12/thys/Consensus_Refined/Voting/OneThirdRule_Proofs.thy:13 | HAVE fact0: "((HOL.eq::nat \<Rightarrow> nat \<Rightarrow> bool) ((Groups.plus_class.plus::nat \<Rightarrow> nat \<Rightarrow> nat) ((Finite_Set.card::'a set \<Rightarrow> nat) (S::'a set)) ((Finite_Set.card::'a set \<Rightarrow> nat) ((Groups.uminus_class.uminus::'a set \<Rightarrow> 'a set) (S::'a set)))) ((Finite_Se... | ./contrib/afp-2025-02-12/thys/Consensus_Refined/Voting/OneThirdRule_Proofs.thy:16:1 | proof-
have "card S + card (-S) = card (UNIV :: 'a set)"
by (rule card_Un_disjoint[of S "-S", simplified Compl_partition, symmetric]) (auto)
thus ?thesis
by simp
qed |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneProps.thy:18 | END WITH finalVertex_def nonFinals_def filter_empty_conv plane_final_facesAt
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneProps.thy:20:1 | by (auto simp add: finalVertex_def nonFinals_def filter_empty_conv plane_final_facesAt) |
./contrib/afp-2025-02-12/thys/AutoCorres2/c-parser/CLanguage.thy:18 | END WITH addr_card_def card_word
| ./contrib/afp-2025-02-12/thys/AutoCorres2/c-parser/CLanguage.thy:19:3 | by (simp add: addr_card_def card_word) |
./contrib/afp-2025-02-12/thys/IMP2/IMP2.thy:14 | END
| ./contrib/afp-2025-02-12/thys/IMP2/IMP2.thy:14:63 | by auto |
./contrib/afp-2025-02-12/thys/Case_Labeling/Examples/Hoare/Labeled_Hoare_Examples.thy:12 | APPLY (vcg_simp)
(*no goal remains!*)
END
| ./contrib/afp-2025-02-12/thys/Case_Labeling/Examples/Hoare/Labeled_Hoare_Examples.thy:19:1 | by vcg_simp |
./contrib/afp-2025-02-12/thys/AutoCorres2/lib/Mutual_CCPO_Recursion.thy:18 | END WITH image_def fun_lub_def fun_eq_iff Inf_apply
| ./contrib/afp-2025-02-12/thys/AutoCorres2/lib/Mutual_CCPO_Recursion.thy:19:3 | unfolding fun_lub_def fun_eq_iff Inf_apply image_def
by simp |
./contrib/afp-2025-02-12/thys/AutoCorres2/AutoCorres.thy:33 | END WITH o_def
| ./contrib/afp-2025-02-12/thys/AutoCorres2/AutoCorres.thy:34:3 | by (simp add: o_def) |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Computation/ArchComp.thy:14 | APPLY (eval)
(*no goal remains!*)
END
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Computation/ArchComp.thy:15:1 | by eval |
./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:24 | END WITH AB AP BP CP DP
| ./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:31:1 | proof -
from AB AP BP CP DP show ?thesis
by blast
qed |
./contrib/afp-2025-02-12/thys/ROBDD/BDD_Code.thy:8 | END WITH blit'_def
| ./contrib/afp-2025-02-12/thys/ROBDD/BDD_Code.thy:11:33 | by (simp add: blit'_def) |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/TameProps.thy:10 | INDUCT xs
(*introduces facts Nil.prems: "\<forall>x::'a\<in>set []. \<not> ((P::'a \<Rightarrow> bool) x \<and> (Q::'a \<Rightarrow> bool) x)" *)
(*goal: "\<forall>x::'a\<in>set []. \<not> ((P::'a \<Rightarrow> bool) x \<and> (Q::'a \<Rightarrow> bool) x) \<Longrightarrow> |filter P []| + |filter Q []| \<le> |[]|" and ... | ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/TameProps.thy:12:1 | by (induct xs) auto |
./contrib/afp-2025-02-12/thys/BurrowsWheeler/counting/Rank_Select.thy:15 | END WITH rank_def count_list_eq_count
| ./contrib/afp-2025-02-12/thys/BurrowsWheeler/counting/Rank_Select.thy:17:3 | by (simp add: count_list_eq_count rank_def) |
./contrib/afp-2025-02-12/thys/Directed_Sets/Well_Order_Connection.thy:11 | END WITH refl_on_def reflexive_def relation_of_def
| ./contrib/afp-2025-02-12/thys/Directed_Sets/Well_Order_Connection.thy:12:3 | by (auto simp: refl_on_def reflexive_def relation_of_def) |
./contrib/afp-2025-02-12/thys/LocalLexing/Limit.thy:9 | INDUCT n
(*introduces facts 0.prems: "setmonotone (f::'a set \<Rightarrow> 'a set)" *)
(*goal: "setmonotone (f::'a set \<Rightarrow> 'a set) \<Longrightarrow> setmonotone (funpower f (0::nat))" and "\<And>n::nat. \<lbrakk>setmonotone (f::'a set \<Rightarrow> 'a set) \<Longrightarrow> setmonotone (funpower f n); setmono... | ./contrib/afp-2025-02-12/thys/LocalLexing/Limit.thy:10:3 | by (induct n, auto simp add: setmonotone_def) |
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_VCG.thy:14 | END WITH HT'_partial_def BB_PROTECT_def assms[unfolded "BB_PROTECT_def" "HT'_partial_def"]
| ./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_VCG.thy:17:5 | using assms
unfolding BB_PROTECT_def HT'_partial_def
by auto |
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_Var_Postprocessor.thy:9 | END WITH assms[unfolded "RENAMING_def"]
| ./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_Var_Postprocessor.thy:11:15 | using assms
unfolding RENAMING_def
by simp |
./contrib/afp-2025-02-12/thys/MonoBoolTranAlgebra/Mono_Bool_Tran.thy:28 | END WITH image_comp fun_eq_iff
| ./contrib/afp-2025-02-12/thys/MonoBoolTranAlgebra/Mono_Bool_Tran.thy:30:3 | by (simp add: fun_eq_iff image_comp) |
./contrib/afp-2025-02-12/thys/CoreC++/CWellForm.thy:20 | END WITH wf_C_mdecl_def
| ./contrib/afp-2025-02-12/thys/CoreC++/CWellForm.thy:27:1 | by (simp add:wf_C_mdecl_def) |
./contrib/afp-2025-02-12/thys/AutoCorres2/c-parser/umm_heap/SepTactic.thy:21 | END WITH sep_conj_extract_def
| ./contrib/afp-2025-02-12/thys/AutoCorres2/c-parser/umm_heap/SepTactic.thy:23:3 | by (simp add: sep_conj_extract_def) |
./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:34 | END WITH ogets_def spec_monad_eqI runs_to_iff(1-31)
| ./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:35:3 | apply (simp add: gets_the_def assert_opt_def[abs_def] ogets_def gets_def)
apply (rule spec_monad_eqI)
apply (auto simp add: runs_to_iff)
done |
./contrib/afp-2025-02-12/thys/AutoCorres2/ExceptionRewrite.thy:19 | END WITH always_fail_def run_bind
| ./contrib/afp-2025-02-12/thys/AutoCorres2/ExceptionRewrite.thy:20:3 | apply (clarsimp simp: always_fail_def run_bind)
done |
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:21 | END
| ./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:22:1 | by simp |
./contrib/afp-2025-02-12/thys/Affine_Arithmetic/Counterclockwise_Vector.thy:13 | END WITH ac_simps(1-48)
| ./contrib/afp-2025-02-12/thys/Affine_Arithmetic/Counterclockwise_Vector.thy:15:3 | by (auto simp: ac_simps) |
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